diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzznehb" "b/data_all_eng_slimpj/shuffled/split2/finalzznehb" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzznehb" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{intro}\nRecently, Dag Normann and the author have established a connection between \\emph{higher-order computability theory} and \\emph{Nonstandard Analysis} (\\cite{dagsam}). \nIn the latter, they investigate the complexity of functionals connected to the \\emph{Heine-Borel compactness} of Cantor space. Surprisingly, this complexity turns out to be \nintimately connected to the \\emph{nonstandard compactness} of Cantor space as given by \\emph{Robinson's theorem} (See \\cite{loeb1}*{p.\\ 42}) in Nonstandard Analysis.\nIn fact, the results in \\cite{dagsam} are `holistic' in nature in that theorems in computability theory give rise to theorems in Nonstandard Analysis, \\emph{and vice versa}. \nWe discuss these results in Section \\ref{knowledge} as they serve as the motivation for this paper. \n \n\\medskip\n\nIn light of the aforementioned connection, it is a natural question which notions from higher-order computability theory have elegant analogues in Nonstandard Analysis, \\emph{and vice versa}.\nThis paper explores one particular case of this question, namely for the technique know as \\emph{Grilliot's trick}, introduced in \\cite{grilling}. \nThe latter `trick' actually constitutes a template for explicitly defining the Turing jump functional $(\\exists^{2})$ in terms of a given effectively discontinuous type two functional. \nBelow, we introduce the \\emph{nonstandard extensionality trick}, which is a technique similar to {Grilliot's trick} {in Nonstandard Analysis}. \nIn this way, we study a new \\emph{computational} aspect of Nonstandard Analysis pertaining to {Reverse Mathematics} (RM), in line with the results in \\cites{samzoo,samzooII, samGH, sambon}.\nWe refer to \\cites{simpson2, simpson1} for an overview to RM. \nWe shall make use of \\emph{internal set theory}, i.e.\\ Nelson's axiomatic Nonstandard Analysis (\\cite{wownelly}). \nWe introduce internal set theory and its fragments from \\cite{brie} in Section~\\ref{WIST}. \n\n\\medskip\n\nThe nonstandard extensionality trick sums up as: From the existence of \\emph{nonstandard discontinuous} functionals, the \\emph{Transfer} principle from Nonstandard analysis (See Section~\\ref{IST}) limited to $\\Pi_{1}^{0}$-formulas is derived; from this (generally ineffective) implication, we obtain an effective implication expressing the Turing jump functional in terms of a discontinuous functional (and no longer involving Nonstandard Analysis). Essential to obtaining this effective implication is the `term extraction theorem' in Theorem \\ref{consresult}, based on \\cite{brie}. \nWe shall apply the nonstandard extensionality trick to \\emph{binary expansion}, the \\emph{intermediate value theorem}, the \\emph{Weierstra\\ss~maximum theorem}, and \\emph{weak weak K\\\"onig's lemma} in Section \\ref{main}. \n\n\\medskip\n\nNow, combining the aforementioned results with similar results from \\cites{samzoo, samzooII} regarding the RM zoo from \\cite{damirzoo}, one gets the idea that the \nhigher-order landscape is not very rich and similar to the second-order framework. \nTo counter this view, we discuss a new class of functionals in Section~\\ref{knowledge} which do not fit the existing categories of RM. \nThese functionals are inspired by the \\emph{Standard Part} axiom of Nonstandard Analysis. \n\n\\medskip\n\nFinally, we hasten to point out that there are well-established techniques for obtaining effective content from classical mathematics, the most prominent one being the \\emph{proof mining} program (\\cite{kohlenbach3}). \nIn particular, the effective results in this paper could be or have been obtained in this way. \nWhat is surprising about the results in this paper (in our opinion) is the emergence of effective content (with relative ease) from Nonstandard Analysis \\emph{despite} claims \nthat the latter is somehow fundamentally non-constructive by e.g.\\ Bishop and Connes (See \\cite{samsynt} for a detailed discussion of the Bishop-Connes critique). \n\n\n\\section{Internal set theory and fragments}\\label{WIST}\nIn this section, we sketch \\emph{internal set theory}, Nelson's \\emph{syntactic} approach to Nonstandard Analysis, first introduced in \\cite{wownelly}, and its fragments from \\cite{brie}. An in-depth and completely elementary introduction to the constructive content of Nonstandard Analysis is \\cite{SB}. \n\\subsection{Introducing internal set theory}\\label{IST}\nNelson's system $\\ensuremath{\\usftext{IST}}$ of internal set theory is defined as follows: The language of $\\ensuremath{\\usftext{IST}}$ consist of the language of $\\ensuremath{\\usftext{ZFC}}$, the `usual' foundations of mathematics, plus a new predicate `st($x$)', read as `$x$ is standard'. \nThe new quantifiers $(\\forall^{\\textup{NSA}^{\\alpha}}x)(\\dots)$ and $(\\exists^{\\textup{NSA}^{\\alpha}}y)(\\dots)$ are short for $(\\forall x)(\\textup{NSA}^{\\alpha}(x)\\rightarrow \\dots)$ and $(\\exists y)(\\textup{NSA}^{\\alpha}(y)\\wedge \\dots)$. \nA formula of $\\ensuremath{\\usftext{IST}}$ is called \\emph{internal} if it does not involve `st', and \\emph{external} otherwise. \n\n\\medskip\n\nThe system $\\ensuremath{\\usftext{IST}}$ is the internal system $\\ensuremath{\\usftext{ZFC}}$ plus the following\\footnote{The `fin' in \\textsf{(I)} means that $x$ is finite, i.e.\\ its number of elements are bounded by a natural number.} three external axioms \\emph{Idealisation}, \\emph{Standard Part}, and \\emph{Transfer} which govern the predicate `st'.\n\\begin{enumerate}\n\\item[\\textsf{(I)}] $(\\forall^{\\textup{NSA}^{\\alpha}~\\textup{fin}}x)(\\exists y)(\\forall z\\in x)\\varphi(z,y)\\rightarrow (\\exists y)(\\forall^{\\textup{NSA}^{\\alpha}}x)\\varphi(x,y)$, for internal $\\varphi$. \n\\item[\\textsf{(S)}] $(\\forall^{\\textup{NSA}^{\\alpha}} x)(\\exists^{\\textup{NSA}^{\\alpha}}y)(\\forall^{\\textup{NSA}^{\\alpha}}z)\\big((z\\in x\\wedge \\varphi(z))\\leftrightarrow z\\in y\\big)$, for any formula $\\varphi$.\n\\item[\\textsf{(T)}] $(\\forall^{\\textup{NSA}^{\\alpha}}t)\\big[(\\forall^{\\textup{NSA}^{\\alpha}}x)\\varphi(x, t)\\rightarrow (\\forall x)\\varphi(x, t)\\big]$, for internal $\\varphi$ and $t, x$ the only free variables. \n\\end{enumerate}\nNelson proves in \\cite{wownelly} that \\textsf{IST} is a conservative extension of \\textsf{ZFC}, i.e.\\ $\\ensuremath{\\usftext{ZFC}}$ and $\\ensuremath{\\usftext{IST}}$ prove the same (internal) sentences. \nVarious fragments of $\\ensuremath{\\usftext{IST}}$ have been studied previously, and we shall make essential use of the system $\\textup{\\textsf{P}}$, a fragment of $\\ensuremath{\\usftext{IST}}$ based on Peano arithmetic, introduced in Section~\\ref{PIPI}. \nThe system $\\textup{\\textsf{P}}$ was first introduced in \\cite{brie} and is exceptional in that it has a `term extraction procedure' \\emph{with a very wide scope}. We discuss this aspect of $\\textup{\\textsf{P}}$ in more detail in Remark~\\ref{firliborn}. \n\n\\subsection{The classical system $\\textup{\\textsf{P}}$}\\label{PIPI}\nIn this section, we introduce the classical system $\\textup{\\textsf{P}}$ which is a conservative extension of Peano arithmetic by Theorem \\ref{consresult}. \nWe refer to \\cite{kohlenbach3}*{\\S3.3} for the detailed definition of the rather mainstream system $\\textsf{E-PA}^{\\omega}$, i.e.\\ \\emph{Peano arithmetic in all finite types with the axiom of extensionality}. \nThe system $\\textup{\\textsf{P}}$ consist of the following axioms, starting with the basic ones. \n\\begin{defi}\\textup{rm}\\label{debs}[Basic axioms of $\\textup{\\textsf{P}}$]\n\\begin{enumerate}\n\\item The system \\textsf{E-PA}$^{\\omega*}$ is the definitional extension of \\textsf{E-PA}$^{\\omega}$ with types for finite sequences as in \\cite{brie}*{\\S2}. \n\\item The set ${\\mathbb T}^{*}$ is the collection of all the constants in the language of $\\textsf{E-PA}^{\\omega*}$. \n\\item The external induction axiom \\textsf{IA}$^{\\textup{NSA}^{\\alpha}}$ is \n\\be\\tag{\\textsf{IA}$^{\\textup{NSA}^{\\alpha}}$}\n\\Phi(0)\\wedge(\\forall^{\\textup{NSA}^{\\alpha}}n^{0})(\\Phi(n) \\rightarrow\\Phi(n+1))\\rightarrow(\\forall^{\\textup{NSA}^{\\alpha}}n^{0})\\Phi(n). \n\\ee\n\\item\\label{krafoi} The system $ \\textsf{E-PA}^{\\omega*}_{\\textup{NSA}^{\\alpha}} $ is defined as $ \\textsf{E-PA}^{\\omega{*}} + {\\mathbb T}^{*}_{\\textup{NSA}^{\\alpha}} + \\textsf{IA}^{\\textup{NSA}^{\\alpha}}$, where ${\\mathbb T}^{*}_{\\textup{NSA}^{\\alpha}}$\nconsists of the following basic axiom schemas.\n\\begin{enumerate}\n\\item The schema\\footnote{The language of $\\textsf{E-PA}_{\\textup{NSA}^{\\alpha}}^{\\omega*}$ contains a symbol $\\textup{NSA}^{\\alpha}_{\\sigma}$ for each finite type $\\sigma$, but the subscript is always omitted. Hence ${\\mathbb T}^{*}_{\\textup{NSA}^{\\alpha}}$ is an \\emph{axiom schema} and not an axiom.\\label{omit}} $\\textup{NSA}^{\\alpha}(x)\\wedge x=y\\rightarrow\\textup{NSA}^{\\alpha}(y)$. \\label{komit}\n\\item The schema providing for each closed term $t\\in {\\mathbb T}^{*}$ the axiom $\\textup{NSA}^{\\alpha}(t)$.\n\\item The schema $\\textup{NSA}^{\\alpha}(f)\\wedge \\textup{NSA}^{\\alpha}(x)\\rightarrow \\textup{NSA}^{\\alpha}(f(x))$.\n\\end{enumerate}\n\\end{enumerate}\nSecondly, Nelson's axiom \\emph{Standard part} is weakened in \\cite{brie} to $\\textup{\\textsf{HAC}}_{\\textup{int}}$:\n\\be\\tag{$\\textup{\\textsf{HAC}}_{\\textup{int}}$}\n(\\forall^{\\textup{NSA}^{\\alpha}}x^{\\rho})(\\exists^{\\textup{NSA}^{\\alpha}}y^{\\tau})\\varphi(x, y)\\rightarrow (\\exists^{\\textup{NSA}^{\\alpha}}F^{\\rho\\rightarrow \\tau^{*}})(\\forall^{\\textup{NSA}^{\\alpha}}x^{\\rho})(\\exists y^{\\tau}\\in F(x))\\varphi(x,y),\n\\ee\nwhere $\\varphi$ is any internal formula and $\\tau^{*}$ is the type of finite sequences of objects of type $\\tau$. Note that $F$ only provides a \\emph{finite sequence} of witnesses to $(\\exists^{\\textup{NSA}^{\\alpha}}y)$, explaining the name \\emph{Herbrandized Axiom of Choice} for $\\textup{\\textsf{HAC}}_{\\textup{int}}$.\n\n\\medskip\n \nThirdly, Nelson's axiom idealisation \\textsf{I} appears in \\cite{brie} as follows: \n\\be\\tag{\\textsf{I}}\n(\\forall^{\\textup{NSA}^{\\alpha}} x^{\\sigma^{*}})(\\exists y^{\\tau} )(\\forall z^{\\sigma}\\in x)\\varphi(z,y)\\rightarrow (\\exists y^{\\tau})(\\forall^{\\textup{NSA}^{\\alpha}} x^{\\sigma})\\varphi(x,y), \n\\ee\nwhere $\\varphi$ is internal and $\\sigma^{*}$ is the type of finite sequences of objects of type $\\sigma$.\n\\end{defi}\nFor $\\textup{\\textsf{P}}\\equiv \\textsf{E-PA}^{\\omega*}_{\\textup{NSA}^{\\alpha}} +\\textup{\\textsf{HAC}}_{\\textup{int}} +\\textsf{I}$, we have the following `term extraction theorem', which is not explicitly formulated or proved in \\cite{brie}. A proof may be found in \\cites{samzoo, sambon}. \n\\begin{thm}[Term extraction]\\label{consresult}\nLet $\\varphi$ be an internal formula and let $\\Delta_{\\ensuremath{{\\usftext{int}}}}$ be a collection of internal formulas. If we have:\n\\be\\label{antecedn}\n\\textup{\\textsf{P}} + \\Delta_{\\ensuremath{{\\usftext{int}}}} \\vdash (\\forall^{\\textup{NSA}^{\\alpha}}\\underline{x})(\\exists^{\\textup{NSA}^{\\alpha}}\\underline{y})\\varphi(\\underline{x}, \\underline{y}, \\underline{a})\n\\ee\nthen one can extract from the proof a sequence of closed terms $t$ in $\\mathcal{T}^{*}$ such that\n\\be\\label{consequalty}\n\\textup{\\textsf{E-PA}}^{\\omega*} + \\Delta_{\\ensuremath{{\\usftext{int}}}} \\vdash ( \\forall \\tup x) (\\exists \\tup y\\in \\tup t(\\tup x)) \\varphi(\\tup x,\\tup y, \\tup a).\n\\ee\n\\end{thm}\n\\begin{proof}\nThe proof of the theorem in a nutshell: A proof interpretation $S_{\\textup{NSA}^{\\alpha}}$ is defined in \\cite{brie}*{Def.\\ 7.1}; a tedious but straightforward verification using the clauses (i)-(v) in \\cite{brie}*{Def.\\ 7.1} establishes that $\\Phi(\\underline{a})^{S_{\\textup{NSA}^{\\alpha}}}\\equiv \\Phi(\\underline{a})$ for $\\Phi(\\underline{a})\\equiv (\\forall^{\\textup{NSA}^{\\alpha}}\\underline{x})(\\exists^{\\textup{NSA}^{\\alpha}}\\underline{y})\\varphi(\\underline{x}, \\underline{y}, \\underline{a})$ and $\\varphi$ internal. The theorem now follows immediately from \\cite{brie}*{Theorem 7.7}. \n\\end{proof}\nThe term $t$ in \\eqref{consequalty} is \\emph{primitive recursive} in the sense of G\\\"odel's system ${T}$. The latter was introduced in \\cite{godel3}, and is also discussed in \\cite{kohlenbach3}*{\\S3}. For the rest of this paper, a `normal form' will refer to a formula as in \\eqref{antecedn}, i.e.\\ of the form $ (\\forall^{\\textup{NSA}^{\\alpha}} \\tup x )(\\exists^{\\textup{NSA}^{\\alpha}} \\tup y) \\varphi(\\tup x,\\tup y, \\tup a)$ for internal $\\varphi$.\n\n\\medskip\n\nAs expected, the previous theorem does not really depend on the presence of full Peano arithmetic. \nIndeed, let \\textsf{E-PRA}$^{\\omega}$ be the system defined in \\cite{kohlenbach2}*{\\S2} and let \\textsf{E-PRA}$^{\\omega*}$ \nbe its definitional extension with types for finite sequences as in \\cite{brie}*{\\S2}. We permit ourselves a slight abuse of notation by not distinguishing between Kohlenbach's $\\textup{\\textsf{RCA}}_{0}^{\\omega}\\equiv \\textup{\\textsf{E-PRA}}^{\\omega}+\\textup{\\textsf{QF-AC}}^{1,0}$ (See \\cite{kohlenbach2}*{\\S2}) and $\\textup{\\textsf{E-PRA}}^{\\omega*}+\\textup{\\textsf{QF-AC}}^{1,0}$. \n\\begin{cor}\\label{consresultcor2}\nThe previous theorem and corollary go through for $\\textup{\\textsf{P}}$ and $\\textup{\\textsf{E-PA}}^{\\omega*}$ replaced by $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}\\equiv \\textsf{\\textup{E-PRA}}^{\\omega*}+{\\mathbb T}_{\\textup{NSA}^{\\alpha}}^{*} +\\textup{\\textsf{HAC}}_{\\textup{int}} +\\textsf{\\textup{I}}+\\textup{\\textsf{QF-AC}}^{1,0}$ and $\\textup{\\textsf{RCA}}_{0}^{\\omega}$. \n\\end{cor}\n\\begin{proof}\nThe proof of \\cite{brie}*{Theorem 7.7} goes through for any fragment of \\textsf{E-PA}$^{\\omega{*}}$ which includes \\textsf{EFA}, sometimes also called $\\textsf{I}\\Delta_{0}+\\textsf{EXP}$. \nIn particular, the exponential function is (all what is) required to `easily' manipulate finite sequences. \n\\end{proof}\nNext, we discuss the vast scope of the term extraction result in Theorem~\\ref{consresult}. \n\\begin{rem}[The scope of term extraction]\\label{firliborn}\\textup{rm}\nFirst of all, there are examples of classically provable sentences (See \\cite{kohlenbach3}*{\\S2.2}) with \\emph{only two} quantifier alternations from which no computational information can be extracted. \nBy contrast, it is shown in \\cite{sambon} that the scope of Theorem~\\ref{consresult} encompasses all theorems of `pure' Nonstandard Analysis, where `pure' means that only \\emph{nonstandard} definitions (of continuity, compactness, differentiability, Riemann integration, et cetera) are used. Indeed, it is easy to prove in $\\textup{\\textsf{P}}$ (or $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$) that these nonstandard definitions have equivalent normal forms, and that an implication between two normal forms is again equivalent to a normal form. In other words, the scope of the term extraction result in Theorem~\\ref{consresult} is vast, as explored in \\cites{sambon, samzoo, samzooII, samGH}. \n\\end{rem}\nFinally, we note that a `constructive' version of $\\textup{\\textsf{P}}$ is introduced in \\cite{brie}*{\\S5}. \nIn particular, the system $\\textup{\\textsf{H}}$ is a conservative extension of Heyting arithmetic $\\textsf{E-HA}^{\\omega}$ for the latter's language, and satisfies \na term extraction theorem similar to Theorem~\\ref{consresult} (See \\cite{brie}*{Theorem~5.9}). We briefly discuss $\\textup{\\textsf{H}}$ in Remark \\ref{reasonsofspace}. \n\n\n\\subsection{Notations}\nWe mostly use the notations from \\cite{brie}, some of which we repeat. \n\n\\medskip\n\nFirst of all, the following notations were sketched in Section \\ref{IST}. \n\\begin{rem}[Notations]\\label{notawin}\\textup{rm}\nWe write $(\\forall^{\\textup{NSA}^{\\alpha}}x^{\\tau})\\Phi(x^{\\tau})$ and $(\\exists^{\\textup{NSA}^{\\alpha}}x^{\\sigma})\\Psi(x^{\\sigma})$ as short for the formula\n$(\\forall x^{\\tau})\\big[\\textup{NSA}^{\\alpha}(x^{\\tau})\\rightarrow \\Phi(x^{\\tau})\\big]$ and $(\\exists x^{\\sigma})\\big[\\textup{NSA}^{\\alpha}(x^{\\sigma})\\wedge \\Psi(x^{\\sigma})\\big]$. \nWe also write $(\\forall x^{0}\\in \\Omega)\\Phi(x^{0})$ and $(\\exists x^{0}\\in \\Omega)\\Psi(x^{0})$ as short for \n$(\\forall x^{0})\\big[\\neg\\textup{NSA}^{\\alpha}(x^{0})\\rightarrow \\Phi(x^{0})\\big]$ and $(\\exists x^{0})\\big[\\neg\\textup{NSA}^{\\alpha}(x^{0})\\wedge \\Psi(x^{0})\\big]$. \nFinally, a formula $A$ is `internal' if it does not involve $\\textup{NSA}^{\\alpha}$. The formula $A^{\\textup{NSA}^{\\alpha}}$ is defined from $A$ by appending `st' to all quantifiers (except bounded number quantifiers). \n\\end{rem}\nSecondly, we use the usual notations for rational and real numbers in $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ as introduced in \\cite{kohlenbach2}*{p.\\ 288-289}. \nWe repeat some of the latter definitions. \n\\begin{defi}[Real numbers and related notions]\\label{keepinitreal}\\textup{rm}~\n\\begin{enumerate}\n\\item A (standard) rational $q^{0}$ is a fraction $\\pm\\frac{m}{n}$ for (standard) $n^{0}>0$ and (standard) $m^{0}$. We write `$q^{0}\\in {\\mathbb Q}$' to denote that $q$ is a rational. \n\\item A (standard) real number $x$ is a (standard) fast-converging Cauchy sequence $q_{(\\cdot)}^{1}$, i.e.\\ $(\\forall n^{0}, i^{0})(|q_{n}-q_{n+i})|<_{0} \\frac{1}{2^{n}})$. \nWe use Kohlenbach's `hat function' from \\cite{kohlenbach2}*{p.\\ 289} to guarantee that every sequence $f^{1}$ is a real. \n\\item We write $[x](k):=q_{k}$ for the $k$-th approximation of a real $x^{1}=(q^{1}_{(\\cdot)})$. \n\\item Two reals $x, y$ represented by $q_{(\\cdot)}$ and $r_{(\\cdot)}$ are \\emph{equal}, denoted $x=_{{\\mathbb{R}}}y$, if $(\\forall n>0)(|q_{n}-r_{n}|\\leq \\frac{1}{2^{n-1}})$. Inequality $x<_{{\\mathbb{R}}}y$ is defined by $(\\exists n>0)(q_{n}+ \\frac{1}{2^{n-1}}< r_{n})$. \n\\item We write $x\\approx y$ if $(\\forall^{\\textup{NSA}^{\\alpha}} n>0)(|q_{n}-r_{n}|\\leq \\frac{1}{2^{n-1}})$ and $x\\gg y$ if $x>_{{\\mathbb{R}}}y\\wedge x\\not\\approx y$. \n\\item Functions $F:{\\mathbb{R}}\\rightarrow {\\mathbb{R}}$ mapping reals to reals are represented by functionals $\\Phi^{1\\rightarrow 1}$ mapping equal reals to equal reals, i.e. \n\\be\\tag{\\textsf{RE}}\\label{furg}\n(\\forall x^{1}, y^{1})(x=_{{\\mathbb{R}}}y\\rightarrow \\Phi(x)=_{{\\mathbb{R}}}\\Phi(y)).\n\\ee \n\\item Sets of objects of type $\\rho$ are denoted $X^{\\rho\\rightarrow 0}, Y^{\\rho\\rightarrow 0}, Z^{\\rho\\rightarrow 0}, \\dots$ and are given by their characteristic functions $f^{\\rho\\rightarrow 0}_{X}$, i.e.\\ $(\\forall x^{\\rho})[x\\in X\\leftrightarrow f_{X}(x)=_{0}1]$, where $f_{X}^{\\rho\\rightarrow 0}$ is assumed to output zero or one. \n\\end{enumerate}\n\\end{defi}\n\nThirdly, we use the usual extensional notion of equality in $\\textup{\\textsf{P}}$. \n\\begin{rem}[Equality in $\\textup{\\textsf{P}}$]\\label{equ}\\textup{rm}\nEquality between natural numbers `$=_{0}$' is a primitive. Equality `$=_{\\tau}$' for type $\\tau$-objects $x,y$ is then defined as follows:\n\\be\\label{aparth}\n[x=_{\\tau}y] \\equiv (\\forall z_{1}^{\\tau_{1}}\\dots z_{k}^{\\tau_{k}})[xz_{1}\\dots z_{k}=_{0}yz_{1}\\dots z_{k}]\n\\ee\nif the type $\\tau$ is composed as $\\tau\\equiv(\\tau_{1}\\rightarrow \\dots\\rightarrow \\tau_{k}\\rightarrow 0)$.\nIn the spirit of Nonstandard Analysis, we define `approximate equality $\\approx_{\\tau}$' as follows:\n\\be\\label{aparth2}\n[x\\approx_{\\tau}y] \\equiv (\\forall^{\\textup{NSA}^{\\alpha}} z_{1}^{\\tau_{1}}\\dots z_{k}^{\\tau_{k}})[xz_{1}\\dots z_{k}=_{0}yz_{1}\\dots z_{k}]\n\\ee\nwith the type $\\tau$ as above. \nThe system $\\textup{\\textsf{P}}$ includes the \\emph{axiom of extensionality}:\n\\be\\label{EXT}\\tag{\\textsf{E}} \n(\\forall x^{\\rho},y^{\\rho}, \\varphi^{\\rho\\rightarrow \\tau}) \\big[x=_{\\rho} y \\rightarrow \\varphi(x)=_{\\tau}\\varphi(y) \\big].\n\\ee\nHowever, as noted in \\cite{brie}*{p.\\ 1973}, the so-called axiom of \\emph{standard} extensionality \\eqref{EXT}$^{\\textup{NSA}^{\\alpha}}$ is not included in $\\textup{\\textsf{P}}$, as this would jeopardise the term extraction property as in Theorem \\ref{consresult}. \nFinally, a functional $\\Xi^{ 1\\rightarrow 0}$ is called an \\emph{extensionality functional} for $\\varphi^{1\\rightarrow 1}$ if \n\\be\\label{turki}\n(\\forall k^{0}, f^{1}, g^{1})\\big[ \\overline{f}\\Xi(f,g, k)=_{0}\\overline{g}\\Xi(f,g,k) \\rightarrow \\overline{\\varphi(f)}k=_{0}\\overline{\\varphi(g)}k \\big], \n\\ee\ni.e.\\ $\\Xi$ witnesses \\eqref{EXT} for $\\varphi$. \nAs will become clear in Section \\ref{main}, $\\eqref{EXT}^{\\textup{NSA}^{\\alpha}}$ is translated to the existence of an extensionality functional when applying Theorem \\ref{consresult}. \n\\end{rem} \n\n\\section{An analogue of Grilliot's trick in Nonstandard Analysis}\\label{main}\nIn this section, we show how from certain equivalences in Nonstandard Analysis involving a fragment of Nelson's \\emph{Transfer}, namely $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, one obtains \\emph{effective} RM-equivalences involving $(\\exists^{2})$ in Kohlenbach's higher-order RM. \n\\be\\tag{$\\exists^{2}$}\n(\\exists \\varphi^{2})(\\forall f^{1})\\big[ (\\exists n)(f(n)=0)\\leftrightarrow \\varphi(f)=0 \\big].\n\\ee\n\\be\\tag{$\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$}\n(\\forall^{\\textup{NSA}^{\\alpha}}f^{1})\\big[(\\forall^{\\textup{NSA}^{\\alpha}}n)f(n)\\ne 0 \\rightarrow (\\forall n)f(n)\\ne0 \\big].\n\\ee\nTo this end, we shall make use of a technique from Nonstandard Analysis we call \\emph{the nonstandard extensionality trick}, and which is similar to \\emph{Grilliot's trick}. We first introduce some of the above italicised notions in the following section.\n\\subsection{Preliminaries}\nIn this section, we introduce the notion of `effective implication' and so-called Grilliot's trick. \nFirst of all, the notion of `effective implication' is defined as one would expect in $\\textup{\\textsf{RCA}}_{0}^{\\omega}$. \n\\begin{defi}\\textup{rm}[Effective implication]\\label{effimp}\nAn implication $(\\exists \\Phi)A(\\Phi)\\rightarrow (\\exists \\Psi)B(\\Psi)$ (proved in $\\textup{\\textsf{RCA}}_{0}$) is \\emph{effective} if there is a term $t$ (in the language of $\\textup{\\textsf{RCA}}_{0}^{\\omega}$) such that additionally $(\\forall \\Phi)[A(\\Phi)\\rightarrow B(t(\\Phi))]$ (proved in $\\textup{\\textsf{RCA}}_{0}^{\\omega}$). \n\\end{defi}\nThe terms obtained using Theorem \\ref{consresult} are \\emph{primitive recursive} in the sense of G\\\"odel's system ${T}$, as discussed in Section \\ref{PIPI}. \nIn light of the elementary nature of an extensionality functional (See Remark~\\ref{equ}), we still refer to an implication as `effective', if the term $t$ as in Definition \\ref{effimp} involves an extensionality functional. \nNote that $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ proves the existence of an extensionality functional thanks to $\\textup{\\textsf{QF-AC}}^{1,0}$ while an unbounded search (available in a more general setting than G\\\"odel's $T$) also yields such a functional. \n\n\\medskip\n\nSecondly, as to methodology, we shall make use of a nonstandard technique, called \\emph{the nonstandard extensionality trick} (See Remark \\ref{trick}), similar to \\emph{Grilliot's trick}. \nNow, the latter trick is in fact an explicit construction to obtain the Turing jump functional $(\\exists^{2})$ from a given effectively discontinuous functional \n(See e.g.\\ \\cite{grilling}, \\cite{kohlenbach2}*{Prop.~3.7}, or \\cite{kooltje}*{Prop.\\ 3.4} for more details). In our \\emph{nonstandard} trick, one obtains $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ from a functional $\\Phi^{1\\rightarrow 1}$ which is \\emph{nonstandard} discontinuous, i.e.\\ there are\n$x_{0}\\approx_{1} x_{1}$ such that $ \\Phi(x_{0})\\not\\approx_{1} \\Phi(x_{1})$. By applying term extraction as in Theorem~\\ref{consresult}, one then obtains an effective implication involving $(\\exists^{2})$.\nAs we will see, the nonstandard proof involving $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ uses proof by contradiction, i.e.\\ \\emph{no attempt to obtain effective content is made in the nonstandard proofs}. \n\n\\medskip\n\nThirdly, in the next sections, we apply the aforementioned nonstandard extensionality trick to \\emph{binary expansion}, the \\emph{intermediate value theorem}, the \\emph{Weierstra\\ss~maximum theorem}, and \\emph{weak weak K\\\"onig's lemma}. We choose these theorems due to their `non-constructive' nature, and as some of the associated uniform versions (sometimes involving sequences) have been studied (\\cites{sayo, polahirst, yamayamaharehare}, \\cite{simpson2}*{IV.2.12}, \\cite{kohlenbach2}*{\\S3}). \nIt is particularly interesting that we can `recycle' the Brouwerian counterexamples to the intermediate value theorem and Weierstra\\ss~maximum theorem (\\cite{beeson1}*{I.7}, \\cite{mandje2}) to obtain nonstandard equivalences. \n\n\\medskip\n\nFinally, it is a natural question \\emph{why} we can obtain computational information from proofs in classical Nonstandard Analysis \\emph{at all}. \nIndeed, Bishop and Connes have made rather strong claims regarding the non-constructive nature of Nonstandard Analysis (See\\footnote{The third reference is Bishop's review of Keisler's introduction to Nonstandard Analysis \\cite{keisler3}.} \\cite{kluut}*{p.\\ 513}, \\cite{bishl}*{p.\\ 1}, \\cite{kuddd}, \\cite{conman2}*{p.\\ 6207} and \\cite{conman}*{p.\\ 26}). Furthermore, there are examples of classically provable sentences (See \\cite{kohlenbach3}*{\\S2.2}) with \\emph{only two} quantifier alternations from which no computational information can be extracted, and the aforementioned theorems involve a lot more quantifier alternations. Moreover, our nonstandard proofs make use of `proof by contradiction', i.e.\\ no attempt at a `constructive' proof is made.\nNonetheless, in Sections \\ref{bincco} to \\ref{hivt}, we shall obtain effective equivalences from certain `non-constructive' nonstandard equivalences. \nFollowing similar results in Section \\ref{carmichael}, we offer an explanation why Nonstandard Analysis contains so much computational information. \n\n\\subsection{Binary conversion}\\label{bincco}\nIn this section, we study the principle of \\emph{binary conversion}, i.e.\\ the statement that every real can be represented in binary as follows:\n\\be\\label{bin}\\tag{$\\textup{\\textsf{BIN}}$}\\textstyle\n(\\forall x\\in [0,1])(\\exists\\alpha^{1}\\leq_{1}1)(x=_{{\\mathbb{R}}}\\sum_{i=1}^{\\infty}\\frac{\\alpha(i)}{2^{i}}).\n\\ee\nHirst shows in \\cite{polahirst} that $\\textup{\\textsf{RCA}}_{0}$ proves $\\textup{\\textsf{BIN}}$, and that a uniform version of the latter \\emph{involving sequences} is equivalent to $\\textup{\\textsf{WKL}}$. \nFurthermore, $\\textup{\\textsf{BIN}}$ is equivalent to ${\\textup{\\textsf{LLPO}}}$ in \\emph{constructive} Reverse Mathematics (\\cite{bridges1}*{p.\\ 10}), while a finer classification may be found in \\cite{bergske}. \nWe study a \\emph{higher-type} {uniform} version of $\\textup{\\textsf{BIN}}$:\n\\be\\label{ubin}\\tag{$\\textup{\\textsf{UBIN}}$}\\textstyle\n(\\exists \\Phi:{\\mathbb{R}}\\rightarrow 1)(\\forall x\\in [0,1])\\big[ \\Phi(x)\\leq_{1}1\\wedge x=_{{\\mathbb{R}}}\\sum_{i=1}^{\\infty}\\frac{\\alpha(i)}{2^{i}}\\big].\n\\ee\nWe shall first establish a particular nonstandard equivalence involving $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, and a nonstandard version of $\\textup{\\textsf{UBIN}}$. As a result of applying Corollary~\\ref{consresultcor2} to this nonstandard equivalence, we obtain an \\emph{effective} equivalence between $\\textup{\\textsf{UBIN}}$ and the following version of arithmetical comprehension. \n\\be\\tag{$\\mu^{2}$}\\label{Frak}\n(\\exists \\mu^{2})\\big[(\\forall f^{1})\\big((\\exists x^{0})f(x)=0 \\rightarrow f(\\mu(f))=0 \\big)\\big]\n\\ee \nThe functional $(\\mu^{2})$ is also known as \\emph{Feferman's non-constructive mu-operator} (See \\cite{avi2}*{\\S8.2}), and is equivalent to $(\\exists^{2})$ in $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ by \\cite{kooltje}*{\\S3}. \nWe denote by $\\textup{\\textsf{MU}}(\\mu)$ the formula in square brackets in $(\\mu^{2})$. \nWe use the following nonstandard version of $\\textup{\\textsf{UBIN}}$, called $\\textup{\\textsf{UBIN}}^{+}$:\n\\[\n(\\exists^{\\textup{NSA}^{\\alpha}} \\Phi:{\\mathbb{R}}\\rightarrow 1)(\\forall^{\\textup{NSA}^{\\alpha}} x\\in [0,1])\\big[\\textup{\\textsf{UBIN}}(\\Phi, x)\\wedge (\\forall^{\\textup{NSA}^{\\alpha}}x, y\\in [0,1])(x\\approx y \\rightarrow \\Phi(x)\\approx_{1}\\Phi(y)) \\big]. \n\\]\nwhere $\\textup{\\textsf{UBIN}}(\\Phi, x)$ is the formula in square brackets in $\\textup{\\textsf{UBIN}}$. Note that the second conjunct of $\\textup{\\textsf{UBIN}}^{+}$ expresses that $\\Phi$ is `standard extensional', i.e.\\ satisfies extensionality as in \\eqref{furg} relative to `st' (and the range is Baire space instead of ${\\mathbb{R}}$). \n\\begin{thm}\\label{proto7}\nFrom a proof in $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$ that $\\textup{\\textsf{UBIN}}^{+}\\leftrightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, terms $s,t$ can be extracted such that $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ proves:\n\\be\\label{frood7}\n(\\forall \\mu^{2})\\big[\\textsf{\\textup{\\textsf{MU}}}(\\mu)\\rightarrow \\textup{\\textsf{UBIN}}(s(\\mu)) \\big] \\wedge (\\forall \\Phi^{1\\rightarrow 1})\\big[ \\textup{\\textsf{UBIN}}(\\Phi)\\rightarrow \\textup{\\textsf{MU}}(t(\\Phi, \\Xi)) \\big].\n\\ee\nwhere $\\Xi$ is an extensionality functional for $\\Phi$ and $\\textup{\\textsf{UBIN}}(\\Phi)$ is $(\\forall x\\in [0,1])\\textup{\\textsf{UBIN}}(\\Phi, x)$. \n\\end{thm}\n\\begin{proof}\nFirst of all, we prove that $\\textup{\\textsf{UBIN}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ in $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$, and obtain the associated second conjunct of \\eqref{frood7}. The remaining results are then sketched.\n\n\\medskip\n\nTo prove $\\textup{\\textsf{UBIN}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ in $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$, assume $\\textup{\\textsf{UBIN}}^{+}$ and suppose that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ is false, i.e.\\ there is \\emph{standard} $g$ such that $(\\forall^{\\textup{NSA}^{\\alpha}}n)g(n)=0 $ but also $ (\\exists m^{0})g(m)\\ne 0$. Now define the \\emph{standard} sequence $\\alpha_{0}$ as follows\n\\be\\label{sly}\n\\alpha_{0}(i):=\n\\begin{cases}\n0 & (\\forall n\\leq i)g(n)=0 \\\\\n1 & \\text{otherwise}\n\\end{cases}.\n\\ee\nFurthermore, define the \\emph{standard} reals $x_{\\pm}:= \\frac{1}{2}\\pm \\sum_{n=1}^{\\infty} \\frac{\\alpha_{0}(n)}{2^{n}}$ and note that $x_{+}\\approx x_{-}$ by the definition of $g$. Since $x_{-}<_{{\\mathbb{R}}} \\frac12 <_{{\\mathbb{R}}}x_{+}$, the binary expansion $\\alpha_{\\pm}$ of $x_{\\pm}$ must be such that $\\alpha_{-}(1)=0$ and $\\alpha_{+}(1)=1$. However, this implies that $\\Phi(x_{-})(1)=0\\ne 1=\\Phi(x_{+})(1)$, and also $\\Phi(x_{-})\\not\\approx_{1}\\Phi(x_{+})$. \nClearly, the latter contradicts the standard extensionality of $\\Phi$ as $x_{+}\\approx x_{-}$ was also proved. \nIn light of this contradiction, we must have $\\textup{\\textsf{UBIN}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. \n\n\\medskip\n\n\nWe now prove the second conjunct in \\eqref{frood7}. \nNote that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ can easily be brought into the following normal form: \n\\be\\label{frux}\n(\\forall^{\\textup{NSA}^{\\alpha}}f^{1})(\\exists^{\\textup{NSA}^{\\alpha}}i^{0})\\big[(\\exists n^{0})f(n)=0\\rightarrow (\\exists m\\leq i)f(m)=0\\big], \n\\ee\nwhere the formula in square brackets is abbreviated by $B(f, i)$. Similarly, the second conjunct of $\\textup{\\textsf{UBIN}}^{+}$ has the following normal form: \n\\be\\label{kurve2}\\textstyle\n(\\forall^{\\textup{NSA}^{\\alpha}} x^{1}, y^{1}\\in [0,1], k^{0})(\\exists^{\\textup{NSA}^{\\alpha}} N)\\big[|x-y|<\\frac{1}{N} \\rightarrow \\overline{\\Phi(x)}k=_{0}\\overline{\\Phi(y)}k\\big],\n\\ee\nwhich is immediate by resolving `$\\approx_{1}$' and `$\\approx$', and bringing standard quantifiers outside. \nWe denote the formula in square brackets in \\eqref{kurve2} by $A(x, y, N, k, \\Phi)$. \nHence, $\\textup{\\textsf{UBIN}}^+\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ now easily yields:\n\\begin{align}\n(\\forall^{\\textup{NSA}^{\\alpha}}\\Phi, \\Xi)\\big[ [(\\forall^{\\textup{NSA}^{\\alpha}}x\\in [0,1])\\textup{\\textsf{UBIN}}(\\Phi, x)&\\wedge (\\forall^{\\textup{NSA}^{\\alpha}} x,y\\in [0,1], k^{0})A(x, y, \\Xi(x, y, k), k, \\Phi)]\\notag\\\\\n& \\rightarrow (\\forall^{\\textup{NSA}^{\\alpha}}f^{1})(\\exists^{\\textup{NSA}^{\\alpha}}n)B(f,n)\\big], \\label{tochtag}\n\\end{align}\nas standard $\\Xi$ as in the antecedent of \\eqref{tochtag} yields standard outputs for standard inputs, and hence \\eqref{kurve2} follows. \nDropping the `st' in the antecedent of \\eqref{tochtag} and bringing out the remaining standard quantifiers, we obtain the normal form:\n\\begin{align}\n(\\forall^{\\textup{NSA}^{\\alpha}}\\Phi, \\Xi ,f^{1})(\\exists^{\\textup{NSA}^{\\alpha}}n)\\big[ [(\\forall x^{1}\\in [0,1])&\\textup{\\textsf{UBIN}}(\\Phi, x)\\wedge\\notag \\\\\n& (\\forall U^{1}, S^{1}, k^{0})A(U, S, \\Xi(U, S, k), k, \\Phi)] \\rightarrow B(f,n)\\big] \\label{enoka}\n\\end{align}\nLet $C(\\Phi, \\Xi, f, n)$ be the formula in big square brackets and apply Corollary~\\ref{consresultcor2} to `$\\textup{\\textsf{RCA}}_{0}^{\\Lambda}\\vdash (\\forall^{\\textup{NSA}^{\\alpha}}\\Phi, \\Xi, f^{1})(\\exists^{\\textup{NSA}^{\\alpha}}n)C(\\Phi, \\Xi, f, n)$' to obtain a term $t$ such that $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ proves \n\\be\\label{drifgs}\n (\\forall \\Phi, \\Xi, f^{1})(\\exists n\\in t(\\Phi, \\Xi, f))C(\\Phi, \\Xi, f, n). \n \\ee\nNow define the term $s(\\Phi, \\Xi, f)$ as $\\max_{i<|t(\\Phi, \\Xi, f)|}t(\\Phi, \\Xi, f)(i)$ and \nnote that the formula $(\\exists n\\in t(\\Phi, \\Xi, f))C(\\Phi, \\Xi, f, n)$ implies $C(\\Phi, \\Xi, f, s(\\Phi, \\Xi, f))$.\nFinally, bring the quantifier involving $f$ inside $C$ to obtain for all $\\Phi, \\Xi$ that\n\\[\n [(\\forall x^{1}\\in[0,1])\\textup{\\textsf{UBIN}}(\\Phi, x)\\wedge (\\forall U^{1}, S^{1}, k^{0})A(U, S, \\Xi(U, S, k), k, \\Phi)] \\rightarrow (\\forall f^{1}) B(f,s(\\Phi, \\Xi, f)).\n\\]\nThus, $s(\\Phi, \\Xi, \\cdot)$ provides the functional $(\\mu^{2})$ if $\\Phi$ satisfies $(\\forall x^{1}\\in[0,1])\\textup{\\textsf{UBIN}}(\\Phi,x )$ and $\\Xi$ is the associated extensionality functional. \n\n\\medskip\n\nFinally, to prove $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}\\rightarrow \\textup{\\textsf{UBIN}}^{+}$, consider \\eqref{frux} and apply $\\textup{\\textsf{HAC}}_{\\textup{int}}$ to the former to obtain $\\nu^{1\\rightarrow 0^{*}}$ such that $(\\forall^{\\textup{NSA}^{\\alpha}}f^{1})(\\exists i^{0}\\in \\nu({f}))A(f, i)$, where $A$ is the formula in square brackets in \\eqref{frux}. Now define the \\emph{standard} functional $\\xi^{2}$ by \n\\[\n\\xi(f):=(\\mu m\\leq \\max_{i<|\\nu(f)|}\\nu(f)(i))(f(m)=0)\n\\]\nand note that $[\\textup{\\textsf{MU}}(\\xi)]^{\\textup{NSA}^{\\alpha}}$, i.e.\\ we have access to Feferman's search operator relative to `st'. In particular, $\\xi^{2}$ provides arithmetical comprehension (and \\emph{Transfer} for $\\Pi_{1}^{0}$-formulas):\n\\be\\label{karmic}\n(\\forall^{\\textup{NSA}^{\\alpha}}f^{1})\\big[( f(\\xi(f))=0) \\leftrightarrow (\\exists m^{0})(f(m)=0)\\leftrightarrow (\\exists^{\\textup{NSA}^{\\alpha}} m^{0})(f(m)=0) \\big]. \n\\ee\nTo define $\\Phi$ as in $\\textup{\\textsf{UBIN}}^{+}$, use $\\xi$ from \\eqref{karmic} to decide if $x\\geq_{{\\mathbb{R}}}\\frac12$ or $x<_{{\\mathbb{R}}}\\frac12$ and define $\\Phi(x)(0)$ as $1$ or $0$ respectively. \nSimilarly, define $\\Phi(x)(n+1)$ as $1$ or $0$ depending on whether $x\\geq_{{\\mathbb{R}}}\\frac{1}{2}(x+\\sum_{i=0}^{n}\\frac{\\Phi(x)(i)}{2^{i}})$ or not, again using $\\xi$. Then $\\Phi$ is standard and \nsatisfies $[\\textup{\\textsf{UBIN}}(\\Phi)]^{\\textup{NSA}^{\\alpha}}$. Now apply $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ to the latter and the axiom of extensionality to obtain $\\textup{\\textsf{UBIN}}^{+}$. The first conjunct of \\eqref{frood7} now follows in the same way as in the first part of the proof. \n\\end{proof}\nNote that the non-computable power of \\emph{uniform} $\\textup{\\textsf{BIN}}$ (both nonstandard and non-nonstandard) arises from the fact that not all reals have a unique binary expansion.\nHence, for small (infinitesimal) variations of the input of the functional in $\\textup{\\textsf{UBIN}}$, we can produce large (standard) variations in the output. This is exploited as follows in the previous proof. \n\\begin{rem}[Nonstandard extensionality trick]\\label{trick}\\textup{rm}\nFirst of all, we note that $\\Phi$ as in $\\textup{\\textsf{UBIN}}(\\Phi)$ is \\emph{nonstandard} discontinuous in that for every $x,y\\in {\\mathbb{R}}$ such that $x<_{{\\mathbb{R}}}\\frac{1}{2}<_{{\\mathbb{R}}}y \\wedge x \\approx \\frac{1}{2}\\approx y$ we have $\\Phi(x)\\not\\approx_{1}\\Phi(y)$, in particular $\\Phi(x)(1)=0\\ne 1=\\Phi(y)(1)$. \nSecondly, we use \\eqref{sly} to define \\emph{standard} points $x_{\\pm}$ at which $\\Phi$ from $\\textup{\\textsf{UBIN}}^{+}$ is \\emph{nonstandard discontinuous}, assuming $\\neg\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. \nThe ensuing contradiction with the standard extensionality of $\\Phi$ yields $\\textup{\\textsf{UBIN}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. Thirdly, applying term extraction to the (normal form \\eqref{enoka} of the) latter implication, we obtain the effective implication \\eqref{frood7}. \n\\end{rem}\nThe previous technique is similar in spirit to Grilliot's trick, but note that our nonstandard technique produces an effective implication, \\emph{without paying attention to effective content}. \nIn particular, we used the non-constructive `proof by contradiction' to establish $\\textup{\\textsf{UBIN}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, and `independence of premises' to obtain the latter's normal form \\eqref{enoka} (See e.g.\\ \\eqref{frux} and \\eqref{kurve2}). \n\n\\medskip\n\nNote that we do not claim that the previous theorem (or the below theorems) is unique or a first in this regard: Kohlenbach's treatment of Grilliot's trick (\\cite{kooltje}) and the \\emph{proof mining} program (\\cite{kohlenbach3}) are well-known to produce effective results from classical mathematics. \nWhat is surprising about results in this paper (in our opinion) is the emergence of effective content (with relative ease) from Nonstandard Analysis \\emph{despite} claims \nthat the latter is somehow fundamentally non-constructive by e.g.\\ Bishop and Connes (See \\cite{samsynt} for a detailed discussion of the Bishop-Connes critique). \n\n\\medskip\n\nSurprisingly, the proof of Theorem \\ref{proto7} goes through constructively, as we discuss now. \n\\begin{rem}[The system $\\textup{\\textsf{H}}$]\\label{reasonsofspace}\\textup{rm}\nThe system $\\textup{\\textsf{H}}$ is a conservative extension of Heyting arithmetic satisfying a term extraction theorem similar to Theorem \\ref{consresult} (See \\cite{brie}*{Theorem~5.9}).\nAlthough $\\textup{\\textsf{H}}$ is based on intuitionistic logic, it does prove the following `standard' version of Markov's principle (See \\cite{brie}*{p.\\ 1978}):\n\\be\\tag{$\\textsf{MP}^{\\textup{NSA}^{\\alpha}}$}\n(\\forall^{\\textup{NSA}^{\\alpha}} f^{1})\\big[ \\neg\\neg[(\\exists^{\\textup{NSA}^{\\alpha}}m)(f(m)=0)] \\rightarrow (\\exists^{\\textup{NSA}^{\\alpha}}n)(f(n)=0) \\big]\n\\ee\nNow, the proof of $ \\textup{\\textsf{UBIN}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ in Theorem \\ref{proto7} easily yields a proof of: \n\\be\\label{lekkerbekske}\n\\textup{\\textsf{UBIN}}^{+} \\rightarrow (\\forall^{\\textup{NSA}^{\\alpha}}f^{1})\\big[(\\exists m)(f(m)=0) \\rightarrow \\neg[(\\forall^{\\textup{NSA}^{\\alpha}}n^{0})(f(n)=0)] \\big]\n\\ee\ninside the system $\\textup{\\textsf{H}}$. \nHowever, combined with $\\textsf{MP}^{\\textup{NSA}^{\\alpha}}$, \\eqref{lekkerbekske} yields that $\\textup{\\textsf{H}}$ also proves the implication $ \\textup{\\textsf{UBIN}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, using the same `proof by contradiction' proof used for Theorem \\ref{proto7}. Furthermore, similar to $\\textsf{MP}^{\\textup{NSA}^{\\alpha}}$, the system $\\textup{\\textsf{H}}$ also contains a `standard' version of the independence of premise schema (in the form of $\\textsf{HIP}_{\\forall^{\\textup{NSA}^{\\alpha}}}$; see \\cite{brie}*{p.\\ 1978}). Thanks to this schema, $\\textup{\\textsf{H}}$ proves $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}} \\rightarrow \\eqref{frux}$ and even that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}} \\rightarrow \\textup{\\textsf{UBIN}}^{+}$ implies its normal form \\eqref{enoka}. Applying the term extraction theorem \\cite{brie}*{Theorem~5.9} for $\\textup{\\textsf{H}}$, a constructive proof of \\eqref{frood7} is established. \n\\end{rem}\n\n\n\\subsection{Weak weak K\\\"onig's lemma}\\label{FUWKL}\nIn this section, we study the principle \\emph{weak weak K\\\"onig's lemma} ($\\textup{\\textsf{WWKL}}$ for short), using the standard extensionality trick in Remark \\ref{trick}.\nNote that $\\textup{\\textsf{WWKL}}$ was not directly studied in \\cites{samzoo, samzooII}. \n\\begin{defi}\\textup{rm}[Weak weak K\\\"onig's lemma]\\label{leipi}~\n\\begin{enumerate}\n\\item We reserve `$T^{1}$' for trees and denote by `$T^{1}\\leq_{1}1$' that $T$ is a \\emph{binary} tree. \n\\item For a binary tree $T$, define $\\nu(T):=\\lim_{n\\rightarrow \\infty}\\frac{\\{\\sigma \\in T: |\\sigma|=n \\}}{2^{n}}$.\n\\item For a binary tree $T$, define `$\\nu(T)>_{{\\mathbb{R}}}a^{1}$' as $(\\exists k^{0})(\\forall n^{0})\\big(\\frac{\\{\\sigma \\in T: |\\sigma|=n \\}}{2^{n}}\\geq a+\\frac{1}{k}\\big)$.\n\\item We define $\\textup{\\textsf{WWKL}}$ as $(\\forall T \\leq_{1}1)\\big[ \\nu(T)>_{{\\mathbb{R}}}0\\rightarrow (\\exists \\beta\\leq_{1}1)(\\forall m)(\\overline{\\beta}m\\in T) \\big]$.\n\\end{enumerate} \n\\end{defi}\nThe principle $\\textup{\\textsf{WWKL}}$ is not part of the `Big Five' of RM, but there are \\emph{some} equivalences involving the former (See \\cite{simpson2}*{X.1}). \nIn this section, we study the following uniform versions:\n\\be\\tag{$\\textup{\\textsf{UWWKL}}$}\n(\\exists \\Phi^{1\\rightarrow 1})(\\forall T \\leq_{1}1)\\big[ \\nu(T)>_{{\\mathbb{R}}}0\\rightarrow (\\forall m)(\\overline{\\Phi(T)}m\\in T) \\big]\n\\ee\nAlso, $\\textup{\\textsf{UWWKL}}(\\Phi(T), T)$ is $\\textup{\\textsf{UWWKL}}$ without the leading quantifiers, and $\\textup{\\textsf{UWWKL}}^{+}$ is\n\\[\n(\\exists^{\\textup{NSA}^{\\alpha}}\\Phi^{1\\rightarrow 1})\\big[(\\forall^{\\textup{NSA}^{\\alpha}}T^{1})\\textup{\\textsf{UWWKL}}(\\Phi(T), T)\\wedge (\\forall^{\\textup{NSA}^{\\alpha}} T^{1}, S^{1})\\big(T\\approx_{1} S \\rightarrow \\Phi(T)\\approx_{1}\\Phi(S) \\big)\\big].\n\\]\nNote that the second conjunct expresses that $\\Phi$ is \\emph{standard extensional}. \nWe have the following theorem, which is the effective version of \\cite{yamayamaharehare}*{Theorem 3.2}. Note that $\\textup{\\textsf{UWWKL}}(\\Phi)$ is $(\\forall T \\leq_{1}1)\\textup{\\textsf{UWWKL}}(\\Phi(T), T)$. \n\\begin{thm}\\label{proto}\nFrom a proof in $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$ that $\\textup{\\textsf{UWWKL}}^{+}\\leftrightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, terms $s,t$ can be extracted such that $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ proves:\n\\be\\label{frood8}\n(\\forall \\mu^{2})\\big[\\textsf{\\textup{\\textsf{MU}}}(\\mu)\\rightarrow \\textup{\\textsf{UWWKL}}(s(\\mu)) \\big] \\wedge (\\forall \\Phi^{1\\rightarrow 1})\\big[ \\textup{\\textsf{UWWKL}}(\\Phi)\\rightarrow \\textup{\\textsf{MU}}(t(\\Phi, \\Xi)) \\big],\n\\ee\nwhere $\\Xi$ is an extensionality functional for $\\Phi$. \n\\end{thm}\n\\begin{proof}\nFirst of all, to prove $\\textup{\\textsf{UWWKL}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ in $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$, assume $\\textup{\\textsf{UWWKL}}^{+}$ and suppose that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ is false, i.e.\\ there is $f$ such that $(\\forall^{\\textup{NSA}^{\\alpha}}n)f(n)=0 \\wedge (\\exists m^{0})f(m)\\ne 0$.\nNow define the trees $T_{i}$ for $i=0,1$ as follows\n\\[\n\\sigma\\in T_{i}\\leftrightarrow \\big[\\sigma(0)=i \\vee \\big[\\sigma(0)=1-i\\wedge (\\forall m\\leq |\\sigma|)f(m)=0\\big] \\big].\n\\]\nBy the definition of $T_{i}$ and the behaviour of $f$, we have $T_{0}\\approx_{1} T_{1}\\approx_{1} 2^{{\\mathbb N}}$, where the latter is the full binary tree and ${\\mathbb N}:=\\{n^{0}:n=_{0}n\\}$. Furthermore, $\\nu(T_{0})=\\nu(T_{1})=\\frac12$ hold, and observe that $T_{0}$ (resp.\\ $T_{1}$) only has paths starting with $0$ (resp.\\ $1$). Hence, we have \n$\\Phi(T_{0})(0)=0\\ne 1= \\Phi(T_{1})(0)$ for $\\Phi$ as in $\\textup{\\textsf{UWWKL}}^{+}$, which yields $\\Phi(T_{0})\\not\\approx_{1} \\Phi(T_{1})$. Clearly, the latter contradicts the standard extensionality of $\\Phi$. \nIn light of this contradiction, we have $\\textup{\\textsf{UWWKL}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. \n\n\\medskip\n\nSecondly, to prove $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}\\rightarrow \\textup{\\textsf{UWWKL}}^{+}$, note that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ implies \\eqref{karmic} as established in the proof of Theorem \\ref{proto7}. \nTo define $\\Phi$ as in $\\textup{\\textsf{UWWKL}}^{+}$, let standard $T^{1}\\leq_{1}1$ be such that $\\nu(T)>_{{\\mathbb{R}}}0$ and use (standard) $\\xi$ from \\eqref{karmic} to decide if\n\\be\\label{contjas}\n(\\forall^{\\textup{NSA}^{\\alpha}} n^{0})(\\exists \\beta^{0^{*}}\\in T)(\\beta(0)=1 \\wedge |\\beta|=n)\\textup{ or } (\\forall^{\\textup{NSA}^{\\alpha}} n^{0})(\\exists \\beta^{0^{*}}\\in T)(\\beta(0)=0 \\wedge |\\beta|=n),\n\\ee\nand define $\\Phi(T)(0)$ as $1$ if the first formula in \\eqref{contjas} holds, and $0$ otherwise. \nSimilarly, for $\\Phi(T)(m+1)$ again use $\\xi$ from \\eqref{karmic} to decide if the following formula holds:\n\\[\n(\\forall^{\\textup{NSA}^{\\alpha}} n^{0}\\geq m+1)(\\exists \\beta^{0^{*}}\\in T)(\\overline{\\beta}m=\\Phi(T)(0)*\\dots* \\Phi(T)(m) \\wedge\\beta(m+1)=1 \\wedge |\\beta|=n) \n\\]\nand define $\\Phi(T)(m+1)$ as $1$ if it does, and zero otherwise. \nThen $\\Phi$ is standard and satisfies $[\\textup{\\textsf{UWWKL}}(\\Phi)]^{\\textup{NSA}^{\\alpha}}$. Now apply $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ to the latter and \\eqref{EXT} to obtain $\\textup{\\textsf{UWWKL}}^{+}$. \n\n\\medskip\n\nThirdly, we now prove the second conjunct in \\eqref{frood8}. \nNote that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ can easily be brought into the normal form \\eqref{frux}\nwhere the formula in square brackets is abbreviated by $B(f, i)$. Similarly, the second conjunct of $\\textup{\\textsf{UWWKL}}^{+}$ has the following normal form: \n\\be\\label{kurve}\n(\\forall^{\\textup{NSA}^{\\alpha}} T^{1}, S^{1}, k^{0})(\\exists^{\\textup{NSA}^{\\alpha}} N)\\big[\\overline{T}N=_{0}\\overline{S}N \\rightarrow \\overline{\\Phi(T)}k=_{0}\\overline{\\Phi(S)}k\\big],\n\\ee\nwhich is immediate by resolving `$\\approx_{1}$' and bringing standard quantifiers outside. \nWe denote the formula in square brackets in \\eqref{kurve} by $A(T, S, N, k, \\Phi)$. \nHence, the implication $\\textup{\\textsf{UWWKL}}^+\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ now immediately yields:\n\\begin{align}\n(\\forall^{\\textup{NSA}^{\\alpha}}\\Phi, \\Xi)\\big[ [(\\forall^{\\textup{NSA}^{\\alpha}}T^{1})\\textup{\\textsf{UWWKL}}(\\Phi(T), T)&\\wedge (\\forall^{\\textup{NSA}^{\\alpha}} U^{1}, S^{1}, k^{0})A(U, S, \\Xi(U, S, k), k, \\Phi)]\\notag\\\\\n& \\rightarrow (\\forall^{\\textup{NSA}^{\\alpha}}f^{1})(\\exists^{\\textup{NSA}^{\\alpha}}n)B(f,n)\\big],\\label{similarluuuuu}\n\\end{align}\nby strengthening the antecedent by introducing $\\Xi$.\nDropping the `st' in the antecedent of the implication and bringing out the remaining standard quantifiers:\n\\[\n(\\forall^{\\textup{NSA}^{\\alpha}}\\Phi, \\Xi ,f^{1})(\\exists^{\\textup{NSA}^{\\alpha}}n)\\big[ [(\\forall T^{1})\\textup{\\textsf{UWWKL}}(\\Phi(T), T)\\wedge (\\forall U, S, k)A(U, S, \\Xi(U, S, k), k, \\Phi)] \\rightarrow B(f,n)\\big] \n\\]\nLet $C(\\Phi, \\Xi, f, n)$ be the formula in big square brackets and apply Corollary~\\ref{consresultcor2} to `$\\textup{\\textsf{RCA}}_{0}^{\\Lambda}\\vdash (\\forall^{\\textup{NSA}^{\\alpha}}\\Phi, \\Xi, f^{1})(\\exists^{\\textup{NSA}^{\\alpha}}n)C(\\Phi, \\Xi, f, n)$' to obtain a term $t$ such that $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ proves \n\\be\\label{drifgs2}\n (\\forall \\Phi, \\Xi, f^{1})(\\exists n\\in t(\\Phi, \\Xi, f))C(\\Phi, \\Xi, f, n). \n \\ee\nNow define the term $s(\\Phi, \\Xi, f)$ as $\\max_{i<|t(\\Phi, \\Xi, f)|}t(\\Phi, \\Xi, f)(i)$ and \nnote that the formula $(\\exists n\\in t(\\Phi, \\Xi, f))C(\\Phi, \\Xi, f, n)$ implies $C(\\Phi, \\Xi, f, s(\\Phi, \\Xi, f))$.\nFinally, bring the quantifier involving $f$ inside $C$ to obtain for all $\\Phi, \\Xi$ that\n\\[\n [(\\forall T^{1})\\textup{\\textsf{UWWKL}}(\\Phi(T), T)\\wedge (\\forall U^{1}, S^{1}, k^{0})A(U, S, \\Xi(U, S, k), k, \\Phi)] \\rightarrow (\\forall f^{1}) B(f,s(\\Phi, \\Xi, f)).\n\\]\nThus, $s(\\Phi, \\Xi, \\cdot)$ provides $(\\mu^{2})$ if $\\Phi$ satisfies $(\\forall T^{1})\\textup{\\textsf{UWWKL}}(\\Phi(T),T )$ and $\\Xi$ is the associated extensionality functional. \n\n\\medskip\n\nFinally, the first conjunct in \\eqref{frood8} is proved as follows: $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}\\rightarrow \\textup{\\textsf{UWWKL}}^{+}$ yields\n\\be\\label{kanttt}\n(\\forall^{\\textup{NSA}^{\\alpha}}f^{1})(\\exists^{\\textup{NSA}^{\\alpha}}n)B(f,n)\\rightarrow (\\forall^{\\textup{NSA}^{\\alpha}}T^{1})(\\exists^{\\textup{NSA}^{\\alpha}}\\alpha^{1}\\leq_{1}1 )\\textup{\\textsf{UWWKL}}(\\alpha, T)\n\\ee\nwhere we used the same notations $ B$ as in \\eqref{similarluuuuu}. Since standard functionals yield standard output for standard input by the basic axioms of Definition \\ref{debs}, \\eqref{kanttt} yields\n\\be\\label{kanttt2}\n(\\forall^{\\textup{NSA}^{\\alpha}} \\mu^{2})\\big[(\\forall^{\\textup{NSA}^{\\alpha}}f^{1})B(f,\\mu(f))\\rightarrow (\\forall^{\\textup{NSA}^{\\alpha}}T^{1})(\\exists^{\\textup{NSA}^{\\alpha}}\\alpha^{1}\\leq_{1}1 )\\textup{\\textsf{UWWKL}}(\\alpha, T)\\big]\n\\ee\nWeakening the antecedent of \\eqref{kanttt2} and bringing outside the standard quantifiers, we obtain\n\\be\\label{kanttt3}\n(\\forall^{\\textup{NSA}^{\\alpha}} \\mu^{2}, T^{1})(\\exists^{\\textup{NSA}^{\\alpha}}\\alpha^{1}\\leq_{1}1 )\\big[(\\forall f^{1})B(f,\\mu(f))\\rightarrow \\textup{\\textsf{UWWKL}}(\\alpha, T)\\big].\n\\ee\nApplying Corollary \\ref{consresultcor2} to `$\\textup{\\textsf{RCA}}_{0}^{\\Lambda}\\vdash \\eqref{kanttt3}$', we obtain a term $t$ such that $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ proves \n\\be\\label{bl;og}\n(\\forall \\mu^{2}, T^{1})(\\exists \\alpha^{1}\\in t(\\mu, T) )\\big[(\\forall f^{1})B(f,\\mu(f))\\rightarrow \\textup{\\textsf{UWWKL}}(\\alpha, T)\\big].\n\\ee\nSince $\\mu^{2}$ satisfying the antecedent of \\eqref{bl;og} is indeed Feferman's mu as in $(\\mu^{2})$, we can select the `right' $\\alpha\\in t(\\mu, T)$, and we have obtained the first conjunct in \\eqref{frood8}.\n\\end{proof}\nAs suggested by its name, $\\textup{\\textsf{WWKL}}$ is a weakening of $\\textup{\\textsf{WKL}}$, namely to binary trees with positive measure. \nHence, the above proof should also go through for $\\textup{\\textsf{UWKL}}^{+}$, which is $\\textup{\\textsf{UWWKL}}^{+}$ \\emph{without} the aforementioned restriction. \nIn particular, the measure does not play any role except saying that the tree is infinite.\nWe hasten to add that the restriction to trees of positive measure \\emph{does} play an important role for $\\textup{\\textsf{STP}}$ and $\\textup{\\textsf{LMP}}$ by Theorem \\ref{komo}. \n\\subsection{The intermediate value theorem}\\label{hivt}\nIn this section, we study the \\emph{intermediate value theorem} \\textsf{IVT}. \nWe will apply the nonstandard extensionality trick from Remark~\\ref{trick} to a nonstandard version \nof the `usual' Brouwerian counterexample to $\\textup{\\textsf{IVT}}$ from \\cite{beeson1}*{I.7}. \n\n\n\\medskip\n\nNow, $\\textup{\\textsf{IVT}}$ has a proof in $\\textup{\\textsf{RCA}}_{0}$ when formulated in second-order arithmetic using so-called RM codes (\\cite{simpson2}*{II.6.2}). \nHowever, $\\textup{\\textsf{IVT}}$ is not constructively true (See \\cite{beeson1}*{I.7}) as the aforementioned proof makes essential use of the law of excluded middle, while a finer classification may be found in \\cite{bergske}. As a consequence of this non-constructive status, there is an equivalence (See \\cite{kohlenbach2}*{Prop.~3.14}) between the Turing jump functional $(\\exists^{2})$ and uniform $\\textup{\\textsf{IVT}}$, the latter defined as $\\textup{\\textsf{UIVT}}$ as follows (See \\cite{kohlenbach2}*{\\S3} for a number of variations):\n\\be\\tag{$\\textup{\\textsf{UIVT}}$}\n(\\exists \\Phi^{(1\\di1)\\di1 })(\\forall f\\in \\overline{C})[f(\\Phi(f))=_{{\\mathbb{R}}}0], \n\\ee\nwhere `$f\\in \\overline{C}$' is short for `$f(0)<_{{\\mathbb{R}}}0 <_{{\\mathbb{R}}} f(1)\\wedge \\eqref{kraakje}$' where the latter is as follows\n\\be\\label{kraakje}\\textstyle\n(\\forall k^{0})(\\forall x^{1}\\in [0,1])(\\exists N^{0})(\\forall y^{1}\\in [0,1])( |x-y|<_{{\\mathbb{R}}}\\frac{1}{N}\\rightarrow |f(x)-f(y)|<_{{\\mathbb{R}}}\\frac1k), \n\\ee\ni.e.\\ the \\emph{internal} `epsilon-delta' definition of (pointwise) continuity on $[0,1]$. We also write `$f\\in C([0,1])$' if $f:{\\mathbb{R}}\\rightarrow {\\mathbb{R}}$ satisfies \\eqref{kraakje}. \nWe define $\\textup{\\textsf{UIVT}}(\\Phi)\\equiv(\\forall f\\in \\overline{C})[f(\\Phi(f))=_{{\\mathbb{R}}}0]$. \nNow consider the following nonstandard and uniform version of $\\textup{\\textsf{UIVT}}$: \n\\be\\tag{$\\textup{\\textsf{UIVT}}^{+}$}\n(\\exists^{\\textup{NSA}^{\\alpha}} \\Phi)\\big[(\\forall^{\\textup{NSA}^{\\alpha}} f\\in \\overline{C})[f(\\Phi(f))=_{{\\mathbb{R}}}0 ]\\wedge (\\forall^{\\textup{NSA}^{\\alpha}}f, g\\in \\overline{C})\\big( f\\approx g\\rightarrow \\Phi(f)\\approx \\Phi(g) \\big) \\big],\n\\ee\n where `$f\\approx g$' is short for $(\\forall^{\\textup{NSA}^{\\alpha}}q^{0}\\in [0,1])(f(q)\\approx g(q))$. \nNote that the second conjunct of $\\textup{\\textsf{UIVT}}^{+}$ expresses that $\\Phi$ is `standard extensional', i.e.\\ satisfies extensionality as in \\eqref{furg} relative to `st'. \nWe have the following theorem. \n\\begin{thm}\\label{proto667}\nFrom a proof in $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$ that $\\textup{\\textsf{UIVT}}^{+}\\leftrightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, terms $s,t$ can be extracted such that $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ proves:\n\\be\\label{frood667}\n(\\forall \\mu^{2})\\big[\\textsf{\\textup{\\textsf{MU}}}(\\mu)\\rightarrow \\textup{\\textsf{UIVT}}(s(\\mu)) \\big] \\wedge (\\forall \\Phi^{1\\rightarrow 1})\\big[ \\textup{\\textsf{UIVT}}(\\Phi)\\rightarrow \\textup{\\textsf{MU}}(t(\\Phi, \\Xi)) \\big],\n\\ee\nwhere $\\Xi$ is an extensionality functional for $\\Phi$.\n\\end{thm}\n\\begin{proof}\nFirst of all, we prove $\\textup{\\textsf{UIVT}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, for which we make use of the `usual' Brouwerian counterexample to $\\textup{\\textsf{IVT}}$ (See e.g.\\ \\cite{beeson1}*{Fig.\\ 2, p.\\ 12}). \nLet $f_{0}:{\\mathbb{R}}\\rightarrow {\\mathbb{R}}$ be the function which is $3x-1$ for $x\\in [0,\\frac13]$, $3x-2$ for $x\\in [\\frac23, 1]$, and zero for $x\\in [\\frac13, \\frac23]$. \nSuppose $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ is false and let standard $g^{1}$ be such that $(\\forall^{\\textup{NSA}^{\\alpha}}n)(g(n)=0)$ and $(\\exists m_{0})g(m_{0})\\ne 0$. Define standard functions $f_{\\pm}(x):= f_{0}(x)\\pm\\sum_{i=0}^{\\infty}\\frac{g(i)}{2^{i}}$ and note that $f_{+}(x)\\approx f_{-}(x)$ for all $x\\in [0,1]$. However, for $\\Phi$ as in $\\textup{\\textsf{UIVT}}^{+}$, we have $\\Phi(f_{+})<_{{\\mathbb{R}}}\\frac13$ and $\\Phi(f_{-})>_{{\\mathbb{R}}}\\frac23$, which contradicts the second conjunct of $\\textup{\\textsf{UIVT}}^{+}$. Hence, $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ follows and we obtain $\\textup{\\textsf{UIVT}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$.\n\n\\medskip\n \nSecondly, for the reversal $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}} \\rightarrow \\textup{\\textsf{UIVT}}^{+}$, fix standard $k^{0}, x^{1}\\in [0,1] $ and $f:{\\mathbb{R}}\\rightarrow{\\mathbb{R}}$ in \\eqref{kraakje} and consider the following $\\Sigma_{2}^{1}$-formula:\n\\be\\label{boal}\\textstyle\n(\\exists N^{0})(\\forall q^{0}\\in [0,1])( |x-q|<_{{\\mathbb{R}}}\\frac{1}{N}\\rightarrow |f(x)-f(q)|\\leq_{{\\mathbb{R}}}\\frac1k).\n\\ee\nApplying\\footnote{\nTo obtain $\\eqref{boal}\\rightarrow\\eqref{aimino}$, note that $\\neg\\eqref{aimino}$ has a normal form; apply $\\textup{\\textsf{HAC}}_{\\textup{int}}$ to remove the existential quantifier and then apply $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ to the resulting formula to obtain a contradiction with \\eqref{boal}.\nIn general, similar to how one `bootstraps' $\\Pi_{1}^{0}$-comprehension from $\\textup{\\textsf{ACA}}_{0}$, the system $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}+\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ proves $\\varphi\\leftrightarrow \\varphi^{\\textup{NSA}^{\\alpha}}$ for \\emph{any} internal arithmetical formula (only involving standard parameters). Indeed, recall that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}\\rightarrow (\\mu^{2})^{\\textup{NSA}^{\\alpha}}$ (See the proof of Theorem \\ref{proto7}) and use the latter to remove all existential quantifiers (except the leading one, if such there is) from $\\varphi^{\\textup{NSA}^{\\alpha}}$; applying $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ to the resulting universal formula yields $\\varphi$. The implication $\\varphi\\rightarrow \\varphi^{\\textup{NSA}^{\\alpha}}$ follows from the previous, $ \\neg\\varphi^{\\textup{NSA}^{\\alpha}}\\rightarrow \\neg\\varphi$ in particular.} $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ to the formula \\eqref{boal} yields:\n\\be\\label{aimino}\\textstyle\n(\\exists^{\\textup{NSA}^{\\alpha}} N^{0})(\\forall^{\\textup{NSA}^{\\alpha}} q^{0}\\in [0,1])( |x-q|\\ll \\frac{1}{N}\\rightarrow |f(x)-f(q)|\\lessapprox \\frac1k).\n\\ee\nApplying $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ to the universal formula in \\eqref{aimino}, we obtain for standard $f$ that\n\\be\\label{opa}\\textstyle\n(\\forall^{\\textup{NSA}^{\\alpha}}x\\in [0,1], k^{0})(\\exists^{\\textup{NSA}^{\\alpha}} N^{0})(\\forall q^{0}\\in [0,1])( |x-q|<_{{\\mathbb{R}}} \\frac{1}{N}\\rightarrow |f(x)-f(q)|\\leq_{{\\mathbb{R}}} \\frac1k),\n\\ee\nand applying $\\textup{\\textsf{HAC}}_{\\textup{int}}$ to the previous formula yields standard $\\Phi^{(1\\times 0)\\rightarrow 0^{*}}$ such that \n\\[\\textstyle\n(\\forall^{\\textup{NSA}^{\\alpha}}x\\in [0,1], k^{0})(\\exists N^{0}\\in \\Phi(x, k))(\\forall q^{0}\\in [0,1])( |x-q|<_{{\\mathbb{R}}} \\frac{1}{N}\\rightarrow |f(x)-f(q)|\\leq_{{\\mathbb{R}}} \\frac1k),\n\\]\nDefine $\\Psi(x, k)$ as $\\max_{i<|\\Phi(x, k)|}\\Phi(x, k)(i)$ and note that $\\Psi$ provides a kind of \\emph{modulus of continuity} for (standard) $f$. \nWith this continuity in place, we can follow (a variation of) the classical proof of $\\textup{\\textsf{IVT}}$ in \\cite{simpson2}*{II.6} to define {standard} $\\Phi$ as in $\\textup{\\textsf{UIVT}}^{+}$ as follows.\n\n\\medskip\n\nTo this end, recall that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ implies the existence of standard $\\xi$ as in \\eqref{karmic}, which allows us to decide for standard $f\\in \\overline{C}$ if $(\\exists^{\\textup{NSA}^{\\alpha}}q^{0}\\in [0,1])(f(q)\\approx 0)$ holds or not. \nIf the latter holds, use $\\xi$ to find such $q^{0}$ and define $\\Phi(f):=q$, which also yields $f(\\Phi(f))=_{{\\mathbb{R}}}0$ by $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. \nIf however $(\\forall^{\\textup{NSA}^{\\alpha}}q^{0}\\in [0,1])(f(q)\\not\\approx 0)$, define $\\Phi(f)(0)$ as $0$ or $\\frac{1}{2}$ depending on whether $f(\\frac12)\\ll 0$ or $f(\\frac12)\\gg 0$ ($\\xi$ as in \\eqref{karmic} decides which disjunct holds). \nSimilarly, define $\\Phi(f)(n+1)$ as $\\Phi(f)(n)$ or $\\Phi(f)(n)+\\frac{1}{2^{n+2}}$ depending on $f(\\Phi(f)(n)+\\frac{1}{2^{n+2}})\\ll0$ or $f(\\Phi(f)(n)+\\frac{1}{2^{n+2}})\\gg0$ ($\\xi$ as in \\eqref{karmic} again decides which disjunct holds). \n\n\\medskip\n\nNote that by $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, we have $f(z)\\gg 0 \\leftrightarrow f(z)>_{{\\mathbb{R}}}0$ for any standard $z^{1}\\in {\\mathbb{R}} $ and $f:{\\mathbb{R}}\\rightarrow {\\mathbb{R}}$. \nIn light of \\eqref{karmic}, we also have access to $I\\Sigma_{1}$ relative to `st', assuming all parameters involved are standard. \nHence, it is easy to prove that $\\Phi(f)$ is a real number such that $(\\forall^{\\textup{NSA}^{\\alpha}}n)(\\Phi(f)(n)<_{{\\mathbb{R}}}\\Phi(f)<_{{\\mathbb{R}}}\\Phi(f)(n)+\\frac{1}{2^{n+1}})$ and $f(\\Phi(f)(n))\\ll 0$ and $f(\\Phi(f)(n)+\\frac{1}{2^{n+1}})\\gg 0$. In light of these facts and the continuity of $f$ as in \\eqref{opa} for $x=\\Phi(f)$, we obtain $f(\\Phi(f))\\approx 0$ and also $f(\\Phi(f))=_{{\\mathbb{R}}}0$ by $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. \nFurthermore, $\\Phi$ only invokes $f$ on rational numbers and $(\\forall^{\\textup{NSA}^{\\alpha}} q^{0}\\in [0,1])(f(q)\\approx g(q))$ for standard $f, g\\in \\overline{C}$ thus implies $(\\forall q^{0}\\in [0,1])(f(q)=_{{\\mathbb{R}}} g(q))$ by $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. By the previous property and the extensionality of $f,g$, we have $\\Phi(f)=_{{\\mathbb{R}}}\\Phi(g)$, and hence the second conjunct of $\\textup{\\textsf{UIVT}}^{+}$. \n \n\\medskip\n\nThirdly, one obtains a normal form for $\\textup{\\textsf{UIVT}}^{+}\\leftrightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ in the same way as in the proofs of Theorems \\ref{proto7} and \\ref{proto}. \nApplying term extraction as in Theorem~\\ref{consresult}, one then readily obtains \\eqref{frood667}. For completeness, we mention the two normal forms corresponding to $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}\\rightarrow \\textup{\\textsf{UIVT}}^{+}$ and $\\textup{\\textsf{UIVT}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, namely as follows:\n\\[\n(\\forall^{\\textup{NSA}^{\\alpha}} f\\in \\overline{C},\\mu^{2})(\\exists^{\\textup{NSA}^{\\alpha}}x\\in [0,1]) \\big[ (\\forall g^{1})B(g,\\mu(g))\\rightarrow (f(x)=_{{\\mathbb{R}}}0 )\\big].\n\\]\n\\begin{align*}\\textstyle\n(\\forall^{\\textup{NSA}^{\\alpha}} \\Phi, \\Xi, g^{1})(\\exists^{\\textup{NSA}^{\\alpha}}n^{0})\\big[\\big((\\forall z\\in \\overline{C})&(z(\\Phi(z))=_{{\\mathbb{R}}}0 )\\textstyle\\wedge (\\forall x, y \\in \\overline{C}, k^{0})(\\exists q^{0},n^{0}\\in \\Xi(x,y,k))\\\\\n&\\textstyle\\big( |x(q) -y(q)|<\\frac{1}{n}\\rightarrow |\\Phi(x)- \\Phi(y)|<\\frac{1}{k} \\big)\\big) \\rightarrow B(g, n)\\big],\n\\end{align*}\nwhere we used the notation $B$ from \\eqref{similarluuuuu}. \nTo be absolutely clear, we point out that the formula `$(\\forall^{\\textup{NSA}^{\\alpha}} f\\in \\overline{C})A(f)$' is short for $(\\forall f)([\\textup{NSA}^{\\alpha}(f)\\wedge f\\in \\overline{C}]\\rightarrow A(f))$. \nIn particular, `$f\\in \\overline{C}$' is the \\emph{internal} formula `$f(1)>_{{\\mathbb{R}}}0>_{{\\mathbb{R}}} f(0)\\wedge \\eqref{kraakje}$' which remains \\emph{untouched} by the addition of `st' to the quantifier over the variable $f$. \nFor this reason, the formula `$f\\in \\overline{C}$' does not play any role in obtaining the normal form of $\\textup{\\textsf{UIVT}}^{+}\\leftrightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ and the associated term extraction via Theorem \\ref{consresult}. \n\\end{proof}\nThe previous proof is easily adapted to the Brouwerian counterexample to the Weierstra\\ss~maximum theorem, as follows. \n\\begin{rem}[Weierstra\\ss~maximum theorem]\\textup{rm}\nThe Brouwerian counterexample to the Weierstra\\ss~maximum theorem is given in \\cite{beeson1}*{I.6} by a function with two relative maxima. \nFor instance, one can use $f_{\\pm}:{\\mathbb{R}}\\rightarrow {\\mathbb{R}}$ which is $(1\\pm x_{0})|\\sin 7x |$ for $x\\in [0, \\frac{1}{7}\\pi]$ and $(1\\mp x_{0})|\\sin 7x |$ for $x\\in [\\frac{1}{7}\\pi, 1]$, where $x_{0}:=\\sum_{i=0}^{\\infty}\\frac{g(i)}{2^{i}}$ and $g$ as in the previous proof. A functional witnessing the Weierstra\\ss~maximum theorem will map $f_{+}$ to $\\frac{\\pi}{14}$ and $f_{-}$ to $\\frac{3\\pi}{14}$ while $f_{+}\\approx f_{-}$ under the same assumptions as in the previous proof.\n\\end{rem}\n\n\\subsection{Rational numbers and the law of excluded middle}\\label{carmichael}\nIn this section, we study the classical dichotomy that every real number is either rational or not, i.e.\\ \n\\be\\label{LEM2}\\tag{\\textsf{DQ}}\n(\\forall x\\in {\\mathbb{R}})\\big[ (\\exists q\\in {\\mathbb Q})(q=_{{\\mathbb{R}}}x)\\vee (\\forall r\\in {\\mathbb Q})(r\\ne_{{\\mathbb{R}}}x) \\big]. \n\\ee\nOur study will yield interesting insights into the role of the law of excluded middle in Nonstandard Analysis. \nIn particular, we will offer a partial explanation \\emph{why} Nonstandard Analysis can produce computational information, as observed in the previous sections (and the aforementioned references). \n\n\\medskip\n\nFirst of all, \\ref{LEM2} is a trivial consequence of the law of excluded middle, and the former is indeed equivalent to \\textsf{LPO} in constructive mathematics (\\cite{brich}*{p.\\ 5}). \nWe will study the \\emph{nonstandard} version of \\ref{LEM2}, defined as follows:\n\\be\\label{LEM1}\\tag{$\\textup{\\textsf{DQ}}_{\\textup{\\textsf{ns}}}$}\n(\\forall^{\\textup{NSA}^{\\alpha}} x\\in {\\mathbb{R}})\\big[ (\\exists^{\\textup{NSA}^{\\alpha}}q\\in {\\mathbb Q})(q=_{{\\mathbb{R}}}x)\\vee (\\forall r\\in {\\mathbb Q})(r\\ne_{{\\mathbb{R}}}x) \\big],\n\\ee\nThe uniform version of \\ref{LEM2} is also obvious:\n\\be\\label{LEM3}\\tag{\\textsf{UDQ}}\n(\\exists \\Phi^{2})(\\forall x\\in {\\mathbb{R}})\\big[ (\\Phi(x)\\in {\\mathbb Q} \\wedge q=_{{\\mathbb{R}}}x)\\vee (\\forall r\\in {\\mathbb Q})(r\\ne_{{\\mathbb{R}}}x) \\big]. \n\\ee\nLet $\\textsf{UDQ}(\\Phi)$ be the previous with the leading quantifier dropped.\n\\begin{thm}\\label{krefje}\nIn $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$, we have $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}\\leftrightarrow \\textsf{\\textup{DQ}}_{\\textup{\\textsf{ns}}}$. From the latter proof, we can extract terms $s$ and $t$ such that\n\\be\\label{unik}\n(\\forall \\mu^{2})\\big[\\textsf{\\textup{\\textsf{MU}}}(\\mu)\\rightarrow \\textup{\\textsf{UDQ}}(s(\\mu)) \\big] \\wedge (\\forall \\Phi^{2})\\big[ \\textup{\\textsf{UDQ}}(\\Phi)\\rightarrow \\textup{\\textsf{MU}}(t(\\Phi)) \\big].\n\\ee\n\\end{thm}\n\\begin{proof}\nThe first forward implication is trivial while the first reverse equivalence follows easily: Suppose $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ is false and consider $f^{1}$ such that $(\\forall^{\\textup{NSA}^{\\alpha}}n)(f(n)=0)\\wedge (\\exists m)(f(m)\\ne 0)$.\nDefine the real $x:=\\sum_{i=0}^{\\infty}\\frac{h(n)}{2^{n}}$ where $h(n)=1$ if $(\\forall i\\leq n)(f(n)=0)$, and zero otherwise. Note that $x$ is a rational number, namely $x=_{{\\mathbb{R}}} \\sum_{i=0}^{m_{0}}\\frac{h(n)}{2^{n}} $, where $m_{0}$ is the last $m$ such that $f(m)\\ne0$. However, we also have $x\\ne_{{\\mathbb{R}}} q$ for every standard rational, and this contradiction yields $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. \n\n\\medskip\n\nSince $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ has a normal form \\eqref{frux} and $\\textsf{DQ}_{\\textup{\\textsf{ns}}}$ has an obvious normal form, \\eqref{unik} follows in the same way as in the second part of the proof of Theorem \\ref{proto7}.\n\\end{proof}\nSecondly, we discuss an apparent (but not actual) contradiction regarding $\\textsf{DQ}_{\\textup{\\textsf{ns}}}$ and the previous theorem, as follows.\n\\begin{rem}[Equality in $\\textup{\\textsf{P}}$]\\label{druj}\\textup{rm}\nFirst of all, $\\textup{\\textsf{P}}$ proves \\ref{LEM2} via the law of excluded middle. \nHence for a \\emph{standard} real $x$ which does not satisfy the second disjunct of \\ref{LEM2}, we may conclude the existence of a rational $q^{0}$ such that $x$ equals $q$. \nSecondly, in light of the first basic axiom of $\\textup{\\textsf{P}}$ (See item (\\ref{komit}) of Definition~\\ref{debs}), $q$ must be standard as $x$ is standard, and $x$ equals $q$. \nHowever, this means that $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$ proves $\\textsf{DQ}_{\\textup{\\textsf{ns}}}$, which is impossible in light of Theorem \\ref{krefje}. \nThirdly, this apparent contradiction is easily explained by noting that `$=_{{\\mathbb{R}}}$' as defined in $\\textup{\\textsf{P}}$ (See Definition \\ref{keepinitreal}) does not fall under item (\\ref{komit}) of Definition \\ref{debs}. \n\\end{rem}\nThirdly, while Theorem \\ref{krefje} is not particularly deep, this theorem inspires Remark \\ref{druj}, which in turn gives rise to the following observation: \nIn $\\textup{\\textsf{RCA}}_{0}^{\\omega}$, either a real is rational or not because of the law of excluded middle \\ref{LEM2}. \nBy contrast, in $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$, there are three possibilities for every standard real:\n\\begin{enumerate}\n\\item $x$ is a standard rational;\n\\item $x$ is not a rational; \n\\item $x$ is a rational, but not standard;\\label{trunki}\n\\end{enumerate}\nand the third possibility \\eqref{trunki} only disappears given $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ by Theorem \\ref{krefje}. Similarly, again over $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$, for a standard function $f^{1}$, there are three possibilities:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\item there is standard $n^{0}$ such that $f(n)=0$;\\label{work}\n\\item for all $m^{0}$ we have $f(m)\\ne0$;\n\\item for all standard $n^{0}$ we have $f(n)\\ne0$ while there is $m^{0}$ with $f(m)= 0$;\\label{kraft}\n\\end{enumerate}\nand the third possibility again only disappears given $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. In particular, \\emph{in the extended language of $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$}, $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ (and not \\ref{LEM2}) is the principle which excludes the third option \\eqref{trunki} and \\eqref{kraft}. In other words, it seems that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ plays the role of the law of excluded middle\/third \\emph{in the extended language of $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$}. \n\n\\medskip\n\nFurthermore, it has been suggested that the predicate `$\\textup{NSA}^{\\alpha}(n^{0})$' can be read as `$n^{0}$ is computationally relevant' or `$n^{0}$ is calculable' in \\cite{brie}*{p.\\ 1963}, \\cite{benno2}*{\\S4}, and \\cite{sambon}*{\\S3.4}. \nIf we read the previous items \\eqref{work}-\\eqref{kraft} through this filter, they reflect three well-known possibilities suggested by the BHK interpretation (See \\cite{troeleke1}*{\\S3.1}): \n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\Roman{enumi}}\n\\item we can compute $n^{0}$ s.t.\\ $f(n)=0$ (constructive existence);\\label{kracht}\n\\item for all $m^{0}$ we have $f(m)\\ne0$;\n\\item $\\neg[(\\forall m^{0})(f(m) \\ne0)]$;\\label{werk} \n\\end{enumerate}\nIn conclusion, we have observed that the role of the law of excluded middle \\emph{in the extended language of $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$} is played by Nelson's axiom \\emph{Transfer}, which is however absent from $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$. Due to this absence, there are \\emph{three} possibilities as in items \\eqref{work}-\\eqref{kraft}, similar to the possibilities in items \\eqref{kracht}-\\eqref{werk} in constructive mathematics. \nIn other words, the systems $\\textup{\\textsf{P}}$ and $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$ are constructive in that they lack \\emph{Transfer}, which is the law of excluded middle \\emph{for the extended language of internal set theory}. \nWe believe this to be a partial explanation of the vast computational content of Nonstandard Analysis established above and in \\cites{sambon, samGH, samzoo, samzooII}. \n\n\n\n\\section{A new class of functionals}\\label{knowledge}\nWe discuss the results from \\cite{dagsam}, some of which were announced in \\cite{samGT2}, and related results. \nThe associated connection between Nonstandard Analysis and computability theory forms the motivation for this paper, as discussed in Section \\ref{intro}. \n\n\\subsection{Nonstandard Analysis and Computability Theory: an introduction}\\label{connexis}\nThe connection between \\emph{computability theory} and \\emph{Nonstandard Analysis} is investigated \\cite{dagsam}. The two following topics are investigated \\emph{and} shown to be intimately related. \n\\begin{enumerate}\n\\item[\\textsf{(T.1)}] A basic property of \\emph{Cantor space} $2^{{\\mathbb N}}$ is \\emph{Heine-Borel compactness}: For any open cover of $2^{{\\mathbb N}}$, there is a \\emph{finite} sub-cover. \nA natural question is: \\emph{How hard is it to {compute} such a finite sub-cover}? This is made precise in \\cite{dagsam} by analysing the complexity of functionals that for $g:2^{{\\mathbb N}}\\rightarrow {\\mathbb N}$, \noutput a finite sequence $\\langle f_0 , \\dots, f_n\\rangle $ in $2^{{\\mathbb N}}$ such that the neighbourhoods defined from $\\overline{f_i}g(f_i)$ for $i\\leq n$ form a cover of Cantor space.\n\\item[\\textsf{(T.2)}] A basic property of Cantor space in \\emph{Nonstandard Analysis} is Abraham Robinson's \\emph{nonstandard compactness} (See \\cite{loeb1}*{p.\\ 42}), i.e.\\ that every binary sequence is `infinitely close' to a \\emph{standard} binary sequence. The strength of this nonstandard compactness property of Cantor space is analysed in \\cite{dagsam} and compared to the other axioms of Nonstandard Analysis and usual mathematics.\n\\end{enumerate}\nThe study of \\textsf{(T.1)} in \\cite{dagsam} involves the \\emph{special fan functional} $\\Theta$, discussed in Section \\ref{thespecial} below and first introduced in \\cite{samGH}. \nClearly, Tait's \\emph{fan functional} (\\cite{noortje}) computes\\footnote{Tait's fan functional $\\Phi$ computes a modulus of \\emph{uniform} continuity $N^{0}=\\Phi(g)$ for any continuous functional $g:2^{{\\mathbb N}}\\rightarrow {\\mathbb N}$. The modulus $N^{0}$ yields a supremum for $g$ by computing the maximum of $g(\\sigma*00\\dots)$ for all binary sequences $\\sigma$ of length $N$.} a sequence $\\langle f_0 , \\dots, f_n\\rangle $ as in \\textsf{(T.1)} for \\emph{continuous} $g:2^{{\\mathbb N}}\\rightarrow {\\mathbb N}$, while the special fan functional does so \\emph{for any} $g:2^{{\\mathbb N}}\\rightarrow {\\mathbb N}$. This generalisation from continuous to general inputs is interesting (and even necessary) in our opinion as mathematics restricted to e.g.\\ only recursive objects, like the Russian school of recursive mathematics, can be strange and counter-intuitive (See \\cite{beeson1}*{Chapter IV} for this opinion). \nSome of the (highly surprising) computational properties of $\\Theta$ established in \\cite{dagsam} are discussed in Section \\ref{thespecial}. In particular, $\\Theta$ seems extremely hard to compute (as in Kleene's S1-S9 from \\cite{longmann}*{\\S5.1}) as no type two functional can compute it. \n\n\\medskip\n\nThe study of \\textsf{(T.2)} in \\cite{dagsam} amounts to developing the Reverse Mathematics of Nonstandard Analysis. For instance, the nonstandard counterparts of $\\textup{\\textsf{WKL}}_{0}$ and $\\textup{\\textsf{WWKL}}_{0}$ are $\\textup{\\textsf{STP}}$ and $\\textup{\\textsf{LMP}}$ (See Section \\ref{thespecial} for definitions), \neach expressing a nonstandard kind of compactness. On the other hand, the nonstandard counterpart of $\\textup{\\textsf{ACA}}_{0}$ is $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ introduced above. While we have $\\textup{\\textsf{ACA}}_{0}\\rightarrow \\textup{\\textsf{WKL}}_{0}\\rightarrow \\textup{\\textsf{WWKL}}_{0}$ in RM, the nonstandard counterparts behave quite differently, namely \nwe have $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}} \\not\\rightarrow \\textup{\\textsf{STP}}$ and $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}} \\not\\rightarrow\\textup{\\textsf{LMP}}$, and much stronger non-implications involving the nonstandard counterpart of $\\Pi_{1}^{1}\\text{-\\textsf{CA}}_{0}$, the strongest `Big Five' system. \n\n\\medskip\n\nWe stress that \\textsf{(T.1)} and \\textsf{(T.2)} are highly intertwined and that the study of these topics in \\cite{dagsam} is `holistic' in nature: \nresults in computability theory give rise to results in Nonstandard Analysis \\emph{and vice versa}, as discussed in the next section. By way of a basic example, consider $\\Theta$ as in \\textsf{(T.1)} and recall that the output of $\\Theta$ is readily computed in terms of Tait's fan functional if $g^{2}$ is continuous on Cantor space. Experience bears out that the uninitiated express extreme skepticism about the fact that $\\Theta$ is also well-defined for \\emph{discontinuous} inputs $g^{2}$. However, $\\Theta$ almost trivially emerges form the nonstandard compactness of Cantor space, i.e.\\ Nonstandard Analysis tells us that the special fan functional $\\Theta$ exists and is well-defined. Furthermore, the fact that the Turing jump functional from $(\\exists^{2})$ cannot compute $\\Theta$ as mentioned in \\textsf{(T.1)} readily implies the non-implication $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}\\not \\rightarrow \\textup{\\textsf{STP}}$ from \\textsf{(T.2)}. \nMore examples are discussed in the next section, and of course \\cite{dagsam}. \n\n\\medskip\n\nFinally, we have sketched a connection between Nonstandard Analysis and computability theory. However, the better part of the latter does not obviously have a counterpart in the former and vice versa. \nWe list two examples: First of all, the fact that no type two functional computes $\\Theta$ is proved in computability theory (See \\cite{dagsam}*{\\S3}) using \\emph{Gandy selection} (See \\cite{longmann}*{Theorem 5.4.5}), but what is the nonstandard counterpart of the latter theorem? \nSecondly, the \\emph{Loeb measure} is one of the crown jewels of Nonstandard Analysis (\\cite{loeb1}), but what is the computability theoretic counterpart of this measure? Note that first steps in this direction have been taken in \\cite{samnewarix}. \nIn the paper at hand, we have formulated a nonstandard counterpart of \\emph{Grilliot's trick} inspired by the above connection between Nonstandard Analysis and computability theory. \n\n\\subsection{The special fan functional and related topics}\\label{thespecial}\nWe introduce the \\emph{special fan functional} and discuss how it derives from the \\emph{Standard Part} axiom of Nonstandard Analysis and why it does not belong to any existing category in RM. \n\n\\medskip\n\nOur motivation for this study is the following discrepancy: On one hand, there is literally a `zoo' of theorems in RM (\\cite{damirzoo}) which do fit into the `Big Five classification' of RM. On the other hand, as shown above and in \\cites{samzoo, samzooII}, uniform theorems are mostly equivalent to $(\\exists^{2})$, with some exceptions based on the contraposition of $\\textup{\\textsf{WKL}}$, i.e.\\ the fan theorem. Thus, the `higher-order RM zoo' consisting of uniform theorems is still rather sparse compared to the original RM zoo. \nIn this light, it is a natural question whether the higher-order RM zoo can be made as populous as the original RM zoo. \n\n\\medskip\n\nAs a first step towards an answer to the aforementioned question, we discuss the following functional. Note that $1^{*}$ is the type of finite sequences of type $1$\n\\begin{defi}\\textup{rm}[Special fan functional]\\label{special}\nWe define $\\textup{\\textsf{SCF}}(\\Theta)$ as follows for $\\Theta^{(2\\rightarrow (0\\times 1^{*}))}$:\n\\[\n(\\forall g^{2}, T^{1}\\leq_{1}1)\\big[(\\forall \\alpha \\in \\Theta(g)(2)) (\\overline{\\alpha}g(\\alpha)\\not\\in T)\n\\rightarrow(\\forall \\beta\\leq_{1}1)(\\exists i\\leq \\Theta(g)(1))(\\overline{\\beta}i\\not\\in T) \\big]. \n\\]\nAny functional $\\Theta$ satisfying $\\textup{\\textsf{SCF}}(\\Theta)$ is referred to as a \\emph{special fan functional}.\n\\end{defi}\nAs noted in \\cite{dagsam} and above, from a computability theoretic perspective, the main property of the special fan functional $\\Theta$ is the selection of $\\Theta(g)(2)$ as a finite sequence of binary sequences $\\langle f_0 , \\dots, f_n\\rangle $ such that the neighbourhoods defined from $\\overline{f_i}g(f_i)$ for $i\\leq n$ form a cover of Cantor space; almost as a by-product, $\\Theta(g)(1)$ can then be chosen to be the maximal value of $g(f_i) + 1$ for $i\\leq n$. \nWe stress that $g^{2}$ in $\\textup{\\textsf{SCF}}(\\Theta)$ may be \\emph{discontinuous} and that Kohlenbach has argued for the study of discontinuous functionals in RM (\\cite{kohlenbach2}).\n\n\\medskip\n\nThe name of $\\Theta$ from the previous definition is due to the fact that a special fan functional may be computed from the intuitionistic fan functional $\\Omega^{3}$, as in Theorem \\ref{kinkel}.\n\\be\\tag{$\\textup{\\textsf{MUC}}(\\Omega)$}\n(\\forall Y^{2}) (\\forall f, g\\leq_{1}1)(\\overline{f}\\Omega(Y)=\\overline{g}\\Omega(Y)\\notag \\rightarrow Y(f)=Y(g)), \n\\ee\nAs to the logical strength of $(\\exists \\Omega)\\textup{\\textsf{MUC}}(\\Omega)$, the latter gives \nrise to a conservative extension of the system $\\textup{\\textsf{WKL}}_{0}$ by \\cite{kohlenbach2}*{Prop.\\ 3.15}. \n\\begin{thm}\\label{kinkel}\nThere is a term $t$ such that $\\textsf{\\textup{E-PA}}^{\\omega}$ proves $(\\forall \\Omega^{3})(\\textup{\\textsf{MUC}}(\\Omega)\\rightarrow \\textup{\\textsf{SCF}}(t(\\Omega))) $. \n\\end{thm}\n\\begin{proof}\nThe theorem was first proved \\emph{indirectly} in \\cite{samGH}*{\\S3} by applying Theorem~\\ref{consresult} to a suitable nonstandard implication. \nFor completeness, we include the following direct proof which can also be found in \\cite{dagsam}. \nNote that $\\Theta(g)$ as in $\\textup{\\textsf{SCF}}(\\Theta)$ has to provide a natural number and a finite sequence of binary sequences.\nThe number $\\Theta(g)(1)$ is defined as $\\max_{|\\sigma|=\\Omega(g)\\wedge \\sigma\\leq_{0^{*}}1}g(\\sigma*00\\dots)$ and the finite sequence of binary sequences $\\Theta(g)(2)$\nconsists of all $\\tau*00\\dots$ where $|\\tau|=\\Theta(g)(1)\\wedge \\tau\\leq_{0^{*}}1$. \nWe have for all $g^{2}$ and $T^{1}\\leq_{1}1$:\n\\be\\label{difffff}\n (\\forall \\beta\\leq_{1}1)(\\beta \\in \\Theta(g)(2)\\rightarrow \\overline{\\beta}{g}(\\beta)\\not \\in T)\\rightarrow (\\forall \\gamma\\leq_{1}1)(\\exists i\\leq \\Theta(g)(1))(\\overline{\\gamma}i\\not \\in T).\n\\ee\nIndeed, suppose the antecendent of \\eqref{difffff} holds. Now take $\\gamma_{0}\\leq_{1}1$, and note that $\\beta_{0}=\\overline{\\gamma_{0}}\\Theta(g)(1)*00\\dots \\in \\Theta(g)(2)$, implying \n$\\overline{\\beta_{0}}{g}(\\beta_{0})\\not \\in T$. But $g(\\alpha)\\leq \\Theta(g)(1)$ for all $\\alpha\\leq_{1}1$, by the definition of $\\Omega$, implying that $\\overline{\\gamma_{0}}{g}(\\beta_{0})=\\overline{\\beta_{0}}{g}(\\beta_{0})\\not \\in T$ by the definition of $\\beta_{0}$, and the consequent of \\eqref{difffff} follows. \n\\end{proof}\nIn light of the previous, $\\Theta$ exists at the level of $\\textup{\\textsf{WKL}}_{0}$ and it therefore stands to reason that it would be \\emph{easy} to compute. \nWe have the following surprising theorem where `computable' should be once again interpreted in the sense of Kleene's S1-S9 (See \\cite{longmann}*{5.1.1}).\nThe metatheory is -as always- $\\ensuremath{\\usftext{ZFC}}$ set theory. \n\\begin{thm}\\label{import}\nLet $\\varphi^{2}$ be any functional of type two. \nAny functional $\\Theta^{3}$ as in $\\textup{\\textsf{SCF}}(\\Theta)$ is not computable in $\\varphi$. \nAny functional $\\Theta^{3}$ as in $\\textup{\\textsf{SCF}}(\\Theta)$ is computable in $(\\exists^{3})$ as follows\n\\be\\tag{$\\exists^{3}$}\n(\\exists E_{3})(\\forall \\varphi^{2})\\big[ (\\exists f^{1})(\\varphi(f)=0)\\leftrightarrow E_{3}(\\varphi)=0 \\big]\n\\ee\n\\end{thm}\n\\begin{proof}\nA proof may be found in \\cite{dagsam}.\n\\end{proof}\nBy the previous, $\\Theta$ is quite different from the usual\\footnote{Note that the connection between $\\Theta$ and Kohlenbach's generalisations of $\\textup{\\textsf{WKL}}$ from \\cite{kohlenbach4}*{\\S5-6} is discussed in \\cite{dagsam}*{\\S4}. This connection turns out to be quite non-trivial and interesting.} objects studied in (higher-order) RM. \nAn obvious question is: Where does the special fan functional and its behaviour come from? \nThe answer is as follows: The nonstandard counterpart of $\\textup{\\textsf{WKL}}_{0}$ is defined as:\n\\be\\tag{$\\textup{\\textsf{STP}}$}\n(\\forall \\alpha^{1}\\leq_{1}1)(\\exists^{\\textup{NSA}^{\\alpha}}\\beta^{1}\\leq_{1}1)(\\alpha\\approx_{1}\\beta), \n\\ee \nwhich has the following normal form, already reminiscent of $\\Theta$. \n\\begin{thm}\\label{lapdog}\nIn $\\textup{\\textsf{P}}$, $\\textup{\\textsf{STP}}$ is equivalent to the following normal form:\n\\begin{align}\\label{frukkklk}\n(\\forall^{\\textup{NSA}^{\\alpha}}g^{2})(\\exists^{\\textup{NSA}^{\\alpha}}w^{1^{*}})\\big[(\\forall T^{1}\\leq_{1}1)(\\exists ( \\alpha^{1}\\leq_{1}1, &~k^{0}) \\in w)\\big((\\overline{\\alpha}g(\\alpha)\\not\\in T)\\\\\n&\\rightarrow(\\forall \\beta\\leq_{1}1)(\\exists i\\leq k)(\\overline{\\beta}i\\not\\in T) \\big)\\big]. \\notag\n\\end{align} \nThe system $\\textup{\\textsf{P}}+(\\exists^{\\textup{NSA}^{\\alpha}}\\Theta)\\textup{\\textsf{SCF}}(\\Theta)$ proves $\\textup{\\textsf{STP}}$. \n\\end{thm}\n\\begin{proof} \nThe following proof is implicit in the results in \\cite{samGH}*{\\S3} and is added for completeness. \nFirst of all, $\\textup{\\textsf{STP}}$ is easily seen to be equivalent to \n\\begin{align}\\label{fanns}\n(\\forall T^{1}\\leq_{1}1)\\big[(\\forall^{\\textup{NSA}^{\\alpha}}n)(\\exists \\beta^{0})&(|\\beta|=n \\wedge \\beta\\in T ) \\rightarrow (\\exists^{\\textup{NSA}^{\\alpha}}\\alpha^{1}\\leq_{1}1)(\\forall^{\\textup{NSA}^{\\alpha}}n^{0})(\\overline{\\alpha}n\\in T) \\big],\n\\end{align}\nand this equivalence may also be found in \\cite{samGH}*{Theorem 3.2}.\nFor \\eqref{frukkklk}$\\rightarrow$\\eqref{fanns}, note that \\eqref{frukkklk} implies for all standard $g^{2}$\n\\begin{align}\\label{frukkklk2}\n(\\forall T^{1}\\leq_{1}1)(\\exists^{\\textup{NSA}^{\\alpha}} ( \\alpha^{1}\\leq_{1}1, &~k^{0})\\big[(\\overline{\\alpha}g(\\alpha)\\not\\in T)\n\\rightarrow(\\forall \\beta\\leq_{1}1)(\\exists i\\leq k)(\\overline{\\beta}i\\not\\in T) \\big], \n\\end{align} \nwhich in turn yields, by bringing all standard quantifiers inside again, that:\n\\begin{align}\\label{frukkklk3}\n(\\forall T\\leq_{1}1) \\big[(\\exists^{\\textup{NSA}^{\\alpha}}g^{2})(\\forall^{\\textup{NSA}^{\\alpha}}\\alpha \\leq_{1}1)(\\overline{\\alpha}g(\\alpha)\\not\\in T)\\rightarrow(\\exists^{\\textup{NSA}^{\\alpha}}k)(\\forall \\beta\\leq_{1}1)(\\overline{\\beta}k\\not\\in T) \\big], \n\\end{align} \nTo obtain \\eqref{fanns} from \\eqref{frukkklk3}, apply $\\ensuremath{\\usftext{HAC}_\\ensuremath{{\\usftext{int}}}}$ to $(\\forall^{\\textup{NSA}^{\\alpha}}\\alpha^{1}\\leq_{1}1)(\\exists^{\\textup{NSA}^{\\alpha}}n)(\\overline{\\alpha}n\\not\\in T)$ to obtain standard $\\Psi^{1\\rightarrow 0^{*}}$ such that \n$(\\forall^{\\textup{NSA}^{\\alpha}}\\alpha^{1}\\leq_{1}1)(\\exists n\\in \\Psi(\\alpha))(\\overline{\\alpha}n\\not\\in T)$, and defining $g(\\alpha):=\\max_{i<|\\Psi|}\\Psi(\\alpha)(i)$ we obtain $g$ as in the antecedent of \\eqref{frukkklk3}. The previous implies \n\\be\\label{gundark}\n(\\forall T^{1}\\leq_{1}1) \\big[(\\forall^{\\textup{NSA}^{\\alpha}}\\alpha^{1}\\leq_{1}1)(\\exists^{\\textup{NSA}^{\\alpha}}n)(\\overline{\\alpha}n\\not\\in T)\\rightarrow (\\exists^{\\textup{NSA}^{\\alpha}}k)(\\forall \\beta\\leq_{1}1)(\\overline{\\beta}i\\not\\in T) \\big], \n\\ee\nwhich is the contraposition of \\eqref{fanns}, using classical logic. For the implication $\\eqref{fanns} \\rightarrow \\eqref{frukkklk}$, consider the contraposition of \\eqref{fanns}, i.e.\\ \\eqref{gundark}, and note that the latter implies \\eqref{frukkklk3}. Now push all standard quantifiers outside as follows:\n\\[\n(\\forall^{\\textup{NSA}^{\\alpha}}g^{2})(\\forall T^{1}\\leq_{1}1)(\\exists^{\\textup{NSA}^{\\alpha}} ( \\alpha^{1}\\leq_{1}1, ~k^{0})\\big[(\\overline{\\alpha}g(\\alpha)\\not\\in T)\n\\rightarrow(\\forall \\beta\\leq_{1}1)(\\exists i\\leq k)(\\overline{\\beta}i\\not\\in T) \\big], \n\\]\nand applying idealisation \\textsf{I} yields \\eqref{frukkklk}. The final part now follows immediately in light of the basic axioms of $\\textup{\\textsf{P}}$ in Definition \\ref{debs}. \n\\end{proof}\nBy the previous theorem $\\Theta$ emerges from Nonstandard Analysis, and the behaviour of $\\Theta$ as in Theorem \\ref{import} can be explained similarly: \nIt is part of the folklore of Nonstandard Analysis that \\emph{Transfer} does not imply \\emph{Standard Part}. The same apparently holds for fragments: $\\textup{\\textsf{P}}+\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ does not prove $\\textup{\\textsf{STP}}$ by the results in \\cite{dagsam}*{\\S6}. \nAs a result of applying Theorem \\ref{consresult}, there is no term of G\\\"odel's $T$ which computes $\\Theta$ in terms of $(\\mu^{2})$. A stronger result as in Theorem \\ref{import} apparently can be obtained. \n\n\\medskip\n\nNext, we discuss a nonstandard version of $\\textup{\\textsf{WWKL}}$, introduced in \\cite{pimpson}, as follows:\n\\be\\tag{$\\textup{\\textsf{LMP}}$}\n(\\forall T \\leq_{1}1)\\big[ \\nu(T)\\gg 0 \\rightarrow (\\exists^{\\textup{NSA}^{\\alpha}} \\beta\\leq_{1}1)(\\forall^{\\textup{NSA}^{\\alpha}}n)(\\overline{\\beta}n\\in T)\\big], \n\\ee\nand one obtains a normal form of $\\textup{\\textsf{LMP}}$ similar to \\eqref{frukkklk}. As for the latter, this normal form gives rise to a \\emph{weak fan functional} $\\Lambda$, first introduced in \\cite{dagsam}*{\\S3}. \nWe have the following theorem where $\\textup{\\textsf{ATR}}_{0}$ is the fourth `Big Five' system of RM (See \\cite{simpson2}*{V}). \n\\begin{thm}\nThe system $\\textup{\\textsf{P}}+\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}+\\textup{\\textsf{STP}}$ proves the consistency of $\\textup{\\textsf{ATR}}_{0}$ while $\\textup{\\textsf{P}}+\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}+\\textup{\\textsf{LMP}}$ does not. \n\\end{thm}\n\\begin{proof}\nSee \\cite{dagsam}*{\\S6}.\n\\end{proof}\nThe previous result is referred to as a `phase transition' in \\cite{dagsam} as there is (currently) nothing in between $\\textup{\\textsf{WKL}}_{0}$ and $\\textup{\\textsf{WWKL}}_{0}$ in the RM zoo. \n\\begin{cor}\nThe system $\\textup{\\textsf{P}}+\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ does not prove $\\textup{\\textsf{STP}}$. \n\\end{cor}\n\\begin{proof}\nSuppose $\\textup{\\textsf{P}}+\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ proves $\\textup{\\textsf{STP}}$ and note that \n$\\textup{\\textsf{P}}+\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ then proves the consistency of $\\textup{\\textsf{ATR}}_{0}$ by the theorem. By Theorem \\ref{consresult}, $\\textsf{E-PA}^{\\omega*}+(\\mu^{2})$ also proves the consistency of $\\textup{\\textsf{ATR}}_{0}$, which contradicts G\\\"odel's incompleteness theorems.\n\\end{proof}\nAt the end of Section \\ref{FUWKL}, we noted that Theorem \\ref{proto} should go through for $\\textup{\\textsf{WKL}}$ instead of $\\textup{\\textsf{WWKL}}$. \nIn particular, the restriction to trees of positive measure (which is part of $\\textup{\\textsf{WWKL}}$) can be lifted while still obtaining the same equivalence as in \\eqref{frood8}. \nHence, these results are \\emph{robust}, i.e.\\ equivalent to small perturbations of themselves (See \\cite{montahue}*{p.\\ 432}). We now provide an example where the notion of `tree with positive measure' yields \\emph{non-robust} results. \nTo this end, consider the following strengthening of $\\textup{\\textsf{LMP}}$:\n\\be\\tag{$\\textup{\\textsf{LMP}}^{+}$}\n(\\forall T \\leq_{1}1)\\big[ \\mu(T)>_{{\\mathbb{R}}}0\\rightarrow (\\exists^{\\textup{NSA}^{\\alpha}} \\beta\\leq_{1}1)(\\forall^{\\textup{NSA}^{\\alpha}} m)(\\overline{\\beta}m\\in T) \\big],\n\\ee\nand the following weakening of $\\textup{\\textsf{STP}}$:\n\\be\\tag{$\\textup{\\textsf{STP}}^{-}$}\n(\\forall T \\leq_{1}1)\\big[(\\forall n^{0})(\\exists \\beta^{0^{*}})(\\beta\\in T\\wedge |\\beta|=n)\\rightarrow (\\exists^{\\textup{NSA}^{\\alpha}} \\beta\\leq_{1}1)(\\forall^{\\textup{NSA}^{\\alpha}} m)(\\overline{\\beta}m\\in T) \\big],\n\\ee\n\\begin{thm}\\label{komo}\nIn $\\textup{\\textsf{P}}_{0}+\\textup{\\textsf{WWKL}}$, we have $\\textup{\\textsf{STP}}\\leftrightarrow \\textup{\\textsf{LMP}}^{+}\\leftrightarrow \\textup{\\textsf{STP}}^{-}$.\n\\end{thm}\n\\begin{proof}\nThe implication $\\textup{\\textsf{STP}}\\rightarrow \\textup{\\textsf{STP}}^{-}$ is immediate; for the reverse implication, apply overspill to the antecedent of \\eqref{fanns} to obtain a sequence $\\beta_{0}^{0^{*}}$ of nonstandard length in $T$. Extend the latter to an infinite tree by including $\\beta_{0}*00\\dots$, which is nonstandard. \nApplying $\\textup{\\textsf{STP}}^{-}$ to this extended tree yields the consequence of \\eqref{fanns}, and hence $\\textup{\\textsf{STP}}$.\nFor $\\textup{\\textsf{STP}}\\rightarrow \\textup{\\textsf{LMP}}^{+}$, apply $\\textup{\\textsf{STP}}$ to the path claimed to exist by $\\textup{\\textsf{WWKL}}$ and note that we obtain $\\textup{\\textsf{LMP}}^{+}$. \nFor $\\textup{\\textsf{LMP}}^{+}\\rightarrow\\textup{\\textsf{STP}}$, fix $f^{1}\\leq_{1}1$ and nonstandard $N$. Define the tree $T\\leq_{1}1$ which is $f$ until height $N$, followed by the full binary tree. \nThen $\\mu(T)>_{{\\mathbb{R}}}0$ and let standard $g^{1}\\leq_{1}1$ be such that $(\\forall^{\\textup{NSA}^{\\alpha}}n)(\\overline{g}n\\in T)$. By definition, we have $f\\approx_{1}g$, and we are done.\n\\end{proof}\nIn light of the theorem, $\\textup{\\textsf{STP}}$ and $\\Theta$ seem fairly robust, while $\\textup{\\textsf{LMP}}$ and $\\Lambda$ are not. \n\n\n\\medskip\n\nFinally, $\\textup{\\textsf{STP}}$ and $\\textup{\\textsf{LMP}}$ are not unique: Similar nonstandard (and functional) versions exist for most of the theorems in the RM zoo. Indeed, since every theorem $T$ in the RM zoo follows from arithmetical comprehension, \nwe can prove $T^{\\textup{NSA}^{\\alpha}}$ in $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}+\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}+\\textup{\\textsf{STP}}$, and use $\\textup{\\textsf{STP}}$ to drop the `st' in the leading quantifier as in \\eqref{fanns} for $\\textup{\\textsf{WKL}}^{\\textup{NSA}^{\\alpha}}$. \nThe author and Dag Normann are currently investigating the exact power of $\\Theta$ and $\\Lambda$ and the strength of the associated nonstandard axioms. \n\n\n\\begin{bibdiv}\n\\begin{biblist}\n\\bib{avi2}{article}{\n author={Avigad, Jeremy},\n author={Feferman, Solomon},\n title={G\\\"odel's functional \\(``Dialectica''\\) interpretation},\n conference={ title={Handbook of proof theory}, },\n book={ publisher={North-Holland}, },\n date={1998},\n pages={337--405},\n}\n\n\\bib{beeson1}{book}{\n author={Beeson, Michael J.},\n title={Foundations of constructive mathematics},\n series={Ergebnisse der Mathematik und ihrer Grenzgebiete},\n volume={6},\n note={Metamathematical studies},\n publisher={Springer},\n date={1985},\n pages={xxiii+466},\n}\n\n\\bib{brie}{article}{\n author={van den Berg, Benno},\n author={Briseid, Eyvind},\n author={Safarik, Pavol},\n title={A functional interpretation for nonstandard arithmetic},\n journal={Ann. Pure Appl. Logic},\n volume={163},\n date={2012},\n pages={1962--1994},\n}\n\n\\bib{bergske}{article}{\n author={Berger, Josef},\n author={Ishihara, Hajime},\n author={Takayuki, Kihara},\n author={Nemoto, Takako},\n title={The binary expansion and the intermediate value theorem in constructive reverse mathematics},\n journal={Available from \\url {http:\/\/www.jaist.ac.jp\/~t-nemoto\/beivt.pdf}},\n}\n\n\\bib{bishl}{book}{\n author={Bishop, Errett},\n title={Aspects of constructivism},\n publisher={Notes on the lectures delivered at the Tenth Holiday Mathematics Symposium},\n place={New Mexico State University, Las Cruces, December 27-31},\n date={1972},\n pages={pp.\\ 37},\n}\n\n\\bib{kuddd}{article}{\n author={Bishop, Errett},\n title={Review of \\cite {keisler3}},\n year={1977},\n journal={Bull. Amer. Math. Soc},\n volume={81},\n number={2},\n pages={205-208},\n}\n\n\\bib{kluut}{article}{\n author={Bishop, Errett},\n title={The crisis in contemporary mathematics},\n booktitle={Proceedings of the American Academy Workshop on the Evolution of Modern Mathematics},\n journal={Historia Math.},\n volume={2},\n date={1975},\n number={4},\n pages={507--517},\n}\n\n\\bib{brich}{book}{\n author={Bridges, Douglas},\n author={Richman, Fred},\n title={Varieties of constructive mathematics},\n series={London Mathematical Society Lecture Note Series},\n volume={97},\n publisher={Cambridge University Press},\n place={Cambridge},\n date={1987},\n pages={x+149},\n}\n\n\\bib{bridges1}{book}{\n author={Bridges, Douglas S.},\n author={V{\\^{\\i }}{\\c {t}}{\\u {a}}, Lumini{\\c {t}}a Simona},\n title={Techniques of constructive analysis},\n series={Universitext},\n publisher={Springer},\n place={New York},\n date={2006},\n pages={xvi+213},\n}\n\n\\bib{conman}{article}{\n author={Connes, Alain},\n title={An interview with Alain Connes, Part I},\n year={2007},\n journal={EMS Newsletter},\n note={\\url {http:\/\/www.mathematics-in-europe.eu\/maths-as-a-profession\/interviews}},\n volume={63},\n pages={25-30},\n}\n\n\\bib{conman2}{article}{\n author={Connes, Alain},\n title={Noncommutative geometry and reality},\n journal={J. Math. Phys.},\n volume={36},\n date={1995},\n number={11},\n pages={6194--6231},\n}\n\n\\bib{damirzoo}{misc}{\n author={Dzhafarov, Damir D.},\n title={Reverse Mathematics Zoo},\n note={\\url {http:\/\/rmzoo.uconn.edu\/}},\n}\n\n\\bib{godel3}{article}{\n author={G{\\\"o}del, Kurt},\n title={\\\"Uber eine bisher noch nicht ben\\\"utzte Erweiterung des finiten Standpunktes},\n language={German, with English summary},\n journal={Dialectica},\n volume={12},\n date={1958},\n pages={280--287},\n}\n\n\\bib{grilling}{article}{\n author={Grilliot, Thomas J.},\n title={On effectively discontinuous type-$2$ objects},\n journal={J. Sym. Logic},\n volume={36},\n date={1971},\n}\n\n\\bib{benno2}{article}{\n author={Hadzihasanovic, Amar},\n author={van den Berg, Benno},\n title={Nonstandard functional interpretations and models},\n journal={To appear in Notre Dame Journal for Formal Logic},\n volume={},\n date={2016},\n number={},\n pages={},\n}\n\n\\bib{polahirst}{article}{\n author={Hirst, Jeffry L.},\n title={Representations of reals in reverse mathematics},\n journal={Bull. Pol. Acad. Sci. Math.},\n volume={55},\n date={2007},\n number={4},\n pages={303--316},\n}\n\n\\bib{loeb1}{book}{\n author={Hurd, Albert E.},\n author={Loeb, Peter A.},\n title={An introduction to nonstandard real analysis},\n series={Pure and Applied Mathematics},\n volume={118},\n publisher={Academic Press Inc.},\n place={Orlando, FL},\n date={1985},\n pages={xii+232},\n}\n\n\\bib{keisler3}{book}{\n author={Keisler, H. Jerome},\n title={Elementary Calculus},\n publisher={Prindle, Weber and Schmidt},\n date={1976},\n pages={xviii + 880 + 61 (appendix)},\n place={Boston},\n}\n\n\\bib{kohlenbach3}{book}{\n author={Kohlenbach, Ulrich},\n title={Applied proof theory: proof interpretations and their use in mathematics},\n series={Springer Monographs in Mathematics},\n publisher={Springer-Verlag},\n place={Berlin},\n date={2008},\n pages={xx+532},\n}\n\n\\bib{kohlenbach2}{article}{\n author={Kohlenbach, Ulrich},\n title={Higher order reverse mathematics},\n conference={ title={Reverse mathematics 2001}, },\n book={ series={Lect. Notes Log.}, volume={21}, publisher={ASL}, },\n date={2005},\n pages={281--295},\n}\n\n\\bib{kohlenbach4}{article}{\n author={Kohlenbach, Ulrich},\n title={Foundational and mathematical uses of higher types},\n conference={ title={Reflections on the foundations of mathematics (Stanford, CA, 1998)}, },\n book={ series={Lect. Notes Log.}, volume={15}, publisher={ASL}, },\n date={2002},\n pages={92--116},\n}\n\n\\bib{kooltje}{article}{\n author={Kohlenbach, Ulrich},\n title={On uniform weak K\\\"onig's lemma},\n journal={Ann. Pure Appl. Logic},\n volume={114},\n date={2002},\n pages={103--116},\n}\n\n\\bib{longmann}{book}{\n author={Longley, John},\n author={Normann, Dag},\n title={Higher-order Computability},\n year={2015},\n publisher={Springer},\n series={Theory and Applications of Computability},\n}\n\n\\bib{mandje2}{article}{\n author={Mandelkern, Mark},\n title={Brouwerian counterexamples},\n journal={Math. Mag.},\n volume={62},\n date={1989},\n number={1},\n pages={3--27},\n}\n\n\\bib{montahue}{article}{\n author={Montalb{\\'a}n, Antonio},\n title={Open questions in reverse mathematics},\n journal={Bull. Symbolic Logic},\n volume={17},\n date={2011},\n number={3},\n pages={431--454},\n}\n\n\\bib{wownelly}{article}{\n author={Nelson, Edward},\n title={Internal set theory: a new approach to nonstandard analysis},\n journal={Bull. Amer. Math. Soc.},\n volume={83},\n date={1977},\n number={6},\n pages={1165--1198},\n}\n\n\\bib{noortje}{book}{\n author={Normann, Dag},\n title={Recursion on the countable functionals},\n series={LNM 811},\n volume={811},\n publisher={Springer},\n date={1980},\n pages={viii+191},\n}\n\n\\bib{dagsam}{article}{\n author={Normann, Dag},\n author={Sanders, Sam},\n title={Nonstandard Analysis, Computability Theory, and their connections},\n journal={Submitted, Available from \\url {https:\/\/arxiv.org\/abs\/1702.06556}},\n date={2017},\n}\n\n\\bib{yamayamaharehare}{article}{\n author={Sakamoto, Nobuyuki},\n author={Yamazaki, Takeshi},\n title={Uniform versions of some axioms of second order arithmetic},\n journal={MLQ Math. Log. Q.},\n volume={50},\n date={2004},\n number={6},\n pages={587--593},\n}\n\n\\bib{sayo}{article}{\n author={Sanders, Sam},\n author={Yokoyama, Keita},\n title={The {D}irac delta function in two settings of {R}everse {M}athematics},\n year={2012},\n journal={Archive for Mathematical Logic},\n volume={51},\n number={1},\n pages={99-121},\n}\n\n\\bib{samGH}{article}{\n author={Sanders, Sam},\n title={The Gandy-Hyland functional and a hitherto unknown computational aspect of Nonstandard Analysis},\n year={2017},\n journal={To appear in \\emph {Computability}, \\url {http:\/\/arxiv.org\/abs\/1502.03622}},\n}\n\n\\bib{samzoo}{article}{\n author={Sanders, Sam},\n title={The taming of the Reverse Mathematics zoo},\n year={2015},\n journal={Submitted, \\url {http:\/\/arxiv.org\/abs\/1412.2022}},\n}\n\n\\bib{samzooII}{article}{\n author={Sanders, Sam},\n title={The refining of the taming of the Reverse Mathematics zoo},\n year={2016},\n journal={To appear in Notre Dame Journal for Formal Logic, \\url {http:\/\/arxiv.org\/abs\/1602.02270}},\n}\n\n\\bib{sambon}{article}{\n author={Sanders, Sam},\n title={The unreasonable effectiveness of Nonstandard Analysis},\n year={2015},\n journal={Submitted, \\url {http:\/\/arxiv.org\/abs\/1508.07434}},\n}\n\n\\bib{SB}{article}{\n author={Sanders, Sam},\n title={To be or not to be constructive},\n journal={To appear in \\emph {Indagationes Mathematicae} and the Brouwer volume \\emph {L.E.J. Brouwer, fifty years later}, arXiv: \\url {https:\/\/arxiv.org\/abs\/1704.00462}},\n date={2017 and 2018},\n pages={pp.\\ 68},\n}\n\n\\bib{samnewarix}{article}{\n author={Sanders, Sam},\n title={The computational content of the Loeb measure},\n year={2016},\n journal={Available from arXiv: \\url {https:\/\/arxiv.org\/abs\/1609.01945}},\n}\n\n\\bib{samsynt}{article}{\n author={Sanders, Sam},\n title={Reverse Formalism 16},\n year={2017},\n journal={To appear in Synthese, \\url {https:\/\/arxiv.org\/abs\/1701.05066}},\n}\n\n\\bib{samGT2}{article}{\n author={Sanders, Sam},\n title={The computational content of Nonstandard Analysis},\n date={2016},\n journal={Electronic Proceedings in Computer Science 213, \\emph {Classic Logic and Computation}, Porto (CL\\&C2016)},\n pages={21-40},\n}\n\n\\bib{simpson1}{collection}{\n title={Reverse mathematics 2001},\n series={Lecture Notes in Logic},\n volume={21},\n editor={Simpson, Stephen G.},\n publisher={ASL},\n place={La Jolla, CA},\n date={2005},\n pages={x+401},\n}\n\n\\bib{simpson2}{book}{\n author={Simpson, Stephen G.},\n title={Subsystems of second order arithmetic},\n series={Perspectives in Logic},\n publisher={CUP},\n date={2009},\n pages={xvi+444},\n}\n\n\\bib{pimpson}{article}{\n author={Simpson, Stephen G.},\n author={Yokoyama, Keita},\n title={A nonstandard counterpart of \\textsf {\\textup {WWKL}}},\n journal={Notre Dame J. Form. Log.},\n volume={52},\n date={2011},\n number={3},\n pages={229--243},\n}\n\n\\bib{troeleke1}{book}{\n author={Troelstra, Anne Sjerp},\n author={van Dalen, Dirk},\n title={Constructivism in mathematics. Vol. I},\n series={Studies in Logic and the Foundations of Mathematics},\n volume={121},\n publisher={North-Holland},\n date={1988},\n pages={xx+342+XIV},\n}\n\n\n\\end{biblist}\n\\end{bibdiv}\n\n\\end{document}\n\n\n\\section{Introduction}\\label{intro}\nRecently, Dag Normann and the author have established a connection between \\emph{higher-order computability theory} and \\emph{Nonstandard Analysis} (\\cite{dagsam}). \nIn the latter, they investigate the complexity of functionals connected to the \\emph{Heine-Borel compactness} of Cantor space. Surprisingly, this complexity turns out to be \nintimately connected to the \\emph{nonstandard compactness} of Cantor space as given by \\emph{Robinson's theorem} (See \\cite{loeb1}*{p.\\ 42}) in Nonstandard Analysis.\nIn fact, the results in \\cite{dagsam} are `holistic' in nature in that theorems in computability theory give rise to theorems in Nonstandard Analysis, \\emph{and vice versa}. \nWe discuss these results in Section \\ref{knowledge} as they serve as the motivation for this paper. \n \n\\medskip\n\nIn light of the aforementioned connection, it is a natural question which notions from higher-order computability theory have elegant analogues in Nonstandard Analysis, \\emph{and vice versa}.\nThis paper explores one particular case of this question, namely for the technique know as \\emph{Grilliot's trick}, introduced in \\cite{grilling}. \nThe latter `trick' actually constitutes a template for explicitly defining the Turing jump functional $(\\exists^{2})$ in terms of a given effectively discontinuous type two functional. \nBelow, we introduce the \\emph{nonstandard extensionality trick}, which is a technique similar to {Grilliot's trick} {in Nonstandard Analysis}. \nIn this way, we study a new \\emph{computational} aspect of Nonstandard Analysis pertaining to {Reverse Mathematics} (RM), in line with the results in \\cites{samzoo,samzooII, samGH, sambon}.\nWe refer to \\cites{simpson2, simpson1} for an overview to RM. \nWe shall make use of \\emph{internal set theory}, i.e.\\ Nelson's axiomatic Nonstandard Analysis (\\cite{wownelly}). \nWe introduce internal set theory and its fragments from \\cite{brie} in Section~\\ref{WIST}. \n\n\\medskip\n\nThe nonstandard extensionality trick sums up as: From the existence of \\emph{nonstandard discontinuous} functionals, the \\emph{Transfer} principle from Nonstandard analysis (See Section~\\ref{IST}) limited to $\\Pi_{1}^{0}$-formulas is derived; from this (generally ineffective) implication, we obtain an effective implication expressing the Turing jump functional in terms of a discontinuous functional (and no longer involving Nonstandard Analysis). Essential to obtaining this effective implication is the `term extraction theorem' in Theorem \\ref{consresult}, based on \\cite{brie}. \nWe shall apply the nonstandard extensionality trick to \\emph{binary expansion}, the \\emph{intermediate value theorem}, the \\emph{Weierstra\\ss~maximum theorem}, and \\emph{weak weak K\\\"onig's lemma} in Section \\ref{main}. \n\n\\medskip\n\nNow, combining the aforementioned results with similar results from \\cites{samzoo, samzooII} regarding the RM zoo from \\cite{damirzoo}, one gets the idea that the \nhigher-order landscape is not very rich and similar to the second-order framework. \nTo counter this view, we discuss a new class of functionals in Section~\\ref{knowledge} which do not fit the existing categories of RM. \nThese functionals are inspired by the \\emph{Standard Part} axiom of Nonstandard Analysis. \n\n\\medskip\n\nFinally, we hasten to point out that there are well-established techniques for obtaining effective content from classical mathematics, the most prominent one being the \\emph{proof mining} program (\\cite{kohlenbach3}). \nIn particular, the effective results in this paper could be or have been obtained in this way. \nWhat is surprising about the results in this paper (in our opinion) is the emergence of effective content (with relative ease) from Nonstandard Analysis \\emph{despite} claims \nthat the latter is somehow fundamentally non-constructive by e.g.\\ Bishop and Connes (See \\cite{samsynt} for a detailed discussion of the Bishop-Connes critique). \n\n\n\\section{Internal set theory and fragments}\\label{WIST}\nIn this section, we sketch \\emph{internal set theory}, Nelson's \\emph{syntactic} approach to Nonstandard Analysis, first introduced in \\cite{wownelly}, and its fragments from \\cite{brie}. An in-depth and completely elementary introduction to the constructive content of Nonstandard Analysis is \\cite{SB}. \n\\subsection{Introducing internal set theory}\\label{IST}\nNelson's system $\\ensuremath{\\usftext{IST}}$ of internal set theory is defined as follows: The language of $\\ensuremath{\\usftext{IST}}$ consist of the language of $\\ensuremath{\\usftext{ZFC}}$, the `usual' foundations of mathematics, plus a new predicate `st($x$)', read as `$x$ is standard'. \nThe new quantifiers $(\\forall^{\\textup{NSA}^{\\alpha}}x)(\\dots)$ and $(\\exists^{\\textup{NSA}^{\\alpha}}y)(\\dots)$ are short for $(\\forall x)(\\textup{NSA}^{\\alpha}(x)\\rightarrow \\dots)$ and $(\\exists y)(\\textup{NSA}^{\\alpha}(y)\\wedge \\dots)$. \nA formula of $\\ensuremath{\\usftext{IST}}$ is called \\emph{internal} if it does not involve `st', and \\emph{external} otherwise. \n\n\\medskip\n\nThe system $\\ensuremath{\\usftext{IST}}$ is the internal system $\\ensuremath{\\usftext{ZFC}}$ plus the\nfollowing\\footnote{The `fin' in \\textsf{(I)} means that $x$ is finite,\n i.e.\\ its number of elements are bounded by a natural number.} three\nexternal axioms \\emph{Idealisation}, \\emph{Standard Part}, and\n\\emph{Transfer} which govern the predicate `st'.\n\n\\begin{itemize}[label=\\textsf{(S)}]\n\\item[\\textsf{(I)}] $(\\forall^{\\textup{NSA}^{\\alpha}~\\textup{fin}}x)(\\exists y)(\\forall z\\in x)\\varphi(z,y)\\rightarrow (\\exists y)(\\forall^{\\textup{NSA}^{\\alpha}}x)\\varphi(x,y)$, for internal $\\varphi$. \n\\item[\\textsf{(S)}] $(\\forall^{\\textup{NSA}^{\\alpha}} x)(\\exists^{\\textup{NSA}^{\\alpha}}y)(\\forall^{\\textup{NSA}^{\\alpha}}z)\\big((z\\in x\\wedge \\varphi(z))\\leftrightarrow z\\in y\\big)$, for any formula $\\varphi$.\n\\item[\\textsf{(T)}] $(\\forall^{\\textup{NSA}^{\\alpha}}t)\\big[(\\forall^{\\textup{NSA}^{\\alpha}}x)\\varphi(x, t)\\rightarrow (\\forall x)\\varphi(x, t)\\big]$, for internal $\\varphi$ and $t, x$ the only free variables. \n\\end{itemize}\nNelson proves in \\cite{wownelly} that \\textsf{IST} is a conservative extension of \\textsf{ZFC}, i.e.\\ $\\ensuremath{\\usftext{ZFC}}$ and $\\ensuremath{\\usftext{IST}}$ prove the same (internal) sentences. \nVarious fragments of $\\ensuremath{\\usftext{IST}}$ have been studied previously, and we shall make essential use of the system $\\textup{\\textsf{P}}$, a fragment of $\\ensuremath{\\usftext{IST}}$ based on Peano arithmetic, introduced in Section~\\ref{PIPI}. \nThe system $\\textup{\\textsf{P}}$ was first introduced in \\cite{brie} and is exceptional in that it has a `term extraction procedure' \\emph{with a very wide scope}. We discuss this aspect of $\\textup{\\textsf{P}}$ in more detail in Remark~\\ref{firliborn}. \n\n\\subsection{The classical system $\\textup{\\textsf{P}}$}\\label{PIPI}\nIn this section, we introduce the classical system $\\textup{\\textsf{P}}$ which is a conservative extension of Peano arithmetic by Theorem \\ref{consresult}. \nWe refer to \\cite{kohlenbach3}*{\\S3.3} for the detailed definition of the rather mainstream system $\\textsf{E-PA}^{\\omega}$, i.e.\\ \\emph{Peano arithmetic in all finite types with the axiom of extensionality}. \nThe system $\\textup{\\textsf{P}}$ consist of the following axioms, starting with the basic ones. \n\\begin{defi}\\textup{rm}\\label{debs}[Basic axioms of $\\textup{\\textsf{P}}$]\n\\begin{enumerate}\n\\item The system \\textsf{E-PA}$^{\\omega*}$ is the definitional extension of \\textsf{E-PA}$^{\\omega}$ with types for finite sequences as in \\cite{brie}*{\\S2}. \n\\item The set ${\\mathbb T}^{*}$ is the collection of all the constants in the language of $\\textsf{E-PA}^{\\omega*}$. \n\\item The external induction axiom \\textsf{IA}$^{\\textup{NSA}^{\\alpha}}$ is \n\\be\\tag{\\textsf{IA}$^{\\textup{NSA}^{\\alpha}}$}\n\\Phi(0)\\wedge(\\forall^{\\textup{NSA}^{\\alpha}}n^{0})(\\Phi(n) \\rightarrow\\Phi(n+1))\\rightarrow(\\forall^{\\textup{NSA}^{\\alpha}}n^{0})\\Phi(n). \n\\ee\n\\item\\label{krafoi} The system $ \\textsf{E-PA}^{\\omega*}_{\\textup{NSA}^{\\alpha}} $ is defined as $ \\textsf{E-PA}^{\\omega{*}} + {\\mathbb T}^{*}_{\\textup{NSA}^{\\alpha}} + \\textsf{IA}^{\\textup{NSA}^{\\alpha}}$, where ${\\mathbb T}^{*}_{\\textup{NSA}^{\\alpha}}$\nconsists of the following basic axiom schemas.\n\\begin{enumerate}\n\\item The schema\\footnote{The language of $\\textsf{E-PA}_{\\textup{NSA}^{\\alpha}}^{\\omega*}$ contains a symbol $\\textup{NSA}^{\\alpha}_{\\sigma}$ for each finite type $\\sigma$, but the subscript is always omitted. Hence ${\\mathbb T}^{*}_{\\textup{NSA}^{\\alpha}}$ is an \\emph{axiom schema} and not an axiom.\\label{omit}} $\\textup{NSA}^{\\alpha}(x)\\wedge x=y\\rightarrow\\textup{NSA}^{\\alpha}(y)$. \\label{komit}\n\\item The schema providing for each closed term $t\\in {\\mathbb T}^{*}$ the axiom $\\textup{NSA}^{\\alpha}(t)$.\n\\item The schema $\\textup{NSA}^{\\alpha}(f)\\wedge \\textup{NSA}^{\\alpha}(x)\\rightarrow \\textup{NSA}^{\\alpha}(f(x))$.\n\\end{enumerate}\n\\end{enumerate}\nSecondly, Nelson's axiom \\emph{Standard part} is weakened in \\cite{brie} to $\\textup{\\textsf{HAC}}_{\\textup{int}}$:\n\\be\\tag{$\\textup{\\textsf{HAC}}_{\\textup{int}}$}\n(\\forall^{\\textup{NSA}^{\\alpha}}x^{\\rho})(\\exists^{\\textup{NSA}^{\\alpha}}y^{\\tau})\\varphi(x, y)\\rightarrow (\\exists^{\\textup{NSA}^{\\alpha}}F^{\\rho\\rightarrow \\tau^{*}})(\\forall^{\\textup{NSA}^{\\alpha}}x^{\\rho})(\\exists y^{\\tau}\\in F(x))\\varphi(x,y),\n\\ee\nwhere $\\varphi$ is any internal formula and $\\tau^{*}$ is the type of finite sequences of objects of type $\\tau$. Note that $F$ only provides a \\emph{finite sequence} of witnesses to $(\\exists^{\\textup{NSA}^{\\alpha}}y)$, explaining the name \\emph{Herbrandized Axiom of Choice} for $\\textup{\\textsf{HAC}}_{\\textup{int}}$.\n\n\\medskip\n \nThirdly, Nelson's axiom idealisation \\textsf{I} appears in \\cite{brie} as follows: \n\\be\\tag{\\textsf{I}}\n(\\forall^{\\textup{NSA}^{\\alpha}} x^{\\sigma^{*}})(\\exists y^{\\tau} )(\\forall z^{\\sigma}\\in x)\\varphi(z,y)\\rightarrow (\\exists y^{\\tau})(\\forall^{\\textup{NSA}^{\\alpha}} x^{\\sigma})\\varphi(x,y), \n\\ee\nwhere $\\varphi$ is internal and $\\sigma^{*}$ is the type of finite sequences of objects of type $\\sigma$.\n\\end{defi}\nFor $\\textup{\\textsf{P}}\\equiv \\textsf{E-PA}^{\\omega*}_{\\textup{NSA}^{\\alpha}} +\\textup{\\textsf{HAC}}_{\\textup{int}} +\\textsf{I}$, we have the following `term extraction theorem', which is not explicitly formulated or proved in \\cite{brie}. A proof may be found in \\cites{samzoo, sambon}. \n\\begin{thm}[Term extraction]\\label{consresult}\nLet $\\varphi$ be an internal formula and let $\\Delta_{\\ensuremath{{\\usftext{int}}}}$ be a collection of internal formulas. If we have:\n\\be\\label{antecedn}\n\\textup{\\textsf{P}} + \\Delta_{\\ensuremath{{\\usftext{int}}}} \\vdash (\\forall^{\\textup{NSA}^{\\alpha}}\\underline{x})(\\exists^{\\textup{NSA}^{\\alpha}}\\underline{y})\\varphi(\\underline{x}, \\underline{y}, \\underline{a})\n\\ee\nthen one can extract from the proof a sequence of closed terms $t$ in $\\mathcal{T}^{*}$ such that\n\\be\\label{consequalty}\n\\textup{\\textsf{E-PA}}^{\\omega*} + \\Delta_{\\ensuremath{{\\usftext{int}}}} \\vdash ( \\forall \\tup x) (\\exists \\tup y\\in \\tup t(\\tup x)) \\varphi(\\tup x,\\tup y, \\tup a).\n\\ee\n\\end{thm}\n\\begin{proof}\nThe proof of the theorem in a nutshell: A proof interpretation $S_{\\textup{NSA}^{\\alpha}}$ is defined in \\cite{brie}*{Def.\\ 7.1}; a tedious but straightforward verification using the clauses (i)-(v) in \\cite{brie}*{Def.\\ 7.1} establishes that $\\Phi(\\underline{a})^{S_{\\textup{NSA}^{\\alpha}}}\\equiv \\Phi(\\underline{a})$ for $\\Phi(\\underline{a})\\equiv (\\forall^{\\textup{NSA}^{\\alpha}}\\underline{x})(\\exists^{\\textup{NSA}^{\\alpha}}\\underline{y})\\varphi(\\underline{x}, \\underline{y}, \\underline{a})$ and $\\varphi$ internal. The theorem now follows immediately from \\cite{brie}*{Theorem 7.7}. \n\\end{proof}\nThe term $t$ in \\eqref{consequalty} is \\emph{primitive recursive} in the sense of G\\\"odel's system ${T}$. The latter was introduced in \\cite{godel3}, and is also discussed in \\cite{kohlenbach3}*{\\S3}. For the rest of this paper, a `normal form' will refer to a formula as in \\eqref{antecedn}, i.e.\\ of the form $ (\\forall^{\\textup{NSA}^{\\alpha}} \\tup x )(\\exists^{\\textup{NSA}^{\\alpha}} \\tup y) \\varphi(\\tup x,\\tup y, \\tup a)$ for internal $\\varphi$.\n\n\\medskip\n\nAs expected, the previous theorem does not really depend on the presence of full Peano arithmetic. \nIndeed, let \\textsf{E-PRA}$^{\\omega}$ be the system defined in \\cite{kohlenbach2}*{\\S2} and let \\textsf{E-PRA}$^{\\omega*}$ \nbe its definitional extension with types for finite sequences as in \\cite{brie}*{\\S2}. We permit ourselves a slight abuse of notation by not distinguishing between Kohlenbach's $\\textup{\\textsf{RCA}}_{0}^{\\omega}\\equiv \\textup{\\textsf{E-PRA}}^{\\omega}+\\textup{\\textsf{QF-AC}}^{1,0}$ (See \\cite{kohlenbach2}*{\\S2}) and $\\textup{\\textsf{E-PRA}}^{\\omega*}+\\textup{\\textsf{QF-AC}}^{1,0}$. \n\\begin{cor}\\label{consresultcor2}\nThe previous theorem and corollary go through for $\\textup{\\textsf{P}}$ and $\\textup{\\textsf{E-PA}}^{\\omega*}$ replaced by $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}\\equiv \\textsf{\\textup{E-PRA}}^{\\omega*}+{\\mathbb T}_{\\textup{NSA}^{\\alpha}}^{*} +\\textup{\\textsf{HAC}}_{\\textup{int}} +\\textsf{\\textup{I}}+\\textup{\\textsf{QF-AC}}^{1,0}$ and $\\textup{\\textsf{RCA}}_{0}^{\\omega}$. \n\\end{cor}\n\\begin{proof}\nThe proof of \\cite{brie}*{Theorem 7.7} goes through for any fragment of \\textsf{E-PA}$^{\\omega{*}}$ which includes \\textsf{EFA}, sometimes also called $\\textsf{I}\\Delta_{0}+\\textsf{EXP}$. \nIn particular, the exponential function is (all what is) required to `easily' manipulate finite sequences. \n\\end{proof}\nNext, we discuss the vast scope of the term extraction result in Theorem~\\ref{consresult}. \n\\begin{rem}[The scope of term extraction]\\label{firliborn}\\textup{rm}\nFirst of all, there are examples of classically provable sentences (See \\cite{kohlenbach3}*{\\S2.2}) with \\emph{only two} quantifier alternations from which no computational information can be extracted. \nBy contrast, it is shown in \\cite{sambon} that the scope of Theorem~\\ref{consresult} encompasses all theorems of `pure' Nonstandard Analysis, where `pure' means that only \\emph{nonstandard} definitions (of continuity, compactness, differentiability, Riemann integration, et cetera) are used. Indeed, it is easy to prove in $\\textup{\\textsf{P}}$ (or $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$) that these nonstandard definitions have equivalent normal forms, and that an implication between two normal forms is again equivalent to a normal form. In other words, the scope of the term extraction result in Theorem~\\ref{consresult} is vast, as explored in \\cites{sambon, samzoo, samzooII, samGH}. \n\\end{rem}\nFinally, we note that a `constructive' version of $\\textup{\\textsf{P}}$ is introduced in \\cite{brie}*{\\S5}. \nIn particular, the system $\\textup{\\textsf{H}}$ is a conservative extension of Heyting arithmetic $\\textsf{E-HA}^{\\omega}$ for the latter's language, and satisfies \na term extraction theorem similar to Theorem~\\ref{consresult} (See \\cite{brie}*{Theorem~5.9}). We briefly discuss $\\textup{\\textsf{H}}$ in Remark \\ref{reasonsofspace}. \n\n\n\\subsection{Notations}\nWe mostly use the notations from \\cite{brie}, some of which we repeat. \n\n\\medskip\n\nFirst of all, the following notations were sketched in Section \\ref{IST}. \n\\begin{rem}[Notations]\\label{notawin}\\textup{rm}\nWe write $(\\forall^{\\textup{NSA}^{\\alpha}}x^{\\tau})\\Phi(x^{\\tau})$ and $(\\exists^{\\textup{NSA}^{\\alpha}}x^{\\sigma})\\Psi(x^{\\sigma})$ as short for the formula\n$(\\forall x^{\\tau})\\big[\\textup{NSA}^{\\alpha}(x^{\\tau})\\rightarrow \\Phi(x^{\\tau})\\big]$ and $(\\exists x^{\\sigma})\\big[\\textup{NSA}^{\\alpha}(x^{\\sigma})\\wedge \\Psi(x^{\\sigma})\\big]$. \nWe also write $(\\forall x^{0}\\in \\Omega)\\Phi(x^{0})$ and $(\\exists x^{0}\\in \\Omega)\\Psi(x^{0})$ as short for \n$(\\forall x^{0})\\big[\\neg\\textup{NSA}^{\\alpha}(x^{0})\\rightarrow \\Phi(x^{0})\\big]$ and $(\\exists x^{0})\\big[\\neg\\textup{NSA}^{\\alpha}(x^{0})\\wedge \\Psi(x^{0})\\big]$. \nFinally, a formula $A$ is `internal' if it does not involve $\\textup{NSA}^{\\alpha}$. The formula $A^{\\textup{NSA}^{\\alpha}}$ is defined from $A$ by appending `st' to all quantifiers (except bounded number quantifiers). \n\\end{rem}\nSecondly, we use the usual notations for rational and real numbers in $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ as introduced in \\cite{kohlenbach2}*{p.\\ 288-289}. \nWe repeat some of the latter definitions. \n\\begin{defi}[Real numbers and related notions]\\label{keepinitreal}\\textup{rm}~\n\\begin{enumerate}\n\\item A (standard) rational $q^{0}$ is a fraction $\\pm\\frac{m}{n}$ for (standard) $n^{0}>0$ and (standard) $m^{0}$. We write `$q^{0}\\in {\\mathbb Q}$' to denote that $q$ is a rational. \n\\item A (standard) real number $x$ is a (standard) fast-converging Cauchy sequence $q_{(\\cdot)}^{1}$, i.e.\\ $(\\forall n^{0}, i^{0})(|q_{n}-q_{n+i})|<_{0} \\frac{1}{2^{n}})$. \nWe use Kohlenbach's `hat function' from \\cite{kohlenbach2}*{p.\\ 289} to guarantee that every sequence $f^{1}$ is a real. \n\\item We write $[x](k):=q_{k}$ for the $k$-th approximation of a real $x^{1}=(q^{1}_{(\\cdot)})$. \n\\item Two reals $x, y$ represented by $q_{(\\cdot)}$ and $r_{(\\cdot)}$ are \\emph{equal}, denoted $x=_{{\\mathbb{R}}}y$, if $(\\forall n>0)(|q_{n}-r_{n}|\\leq \\frac{1}{2^{n-1}})$. Inequality $x<_{{\\mathbb{R}}}y$ is defined by $(\\exists n>0)(q_{n}+ \\frac{1}{2^{n-1}}< r_{n})$. \n\\item We write $x\\approx y$ if $(\\forall^{\\textup{NSA}^{\\alpha}} n>0)(|q_{n}-r_{n}|\\leq \\frac{1}{2^{n-1}})$ and $x\\gg y$ if $x>_{{\\mathbb{R}}}y\\wedge x\\not\\approx y$. \n\\item Functions $F:{\\mathbb{R}}\\rightarrow {\\mathbb{R}}$ mapping reals to reals are represented by functionals $\\Phi^{1\\rightarrow 1}$ mapping equal reals to equal reals, i.e. \n\\be\\tag{\\textsf{RE}}\\label{furg}\n(\\forall x^{1}, y^{1})(x=_{{\\mathbb{R}}}y\\rightarrow \\Phi(x)=_{{\\mathbb{R}}}\\Phi(y)).\n\\ee \n\\item Sets of objects of type $\\rho$ are denoted $X^{\\rho\\rightarrow 0}, Y^{\\rho\\rightarrow 0}, Z^{\\rho\\rightarrow 0}, \\dots$ and are given by their characteristic functions $f^{\\rho\\rightarrow 0}_{X}$, i.e.\\ $(\\forall x^{\\rho})[x\\in X\\leftrightarrow f_{X}(x)=_{0}1]$, where $f_{X}^{\\rho\\rightarrow 0}$ is assumed to output zero or one. \n\\end{enumerate}\n\\end{defi}\n\n\\noindent Thirdly, we use the usual extensional notion of equality in $\\textup{\\textsf{P}}$. \n\\begin{rem}[Equality in $\\textup{\\textsf{P}}$]\\label{equ}\\textup{rm}\nEquality between natural numbers `$=_{0}$' is a primitive. Equality `$=_{\\tau}$' for type $\\tau$-objects $x,y$ is then defined as follows:\n\\be\\label{aparth}\n[x=_{\\tau}y] \\equiv (\\forall z_{1}^{\\tau_{1}}\\dots z_{k}^{\\tau_{k}})[xz_{1}\\dots z_{k}=_{0}yz_{1}\\dots z_{k}]\n\\ee\nif the type $\\tau$ is composed as $\\tau\\equiv(\\tau_{1}\\rightarrow \\dots\\rightarrow \\tau_{k}\\rightarrow 0)$.\nIn the spirit of Nonstandard Analysis, we define `approximate equality $\\approx_{\\tau}$' as follows:\n\\be\\label{aparth2}\n[x\\approx_{\\tau}y] \\equiv (\\forall^{\\textup{NSA}^{\\alpha}} z_{1}^{\\tau_{1}}\\dots z_{k}^{\\tau_{k}})[xz_{1}\\dots z_{k}=_{0}yz_{1}\\dots z_{k}]\n\\ee\nwith the type $\\tau$ as above. \nThe system $\\textup{\\textsf{P}}$ includes the \\emph{axiom of extensionality}:\n\\be\\label{EXT}\\tag{\\textsf{E}} \n(\\forall x^{\\rho},y^{\\rho}, \\varphi^{\\rho\\rightarrow \\tau}) \\big[x=_{\\rho} y \\rightarrow \\varphi(x)=_{\\tau}\\varphi(y) \\big].\n\\ee\nHowever, as noted in \\cite{brie}*{p.\\ 1973}, the so-called axiom of \\emph{standard} extensionality \\eqref{EXT}$^{\\textup{NSA}^{\\alpha}}$ is not included in $\\textup{\\textsf{P}}$, as this would jeopardise the term extraction property as in Theorem \\ref{consresult}. \nFinally, a functional $\\Xi^{ 1\\rightarrow 0}$ is called an \\emph{extensionality functional} for $\\varphi^{1\\rightarrow 1}$ if \n\\be\\label{turki}\n(\\forall k^{0}, f^{1}, g^{1})\\big[ \\overline{f}\\Xi(f,g, k)=_{0}\\overline{g}\\Xi(f,g,k) \\rightarrow \\overline{\\varphi(f)}k=_{0}\\overline{\\varphi(g)}k \\big], \n\\ee\ni.e.\\ $\\Xi$ witnesses \\eqref{EXT} for $\\varphi$. \nAs will become clear in Section \\ref{main}, $\\eqref{EXT}^{\\textup{NSA}^{\\alpha}}$ is translated to the existence of an extensionality functional when applying Theorem \\ref{consresult}. \n\\end{rem} \n\n\\section{An analogue of Grilliot's trick in Nonstandard Analysis}\\label{main}\nIn this section, we show how from certain equivalences in Nonstandard Analysis involving a fragment of Nelson's \\emph{Transfer}, namely $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, one obtains \\emph{effective} RM-equivalences involving $(\\exists^{2})$ in Kohlenbach's higher-order RM. \n\\be\\tag{$\\exists^{2}$}\n(\\exists \\varphi^{2})(\\forall f^{1})\\big[ (\\exists n)(f(n)=0)\\leftrightarrow \\varphi(f)=0 \\big].\n\\ee\n\\be\\tag{$\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$}\n\\qquad\\qquad\\quad\\enspace(\\forall^{\\textup{NSA}^{\\alpha}}f^{1})\\big[(\\forall^{\\textup{NSA}^{\\alpha}}n)f(n)\\ne 0 \\rightarrow (\\forall n)f(n)\\ne0 \\big].\n\\ee\nTo this end, we shall make use of a technique from Nonstandard Analysis we call \\emph{the nonstandard extensionality trick}, and which is similar to \\emph{Grilliot's trick}. We first introduce some of the above italicised notions in the following section.\n\\subsection{Preliminaries}\nIn this section, we introduce the notion of `effective implication' and so-called Grilliot's trick. \nFirst of all, the notion of `effective implication' is defined as one would expect in $\\textup{\\textsf{RCA}}_{0}^{\\omega}$. \n\\begin{defi}\\textup{rm}[Effective implication]\\label{effimp}\nAn implication $(\\exists \\Phi)A(\\Phi)\\rightarrow (\\exists \\Psi)B(\\Psi)$ (proved in $\\textup{\\textsf{RCA}}_{0}$) is \\emph{effective} if there is a term $t$ (in the language of $\\textup{\\textsf{RCA}}_{0}^{\\omega}$) such that additionally $(\\forall \\Phi)[A(\\Phi)\\rightarrow B(t(\\Phi))]$ (proved in $\\textup{\\textsf{RCA}}_{0}^{\\omega}$). \n\\end{defi}\nThe terms obtained using Theorem \\ref{consresult} are \\emph{primitive recursive} in the sense of G\\\"odel's system ${T}$, as discussed in Section \\ref{PIPI}. \nIn light of the elementary nature of an extensionality functional (See Remark~\\ref{equ}), we still refer to an implication as `effective', if the term $t$ as in Definition \\ref{effimp} involves an extensionality functional. \nNote that $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ proves the existence of an extensionality functional thanks to $\\textup{\\textsf{QF-AC}}^{1,0}$ while an unbounded search (available in a more general setting than G\\\"odel's $T$) also yields such a functional. \n\n\\medskip\n\nSecondly, as to methodology, we shall make use of a nonstandard technique, called \\emph{the nonstandard extensionality trick} (See Remark \\ref{trick}), similar to \\emph{Grilliot's trick}. \nNow, the latter trick is in fact an explicit construction to obtain the Turing jump functional $(\\exists^{2})$ from a given effectively discontinuous functional \n(See e.g.\\ \\cite{grilling}, \\cite{kohlenbach2}*{Prop.~3.7}, or \\cite{kooltje}*{Prop.\\ 3.4} for more details). In our \\emph{nonstandard} trick, one obtains $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ from a functional $\\Phi^{1\\rightarrow 1}$ which is \\emph{nonstandard} discontinuous, i.e.\\ there are\n$x_{0}\\approx_{1} x_{1}$ such that $ \\Phi(x_{0})\\not\\approx_{1} \\Phi(x_{1})$. By applying term extraction as in Theorem~\\ref{consresult}, one then obtains an effective implication involving $(\\exists^{2})$.\nAs we will see, the nonstandard proof involving $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ uses proof by contradiction, i.e.\\ \\emph{no attempt to obtain effective content is made in the nonstandard proofs}. \n\n\\medskip\n\nThirdly, in the next sections, we apply the aforementioned nonstandard extensionality trick to \\emph{binary expansion}, the \\emph{intermediate value theorem}, the \\emph{Weierstra\\ss~maximum theorem}, and \\emph{weak weak K\\\"onig's lemma}. We choose these theorems due to their `non-constructive' nature, and as some of the associated uniform versions (sometimes involving sequences) have been studied (\\cites{sayo, polahirst, yamayamaharehare}, \\cite{simpson2}*{IV.2.12}, \\cite{kohlenbach2}*{\\S3}). \nIt is particularly interesting that we can `recycle' the Brouwerian counterexamples to the intermediate value theorem and Weierstra\\ss~maximum theorem (\\cite{beeson1}*{I.7}, \\cite{mandje2}) to obtain nonstandard equivalences. \n\n\\medskip\n\nFinally, it is a natural question \\emph{why} we can obtain computational information from proofs in classical Nonstandard Analysis \\emph{at all}. \nIndeed, Bishop and Connes have made rather strong claims regarding the non-constructive nature of Nonstandard Analysis (See\\footnote{The third reference is Bishop's review of Keisler's introduction to Nonstandard Analysis \\cite{keisler3}.} \\cite{kluut}*{p.\\ 513}, \\cite{bishl}*{p.\\ 1}, \\cite{kuddd}, \\cite{conman2}*{p.\\ 6207} and \\cite{conman}*{p.\\ 26}). Furthermore, there are examples of classically provable sentences (See \\cite{kohlenbach3}*{\\S2.2}) with \\emph{only two} quantifier alternations from which no computational information can be extracted, and the aforementioned theorems involve a lot more quantifier alternations. Moreover, our nonstandard proofs make use of `proof by contradiction', i.e.\\ no attempt at a `constructive' proof is made.\nNonetheless, in Sections \\ref{bincco} to \\ref{hivt}, we shall obtain effective equivalences from certain `non-constructive' nonstandard equivalences. \nFollowing similar results in Section \\ref{carmichael}, we offer an explanation why Nonstandard Analysis contains so much computational information. \n\n\\subsection{Binary conversion}\\label{bincco}\nIn this section, we study the principle of \\emph{binary conversion}, i.e.\\ the statement that every real can be represented in binary as follows:\n\\be\\label{bin}\\tag{$\\textup{\\textsf{BIN}}$}\\textstyle\n(\\forall x\\in [0,1])(\\exists\\alpha^{1}\\leq_{1}1)(x=_{{\\mathbb{R}}}\\sum_{i=1}^{\\infty}\\frac{\\alpha(i)}{2^{i}}).\n\\ee\nHirst shows in \\cite{polahirst} that $\\textup{\\textsf{RCA}}_{0}$ proves $\\textup{\\textsf{BIN}}$, and that a uniform version of the latter \\emph{involving sequences} is equivalent to $\\textup{\\textsf{WKL}}$. \nFurthermore, $\\textup{\\textsf{BIN}}$ is equivalent to ${\\textup{\\textsf{LLPO}}}$ in \\emph{constructive} Reverse Mathematics (\\cite{bridges1}*{p.\\ 10}), while a finer classification may be found in \\cite{bergske}. \nWe study a \\emph{higher-type} {uniform} version of $\\textup{\\textsf{BIN}}$:\n\\be\\label{ubin}\\tag{$\\textup{\\textsf{UBIN}}$}\\textstyle\n(\\exists \\Phi:{\\mathbb{R}}\\rightarrow 1)(\\forall x\\in [0,1])\\big[ \\Phi(x)\\leq_{1}1\\wedge x=_{{\\mathbb{R}}}\\sum_{i=1}^{\\infty}\\frac{\\alpha(i)}{2^{i}}\\big].\n\\ee\nWe shall first establish a particular nonstandard equivalence involving $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, and a nonstandard version of $\\textup{\\textsf{UBIN}}$. As a result of applying Corollary~\\ref{consresultcor2} to this nonstandard equivalence, we obtain an \\emph{effective} equivalence between $\\textup{\\textsf{UBIN}}$ and the following version of arithmetical comprehension. \n\\be\\tag{$\\mu^{2}$}\\label{Frak}\n(\\exists \\mu^{2})\\big[(\\forall f^{1})\\big((\\exists x^{0})f(x)=0 \\rightarrow f(\\mu(f))=0 \\big)\\big].\n\\ee \nThe functional $(\\mu^{2})$ is also known as \\emph{Feferman's non-constructive mu-operator} (See \\cite{avi2}*{\\S8.2}), and is equivalent to $(\\exists^{2})$ in $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ by \\cite{kooltje}*{\\S3}. \nWe denote by $\\textup{\\textsf{MU}}(\\mu)$ the formula in square brackets in $(\\mu^{2})$. \nWe use the following nonstandard version of $\\textup{\\textsf{UBIN}}$, called $\\textup{\\textsf{UBIN}}^{+}$:\n\\[\n(\\exists^{\\textup{NSA}^{\\alpha}} \\Phi:{\\mathbb{R}}\\rightarrow 1)(\\forall^{\\textup{NSA}^{\\alpha}} x\\in [0,1])\\big[\\textup{\\textsf{UBIN}}(\\Phi, x)\\wedge (\\forall^{\\textup{NSA}^{\\alpha}}x, y\\in [0,1])(x\\approx y \\rightarrow \\Phi(x)\\approx_{1}\\Phi(y)) \\big]. \n\\]\nwhere $\\textup{\\textsf{UBIN}}(\\Phi, x)$ is the formula in square brackets in $\\textup{\\textsf{UBIN}}$. Note that the second conjunct of $\\textup{\\textsf{UBIN}}^{+}$ expresses that $\\Phi$ is `standard extensional', i.e.\\ satisfies extensionality as in \\eqref{furg} relative to `st' (and the range is Baire space instead of ${\\mathbb{R}}$). \n\\begin{thm}\\label{proto7}\nFrom a proof in $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$ that $\\textup{\\textsf{UBIN}}^{+}\\leftrightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, terms $s,t$ can be extracted such that $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ proves:\n\\be\\label{frood7}\n(\\forall \\mu^{2})\\big[\\textsf{\\textup{\\textsf{MU}}}(\\mu)\\rightarrow \\textup{\\textsf{UBIN}}(s(\\mu)) \\big] \\wedge (\\forall \\Phi^{1\\rightarrow 1})\\big[ \\textup{\\textsf{UBIN}}(\\Phi)\\rightarrow \\textup{\\textsf{MU}}(t(\\Phi, \\Xi)) \\big].\n\\ee\nwhere $\\Xi$ is an extensionality functional for $\\Phi$ and $\\textup{\\textsf{UBIN}}(\\Phi)$ is $(\\forall x\\in [0,1])\\textup{\\textsf{UBIN}}(\\Phi, x)$. \n\\end{thm}\n\\begin{proof}\nFirst of all, we prove that $\\textup{\\textsf{UBIN}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ in $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$, and obtain the associated second conjunct of \\eqref{frood7}. The remaining results are then sketched.\n\n\\medskip\n\nTo prove $\\textup{\\textsf{UBIN}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ in $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$, assume $\\textup{\\textsf{UBIN}}^{+}$ and suppose that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ is false, i.e.\\ there is \\emph{standard} $g$ such that $(\\forall^{\\textup{NSA}^{\\alpha}}n)g(n)=0 $ but also $ (\\exists m^{0})g(m)\\ne 0$. Now define the \\emph{standard} sequence $\\alpha_{0}$ as follows\n\\be\\label{sly}\n\\alpha_{0}(i):=\n\\begin{cases}\n0 & (\\forall n\\leq i)g(n)=0 \\\\\n1 & \\text{otherwise}\n\\end{cases}.\n\\ee\nFurthermore, define the \\emph{standard} reals $x_{\\pm}:= \\frac{1}{2}\\pm \\sum_{n=1}^{\\infty} \\frac{\\alpha_{0}(n)}{2^{n}}$ and note that $x_{+}\\approx x_{-}$ by the definition of $g$. Since $x_{-}<_{{\\mathbb{R}}} \\frac12 <_{{\\mathbb{R}}}x_{+}$, the binary expansion $\\alpha_{\\pm}$ of $x_{\\pm}$ must be such that $\\alpha_{-}(1)=0$ and $\\alpha_{+}(1)=1$. However, this implies that $\\Phi(x_{-})(1)=0\\ne 1=\\Phi(x_{+})(1)$, and also $\\Phi(x_{-})\\not\\approx_{1}\\Phi(x_{+})$. \nClearly, the latter contradicts the standard extensionality of $\\Phi$ as $x_{+}\\approx x_{-}$ was also proved. \nIn light of this contradiction, we must have $\\textup{\\textsf{UBIN}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. \n\n\\medskip\n\n\nWe now prove the second conjunct in \\eqref{frood7}. \nNote that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ can easily be brought into the following normal form: \n\\be\\label{frux}\n(\\forall^{\\textup{NSA}^{\\alpha}}f^{1})(\\exists^{\\textup{NSA}^{\\alpha}}i^{0})\\big[(\\exists n^{0})f(n)=0\\rightarrow (\\exists m\\leq i)f(m)=0\\big], \n\\ee\nwhere the formula in square brackets is abbreviated by $B(f, i)$. Similarly, the second conjunct of $\\textup{\\textsf{UBIN}}^{+}$ has the following normal form: \n\\be\\label{kurve2}\\textstyle\n(\\forall^{\\textup{NSA}^{\\alpha}} x^{1}, y^{1}\\in [0,1], k^{0})(\\exists^{\\textup{NSA}^{\\alpha}} N)\\big[|x-y|<\\frac{1}{N} \\rightarrow \\overline{\\Phi(x)}k=_{0}\\overline{\\Phi(y)}k\\big],\n\\ee\nwhich is immediate by resolving `$\\approx_{1}$' and `$\\approx$', and bringing standard quantifiers outside. \nWe denote the formula in square brackets in \\eqref{kurve2} by $A(x, y, N, k, \\Phi)$. \nHence, $\\textup{\\textsf{UBIN}}^+\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ now easily yields:\n\\begin{align}\n(\\forall^{\\textup{NSA}^{\\alpha}}\\Phi, \\Xi)\\big[ [(\\forall^{\\textup{NSA}^{\\alpha}}x\\in [0,1])\\textup{\\textsf{UBIN}}(\\Phi, x)&\\wedge (\\forall^{\\textup{NSA}^{\\alpha}} x,y\\in [0,1], k^{0})A(x, y, \\Xi(x, y, k), k, \\Phi)]\\notag\\\\\n& \\rightarrow (\\forall^{\\textup{NSA}^{\\alpha}}f^{1})(\\exists^{\\textup{NSA}^{\\alpha}}n)B(f,n)\\big], \\label{tochtag}\n\\end{align}\nas standard $\\Xi$ as in the antecedent of \\eqref{tochtag} yields standard outputs for standard inputs, and hence \\eqref{kurve2} follows. \nDropping the `st' in the antecedent of \\eqref{tochtag} and bringing out the remaining standard quantifiers, we obtain the normal form:\n\\begin{align}\n(\\forall^{\\textup{NSA}^{\\alpha}}\\Phi, \\Xi ,f^{1})(\\exists^{\\textup{NSA}^{\\alpha}}n)\\big[ [(\\forall x^{1}\\in [0,1])&\\textup{\\textsf{UBIN}}(\\Phi, x)\\wedge\\notag \\\\\n& (\\forall U^{1}, S^{1}, k^{0})A(U, S, \\Xi(U, S, k), k, \\Phi)] \\rightarrow B(f,n)\\big]. \\label{enoka}\n\\end{align}\nLet $C(\\Phi, \\Xi, f, n)$ be the formula in big square brackets and apply Corollary~\\ref{consresultcor2} to `$\\textup{\\textsf{RCA}}_{0}^{\\Lambda}\\vdash (\\forall^{\\textup{NSA}^{\\alpha}}\\Phi, \\Xi, f^{1})(\\exists^{\\textup{NSA}^{\\alpha}}n)C(\\Phi, \\Xi, f, n)$' to obtain a term $t$ such that $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ proves \n\\be\\label{drifgs}\n (\\forall \\Phi, \\Xi, f^{1})(\\exists n\\in t(\\Phi, \\Xi, f))C(\\Phi, \\Xi, f, n). \n \\ee\nNow define the term $s(\\Phi, \\Xi, f)$ as $\\max_{i<|t(\\Phi, \\Xi, f)|}t(\\Phi, \\Xi, f)(i)$ and \nnote that the formula $(\\exists n\\in t(\\Phi, \\Xi, f))C(\\Phi, \\Xi, f, n)$ implies $C(\\Phi, \\Xi, f, s(\\Phi, \\Xi, f))$.\nFinally, bring the quantifier involving $f$ inside $C$ to obtain for all $\\Phi, \\Xi$ that\n\\[\n [(\\forall x^{1}\\in[0,1])\\textup{\\textsf{UBIN}}(\\Phi, x)\\wedge (\\forall U^{1}, S^{1}, k^{0})A(U, S, \\Xi(U, S, k), k, \\Phi)] \\rightarrow (\\forall f^{1}) B(f,s(\\Phi, \\Xi, f)).\n\\]\nThus, $s(\\Phi, \\Xi, \\cdot)$ provides the functional $(\\mu^{2})$ if $\\Phi$ satisfies $(\\forall x^{1}\\in[0,1])\\textup{\\textsf{UBIN}}(\\Phi,x )$ and $\\Xi$ is the associated extensionality functional. \n\n\\medskip\n\nFinally, to prove $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}\\rightarrow \\textup{\\textsf{UBIN}}^{+}$, consider \\eqref{frux} and apply $\\textup{\\textsf{HAC}}_{\\textup{int}}$ to the former to obtain $\\nu^{1\\rightarrow 0^{*}}$ such that $(\\forall^{\\textup{NSA}^{\\alpha}}f^{1})(\\exists i^{0}\\in \\nu({f}))A(f, i)$, where $A$ is the formula in square brackets in \\eqref{frux}. Now define the \\emph{standard} functional $\\xi^{2}$ by \n\\[\n\\xi(f):=(\\mu m\\leq \\max_{i<|\\nu(f)|}\\nu(f)(i))(f(m)=0)\n\\]\nand note that $[\\textup{\\textsf{MU}}(\\xi)]^{\\textup{NSA}^{\\alpha}}$, i.e.\\ we have access to Feferman's search operator relative to `st'. In particular, $\\xi^{2}$ provides arithmetical comprehension (and \\emph{Transfer} for $\\Pi_{1}^{0}$-formulas):\n\\be\\label{karmic}\n(\\forall^{\\textup{NSA}^{\\alpha}}f^{1})\\big[( f(\\xi(f))=0) \\leftrightarrow (\\exists m^{0})(f(m)=0)\\leftrightarrow (\\exists^{\\textup{NSA}^{\\alpha}} m^{0})(f(m)=0) \\big]. \n\\ee\nTo define $\\Phi$ as in $\\textup{\\textsf{UBIN}}^{+}$, use $\\xi$ from \\eqref{karmic} to decide if $x\\geq_{{\\mathbb{R}}}\\frac12$ or $x<_{{\\mathbb{R}}}\\frac12$ and define $\\Phi(x)(0)$ as $1$ or $0$ respectively. \nSimilarly, define $\\Phi(x)(n+1)$ as $1$ or $0$ depending on whether $x\\geq_{{\\mathbb{R}}}\\frac{1}{2}(x+\\sum_{i=0}^{n}\\frac{\\Phi(x)(i)}{2^{i}})$ or not, again using $\\xi$. Then $\\Phi$ is standard and \nsatisfies $[\\textup{\\textsf{UBIN}}(\\Phi)]^{\\textup{NSA}^{\\alpha}}$. Now apply $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ to the latter and the axiom of extensionality to obtain $\\textup{\\textsf{UBIN}}^{+}$. The first conjunct of \\eqref{frood7} now follows in the same way as in the first part of the proof. \n\\end{proof}\nNote that the non-computable power of \\emph{uniform} $\\textup{\\textsf{BIN}}$ (both nonstandard and non-nonstandard) arises from the fact that not all reals have a unique binary expansion.\nHence, for small (infinitesimal) variations of the input of the functional in $\\textup{\\textsf{UBIN}}$, we can produce large (standard) variations in the output. This is exploited as follows in the previous proof. \n\\begin{rem}[Nonstandard extensionality trick]\\label{trick}\\textup{rm}\nFirst of all, we note that $\\Phi$ as in $\\textup{\\textsf{UBIN}}(\\Phi)$ is \\emph{nonstandard} discontinuous in that for every $x,y\\in {\\mathbb{R}}$ such that $x<_{{\\mathbb{R}}}\\frac{1}{2}<_{{\\mathbb{R}}}y \\wedge x \\approx \\frac{1}{2}\\approx y$ we have $\\Phi(x)\\not\\approx_{1}\\Phi(y)$, in particular $\\Phi(x)(1)=0\\ne 1=\\Phi(y)(1)$. \nSecondly, we use \\eqref{sly} to define \\emph{standard} points $x_{\\pm}$ at which $\\Phi$ from $\\textup{\\textsf{UBIN}}^{+}$ is \\emph{nonstandard discontinuous}, assuming $\\neg\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. \nThe ensuing contradiction with the standard extensionality of $\\Phi$ yields $\\textup{\\textsf{UBIN}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. Thirdly, applying term extraction to the (normal form \\eqref{enoka} of the) latter implication, we obtain the effective implication \\eqref{frood7}. \n\\end{rem}\nThe previous technique is similar in spirit to Grilliot's trick, but note that our nonstandard technique produces an effective implication, \\emph{without paying attention to effective content}. \nIn particular, we used the non-constructive `proof by contradiction' to establish $\\textup{\\textsf{UBIN}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, and `independence of premises' to obtain the latter's normal form \\eqref{enoka} (See e.g.\\ \\eqref{frux} and \\eqref{kurve2}). \n\n\\medskip\n\nNote that we do not claim that the previous theorem (or the below theorems) is unique or a first in this regard: Kohlenbach's treatment of Grilliot's trick (\\cite{kooltje}) and the \\emph{proof mining} program (\\cite{kohlenbach3}) are well-known to produce effective results from classical mathematics. \nWhat is surprising about results in this paper (in our opinion) is the emergence of effective content (with relative ease) from Nonstandard Analysis \\emph{despite} claims \nthat the latter is somehow fundamentally non-constructive by e.g.\\ Bishop and Connes (See \\cite{samsynt} for a detailed discussion of the Bishop-Connes critique). \n\n\\medskip\n\nSurprisingly, the proof of Theorem \\ref{proto7} goes through constructively, as we discuss now. \n\\begin{rem}[The system $\\textup{\\textsf{H}}$]\\label{reasonsofspace}\\textup{rm}\nThe system $\\textup{\\textsf{H}}$ is a conservative extension of Heyting arithmetic satisfying a term extraction theorem similar to Theorem \\ref{consresult} (See \\cite{brie}*{Theorem~5.9}).\nAlthough $\\textup{\\textsf{H}}$ is based on intuitionistic logic, it does prove the following `standard' version of Markov's principle (See \\cite{brie}*{p.\\ 1978}):\n\\be\\tag{$\\textsf{MP}^{\\textup{NSA}^{\\alpha}}$}\n(\\forall^{\\textup{NSA}^{\\alpha}} f^{1})\\big[ \\neg\\neg[(\\exists^{\\textup{NSA}^{\\alpha}}m)(f(m)=0)] \\rightarrow (\\exists^{\\textup{NSA}^{\\alpha}}n)(f(n)=0) \\big].\n\\ee\nNow, the proof of $ \\textup{\\textsf{UBIN}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ in Theorem \\ref{proto7} easily yields a proof of: \n\\be\\label{lekkerbekske}\n\\textup{\\textsf{UBIN}}^{+} \\rightarrow (\\forall^{\\textup{NSA}^{\\alpha}}f^{1})\\big[(\\exists m)(f(m)=0) \\rightarrow \\neg[(\\forall^{\\textup{NSA}^{\\alpha}}n^{0})(f(n)=0)] \\big].\n\\ee\ninside the system $\\textup{\\textsf{H}}$. \nHowever, combined with $\\textsf{MP}^{\\textup{NSA}^{\\alpha}}$, \\eqref{lekkerbekske} yields that $\\textup{\\textsf{H}}$ also proves the implication $ \\textup{\\textsf{UBIN}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, using the same `proof by contradiction' proof used for Theorem \\ref{proto7}. Furthermore, similar to $\\textsf{MP}^{\\textup{NSA}^{\\alpha}}$, the system $\\textup{\\textsf{H}}$ also contains a `standard' version of the independence of premise schema (in the form of $\\textsf{HIP}_{\\forall^{\\textup{NSA}^{\\alpha}}}$; see \\cite{brie}*{p.\\ 1978}). Thanks to this schema, $\\textup{\\textsf{H}}$ proves $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}} \\rightarrow \\eqref{frux}$ and even that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}} \\rightarrow \\textup{\\textsf{UBIN}}^{+}$ implies its normal form \\eqref{enoka}. Applying the term extraction theorem \\cite{brie}*{Theorem~5.9} for $\\textup{\\textsf{H}}$, a constructive proof of \\eqref{frood7} is established. \n\\end{rem}\n\n\n\\subsection{Weak weak K\\\"onig's lemma}\\label{FUWKL}\nIn this section, we study the principle \\emph{weak weak K\\\"onig's lemma} ($\\textup{\\textsf{WWKL}}$ for short), using the standard extensionality trick in Remark \\ref{trick}.\nNote that $\\textup{\\textsf{WWKL}}$ was not directly studied in \\cites{samzoo, samzooII}. \n\\begin{defi}\\textup{rm}[Weak weak K\\\"onig's lemma]\\label{leipi}~\n\\begin{enumerate}\n\\item We reserve `$T^{1}$' for trees and denote by `$T^{1}\\leq_{1}1$' that $T$ is a \\emph{binary} tree. \n\\item For a binary tree $T$, define $\\nu(T):=\\lim_{n\\rightarrow \\infty}\\frac{\\{\\sigma \\in T: |\\sigma|=n \\}}{2^{n}}$.\n\\item For a binary tree $T$, define `$\\nu(T)>_{{\\mathbb{R}}}a^{1}$' as $(\\exists k^{0})(\\forall n^{0})\\big(\\frac{\\{\\sigma \\in T: |\\sigma|=n \\}}{2^{n}}\\geq a+\\frac{1}{k}\\big)$.\n\\item We define $\\textup{\\textsf{WWKL}}$ as $(\\forall T \\leq_{1}1)\\big[ \\nu(T)>_{{\\mathbb{R}}}0\\rightarrow (\\exists \\beta\\leq_{1}1)(\\forall m)(\\overline{\\beta}m\\in T) \\big]$.\n\\end{enumerate} \n\\end{defi}\nThe principle $\\textup{\\textsf{WWKL}}$ is not part of the `Big Five' of RM, but there are \\emph{some} equivalences involving the former (See \\cite{simpson2}*{X.1}). \nIn this section, we study the following uniform versions:\n\\be\\tag{$\\textup{\\textsf{UWWKL}}$}\n(\\exists \\Phi^{1\\rightarrow 1})(\\forall T \\leq_{1}1)\\big[ \\nu(T)>_{{\\mathbb{R}}}0\\rightarrow (\\forall m)(\\overline{\\Phi(T)}m\\in T) \\big]\n\\ee\nAlso, $\\textup{\\textsf{UWWKL}}(\\Phi(T), T)$ is $\\textup{\\textsf{UWWKL}}$ without the leading quantifiers, and $\\textup{\\textsf{UWWKL}}^{+}$ is\n\\[\n(\\exists^{\\textup{NSA}^{\\alpha}}\\Phi^{1\\rightarrow 1})\\big[(\\forall^{\\textup{NSA}^{\\alpha}}T^{1})\\textup{\\textsf{UWWKL}}(\\Phi(T), T)\\wedge (\\forall^{\\textup{NSA}^{\\alpha}} T^{1}, S^{1})\\big(T\\approx_{1} S \\rightarrow \\Phi(T)\\approx_{1}\\Phi(S) \\big)\\big].\n\\]\nNote that the second conjunct expresses that $\\Phi$ is \\emph{standard extensional}. \nWe have the following theorem, which is the effective version of \\cite{yamayamaharehare}*{Theorem 3.2}. Note that $\\textup{\\textsf{UWWKL}}(\\Phi)$ is $(\\forall T \\leq_{1}1)\\textup{\\textsf{UWWKL}}(\\Phi(T), T)$. \n\\begin{thm}\\label{proto}\nFrom a proof in $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$ that $\\textup{\\textsf{UWWKL}}^{+}\\leftrightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, terms $s,t$ can be extracted such that $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ proves:\n\\be\\label{frood8}\n(\\forall \\mu^{2})\\big[\\textsf{\\textup{\\textsf{MU}}}(\\mu)\\rightarrow \\textup{\\textsf{UWWKL}}(s(\\mu)) \\big] \\wedge (\\forall \\Phi^{1\\rightarrow 1})\\big[ \\textup{\\textsf{UWWKL}}(\\Phi)\\rightarrow \\textup{\\textsf{MU}}(t(\\Phi, \\Xi)) \\big],\n\\ee\nwhere $\\Xi$ is an extensionality functional for $\\Phi$. \n\\end{thm}\n\\begin{proof}\nFirst of all, to prove $\\textup{\\textsf{UWWKL}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ in $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$, assume $\\textup{\\textsf{UWWKL}}^{+}$ and suppose that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ is false, i.e.\\ there is $f$ such that $(\\forall^{\\textup{NSA}^{\\alpha}}n)f(n)=0 \\wedge (\\exists m^{0})f(m)\\ne 0$.\nNow define the trees $T_{i}$ for $i=0,1$ as follows\n\\[\n\\sigma\\in T_{i}\\leftrightarrow \\big[\\sigma(0)=i \\vee \\big[\\sigma(0)=1-i\\wedge (\\forall m\\leq |\\sigma|)f(m)=0\\big] \\big].\n\\]\nBy the definition of $T_{i}$ and the behaviour of $f$, we have $T_{0}\\approx_{1} T_{1}\\approx_{1} 2^{{\\mathbb N}}$, where the latter is the full binary tree and ${\\mathbb N}:=\\{n^{0}:n=_{0}n\\}$. Furthermore, $\\nu(T_{0})=\\nu(T_{1})=\\frac12$ hold, and observe that $T_{0}$ (resp.\\ $T_{1}$) only has paths starting with $0$ (resp.\\ $1$). Hence, we have \n$\\Phi(T_{0})(0)=0\\ne 1= \\Phi(T_{1})(0)$ for $\\Phi$ as in $\\textup{\\textsf{UWWKL}}^{+}$, which yields $\\Phi(T_{0})\\not\\approx_{1} \\Phi(T_{1})$. Clearly, the latter contradicts the standard extensionality of $\\Phi$. \nIn light of this contradiction, we have $\\textup{\\textsf{UWWKL}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. \n\n\\medskip\n\nSecondly, to prove $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}\\rightarrow \\textup{\\textsf{UWWKL}}^{+}$, note that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ implies \\eqref{karmic} as established in the proof of Theorem \\ref{proto7}. \nTo define $\\Phi$ as in $\\textup{\\textsf{UWWKL}}^{+}$, let standard $T^{1}\\leq_{1}1$ be such that $\\nu(T)>_{{\\mathbb{R}}}0$ and use (standard) $\\xi$ from \\eqref{karmic} to decide if\n\\be\\label{contjas}\n(\\forall^{\\textup{NSA}^{\\alpha}} n^{0})(\\exists \\beta^{0^{*}}\\in T)(\\beta(0)=1 \\wedge |\\beta|=n)\\textup{ or } (\\forall^{\\textup{NSA}^{\\alpha}} n^{0})(\\exists \\beta^{0^{*}}\\in T)(\\beta(0)=0 \\wedge |\\beta|=n),\n\\ee\nand define $\\Phi(T)(0)$ as $1$ if the first formula in \\eqref{contjas} holds, and $0$ otherwise. \nSimilarly, for $\\Phi(T)(m+1)$ again use $\\xi$ from \\eqref{karmic} to decide if the following formula holds:\n\\[\n(\\forall^{\\textup{NSA}^{\\alpha}} n^{0}\\geq m+1)(\\exists \\beta^{0^{*}}\\in T)(\\overline{\\beta}m=\\Phi(T)(0)*\\dots* \\Phi(T)(m) \\wedge\\beta(m+1)=1 \\wedge |\\beta|=n) \n\\]\nand define $\\Phi(T)(m+1)$ as $1$ if it does, and zero otherwise. \nThen $\\Phi$ is standard and satisfies $[\\textup{\\textsf{UWWKL}}(\\Phi)]^{\\textup{NSA}^{\\alpha}}$. Now apply $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ to the latter and \\eqref{EXT} to obtain $\\textup{\\textsf{UWWKL}}^{+}$. \n\n\\medskip\n\nThirdly, we now prove the second conjunct in \\eqref{frood8}. \nNote that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ can easily be brought into the normal form \\eqref{frux}\nwhere the formula in square brackets is abbreviated by $B(f, i)$. Similarly, the second conjunct of $\\textup{\\textsf{UWWKL}}^{+}$ has the following normal form: \n\\be\\label{kurve}\n(\\forall^{\\textup{NSA}^{\\alpha}} T^{1}, S^{1}, k^{0})(\\exists^{\\textup{NSA}^{\\alpha}} N)\\big[\\overline{T}N=_{0}\\overline{S}N \\rightarrow \\overline{\\Phi(T)}k=_{0}\\overline{\\Phi(S)}k\\big],\n\\ee\nwhich is immediate by resolving `$\\approx_{1}$' and bringing standard quantifiers outside. \nWe denote the formula in square brackets in \\eqref{kurve} by $A(T, S, N, k, \\Phi)$. \nHence, the implication $\\textup{\\textsf{UWWKL}}^+\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ now immediately yields:\n\\begin{align}\n(\\forall^{\\textup{NSA}^{\\alpha}}\\Phi, \\Xi)\\big[ [(\\forall^{\\textup{NSA}^{\\alpha}}T^{1})\\textup{\\textsf{UWWKL}}(\\Phi(T), T)&\\wedge (\\forall^{\\textup{NSA}^{\\alpha}} U^{1}, S^{1}, k^{0})A(U, S, \\Xi(U, S, k), k, \\Phi)]\\notag\\\\\n& \\rightarrow (\\forall^{\\textup{NSA}^{\\alpha}}f^{1})(\\exists^{\\textup{NSA}^{\\alpha}}n)B(f,n)\\big],\\label{similarluuuuu}\n\\end{align}\nby strengthening the antecedent by introducing $\\Xi$.\nDropping the `st' in the antecedent of the implication and bringing out the remaining standard quantifiers:\n\\[\n(\\forall^{\\textup{NSA}^{\\alpha}}\\Phi, \\Xi ,f^{1})(\\exists^{\\textup{NSA}^{\\alpha}}n)\\big[ [(\\forall T^{1})\\textup{\\textsf{UWWKL}}(\\Phi(T), T)\\wedge (\\forall U, S, k)A(U, S, \\Xi(U, S, k), k, \\Phi)] \\rightarrow B(f,n)\\big]. \n\\]\nLet $C(\\Phi, \\Xi, f, n)$ be the formula in big square brackets and apply Corollary~\\ref{consresultcor2} to `$\\textup{\\textsf{RCA}}_{0}^{\\Lambda}\\vdash (\\forall^{\\textup{NSA}^{\\alpha}}\\Phi, \\Xi, f^{1})(\\exists^{\\textup{NSA}^{\\alpha}}n)C(\\Phi, \\Xi, f, n)$' to obtain a term $t$ such that $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ proves \n\\be\\label{drifgs2}\n (\\forall \\Phi, \\Xi, f^{1})(\\exists n\\in t(\\Phi, \\Xi, f))C(\\Phi, \\Xi, f, n). \n \\ee\nNow define the term $s(\\Phi, \\Xi, f)$ as $\\max_{i<|t(\\Phi, \\Xi, f)|}t(\\Phi, \\Xi, f)(i)$ and \nnote that the formula $(\\exists n\\in t(\\Phi, \\Xi, f))C(\\Phi, \\Xi, f, n)$ implies $C(\\Phi, \\Xi, f, s(\\Phi, \\Xi, f))$.\nFinally, bring the quantifier involving $f$ inside $C$ to obtain for all $\\Phi, \\Xi$ that\n\\[\n [(\\forall T^{1})\\textup{\\textsf{UWWKL}}(\\Phi(T), T)\\wedge (\\forall U^{1}, S^{1}, k^{0})A(U, S, \\Xi(U, S, k), k, \\Phi)] \\rightarrow (\\forall f^{1}) B(f,s(\\Phi, \\Xi, f)).\n\\]\nThus, $s(\\Phi, \\Xi, \\cdot)$ provides $(\\mu^{2})$ if $\\Phi$ satisfies $(\\forall T^{1})\\textup{\\textsf{UWWKL}}(\\Phi(T),T )$ and $\\Xi$ is the associated extensionality functional. \n\n\\medskip\n\nFinally, the first conjunct in \\eqref{frood8} is proved as follows: $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}\\rightarrow \\textup{\\textsf{UWWKL}}^{+}$ yields\n\\be\\label{kanttt}\n(\\forall^{\\textup{NSA}^{\\alpha}}f^{1})(\\exists^{\\textup{NSA}^{\\alpha}}n)B(f,n)\\rightarrow (\\forall^{\\textup{NSA}^{\\alpha}}T^{1})(\\exists^{\\textup{NSA}^{\\alpha}}\\alpha^{1}\\leq_{1}1 )\\textup{\\textsf{UWWKL}}(\\alpha, T),\n\\ee\nwhere we used the same notations $ B$ as in \\eqref{similarluuuuu}. Since standard functionals yield standard output for standard input by the basic axioms of Definition \\ref{debs}, \\eqref{kanttt} yields\n\\be\\label{kanttt2}\n(\\forall^{\\textup{NSA}^{\\alpha}} \\mu^{2})\\big[(\\forall^{\\textup{NSA}^{\\alpha}}f^{1})B(f,\\mu(f))\\rightarrow (\\forall^{\\textup{NSA}^{\\alpha}}T^{1})(\\exists^{\\textup{NSA}^{\\alpha}}\\alpha^{1}\\leq_{1}1 )\\textup{\\textsf{UWWKL}}(\\alpha, T)\\big].\n\\ee\nWeakening the antecedent of \\eqref{kanttt2} and bringing outside the standard quantifiers, we obtain\n\\be\\label{kanttt3}\n(\\forall^{\\textup{NSA}^{\\alpha}} \\mu^{2}, T^{1})(\\exists^{\\textup{NSA}^{\\alpha}}\\alpha^{1}\\leq_{1}1 )\\big[(\\forall f^{1})B(f,\\mu(f))\\rightarrow \\textup{\\textsf{UWWKL}}(\\alpha, T)\\big].\n\\ee\nApplying Corollary \\ref{consresultcor2} to `$\\textup{\\textsf{RCA}}_{0}^{\\Lambda}\\vdash \\eqref{kanttt3}$', we obtain a term $t$ such that $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ proves \n\\be\\label{bl;og}\n(\\forall \\mu^{2}, T^{1})(\\exists \\alpha^{1}\\in t(\\mu, T) )\\big[(\\forall f^{1})B(f,\\mu(f))\\rightarrow \\textup{\\textsf{UWWKL}}(\\alpha, T)\\big].\n\\ee\nSince $\\mu^{2}$ satisfying the antecedent of \\eqref{bl;og} is indeed Feferman's mu as in $(\\mu^{2})$, we can select the `right' $\\alpha\\in t(\\mu, T)$, and we have obtained the first conjunct in \\eqref{frood8}.\n\\end{proof}\nAs suggested by its name, $\\textup{\\textsf{WWKL}}$ is a weakening of $\\textup{\\textsf{WKL}}$, namely to binary trees with positive measure. \nHence, the above proof should also go through for $\\textup{\\textsf{UWKL}}^{+}$, which is $\\textup{\\textsf{UWWKL}}^{+}$ \\emph{without} the aforementioned restriction. \nIn particular, the measure does not play any role except saying that the tree is infinite.\nWe hasten to add that the restriction to trees of positive measure \\emph{does} play an important role for $\\textup{\\textsf{STP}}$ and $\\textup{\\textsf{LMP}}$ by Theorem \\ref{komo}. \n\\subsection{The intermediate value theorem}\\label{hivt}\nIn this section, we study the \\emph{intermediate value theorem} \\textsf{IVT}. \nWe will apply the nonstandard extensionality trick from Remark~\\ref{trick} to a nonstandard version \nof the `usual' Brouwerian counterexample to $\\textup{\\textsf{IVT}}$ from \\cite{beeson1}*{I.7}. \n\n\n\\medskip\n\nNow, $\\textup{\\textsf{IVT}}$ has a proof in $\\textup{\\textsf{RCA}}_{0}$ when formulated in second-order arithmetic using so-called RM codes (\\cite{simpson2}*{II.6.2}). \nHowever, $\\textup{\\textsf{IVT}}$ is not constructively true (See \\cite{beeson1}*{I.7}) as the aforementioned proof makes essential use of the law of excluded middle, while a finer classification may be found in \\cite{bergske}. As a consequence of this non-constructive status, there is an equivalence (See \\cite{kohlenbach2}*{Prop.~3.14}) between the Turing jump functional $(\\exists^{2})$ and uniform $\\textup{\\textsf{IVT}}$, the latter defined as $\\textup{\\textsf{UIVT}}$ as follows (See \\cite{kohlenbach2}*{\\S3} for a number of variations):\n\\be\\tag{$\\textup{\\textsf{UIVT}}$}\n(\\exists \\Phi^{(1\\di1)\\di1 })(\\forall f\\in \\overline{C})[f(\\Phi(f))=_{{\\mathbb{R}}}0], \n\\ee\nwhere `$f\\in \\overline{C}$' is short for `$f(0)<_{{\\mathbb{R}}}0 <_{{\\mathbb{R}}} f(1)\\wedge \\eqref{kraakje}$' where the latter is as follows\n\\be\\label{kraakje}\\textstyle\n(\\forall k^{0})(\\forall x^{1}\\in [0,1])(\\exists N^{0})(\\forall y^{1}\\in [0,1])( |x-y|<_{{\\mathbb{R}}}\\frac{1}{N}\\rightarrow |f(x)-f(y)|<_{{\\mathbb{R}}}\\frac1k), \n\\ee\ni.e.\\ the \\emph{internal} `epsilon-delta' definition of (pointwise) continuity on $[0,1]$. We also write `$f\\in C([0,1])$' if $f:{\\mathbb{R}}\\rightarrow {\\mathbb{R}}$ satisfies \\eqref{kraakje}. \nWe define $\\textup{\\textsf{UIVT}}(\\Phi)\\equiv(\\forall f\\in \\overline{C})[f(\\Phi(f))=_{{\\mathbb{R}}}0]$. \nNow consider the following nonstandard and uniform version of $\\textup{\\textsf{UIVT}}$: \n\\be\\tag{$\\textup{\\textsf{UIVT}}^{+}$}\n(\\exists^{\\textup{NSA}^{\\alpha}} \\Phi)\\big[(\\forall^{\\textup{NSA}^{\\alpha}} f\\in \\overline{C})[f(\\Phi(f))=_{{\\mathbb{R}}}0 ]\\wedge (\\forall^{\\textup{NSA}^{\\alpha}}f, g\\in \\overline{C})\\big( f\\approx g\\rightarrow \\Phi(f)\\approx \\Phi(g) \\big) \\big],\n\\ee\n where `$f\\approx g$' is short for $(\\forall^{\\textup{NSA}^{\\alpha}}q^{0}\\in [0,1])(f(q)\\approx g(q))$. \nNote that the second conjunct of $\\textup{\\textsf{UIVT}}^{+}$ expresses that $\\Phi$ is `standard extensional', i.e.\\ satisfies extensionality as in \\eqref{furg} relative to `st'. \nWe have the following theorem. \n\\begin{thm}\\label{proto667}\nFrom a proof in $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$ that $\\textup{\\textsf{UIVT}}^{+}\\leftrightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, terms $s,t$ can be extracted such that $\\textup{\\textsf{RCA}}_{0}^{\\omega}$ proves:\n\\be\\label{frood667}\n(\\forall \\mu^{2})\\big[\\textsf{\\textup{\\textsf{MU}}}(\\mu)\\rightarrow \\textup{\\textsf{UIVT}}(s(\\mu)) \\big] \\wedge (\\forall \\Phi^{1\\rightarrow 1})\\big[ \\textup{\\textsf{UIVT}}(\\Phi)\\rightarrow \\textup{\\textsf{MU}}(t(\\Phi, \\Xi)) \\big],\n\\ee\nwhere $\\Xi$ is an extensionality functional for $\\Phi$.\n\\end{thm}\n\\begin{proof}\nFirst of all, we prove $\\textup{\\textsf{UIVT}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, for which we make use of the `usual' Brouwerian counterexample to $\\textup{\\textsf{IVT}}$ (See e.g.\\ \\cite{beeson1}*{Fig.\\ 2, p.\\ 12}). \nLet $f_{0}:{\\mathbb{R}}\\rightarrow {\\mathbb{R}}$ be the function which is $3x-1$ for $x\\in [0,\\frac13]$, $3x-2$ for $x\\in [\\frac23, 1]$, and zero for $x\\in [\\frac13, \\frac23]$. \nSuppose $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ is false and let standard $g^{1}$ be such that $(\\forall^{\\textup{NSA}^{\\alpha}}n)(g(n)=0)$ and $(\\exists m_{0})g(m_{0})\\ne 0$. Define standard functions $f_{\\pm}(x):= f_{0}(x)\\pm\\sum_{i=0}^{\\infty}\\frac{g(i)}{2^{i}}$ and note that $f_{+}(x)\\approx f_{-}(x)$ for all $x\\in [0,1]$. However, for $\\Phi$ as in $\\textup{\\textsf{UIVT}}^{+}$, we have $\\Phi(f_{+})<_{{\\mathbb{R}}}\\frac13$ and $\\Phi(f_{-})>_{{\\mathbb{R}}}\\frac23$, which contradicts the second conjunct of $\\textup{\\textsf{UIVT}}^{+}$. Hence, $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ follows and we obtain $\\textup{\\textsf{UIVT}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$.\n\n\\medskip\n \nSecondly, for the reversal $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}} \\rightarrow \\textup{\\textsf{UIVT}}^{+}$, fix standard $k^{0}, x^{1}\\in [0,1] $ and $f:{\\mathbb{R}}\\rightarrow{\\mathbb{R}}$ in \\eqref{kraakje} and consider the following $\\Sigma_{2}^{1}$-formula:\n\\be\\label{boal}\\textstyle\n(\\exists N^{0})(\\forall q^{0}\\in [0,1])( |x-q|<_{{\\mathbb{R}}}\\frac{1}{N}\\rightarrow |f(x)-f(q)|\\leq_{{\\mathbb{R}}}\\frac1k).\n\\ee\nApplying\\footnote{\nTo obtain $\\eqref{boal}\\rightarrow\\eqref{aimino}$, note that $\\neg\\eqref{aimino}$ has a normal form; apply $\\textup{\\textsf{HAC}}_{\\textup{int}}$ to remove the existential quantifier and then apply $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ to the resulting formula to obtain a contradiction with \\eqref{boal}.\nIn general, similar to how one `bootstraps' $\\Pi_{1}^{0}$-comprehension from $\\textup{\\textsf{ACA}}_{0}$, the system $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}+\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ proves $\\varphi\\leftrightarrow \\varphi^{\\textup{NSA}^{\\alpha}}$ for \\emph{any} internal arithmetical formula (only involving standard parameters). Indeed, recall that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}\\rightarrow (\\mu^{2})^{\\textup{NSA}^{\\alpha}}$ (See the proof of Theorem \\ref{proto7}) and use the latter to remove all existential quantifiers (except the leading one, if such there is) from $\\varphi^{\\textup{NSA}^{\\alpha}}$; applying $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ to the resulting universal formula yields $\\varphi$. The implication $\\varphi\\rightarrow \\varphi^{\\textup{NSA}^{\\alpha}}$ follows from the previous, $ \\neg\\varphi^{\\textup{NSA}^{\\alpha}}\\rightarrow \\neg\\varphi$ in particular.} $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ to the formula \\eqref{boal} yields:\n\\be\\label{aimino}\\textstyle\n(\\exists^{\\textup{NSA}^{\\alpha}} N^{0})(\\forall^{\\textup{NSA}^{\\alpha}} q^{0}\\in [0,1])( |x-q|\\ll \\frac{1}{N}\\rightarrow |f(x)-f(q)|\\lessapprox \\frac1k).\n\\ee\nApplying $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ to the universal formula in \\eqref{aimino}, we obtain for standard $f$ that\n\\be\\label{opa}\\textstyle\n(\\forall^{\\textup{NSA}^{\\alpha}}x\\in [0,1], k^{0})(\\exists^{\\textup{NSA}^{\\alpha}} N^{0})(\\forall q^{0}\\in [0,1])( |x-q|<_{{\\mathbb{R}}} \\frac{1}{N}\\rightarrow |f(x)-f(q)|\\leq_{{\\mathbb{R}}} \\frac1k),\n\\ee\nand applying $\\textup{\\textsf{HAC}}_{\\textup{int}}$ to the previous formula yields standard $\\Phi^{(1\\times 0)\\rightarrow 0^{*}}$ such that \n\\[\\textstyle\n(\\forall^{\\textup{NSA}^{\\alpha}}x\\in [0,1], k^{0})(\\exists N^{0}\\in \\Phi(x, k))(\\forall q^{0}\\in [0,1])( |x-q|<_{{\\mathbb{R}}} \\frac{1}{N}\\rightarrow |f(x)-f(q)|\\leq_{{\\mathbb{R}}} \\frac1k),\n\\]\nDefine $\\Psi(x, k)$ as $\\max_{i<|\\Phi(x, k)|}\\Phi(x, k)(i)$ and note that $\\Psi$ provides a kind of \\emph{modulus of continuity} for (standard) $f$. \nWith this continuity in place, we can follow (a variation of) the classical proof of $\\textup{\\textsf{IVT}}$ in \\cite{simpson2}*{II.6} to define {standard} $\\Phi$ as in $\\textup{\\textsf{UIVT}}^{+}$ as follows.\n\n\\medskip\n\nTo this end, recall that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ implies the existence of standard $\\xi$ as in \\eqref{karmic}, which allows us to decide for standard $f\\in \\overline{C}$ if $(\\exists^{\\textup{NSA}^{\\alpha}}q^{0}\\in [0,1])(f(q)\\approx 0)$ holds or not. \nIf the latter holds, use $\\xi$ to find such $q^{0}$ and define $\\Phi(f):=q$, which also yields $f(\\Phi(f))=_{{\\mathbb{R}}}0$ by $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. \nIf however $(\\forall^{\\textup{NSA}^{\\alpha}}q^{0}\\in [0,1])(f(q)\\not\\approx 0)$, define $\\Phi(f)(0)$ as $0$ or $\\frac{1}{2}$ depending on whether $f(\\frac12)\\ll 0$ or $f(\\frac12)\\gg 0$ ($\\xi$ as in \\eqref{karmic} decides which disjunct holds). \nSimilarly, define $\\Phi(f)(n+1)$ as $\\Phi(f)(n)$ or $\\Phi(f)(n)+\\frac{1}{2^{n+2}}$ depending on $f(\\Phi(f)(n)+\\frac{1}{2^{n+2}})\\ll0$ or $f(\\Phi(f)(n)+\\frac{1}{2^{n+2}})\\gg0$ ($\\xi$ as in \\eqref{karmic} again decides which disjunct holds). \n\n\\medskip\n\nNote that by $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, we have $f(z)\\gg 0 \\leftrightarrow f(z)>_{{\\mathbb{R}}}0$ for any standard $z^{1}\\in {\\mathbb{R}} $ and $f:{\\mathbb{R}}\\rightarrow {\\mathbb{R}}$. \nIn light of \\eqref{karmic}, we also have access to $I\\Sigma_{1}$ relative to `st', assuming all parameters involved are standard. \nHence, it is easy to prove that $\\Phi(f)$ is a real number such that $(\\forall^{\\textup{NSA}^{\\alpha}}n)(\\Phi(f)(n)<_{{\\mathbb{R}}}\\Phi(f)<_{{\\mathbb{R}}}\\Phi(f)(n)+\\frac{1}{2^{n+1}})$ and $f(\\Phi(f)(n))\\ll 0$ and $f(\\Phi(f)(n)+\\frac{1}{2^{n+1}})\\gg 0$. In light of these facts and the continuity of $f$ as in \\eqref{opa} for $x=\\Phi(f)$, we obtain $f(\\Phi(f))\\approx 0$ and also $f(\\Phi(f))=_{{\\mathbb{R}}}0$ by $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. \nFurthermore, $\\Phi$ only invokes $f$ on rational numbers and $(\\forall^{\\textup{NSA}^{\\alpha}} q^{0}\\in [0,1])(f(q)\\approx g(q))$ for standard $f, g\\in \\overline{C}$ thus implies $(\\forall q^{0}\\in [0,1])(f(q)=_{{\\mathbb{R}}} g(q))$ by $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. By the previous property and the extensionality of $f,g$, we have $\\Phi(f)=_{{\\mathbb{R}}}\\Phi(g)$, and hence the second conjunct of $\\textup{\\textsf{UIVT}}^{+}$. \n \n\\medskip\n\nThirdly, one obtains a normal form for $\\textup{\\textsf{UIVT}}^{+}\\leftrightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ in the same way as in the proofs of Theorems \\ref{proto7} and \\ref{proto}. \nApplying term extraction as in Theorem~\\ref{consresult}, one then readily obtains \\eqref{frood667}. For completeness, we mention the two normal forms corresponding to $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}\\rightarrow \\textup{\\textsf{UIVT}}^{+}$ and $\\textup{\\textsf{UIVT}}^{+}\\rightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$, namely as follows:\n\\[\n(\\forall^{\\textup{NSA}^{\\alpha}} f\\in \\overline{C},\\mu^{2})(\\exists^{\\textup{NSA}^{\\alpha}}x\\in [0,1]) \\big[ (\\forall g^{1})B(g,\\mu(g))\\rightarrow (f(x)=_{{\\mathbb{R}}}0 )\\big].\n\\]\n\\begin{align*}\\textstyle\n(\\forall^{\\textup{NSA}^{\\alpha}} \\Phi, \\Xi, g^{1})(\\exists^{\\textup{NSA}^{\\alpha}}n^{0})\\big[\\big((\\forall z\\in \\overline{C})&(z(\\Phi(z))=_{{\\mathbb{R}}}0 )\\textstyle\\wedge (\\forall x, y \\in \\overline{C}, k^{0})(\\exists q^{0},n^{0}\\in \\Xi(x,y,k))\\\\\n&\\textstyle\\big( |x(q) -y(q)|<\\frac{1}{n}\\rightarrow |\\Phi(x)- \\Phi(y)|<\\frac{1}{k} \\big)\\big) \\rightarrow B(g, n)\\big],\n\\end{align*}\nwhere we used the notation $B$ from \\eqref{similarluuuuu}. \nTo be absolutely clear, we point out that the formula `$(\\forall^{\\textup{NSA}^{\\alpha}} f\\in \\overline{C})A(f)$' is short for $(\\forall f)([\\textup{NSA}^{\\alpha}(f)\\wedge f\\in \\overline{C}]\\rightarrow A(f))$. \nIn particular, `$f\\in \\overline{C}$' is the \\emph{internal} formula `$f(1)>_{{\\mathbb{R}}}0>_{{\\mathbb{R}}} f(0)\\wedge \\eqref{kraakje}$' which remains \\emph{untouched} by the addition of `st' to the quantifier over the variable $f$. \nFor this reason, the formula `$f\\in \\overline{C}$' does not play any role in obtaining the normal form of $\\textup{\\textsf{UIVT}}^{+}\\leftrightarrow \\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ and the associated term extraction via Theorem \\ref{consresult}. \n\\end{proof}\nThe previous proof is easily adapted to the Brouwerian counterexample to the Weierstra\\ss~maximum theorem, as follows. \n\\begin{rem}[Weierstra\\ss~maximum theorem]\\textup{rm}\nThe Brouwerian counterexample to the Weierstra\\ss~maximum theorem is given in \\cite{beeson1}*{I.6} by a function with two relative maxima. \nFor instance, one can use $f_{\\pm}:{\\mathbb{R}}\\rightarrow {\\mathbb{R}}$ which is $(1\\pm x_{0})|\\sin 7x |$ for $x\\in [0, \\frac{1}{7}\\pi]$ and $(1\\mp x_{0})|\\sin 7x |$ for $x\\in [\\frac{1}{7}\\pi, 1]$, where $x_{0}:=\\sum_{i=0}^{\\infty}\\frac{g(i)}{2^{i}}$ and $g$ as in the previous proof. A functional witnessing the Weierstra\\ss~maximum theorem will map $f_{+}$ to $\\frac{\\pi}{14}$ and $f_{-}$ to $\\frac{3\\pi}{14}$ while $f_{+}\\approx f_{-}$ under the same assumptions as in the previous proof.\n\\end{rem}\n\n\\subsection{Rational numbers and the law of excluded middle}\\label{carmichael}\nIn this section, we study the classical dichotomy that every real number is either rational or not, i.e.\\ \n\\be\\label{LEM2}\\tag{\\textsf{DQ}}\n(\\forall x\\in {\\mathbb{R}})\\big[ (\\exists q\\in {\\mathbb Q})(q=_{{\\mathbb{R}}}x)\\vee (\\forall r\\in {\\mathbb Q})(r\\ne_{{\\mathbb{R}}}x) \\big]. \n\\ee\nOur study will yield interesting insights into the role of the law of excluded middle in Nonstandard Analysis. \nIn particular, we will offer a partial explanation \\emph{why} Nonstandard Analysis can produce computational information, as observed in the previous sections (and the aforementioned references). \n\n\\medskip\n\nFirst of all, \\ref{LEM2} is a trivial consequence of the law of excluded middle, and the former is indeed equivalent to \\textsf{LPO} in constructive mathematics (\\cite{brich}*{p.\\ 5}). \nWe will study the \\emph{nonstandard} version of \\ref{LEM2}, defined as follows:\n\\be\\label{LEM1}\\tag{$\\textup{\\textsf{DQ}}_{\\textup{\\textsf{ns}}}$}\n(\\forall^{\\textup{NSA}^{\\alpha}} x\\in {\\mathbb{R}})\\big[ (\\exists^{\\textup{NSA}^{\\alpha}}q\\in {\\mathbb Q})(q=_{{\\mathbb{R}}}x)\\vee (\\forall r\\in {\\mathbb Q})(r\\ne_{{\\mathbb{R}}}x) \\big],\n\\ee\nThe uniform version of \\ref{LEM2} is also obvious:\n\\be\\label{LEM3}\\tag{\\textsf{UDQ}}\n(\\exists \\Phi^{2})(\\forall x\\in {\\mathbb{R}})\\big[ (\\Phi(x)\\in {\\mathbb Q} \\wedge q=_{{\\mathbb{R}}}x)\\vee (\\forall r\\in {\\mathbb Q})(r\\ne_{{\\mathbb{R}}}x) \\big]. \n\\ee\nLet $\\textsf{UDQ}(\\Phi)$ be the previous with the leading quantifier dropped.\n\\begin{thm}\\label{krefje}\nIn $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$, we have $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}\\leftrightarrow \\textsf{\\textup{DQ}}_{\\textup{\\textsf{ns}}}$. From the latter proof, we can extract terms $s$ and $t$ such that\n\\be\\label{unik}\n(\\forall \\mu^{2})\\big[\\textsf{\\textup{\\textsf{MU}}}(\\mu)\\rightarrow \\textup{\\textsf{UDQ}}(s(\\mu)) \\big] \\wedge (\\forall \\Phi^{2})\\big[ \\textup{\\textsf{UDQ}}(\\Phi)\\rightarrow \\textup{\\textsf{MU}}(t(\\Phi)) \\big].\n\\ee\n\\end{thm}\n\\begin{proof}\nThe first forward implication is trivial while the first reverse equivalence follows easily: Suppose $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ is false and consider $f^{1}$ such that $(\\forall^{\\textup{NSA}^{\\alpha}}n)(f(n)=0)\\wedge (\\exists m)(f(m)\\ne 0)$.\nDefine the real $x:=\\sum_{i=0}^{\\infty}\\frac{h(n)}{2^{n}}$ where $h(n)=1$ if $(\\forall i\\leq n)(f(n)=0)$, and zero otherwise. Note that $x$ is a rational number, namely $x=_{{\\mathbb{R}}} \\sum_{i=0}^{m_{0}}\\frac{h(n)}{2^{n}} $, where $m_{0}$ is the last $m$ such that $f(m)\\ne0$. However, we also have $x\\ne_{{\\mathbb{R}}} q$ for every standard rational, and this contradiction yields $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. \n\n\\medskip\n\nSince $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ has a normal form \\eqref{frux} and $\\textsf{DQ}_{\\textup{\\textsf{ns}}}$ has an obvious normal form, \\eqref{unik} follows in the same way as in the second part of the proof of Theorem \\ref{proto7}.\n\\end{proof}\nSecondly, we discuss an apparent (but not actual) contradiction regarding $\\textsf{DQ}_{\\textup{\\textsf{ns}}}$ and the previous theorem, as follows.\n\\begin{rem}[Equality in $\\textup{\\textsf{P}}$]\\label{druj}\\textup{rm}\nFirst of all, $\\textup{\\textsf{P}}$ proves \\ref{LEM2} via the law of excluded middle. \nHence for a \\emph{standard} real $x$ which does not satisfy the second disjunct of \\ref{LEM2}, we may conclude the existence of a rational $q^{0}$ such that $x$ equals $q$. \nSecondly, in light of the first basic axiom of $\\textup{\\textsf{P}}$ (See item (\\ref{komit}) of Definition~\\ref{debs}), $q$ must be standard as $x$ is standard, and $x$ equals $q$. \nHowever, this means that $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$ proves $\\textsf{DQ}_{\\textup{\\textsf{ns}}}$, which is impossible in light of Theorem \\ref{krefje}. \nThirdly, this apparent contradiction is easily explained by noting that `$=_{{\\mathbb{R}}}$' as defined in $\\textup{\\textsf{P}}$ (See Definition \\ref{keepinitreal}) does not fall under item (\\ref{komit}) of Definition \\ref{debs}. \n\\end{rem}\nThirdly, while Theorem \\ref{krefje} is not particularly deep, this theorem inspires Remark \\ref{druj}, which in turn gives rise to the following observation: \nIn $\\textup{\\textsf{RCA}}_{0}^{\\omega}$, either a real is rational or not because of the law of excluded middle \\ref{LEM2}. \nBy contrast, in $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$, there are three possibilities for every standard real:\n\\begin{enumerate}\n\\item $x$ is a standard rational;\n\\item $x$ is not a rational; \n\\item $x$ is a rational, but not standard;\\label{trunki}\n\\end{enumerate}\nand the third possibility \\eqref{trunki} only disappears given $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ by Theorem \\ref{krefje}. Similarly, again over $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$, for a standard function $f^{1}$, there are three possibilities:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\item there is standard $n^{0}$ such that $f(n)=0$;\\label{work}\n\\item for all $m^{0}$ we have $f(m)\\ne0$;\n\\item for all standard $n^{0}$ we have $f(n)\\ne0$ while there is $m^{0}$ with $f(m)= 0$;\\label{kraft}\n\\end{enumerate}\nand the third possibility again only disappears given $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$. In particular, \\emph{in the extended language of $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$}, $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ (and not \\ref{LEM2}) is the principle which excludes the third option \\eqref{trunki} and \\eqref{kraft}. In other words, it seems that $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ plays the role of the law of excluded middle\/third \\emph{in the extended language of $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$}. \n\n\\medskip\n\nFurthermore, it has been suggested that the predicate `$\\textup{NSA}^{\\alpha}(n^{0})$' can be read as `$n^{0}$ is computationally relevant' or `$n^{0}$ is calculable' in \\cite{brie}*{p.\\ 1963}, \\cite{benno2}*{\\S4}, and \\cite{sambon}*{\\S3.4}. \nIf we read the previous items \\eqref{work}-\\eqref{kraft} through this filter, they reflect three well-known possibilities suggested by the BHK interpretation (See \\cite{troeleke1}*{\\S3.1}): \n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\Roman{enumi}}\n\\item we can compute $n^{0}$ s.t.\\ $f(n)=0$ (constructive existence);\\label{kracht}\n\\item for all $m^{0}$ we have $f(m)\\ne0$;\n\\item $\\neg[(\\forall m^{0})(f(m) \\ne0)]$;\\label{werk} \n\\end{enumerate}\nIn conclusion, we have observed that the role of the law of excluded middle \\emph{in the extended language of $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$} is played by Nelson's axiom \\emph{Transfer}, which is however absent from $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$. Due to this absence, there are \\emph{three} possibilities as in items \\eqref{work}-\\eqref{kraft}, similar to the possibilities in items \\eqref{kracht}-\\eqref{werk} in constructive mathematics. \nIn other words, the systems $\\textup{\\textsf{P}}$ and $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}$ are constructive in that they lack \\emph{Transfer}, which is the law of excluded middle \\emph{for the extended language of internal set theory}. \nWe believe this to be a partial explanation of the vast computational content of Nonstandard Analysis established above and in \\cites{sambon, samGH, samzoo, samzooII}. \n\n\n\n\\section{A new class of functionals}\\label{knowledge}\nWe discuss the results from \\cite{dagsam}, some of which were announced in \\cite{samGT2}, and related results. \nThe associated connection between Nonstandard Analysis and computability theory forms the motivation for this paper, as discussed in Section \\ref{intro}. \n\n\\subsection{Nonstandard Analysis and Computability Theory: an introduction}\\label{connexis}\nThe connection between \\emph{computability theory} and \\emph{Nonstandard Analysis} is investigated \\cite{dagsam}. The two following topics are investigated \\emph{and} shown to be intimately related. \n\\begin{enumerate}[label=\\textsf{(T.\\arabic*)}]\n\\item[\\textsf{(T.1)}] A basic property of \\emph{Cantor space} $2^{{\\mathbb N}}$ is \\emph{Heine-Borel compactness}: For any open cover of $2^{{\\mathbb N}}$, there is a \\emph{finite} sub-cover. \nA natural question is: \\emph{How hard is it to {compute} such a finite sub-cover}? This is made precise in \\cite{dagsam} by analysing the complexity of functionals that for $g:2^{{\\mathbb N}}\\rightarrow {\\mathbb N}$, \noutput a finite sequence $\\langle f_0 , \\dots, f_n\\rangle $ in $2^{{\\mathbb N}}$ such that the neighbourhoods defined from $\\overline{f_i}g(f_i)$ for $i\\leq n$ form a cover of Cantor space.\n\\item[\\textsf{(T.2)}] A basic property of Cantor space in \\emph{Nonstandard Analysis} is Abraham Robinson's \\emph{nonstandard compactness} (See \\cite{loeb1}*{p.\\ 42}), i.e.\\ that every binary sequence is `infinitely close' to a \\emph{standard} binary sequence. The strength of this nonstandard compactness property of Cantor space is analysed in \\cite{dagsam} and compared to the other axioms of Nonstandard Analysis and usual mathematics.\n\\end{enumerate}\nThe study of \\textsf{(T.1)} in \\cite{dagsam} involves the \\emph{special fan functional} $\\Theta$, discussed in Section \\ref{thespecial} below and first introduced in \\cite{samGH}. \nClearly, Tait's \\emph{fan functional} (\\cite{noortje}) computes\\footnote{Tait's fan functional $\\Phi$ computes a modulus of \\emph{uniform} continuity $N^{0}=\\Phi(g)$ for any continuous functional $g:2^{{\\mathbb N}}\\rightarrow {\\mathbb N}$. The modulus $N^{0}$ yields a supremum for $g$ by computing the maximum of $g(\\sigma*00\\dots)$ for all binary sequences $\\sigma$ of length $N$.} a sequence $\\langle f_0 , \\dots, f_n\\rangle $ as in \\textsf{(T.1)} for \\emph{continuous} $g:2^{{\\mathbb N}}\\rightarrow {\\mathbb N}$, while the special fan functional does so \\emph{for any} $g:2^{{\\mathbb N}}\\rightarrow {\\mathbb N}$. This generalisation from continuous to general inputs is interesting (and even necessary) in our opinion as mathematics restricted to e.g.\\ only recursive objects, like the Russian school of recursive mathematics, can be strange and counter-intuitive (See \\cite{beeson1}*{Chapter IV} for this opinion). \nSome of the (highly surprising) computational properties of $\\Theta$ established in \\cite{dagsam} are discussed in Section \\ref{thespecial}. In particular, $\\Theta$ seems extremely hard to compute (as in Kleene's S1-S9 from \\cite{longmann}*{\\S5.1}) as no type two functional can compute it. \n\n\\medskip\n\nThe study of \\textsf{(T.2)} in \\cite{dagsam} amounts to developing the Reverse Mathematics of Nonstandard Analysis. For instance, the nonstandard counterparts of $\\textup{\\textsf{WKL}}_{0}$ and $\\textup{\\textsf{WWKL}}_{0}$ are $\\textup{\\textsf{STP}}$ and $\\textup{\\textsf{LMP}}$ (See Section \\ref{thespecial} for definitions), \neach expressing a nonstandard kind of compactness. On the other hand, the nonstandard counterpart of $\\textup{\\textsf{ACA}}_{0}$ is $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ introduced above. While we have $\\textup{\\textsf{ACA}}_{0}\\rightarrow \\textup{\\textsf{WKL}}_{0}\\rightarrow \\textup{\\textsf{WWKL}}_{0}$ in RM, the nonstandard counterparts behave quite differently, namely \nwe have $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}} \\not\\rightarrow \\textup{\\textsf{STP}}$ and $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}} \\not\\rightarrow\\textup{\\textsf{LMP}}$, and much stronger non-implications involving the nonstandard counterpart of $\\Pi_{1}^{1}\\text{-\\textsf{CA}}_{0}$, the strongest `Big Five' system. \n\n\\medskip\n\nWe stress that \\textsf{(T.1)} and \\textsf{(T.2)} are highly intertwined and that the study of these topics in \\cite{dagsam} is `holistic' in nature: \nresults in computability theory give rise to results in Nonstandard Analysis \\emph{and vice versa}, as discussed in the next section. By way of a basic example, consider $\\Theta$ as in \\textsf{(T.1)} and recall that the output of $\\Theta$ is readily computed in terms of Tait's fan functional if $g^{2}$ is continuous on Cantor space. Experience bears out that the uninitiated express extreme skepticism about the fact that $\\Theta$ is also well-defined for \\emph{discontinuous} inputs $g^{2}$. However, $\\Theta$ almost trivially emerges form the nonstandard compactness of Cantor space, i.e.\\ Nonstandard Analysis tells us that the special fan functional $\\Theta$ exists and is well-defined. Furthermore, the fact that the Turing jump functional from $(\\exists^{2})$ cannot compute $\\Theta$ as mentioned in \\textsf{(T.1)} readily implies the non-implication $\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}\\not \\rightarrow \\textup{\\textsf{STP}}$ from \\textsf{(T.2)}. \nMore examples are discussed in the next section, and of course \\cite{dagsam}. \n\n\\medskip\n\nFinally, we have sketched a connection between Nonstandard Analysis and computability theory. However, the better part of the latter does not obviously have a counterpart in the former and vice versa. \nWe list two examples: First of all, the fact that no type two functional computes $\\Theta$ is proved in computability theory (See \\cite{dagsam}*{\\S3}) using \\emph{Gandy selection} (See \\cite{longmann}*{Theorem 5.4.5}), but what is the nonstandard counterpart of the latter theorem? \nSecondly, the \\emph{Loeb measure} is one of the crown jewels of Nonstandard Analysis (\\cite{loeb1}), but what is the computability theoretic counterpart of this measure? Note that first steps in this direction have been taken in \\cite{samnewarix}. \nIn the paper at hand, we have formulated a nonstandard counterpart of \\emph{Grilliot's trick} inspired by the above connection between Nonstandard Analysis and computability theory. \n\n\\subsection{The special fan functional and related topics}\\label{thespecial}\nWe introduce the \\emph{special fan functional} and discuss how it derives from the \\emph{Standard Part} axiom of Nonstandard Analysis and why it does not belong to any existing category in RM. \n\n\\medskip\n\nOur motivation for this study is the following discrepancy: On one hand, there is literally a `zoo' of theorems in RM (\\cite{damirzoo}) which do fit into the `Big Five classification' of RM. On the other hand, as shown above and in \\cites{samzoo, samzooII}, uniform theorems are mostly equivalent to $(\\exists^{2})$, with some exceptions based on the contraposition of $\\textup{\\textsf{WKL}}$, i.e.\\ the fan theorem. Thus, the `higher-order RM zoo' consisting of uniform theorems is still rather sparse compared to the original RM zoo. \nIn this light, it is a natural question whether the higher-order RM zoo can be made as populous as the original RM zoo. \n\n\\medskip\n\nAs a first step towards an answer to the aforementioned question, we discuss the following functional. Note that $1^{*}$ is the type of finite sequences of type $1$.\n\\begin{defi}\\textup{rm}[Special fan functional]\\label{special}\nWe define $\\textup{\\textsf{SCF}}(\\Theta)$ as follows for $\\Theta^{(2\\rightarrow (0\\times 1^{*}))}$:\n\\[\n(\\forall g^{2}, T^{1}\\leq_{1}1)\\big[(\\forall \\alpha \\in \\Theta(g)(2)) (\\overline{\\alpha}g(\\alpha)\\not\\in T)\n\\rightarrow(\\forall \\beta\\leq_{1}1)(\\exists i\\leq \\Theta(g)(1))(\\overline{\\beta}i\\not\\in T) \\big]. \n\\]\nAny functional $\\Theta$ satisfying $\\textup{\\textsf{SCF}}(\\Theta)$ is referred to as a \\emph{special fan functional}.\n\\end{defi}\nAs noted in \\cite{dagsam} and above, from a computability theoretic perspective, the main property of the special fan functional $\\Theta$ is the selection of $\\Theta(g)(2)$ as a finite sequence of binary sequences $\\langle f_0 , \\dots, f_n\\rangle $ such that the neighbourhoods defined from $\\overline{f_i}g(f_i)$ for $i\\leq n$ form a cover of Cantor space; almost as a by-product, $\\Theta(g)(1)$ can then be chosen to be the maximal value of $g(f_i) + 1$ for $i\\leq n$. \nWe stress that $g^{2}$ in $\\textup{\\textsf{SCF}}(\\Theta)$ may be \\emph{discontinuous} and that Kohlenbach has argued for the study of discontinuous functionals in RM (\\cite{kohlenbach2}).\n\n\\medskip\n\nThe name of $\\Theta$ from the previous definition is due to the fact that a special fan functional may be computed from the intuitionistic fan functional $\\Omega^{3}$, as in Theorem \\ref{kinkel}.\n\\be\\tag{$\\textup{\\textsf{MUC}}(\\Omega)$}\n(\\forall Y^{2}) (\\forall f, g\\leq_{1}1)(\\overline{f}\\Omega(Y)=\\overline{g}\\Omega(Y)\\notag \\rightarrow Y(f)=Y(g)). \n\\ee\nAs to the logical strength of $(\\exists \\Omega)\\textup{\\textsf{MUC}}(\\Omega)$, the latter gives \nrise to a conservative extension of the system $\\textup{\\textsf{WKL}}_{0}$ by \\cite{kohlenbach2}*{Prop.\\ 3.15}. \n\\begin{thm}\\label{kinkel}\nThere is a term $t$ such that $\\textsf{\\textup{E-PA}}^{\\omega}$ proves $(\\forall \\Omega^{3})(\\textup{\\textsf{MUC}}(\\Omega)\\rightarrow \\textup{\\textsf{SCF}}(t(\\Omega))) $. \n\\end{thm}\n\\begin{proof}\nThe theorem was first proved \\emph{indirectly} in \\cite{samGH}*{\\S3} by applying Theorem~\\ref{consresult} to a suitable nonstandard implication. \nFor completeness, we include the following direct proof which can also be found in \\cite{dagsam}. \nNote that $\\Theta(g)$ as in $\\textup{\\textsf{SCF}}(\\Theta)$ has to provide a natural number and a finite sequence of binary sequences.\nThe number $\\Theta(g)(1)$ is defined as $\\max_{|\\sigma|=\\Omega(g)\\wedge \\sigma\\leq_{0^{*}}1}g(\\sigma*00\\dots)$ and the finite sequence of binary sequences $\\Theta(g)(2)$\nconsists of all $\\tau*00\\dots$ where $|\\tau|=\\Theta(g)(1)\\wedge \\tau\\leq_{0^{*}}1$. \nWe have for all $g^{2}$ and $T^{1}\\leq_{1}1$:\n\\be\\label{difffff}\n (\\forall \\beta\\leq_{1}1)(\\beta \\in \\Theta(g)(2)\\rightarrow \\overline{\\beta}{g}(\\beta)\\not \\in T)\\rightarrow (\\forall \\gamma\\leq_{1}1)(\\exists i\\leq \\Theta(g)(1))(\\overline{\\gamma}i\\not \\in T).\n\\ee\nIndeed, suppose the antecendent of \\eqref{difffff} holds. Now take $\\gamma_{0}\\leq_{1}1$, and note that $\\beta_{0}=\\overline{\\gamma_{0}}\\Theta(g)(1)*00\\dots \\in \\Theta(g)(2)$, implying \n$\\overline{\\beta_{0}}{g}(\\beta_{0})\\not \\in T$. But $g(\\alpha)\\leq \\Theta(g)(1)$ for all $\\alpha\\leq_{1}1$, by the definition of $\\Omega$, implying that $\\overline{\\gamma_{0}}{g}(\\beta_{0})=\\overline{\\beta_{0}}{g}(\\beta_{0})\\not \\in T$ by the definition of $\\beta_{0}$, and the consequent of \\eqref{difffff} follows. \n\\end{proof}\nIn light of the previous, $\\Theta$ exists at the level of $\\textup{\\textsf{WKL}}_{0}$ and it therefore stands to reason that it would be \\emph{easy} to compute. \nWe have the following surprising theorem where `computable' should be once again interpreted in the sense of Kleene's S1-S9 (See \\cite{longmann}*{5.1.1}).\nThe metatheory is -as always- $\\ensuremath{\\usftext{ZFC}}$ set theory. \n\\begin{thm}\\label{import}\nLet $\\varphi^{2}$ be any functional of type two. \nAny functional $\\Theta^{3}$ as in $\\textup{\\textsf{SCF}}(\\Theta)$ is not computable in $\\varphi$. \nAny functional $\\Theta^{3}$ as in $\\textup{\\textsf{SCF}}(\\Theta)$ is computable in $(\\exists^{3})$ as follows\n\\be\\tag{$\\exists^{3}$}\n(\\exists E_{3})(\\forall \\varphi^{2})\\big[ (\\exists f^{1})(\\varphi(f)=0)\\leftrightarrow E_{3}(\\varphi)=0 \\big].\n\\ee\n\\end{thm}\n\\begin{proof}\nA proof may be found in \\cite{dagsam}.\n\\end{proof}\nBy the previous, $\\Theta$ is quite different from the usual\\footnote{Note that the connection between $\\Theta$ and Kohlenbach's generalisations of $\\textup{\\textsf{WKL}}$ from \\cite{kohlenbach4}*{\\S5-6} is discussed in \\cite{dagsam}*{\\S4}. This connection turns out to be quite non-trivial and interesting.} objects studied in (higher-order) RM. \nAn obvious question is: Where does the special fan functional and its behaviour come from? \nThe answer is as follows: The nonstandard counterpart of $\\textup{\\textsf{WKL}}_{0}$ is defined as:\n\\be\\tag{$\\textup{\\textsf{STP}}$}\n(\\forall \\alpha^{1}\\leq_{1}1)(\\exists^{\\textup{NSA}^{\\alpha}}\\beta^{1}\\leq_{1}1)(\\alpha\\approx_{1}\\beta), \n\\ee \nwhich has the following normal form, already reminiscent of $\\Theta$. \n\\begin{thm}\\label{lapdog}\nIn $\\textup{\\textsf{P}}$, $\\textup{\\textsf{STP}}$ is equivalent to the following normal form:\n\\begin{align}\\label{frukkklk}\n(\\forall^{\\textup{NSA}^{\\alpha}}g^{2})(\\exists^{\\textup{NSA}^{\\alpha}}w^{1^{*}})\\big[(\\forall T^{1}\\leq_{1}1)(\\exists ( \\alpha^{1}\\leq_{1}1, &~k^{0}) \\in w)\\big((\\overline{\\alpha}g(\\alpha)\\not\\in T)\\\\\n&\\rightarrow(\\forall \\beta\\leq_{1}1)(\\exists i\\leq k)(\\overline{\\beta}i\\not\\in T) \\big)\\big]. \\notag\n\\end{align} \nThe system $\\textup{\\textsf{P}}+(\\exists^{\\textup{NSA}^{\\alpha}}\\Theta)\\textup{\\textsf{SCF}}(\\Theta)$ proves $\\textup{\\textsf{STP}}$. \n\\end{thm}\n\\begin{proof} \nThe following proof is implicit in the results in \\cite{samGH}*{\\S3} and is added for completeness. \nFirst of all, $\\textup{\\textsf{STP}}$ is easily seen to be equivalent to \n\\begin{align}\\label{fanns}\n(\\forall T^{1}\\leq_{1}1)\\big[(\\forall^{\\textup{NSA}^{\\alpha}}n)(\\exists \\beta^{0})&(|\\beta|=n \\wedge \\beta\\in T ) \\rightarrow (\\exists^{\\textup{NSA}^{\\alpha}}\\alpha^{1}\\leq_{1}1)(\\forall^{\\textup{NSA}^{\\alpha}}n^{0})(\\overline{\\alpha}n\\in T) \\big],\n\\end{align}\nand this equivalence may also be found in \\cite{samGH}*{Theorem 3.2}.\nFor \\eqref{frukkklk}$\\rightarrow$\\eqref{fanns}, note that \\eqref{frukkklk} implies for all standard $g^{2}$\n\\begin{align}\\label{frukkklk2}\n(\\forall T^{1}\\leq_{1}1)(\\exists^{\\textup{NSA}^{\\alpha}} ( \\alpha^{1}\\leq_{1}1, &~k^{0})\\big[(\\overline{\\alpha}g(\\alpha)\\not\\in T)\n\\rightarrow(\\forall \\beta\\leq_{1}1)(\\exists i\\leq k)(\\overline{\\beta}i\\not\\in T) \\big], \n\\end{align} \nwhich in turn yields, by bringing all standard quantifiers inside again, that:\n\\begin{align}\\label{frukkklk3}\n(\\forall T\\leq_{1}1) \\big[(\\exists^{\\textup{NSA}^{\\alpha}}g^{2})(\\forall^{\\textup{NSA}^{\\alpha}}\\alpha \\leq_{1}1)(\\overline{\\alpha}g(\\alpha)\\not\\in T)\\rightarrow(\\exists^{\\textup{NSA}^{\\alpha}}k)(\\forall \\beta\\leq_{1}1)(\\overline{\\beta}k\\not\\in T) \\big], \n\\end{align} \nTo obtain \\eqref{fanns} from \\eqref{frukkklk3}, apply $\\ensuremath{\\usftext{HAC}_\\ensuremath{{\\usftext{int}}}}$ to $(\\forall^{\\textup{NSA}^{\\alpha}}\\alpha^{1}\\leq_{1}1)(\\exists^{\\textup{NSA}^{\\alpha}}n)(\\overline{\\alpha}n\\not\\in T)$ to obtain standard $\\Psi^{1\\rightarrow 0^{*}}$ such that \n$(\\forall^{\\textup{NSA}^{\\alpha}}\\alpha^{1}\\leq_{1}1)(\\exists n\\in \\Psi(\\alpha))(\\overline{\\alpha}n\\not\\in T)$, and defining $g(\\alpha):=\\max_{i<|\\Psi|}\\Psi(\\alpha)(i)$ we obtain $g$ as in the antecedent of \\eqref{frukkklk3}. The previous implies \n\\be\\label{gundark}\n(\\forall T^{1}\\leq_{1}1) \\big[(\\forall^{\\textup{NSA}^{\\alpha}}\\alpha^{1}\\leq_{1}1)(\\exists^{\\textup{NSA}^{\\alpha}}n)(\\overline{\\alpha}n\\not\\in T)\\rightarrow (\\exists^{\\textup{NSA}^{\\alpha}}k)(\\forall \\beta\\leq_{1}1)(\\overline{\\beta}i\\not\\in T) \\big], \n\\ee\nwhich is the contraposition of \\eqref{fanns}, using classical logic. For the implication $\\eqref{fanns} \\rightarrow \\eqref{frukkklk}$, consider the contraposition of \\eqref{fanns}, i.e.\\ \\eqref{gundark}, and note that the latter implies \\eqref{frukkklk3}. Now push all standard quantifiers outside as follows:\n\\[\n(\\forall^{\\textup{NSA}^{\\alpha}}g^{2})(\\forall T^{1}\\leq_{1}1)(\\exists^{\\textup{NSA}^{\\alpha}} ( \\alpha^{1}\\leq_{1}1, ~k^{0})\\big[(\\overline{\\alpha}g(\\alpha)\\not\\in T)\n\\rightarrow(\\forall \\beta\\leq_{1}1)(\\exists i\\leq k)(\\overline{\\beta}i\\not\\in T) \\big], \n\\]\nand applying idealisation \\textsf{I} yields \\eqref{frukkklk}. The final part now follows immediately in light of the basic axioms of $\\textup{\\textsf{P}}$ in Definition \\ref{debs}. \n\\end{proof}\nBy the previous theorem $\\Theta$ emerges from Nonstandard Analysis, and the behaviour of $\\Theta$ as in Theorem \\ref{import} can be explained similarly: \nIt is part of the folklore of Nonstandard Analysis that \\emph{Transfer} does not imply \\emph{Standard Part}. The same apparently holds for fragments: $\\textup{\\textsf{P}}+\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ does not prove $\\textup{\\textsf{STP}}$ by the results in \\cite{dagsam}*{\\S6}. \nAs a result of applying Theorem \\ref{consresult}, there is no term of G\\\"odel's $T$ which computes $\\Theta$ in terms of $(\\mu^{2})$. A stronger result as in Theorem \\ref{import} apparently can be obtained. \n\n\\medskip\n\nNext, we discuss a nonstandard version of $\\textup{\\textsf{WWKL}}$, introduced in \\cite{pimpson}, as follows:\n\\be\\tag{$\\textup{\\textsf{LMP}}$}\n(\\forall T \\leq_{1}1)\\big[ \\nu(T)\\gg 0 \\rightarrow (\\exists^{\\textup{NSA}^{\\alpha}} \\beta\\leq_{1}1)(\\forall^{\\textup{NSA}^{\\alpha}}n)(\\overline{\\beta}n\\in T)\\big], \n\\ee\nand one obtains a normal form of $\\textup{\\textsf{LMP}}$ similar to \\eqref{frukkklk}. As for the latter, this normal form gives rise to a \\emph{weak fan functional} $\\Lambda$, first introduced in \\cite{dagsam}*{\\S3}. \nWe have the following theorem where $\\textup{\\textsf{ATR}}_{0}$ is the fourth `Big Five' system of RM (See \\cite{simpson2}*{V}). \n\\begin{thm}\nThe system $\\textup{\\textsf{P}}+\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}+\\textup{\\textsf{STP}}$ proves the consistency of $\\textup{\\textsf{ATR}}_{0}$ while $\\textup{\\textsf{P}}+\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}+\\textup{\\textsf{LMP}}$ does not. \n\\end{thm}\n\\begin{proof}\nSee \\cite{dagsam}*{\\S6}.\n\\end{proof}\nThe previous result is referred to as a `phase transition' in \\cite{dagsam} as there is (currently) nothing in between $\\textup{\\textsf{WKL}}_{0}$ and $\\textup{\\textsf{WWKL}}_{0}$ in the RM zoo. \n\\begin{cor}\nThe system $\\textup{\\textsf{P}}+\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ does not prove $\\textup{\\textsf{STP}}$. \n\\end{cor}\n\\begin{proof}\nSuppose $\\textup{\\textsf{P}}+\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ proves $\\textup{\\textsf{STP}}$ and note that \n$\\textup{\\textsf{P}}+\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}$ then proves the consistency of $\\textup{\\textsf{ATR}}_{0}$ by the theorem. By Theorem \\ref{consresult}, $\\textsf{E-PA}^{\\omega*}+(\\mu^{2})$ also proves the consistency of $\\textup{\\textsf{ATR}}_{0}$, which contradicts G\\\"odel's incompleteness theorems.\n\\end{proof}\nAt the end of Section \\ref{FUWKL}, we noted that Theorem \\ref{proto} should go through for $\\textup{\\textsf{WKL}}$ instead of $\\textup{\\textsf{WWKL}}$. \nIn particular, the restriction to trees of positive measure (which is part of $\\textup{\\textsf{WWKL}}$) can be lifted while still obtaining the same equivalence as in \\eqref{frood8}. \nHence, these results are \\emph{robust}, i.e.\\ equivalent to small perturbations of themselves (See \\cite{montahue}*{p.\\ 432}). We now provide an example where the notion of `tree with positive measure' yields \\emph{non-robust} results. \nTo this end, consider the following strengthening of $\\textup{\\textsf{LMP}}$:\n\\be\\tag{$\\textup{\\textsf{LMP}}^{+}$}\n(\\forall T \\leq_{1}1)\\big[ \\mu(T)>_{{\\mathbb{R}}}0\\rightarrow (\\exists^{\\textup{NSA}^{\\alpha}} \\beta\\leq_{1}1)(\\forall^{\\textup{NSA}^{\\alpha}} m)(\\overline{\\beta}m\\in T) \\big],\n\\ee\nand the following weakening of $\\textup{\\textsf{STP}}$:\n\\be\\tag{$\\textup{\\textsf{STP}}^{-}$}\n(\\forall T \\leq_{1}1)\\big[(\\forall n^{0})(\\exists \\beta^{0^{*}})(\\beta\\in T\\wedge |\\beta|=n)\\rightarrow (\\exists^{\\textup{NSA}^{\\alpha}} \\beta\\leq_{1}1)(\\forall^{\\textup{NSA}^{\\alpha}} m)(\\overline{\\beta}m\\in T) \\big],\n\\ee\n\\begin{thm}\\label{komo}\nIn $\\textup{\\textsf{P}}_{0}+\\textup{\\textsf{WWKL}}$, we have $\\textup{\\textsf{STP}}\\leftrightarrow \\textup{\\textsf{LMP}}^{+}\\leftrightarrow \\textup{\\textsf{STP}}^{-}$.\n\\end{thm}\n\\begin{proof}\nThe implication $\\textup{\\textsf{STP}}\\rightarrow \\textup{\\textsf{STP}}^{-}$ is immediate; for the reverse implication, apply overspill to the antecedent of \\eqref{fanns} to obtain a sequence $\\beta_{0}^{0^{*}}$ of nonstandard length in $T$. Extend the latter to an infinite tree by including $\\beta_{0}*00\\dots$, which is nonstandard. \nApplying $\\textup{\\textsf{STP}}^{-}$ to this extended tree yields the consequence of \\eqref{fanns}, and hence $\\textup{\\textsf{STP}}$.\nFor $\\textup{\\textsf{STP}}\\rightarrow \\textup{\\textsf{LMP}}^{+}$, apply $\\textup{\\textsf{STP}}$ to the path claimed to exist by $\\textup{\\textsf{WWKL}}$ and note that we obtain $\\textup{\\textsf{LMP}}^{+}$. \nFor $\\textup{\\textsf{LMP}}^{+}\\rightarrow\\textup{\\textsf{STP}}$, fix $f^{1}\\leq_{1}1$ and nonstandard $N$. Define the tree $T\\leq_{1}1$ which is $f$ until height $N$, followed by the full binary tree. \nThen $\\mu(T)>_{{\\mathbb{R}}}0$ and let standard $g^{1}\\leq_{1}1$ be such that $(\\forall^{\\textup{NSA}^{\\alpha}}n)(\\overline{g}n\\in T)$. By definition, we have $f\\approx_{1}g$, and we are done.\n\\end{proof}\nIn light of the theorem, $\\textup{\\textsf{STP}}$ and $\\Theta$ seem fairly robust, while $\\textup{\\textsf{LMP}}$ and $\\Lambda$ are not. \n\n\n\\medskip\n\nFinally, $\\textup{\\textsf{STP}}$ and $\\textup{\\textsf{LMP}}$ are not unique: Similar nonstandard (and functional) versions exist for most of the theorems in the RM zoo. Indeed, since every theorem $T$ in the RM zoo follows from arithmetical comprehension, \nwe can prove $T^{\\textup{NSA}^{\\alpha}}$ in $\\textup{\\textsf{RCA}}_{0}^{\\Lambda}+\\Pi_{1}^{0}\\textup{-\\textsf{TRANS}}+\\textup{\\textsf{STP}}$, and use $\\textup{\\textsf{STP}}$ to drop the `st' in the leading quantifier as in \\eqref{fanns} for $\\textup{\\textsf{WKL}}^{\\textup{NSA}^{\\alpha}}$. \nThe author and Dag Normann are currently investigating the exact power of $\\Theta$ and $\\Lambda$ and the strength of the associated nonstandard axioms. \n\n\n\\begin{bibdiv}\n\\begin{biblist}\n\\bib{avi2}{article}{\n author={Avigad, Jeremy},\n author={Feferman, Solomon},\n title={G\\\"odel's functional \\(``Dialectica''\\) interpretation},\n conference={ title={Handbook of proof theory}, },\n book={ publisher={North-Holland}, },\n date={1998},\n pages={337--405},\n}\n\n\\bib{beeson1}{book}{\n author={Beeson, Michael J.},\n title={Foundations of constructive mathematics},\n series={Ergebnisse der Mathematik und ihrer Grenzgebiete},\n volume={6},\n note={Metamathematical studies},\n publisher={Springer},\n date={1985},\n pages={xxiii+466},\n}\n\n\\bib{brie}{article}{\n author={van den Berg, Benno},\n author={Briseid, Eyvind},\n author={Safarik, Pavol},\n title={A functional interpretation for nonstandard arithmetic},\n journal={Ann. Pure Appl. Logic},\n volume={163},\n date={2012},\n pages={1962--1994},\n}\n\n\\bib{bergske}{article}{\n author={Berger, Josef},\n author={Ishihara, Hajime},\n author={Takayuki, Kihara},\n author={Nemoto, Takako},\n title={The binary expansion and the intermediate value theorem in constructive reverse mathematics},\n journal={Available from \\url {http:\/\/www.jaist.ac.jp\/~t-nemoto\/beivt.pdf}},\n}\n\n\\bib{bishl}{book}{\n author={Bishop, Errett},\n title={Aspects of constructivism},\n publisher={Notes on the lectures delivered at the Tenth Holiday Mathematics Symposium},\n place={New Mexico State University, Las Cruces, December 27-31},\n date={1972},\n pages={pp.\\ 37},\n}\n\n\\bib{kuddd}{article}{\n author={Bishop, Errett},\n title={Review of \\cite {keisler3}},\n year={1977},\n journal={Bull. Amer. Math. Soc},\n volume={81},\n number={2},\n pages={205-208},\n}\n\n\\bib{kluut}{article}{\n author={Bishop, Errett},\n title={The crisis in contemporary mathematics},\n booktitle={Proceedings of the American Academy Workshop on the Evolution of Modern Mathematics},\n journal={Historia Math.},\n volume={2},\n date={1975},\n number={4},\n pages={507--517},\n}\n\n\\bib{brich}{book}{\n author={Bridges, Douglas},\n author={Richman, Fred},\n title={Varieties of constructive mathematics},\n series={London Mathematical Society Lecture Note Series},\n volume={97},\n publisher={Cambridge University Press},\n place={Cambridge},\n date={1987},\n pages={x+149},\n}\n\n\\bib{bridges1}{book}{\n author={Bridges, Douglas S.},\n author={V{\\^{\\i }}{\\c {t}}{\\u {a}}, Lumini{\\c {t}}a Simona},\n title={Techniques of constructive analysis},\n series={Universitext},\n publisher={Springer},\n place={New York},\n date={2006},\n pages={xvi+213},\n}\n\n\\bib{conman}{article}{\n author={Connes, Alain},\n title={An interview with Alain Connes, Part I},\n year={2007},\n journal={EMS Newsletter},\n note={\\url {http:\/\/www.mathematics-in-europe.eu\/maths-as-a-profession\/interviews}},\n volume={63},\n pages={25-30},\n}\n\n\\bib{conman2}{article}{\n author={Connes, Alain},\n title={Noncommutative geometry and reality},\n journal={J. Math. Phys.},\n volume={36},\n date={1995},\n number={11},\n pages={6194--6231},\n}\n\n\\bib{damirzoo}{misc}{\n author={Dzhafarov, Damir D.},\n title={Reverse Mathematics Zoo},\n note={\\url {http:\/\/rmzoo.uconn.edu\/}},\n}\n\n\\bib{godel3}{article}{\n author={G{\\\"o}del, Kurt},\n title={\\\"Uber eine bisher noch nicht ben\\\"utzte Erweiterung des finiten Standpunktes},\n language={German, with English summary},\n journal={Dialectica},\n volume={12},\n date={1958},\n pages={280--287},\n}\n\n\\bib{grilling}{article}{\n author={Grilliot, Thomas J.},\n title={On effectively discontinuous type-$2$ objects},\n journal={J. Sym. Logic},\n volume={36},\n date={1971},\n}\n\n\\bib{benno2}{article}{\n author={Hadzihasanovic, Amar},\n author={van den Berg, Benno},\n title={Nonstandard functional interpretations and models},\n journal={To appear in Notre Dame Journal for Formal Logic},\n volume={},\n date={2016},\n number={},\n pages={},\n}\n\n\\bib{polahirst}{article}{\n author={Hirst, Jeffry L.},\n title={Representations of reals in reverse mathematics},\n journal={Bull. Pol. Acad. Sci. Math.},\n volume={55},\n date={2007},\n number={4},\n pages={303--316},\n}\n\n\\bib{loeb1}{book}{\n author={Hurd, Albert E.},\n author={Loeb, Peter A.},\n title={An introduction to nonstandard real analysis},\n series={Pure and Applied Mathematics},\n volume={118},\n publisher={Academic Press Inc.},\n place={Orlando, FL},\n date={1985},\n pages={xii+232},\n}\n\n\\bib{keisler3}{book}{\n author={Keisler, H. Jerome},\n title={Elementary Calculus},\n publisher={Prindle, Weber and Schmidt},\n date={1976},\n pages={xviii + 880 + 61 (appendix)},\n place={Boston},\n}\n\n\\bib{kohlenbach3}{book}{\n author={Kohlenbach, Ulrich},\n title={Applied proof theory: proof interpretations and their use in mathematics},\n series={Springer Monographs in Mathematics},\n publisher={Springer-Verlag},\n place={Berlin},\n date={2008},\n pages={xx+532},\n}\n\n\\bib{kohlenbach2}{article}{\n author={Kohlenbach, Ulrich},\n title={Higher order reverse mathematics},\n conference={ title={Reverse mathematics 2001}, },\n book={ series={Lect. Notes Log.}, volume={21}, publisher={ASL}, },\n date={2005},\n pages={281--295},\n}\n\n\\bib{kohlenbach4}{article}{\n author={Kohlenbach, Ulrich},\n title={Foundational and mathematical uses of higher types},\n conference={ title={Reflections on the foundations of mathematics (Stanford, CA, 1998)}, },\n book={ series={Lect. Notes Log.}, volume={15}, publisher={ASL}, },\n date={2002},\n pages={92--116},\n}\n\n\\bib{kooltje}{article}{\n author={Kohlenbach, Ulrich},\n title={On uniform weak K\\\"onig's lemma},\n journal={Ann. Pure Appl. Logic},\n volume={114},\n date={2002},\n pages={103--116},\n}\n\n\\bib{longmann}{book}{\n author={Longley, John},\n author={Normann, Dag},\n title={Higher-order Computability},\n year={2015},\n publisher={Springer},\n series={Theory and Applications of Computability},\n}\n\n\\bib{mandje2}{article}{\n author={Mandelkern, Mark},\n title={Brouwerian counterexamples},\n journal={Math. Mag.},\n volume={62},\n date={1989},\n number={1},\n pages={3--27},\n}\n\n\\bib{montahue}{article}{\n author={Montalb{\\'a}n, Antonio},\n title={Open questions in reverse mathematics},\n journal={Bull. Symbolic Logic},\n volume={17},\n date={2011},\n number={3},\n pages={431--454},\n}\n\n\\bib{wownelly}{article}{\n author={Nelson, Edward},\n title={Internal set theory: a new approach to nonstandard analysis},\n journal={Bull. Amer. Math. Soc.},\n volume={83},\n date={1977},\n number={6},\n pages={1165--1198},\n}\n\n\\bib{noortje}{book}{\n author={Normann, Dag},\n title={Recursion on the countable functionals},\n series={LNM 811},\n volume={811},\n publisher={Springer},\n date={1980},\n pages={viii+191},\n}\n\n\\bib{dagsam}{article}{\n author={Normann, Dag},\n author={Sanders, Sam},\n title={Nonstandard Analysis, Computability Theory, and their connections},\n journal={Submitted, Available from \\url {https:\/\/arxiv.org\/abs\/1702.06556}},\n date={2017},\n}\n\n\\bib{yamayamaharehare}{article}{\n author={Sakamoto, Nobuyuki},\n author={Yamazaki, Takeshi},\n title={Uniform versions of some axioms of second order arithmetic},\n journal={MLQ Math. Log. Q.},\n volume={50},\n date={2004},\n number={6},\n pages={587--593},\n}\n\n\\bib{sayo}{article}{\n author={Sanders, Sam},\n author={Yokoyama, Keita},\n title={The {D}irac delta function in two settings of {R}everse {M}athematics},\n year={2012},\n journal={Archive for Mathematical Logic},\n volume={51},\n number={1},\n pages={99-121},\n}\n\n\\bib{samGH}{article}{\n author={Sanders, Sam},\n title={The Gandy-Hyland functional and a hitherto unknown computational aspect of Nonstandard Analysis},\n year={2017},\n journal={To appear in \\emph {Computability}, \\url {http:\/\/arxiv.org\/abs\/1502.03622}},\n}\n\n\\bib{samzoo}{article}{\n author={Sanders, Sam},\n title={The taming of the Reverse Mathematics zoo},\n year={2015},\n journal={Submitted, \\url {http:\/\/arxiv.org\/abs\/1412.2022}},\n}\n\n\\bib{samzooII}{article}{\n author={Sanders, Sam},\n title={The refining of the taming of the Reverse Mathematics zoo},\n year={2016},\n journal={To appear in Notre Dame Journal for Formal Logic, \\url {http:\/\/arxiv.org\/abs\/1602.02270}},\n}\n\n\\bib{sambon}{article}{\n author={Sanders, Sam},\n title={The unreasonable effectiveness of Nonstandard Analysis},\n year={2015},\n journal={Submitted, \\url {http:\/\/arxiv.org\/abs\/1508.07434}},\n}\n\n\\bib{SB}{article}{\n author={Sanders, Sam},\n title={To be or not to be constructive},\n journal={\\emph {Indagationes Mathematicae} and the Brouwer volume \\emph {L.E.J. Brouwer, fifty years later}, \\href{https:\/\/doi.org\/10.1016\/j.indag.2017.05.005}{ScienceDirect}},\n date={2017 and 2018},\n pages={pp.\\ 68},\n}\n\n\\bib{samnewarix}{article}{\n author={Sanders, Sam},\n title={The computational content of the Loeb measure},\n year={2016},\n journal={Available from arXiv: \\url {https:\/\/arxiv.org\/abs\/1609.01945}},\n}\n\n\\bib{samsynt}{article}{\n author={Sanders, Sam},\n title={Reverse Formalism 16},\n year={2017},\n journal={Synthese, \\href{https:\/\/link.springer.com\/article\/10.1007\\%2Fs11229-017-1322-2}{SpringerLink}},\n}\n\n\\bib{samGT2}{article}{\n author={Sanders, Sam},\n title={The computational content of Nonstandard Analysis},\n date={2016},\n journal={Electronic Proceedings in Computer Science 213, \\emph {Classic Logic and Computation}, Porto (CL\\&C2016)},\n pages={21-40},\n}\n\n\\bib{simpson1}{collection}{\n title={Reverse mathematics 2001},\n series={Lecture Notes in Logic},\n volume={21},\n editor={Simpson, Stephen G.},\n publisher={ASL},\n place={La Jolla, CA},\n date={2005},\n pages={x+401},\n}\n\n\\bib{simpson2}{book}{\n author={Simpson, Stephen G.},\n title={Subsystems of second order arithmetic},\n series={Perspectives in Logic},\n publisher={CUP},\n date={2009},\n pages={xvi+444},\n}\n\n\\bib{pimpson}{article}{\n author={Simpson, Stephen G.},\n author={Yokoyama, Keita},\n title={A nonstandard counterpart of \\textsf {\\textup {WWKL}}},\n journal={Notre Dame J. Form. Log.},\n volume={52},\n date={2011},\n number={3},\n pages={229--243},\n}\n\n\\bib{troeleke1}{book}{\n author={Troelstra, Anne Sjerp},\n author={van Dalen, Dirk},\n title={Constructivism in mathematics. Vol. I},\n series={Studies in Logic and the Foundations of Mathematics},\n volume={121},\n publisher={North-Holland},\n date={1988},\n pages={xx+342+XIV},\n}\n\n\n\\end{biblist}\n\\end{bibdiv}\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:intro}\n\nAfter dominating the early days of exoplanetology, the radial velocity (RV) technique continues to play an outstanding role in exoplanet discovery. With observational baselines now extending over several decades, the RV technique is currently the main method for discovering long-period exoplanets \\citep{Gregory10, Marmier13, Kane19, Rickman19}.\n\nThe radial velocity method does however have its limitations, of which the best known is the sin $i$ degeneracy that leaves the orbital inclination unknown. This constrains the determination of mass to a lower limit ($m\\sin i$) and means that the true mass of RV planets can only be determined if the inclination can be constrained through other methods, which remains a challenging endeavour. The most promising way of measuring orbital inclinations is through astrometry, but thus far most astrometric data available have been insufficient to detect signals in the planetary regime. For example, \\citet{Reffert11} used astrometry from the Hipparcos mission \\citep{Hipparcos, HipparcosNew} to constrain the true masses of 310 substellar companion candidates but could confirm only nine as planets, all with upper limits on the masses rather than positive detections. The ongoing Gaia mission \\citep{Gaia} is projected to dramatically improve this state of affairs by detecting many thousands of planets with astrometry \\citep{Perryman14}, but this data will not be made public until Gaia Data Release 4, which is still years away.\n\nA number of authors \\citep[including, and not limited to,][]{Calissendorff18, Snellen18, Brandt19, Kervella19, Feng19} have pioneered the combination of Gaia and Hipparcos proper motion data to produce astrometry that can be used to detect companions on wide orbits. Although this amounts to only three measurements of tangential velocity (Hipparcos, Gaia, and the average motion between those observations), the high precision and long time-scale of these measurements means that, in combination with RV or imaging data, Hipparcos-Gaia astrometry allows for positive detection of planetary-mass companions in favourable cases \\citep[e.g.][]{Dupuy19, DeRosa20, Xuan20, Li21}.\n\n\\citet{Blunt19} presented the discovery of HR~5183~b, a giant planet with a $74^{+43}_{-22}$ year orbital period discovered as a result of its spectacular periastron passage that occurred during 2017-2018. This remarkable companion has among the longest orbital periods of any planet discovered by the RV method, as well as one of the highest orbital eccentricities ($0.84 \\pm 0.04$). Despite the extreme orbit of HR~5183~b, \\citet{Kane19.HR5183} explored dynamical stability in the system and remarkably found that stable orbits in the Habitable Zone of the star are possible. \n\nHR~5183 has a probable stellar companion called HIP~67291 at a large sky separation of 490\" ($\\sim$15400~AU projected separation). The presence of this companion may have dynamically influenced the evolution of the planet HR~5183~b, a possibility that has recently been explored by \\citet{Mustill22}. The authors find that orbital excitation of an initially circular planetary orbit by HIP~67291 is \\boldnew{less likely than planet-planet scattering} to produce the observed eccentricity, but notably the combination of both of these excitation mechanisms results in the highest probability of reproducing the planetary orbit. As a corollary, \\citet{Mustill22} interestingly find that the predicted distribution of mutual inclinations between the stellar and planetary orbits differ depending on the formation scenario, with planet-planet scattering inducing a broader distribution of mutual inclinations, potentially allowing for direct constraints on the planet formation history if the planet-binary mutual inclination can be measured precisely.\n\nIn this work we present the detection of HR~5183~b's astrometric reflex signal with Hipparcos-Gaia astrometry, and use this to provide the first constraints on the orbital inclination of this remarkable planet. We furthermore utilise precise Gaia astrometry of HR~5183 and HIP~67291 to explore the orbit of the binary, and combine the constraints on the inclinations of both orbits in an attempt to constrain the planet-binary mutual inclination. Finally, we demonstrate that there is a \\bmaroon{moderately} significant excess of stellar multiples among systems containing highly eccentric planets, providing observational evidence that dynamical interactions with binary companions plays an important role in the origins of these extreme exoplanets.\n\nIn Section \\ref{sec:method} we outline our method for modelling the planetary and stellar orbits. In Section \\ref{sec:results} we document our results, followed by discussion in Section \\ref{sec:discussion} and concluding remarks in Section \\ref{sec:conclusions}.\n\n\n\\section{Method} \\label{sec:method}\n\n\\begin{table*}\n\t\\centering\n\t\\caption{Parameters of the stellar binary used for the LOFTI model.}\n\t\\label{table:binary_astrometry}\n\t\\begin{tabular}{lcc}\n\t\t\\hline\n\t\tParameter & HR 5183 & HIP 67291 \\\\\n\t\t\\hline\n\t\tGaia EDR3 ID & 3721126409323324416 & 3721114933170707328 \\\\\n\t\t\\hline\n\t\tMass [${M_\\odot}$] & $1.07 \\pm 0.04$ $^a$ & $0.67 \\pm 0.05$ $^a$ \\\\\n\t\tParallax $\\varpi$ [mas] & $31.7806 \\pm 0.0257$ $^b$ & $31.8422 \\pm 0.0157$ $^b$ \\\\\n\t\tRA proper motion $\\mu_{\\text{RA}}$ [\\masyr{}] & \\boldagain{$-510.20\\pm0.05$} $^c$ & $-509.328 \\pm 0.016$ $^b$ \\\\\n\t\tDeclination proper motion $\\mu_{\\text{Dec}}$ [\\masyr{}] & \\boldagain{$-110.40\\pm0.04$} $^c$ & $-110.975 \\pm 0.011$ $^b$ \\\\\n\t\tRadial velocity [$\\text{km}\\,\\text{s}^{-1}$] & \\boldnew{$-30.66 \\pm 0.10$} $^c$ & $-30.67 \\pm 0.15$ $^b$ \\\\\n\t\tProjected separation [arcsec] & -- & $488.53692 \\pm 0.00002$ $^b$ \\\\\n\t\tPosition angle [degrees] & -- & $104.823966 \\pm 0.000002$ $^b$ \\\\\n\t\t\\hline\n\t\t\\multicolumn{3}{l}{$^a$ From \\citet{Blunt19} $^b$ From \\citet{GaiaEDR3} $^c$ This work, after `de-projection' (see text).}\\\\\n\t\\end{tabular}\n\\end{table*}\n\n\\subsection{Stellar parameters}\n\n\\citet{Blunt19} determined precise stellar parameters for HR~5183 and HIP~67291 using a combination of spectroscopic and photometric data plus Gaia DR2 parallaxes. Though updated parallaxes from Gaia EDR3 \\citep{GaiaEDR3} are now available, the difference from the DR2 astrometry for these two stars is slight and we therefore do not expect to make significant improvements in parameter precision over \\citet{Blunt19}. We therefore adopt the stellar parameters from that work, of which the most relevant for our purposes are the stellar masses, $1.07 \\pm 0.04$ and $0.67 \\pm 0.05$ $M_\\odot$ respectively.\n\n\\subsection{The planetary orbit}\n\n\\subsubsection{Data}\n\nIn this work we use the same radial velocity dataset as \\citet{Blunt19}, and we defer to that work for description of the RV data collection. The available RV data amounts to 175 observations with HJST\/Tull, 104 at Lick\/APF, and 78 with Keck\/HIRES, of which 20 predate the 2004 spectrograph upgrade and 58 postdate it; as in \\citet{Blunt19} we treat the pre- and post-upgrade HIRES radial velocities as separate datasets, allowing for a free offset between them.\n\nFor the astrometric data, we make use of the recently updated Hipparcos-Gaia Catalog of Accelerations \\citep[HGCA;][]{Brandt21} based on proper motions from Gaia~EDR3 \\citep{GaiaEDR3}. The data provided by the HGCA consists of three measurements of proper motion, in two co-ordinates each \\citep{Brandt19}. These are the Hipparcos proper motion (\\boldnew{$\\mu_\\text{H}$}) measured at approximately epoch 1991.25, the Gaia EDR3 proper motion (\\boldnew{$\\mu_\\text{G}$}) measured at approximately 2016.0, and the mean Hipparcos-Gaia proper motion (\\boldnew{$\\mu_{\\text{HG}}$}), derived from the change in sky position observed by the two telescopes. This last component has been referred to by a variety of \\boldnew{names}; as in \\citet{Venner21}, we refer to this measurement as the `Hip-Gaia proper motion'.\n\n\\subsubsection{Planetary orbit model} \\label{subsec:planet_method}\n\nThe model used for the planetary orbit in this work is largely identical to the one in \\citet{Venner21}, and we refer the reader to that work for detailed description of the techniques used here. In summary, we jointly fit the radial velocity and astrometric data using a two-body Keplerian model. This involves a total of 19 parameters, of which two are assigned Gaussian priors (the stellar mass $M_*$ and the parallax $\\varpi$, for which we adopt the Gaia EDR3 parallax for HR~5183), seven describe the orbit of HR~5183~b (the orbital period $P$, the RV semi-amplitude $K$, the eccentricity $e$ and argument of periastron $\\omega_1$ parameterised as $\\sqrt{e}\\sin\\omega_1$ and $\\sqrt{e}\\cos\\omega_1$, the time of periastron $T_p$, the orbital inclination $i$, and the longitude of node $\\Omega$), two describe the proper motion of the system barycentre ($\\mu_{\\text{bary,RA}}$, $\\mu_{\\text{bary,Dec}}$), and the remaining eight are normalisation offsets and jitter parameters for each radial velocity dataset. The proper motions from Hipparcos and Gaia, which are effectively averaged over the respective $\\sim$3.36 year and $\\sim$2.76 year observation time-scales for those mission \\citep{Hipparcos, GaiaEDR3astrometry}, are resampled based on their underlying observation times.\\footnote{As the Gaia observation times are \\bmaroon{not} yet available, we use the Gaia Observation Forecast Tool (\\url{https:\/\/gaia.esac.esa.int\/gost\/}) to derive predicted measurement epochs.}\n\nWe use the Markov Chain Monte Carlo (MCMC) ensemble sampler \\texttt{emcee~v3.0.2} \\citep{emcee} to explore our parameter space. A total of 50 walkers were used to sample the 19-parameter model over $6\\times10^{5}$ steps, with confirmation of convergence in the MCMC performed in the same way as in \\citet{Venner21}. To derive our posterior samples, we then discarded 33\\% of the chain as burn-in and saved every hundredth step from the 50 walkers. Using these posteriors, we then extracted the $68.3\\%$ confidence intervals for the model parameters.\n\nThe main area of difference in our model with respect to \\citet{Venner21} is that we adopt the same informed prior on the orbital period as \\citet{Blunt19}:\n\n\\begin{equation}\n\\label{equation:Pprob}\np(P,t_d,B)=\n\\begin{cases}\n1 & \\text{if } (P-t_d)1.4$ as advocated by \\citealt{Lindegren18}, \\bmaroon{though the results of studies such as \\citealt{Belokurov20} and \\citealt{Stassun21} suggest that RUWE values even slightly above $1.0$ can constitute evidence of unresolved companions}) indicates the fit to the Gaia astrometry is in some way subpar, such as for an unresolved close binary or due to significantly non-linear proper motion. In Gaia EDR3, HR~5183 and HIP~67291 have RUWEs of 1.108 and 0.982 respectively, which indicates that the fit to the Gaia EDR3 astrometry is of good quality for both stars and justifies our use of the Gaia astrometric data for modelling the orbit of this wide pair.\n\n\\boldnew{The formalism adopted by LOFTI assumes that the projected positions and velocities of the binary lie on a Cartesian plane. However, this assumption breaks down for large binary separations as a result of the curvature of the celestial sphere. Since the observable sky motions of the stars are projections of their true space velocities onto the celestial sphere, for widely separated stars the spherical nature of that surface means that their space velocities are projected with slightly differing vectors. This effect was explored by \\citet{ElBadry19}, who showed that projection effects can cause two stars with identical space velocities to have significantly differing proper motions and radial velocities from the perspective of the observer. The author further} explored the significance of projection effects \\boldnew{against binary separation} and found that these become non-negligible \\boldnew{at} separations above $\\gtrsim$0.1 pc for binaries within 120 pc\\boldnew{. This} is comparable to the separation between HR~5183 and HIP~67291 \\boldnew{(490\" sky separation = 0.075~pc projected separation), so it is therefore necessary to account for projection effects in the velocity data when modelling the orbit of the binary.}\n\nTo correct for projection effects \\boldnew{in the binary velocities we employ an approach of empirical `de-projection', which was previously used} in \\citet{Venner21}. \\boldnew{This works} by first converting the observed sky position and velocities of \\boldnew{HR~5183} into its space velocities, and then converting these into the \\boldnew{sky velocities} that would \\boldnew{occur if the star instead} lay at the sky position of \\boldnew{HIP~67291. This produces a set of hypothetical proper motions and radial velocities for HR~5183 that lie in the same plane as the observed values for HIP~67291, correcting for perspective effects arising from projection onto the celestial sphere and allowing for the direct comparison of their sky velocities.\n\nTo do this we must first define \\boldnew{the} astrometry and radial velocities for both stars. For HIP~67291 we adopt the parallax, proper motion, and radial velocity from Gaia EDR3 \\citep{GaiaEDR3}. We note that, as a Hipparcos star, HIP~67291 is present in the HGCA \\boldnew{\\citep{Brandt21}}; however \\boldnew{for this star the Hip-Gaia average proper motion is less precise than} the Gaia EDR3 proper motion, so we continue to use the EDR3 proper motion for this star. Notably \\boldnew{all proper motion measurements of HIP~67291 in the HGCA are consistent with constant motion}, suggesting that the star does not have massive interior companions and therefore justifying the use of its astrometry to constrain the orbit of the wide binary. For HR~5183 we adopt the Gaia EDR3 parallax, however instead of the EDR3 proper motion we use the barycentric proper motion of (\\boldnew{$-510.64 \\pm 0.05$}, $-110.41 \\pm 0.04$) \\masyr{} derived from our model of the planetary orbit (see Section \\ref{subsec:planet_results}), as this has been corrected for the perturbatory effects of HR~5183~b. For the adopted radial velocity we make use of the fact that HR~5183 is a Gaia RV standard star and was used as a validator for the Gaia DR2 radial velocities by \\citet{Soubiran18}. Because the Gaia RV zero-point is calibrated to the SOPHIE RVs used in that work, the precise radial velocity of HR~5183 is directly comparable to the Gaia RV of HIP~67291 without instrumental offsets. Subtracting our best-fitting planetary orbit from the SOPHIE radial velocities provided by \\citet{Soubiran18} we measure an absolute RV of $-30.50~\\text{km}\\,\\text{s}^{-1}$ for HR~5183; we adopt an inflated uncertainty of $\\pm 0.1~\\text{km}\\,\\text{s}^{-1}$ for this value to accommodate for any systematics relating to the absolute RV reference frame.\n\nTaking \\boldnew{our} adopted astrometric and radial velocity data for HR~5183, we calculate stellar space velocities \\boldnew{of \\boldagain{$(U,V,W)=(-60.58\\pm0.06, -55.07\\pm0.05, -17.36\\pm0.09)$}~km~s$^{-1}$ using \\boldagain{the} equations of \\citet{Johnson87}. By converting these to the sky velocities that would result at the position of HIP~67291 we \\boldagain{obtain} a `de-projected' proper motion of \\boldagain{($-510.20\\pm0.05, -110.40\\pm0.04$)}~\\masyr{} and a radial velocity of $-30.66\\pm0.10$~km~s$^{-1}$ for HR~5183.\\footnote{\\boldagain{We make an implementation of our `de-projection' technique available at \\url{https:\/\/colab.research.google.com\/drive\/10a8URblzfdStzxNdCAb-Wnep_DV_-txj}.}} The perspective-corrected proper motion of the star differs by \\boldagain{($+0.44, +0.01$)}~\\masyr{} from our input values, a statistically significant correction which demonstrates the importance of accounting for projection effects in this system.}\n\n\\boldnew{We summarise our adopted system data in Table \\ref{table:binary_astrometry}. Using these parameters as inputs for LOFTI, we ran the orbital model as described in \\citet{Pearce20} for a total of $5\\times10^{5}$ samples.}\n\n\n\n\\subsubsection{Binary bound probability} \\label{subsubsec:bound_prob}\n\nHaving revised the astrometric and radial velocity data for the binary, we are now in a position to reappraise the calculation of binding probability for the binary as conducted by \\citet{Blunt19}. Briefly, we performed a Monte Carlo simulation of orbital velocities drawn from the `de-projected' proper motions and radial velocities, and determined the fraction of total velocity vectors that exceed the escape velocity at the current binary separation (v$_{\\rm{esc}} = 0.37$ km~s$^{-1}$ for a test particle at 15400 AU separation from a central mass of 1.74 M$_\\odot$). We report on the results of this simulation in Section \\ref{subsec:binary_results}.\n\n\n\\section{Results} \\label{sec:results}\n\n\\subsection{HR 5183 b} \\label{subsec:planet_results}\n\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{HR5183_RV.pdf}\n\t\\caption{Our radial velocity model (top) and residuals (bottom) for HR~5183, showing the reflex signal of its planetary companion. The black line corresponds to the best-fitting model, while the grey lines are drawn randomly from the posterior distribution and demonstrate the range of plausible models. Our fit to the radial velocities is essentially identical to that of \\citet{Blunt19}.}\n\t\\label{figure:planet_RV}\n\\end{figure}\n\n\\begin{table*}\n\t\\centering\n\t\\caption{Parameters of HR 5183 b. All values are medians and 1$\\sigma$ confidence intervals from the posterior distributions.}\n\t\\label{table:planet_orbit}\n\t\\begin{tabular}{lcc}\n\t\t\\hline\n\t\tParameter & \\citet{Blunt19} & This work \\\\\n\t\t\\hline\n\t\tPeriod $P$ [years] & $74^{+43}_{-22}$ & $102^{+84}_{-34}$ \\\\\n\t\tPeriod $P$ [days] & $\\left(27000^{+16000}_{-8000}\\right)$ & $37200^{+30700}_{-12400}$ \\\\\n\t\tRV semi-amplitude $K$ [\\ms{}] & $38.25^{+0.58}_{-0.55}$ & $38.4 \\pm 0.6$ \\\\\n\t\tEccentricity $e$ & $0.84 \\pm 0.04$ & $0.87 \\pm 0.04$ \\\\\n\t\tArgument of periastron (primary) $\\omega_1$ [degrees] & $\\left(340 \\pm 2\\right)$ & $339.9 \\pm 1.8$ \\\\\n\t\tTime of periastron $T_0$ [JD] & $2458121 \\pm 12$ & $2458122 \\pm 12$ \\\\\n\t\tSecondary minimum mass $m_2\\sin i$ [$M_J$] & $3.23^{+0.15}_{-0.14}$ & $3.24 \\pm 0.12$ \\\\\n\t\tRelative semi-major axis $a$ [AU] & $18^{+6}_{-4}$ & $22.3^{+11.0}_{-5.3}$ \\\\\n\t\tPeriastron distance $a(1-e)$ [AU] & $2.88^{+0.09}_{-0.08}$ & $2.87^{+0.08}_{-0.08}$ \\\\\n\t\tApoastron distance $a(1+e)$ [AU] & -- & $41.8^{+22.1}_{-10.6}$ \\\\\n\t\t\\hline\n\t\tOrbital inclination $i$ [degrees] & -- & $89.9^{+13.3}_{-13.5}$ \\\\\n\t\tLongitude of node $\\Omega$ [degrees] & -- & $224.0^{+18.2}_{-20.3}$ \\\\\n\t\t$\\sin i$ & -- & $0.987^{+0.012}_{-0.039}$ \\\\\n\t\tOrbital velocity semi-amplitude $\\frac{K}{\\sin i}$ [\\ms{}] & -- & $39.1^{+1.6}_{-0.9}$ \\\\\n\t\tSecondary mass $m_2$ [$M_J$] & -- & $3.31^{+0.18}_{-0.14}$ \\\\\n\t\t\\hline\n\t\tBarycentric RA proper motion $\\mu_{\\text{bary,RA}}$ [\\masyr{}] & -- & \\boldnew{$-510.64 \\pm 0.05$} \\\\\n\t\tBarycentric declination proper motion $\\mu_{\\text{bary,Dec}}$ [\\masyr{}] & -- & $-110.41 \\pm 0.04$ \\\\\n\t\t\\hline\n\t\tTull RV offset $\\gamma_{\\text{T}}$ [\\ms{}] & $-19.2^{+1.9}_{-2.1}$ & $-20.6^{+2.0}_{-2.2}$ \\\\\n\t\tAPF RV offset $\\gamma_{\\text{APF}}$ [\\ms{}] & $-47.2^{+2.0}_{-2.2}$ & $-48.6^{+2.1}_{-2.4}$ \\\\\n\t\tHIRES pre-upgrade RV offset $\\gamma_{\\text{H,pre}}$ [\\ms{}] & $-52.6^{+1.3}_{-1.5}$ & $-53.6^{+1.5}_{-1.7}$ \\\\\n\t\tHIRES post-upgrade RV offset $\\gamma_{\\text{H,post}}$ [\\ms{}] & $-52.4^{+2.0}_{-2.1}$ & $-53.8^{+2.1}_{-2.3}$ \\\\\n\t\t\\hline\n\t\tTull RV jitter $\\sigma_{\\text{T}}$ [\\ms{}] & $5.8^{+0.6}_{-0.5}$ & $5.8^{+0.6}_{-0.5}$ \\\\\n\t\tAPF RV jitter $\\sigma_{\\text{APF}}$ [\\ms{}] & $3.7^{+0.5}_{-0.4}$ & $3.7^{+0.5}_{-0.4}$ \\\\\n\t\tHIRES pre-upgrade RV jitter $\\sigma_{\\text{H,pre}}$ [\\ms{}] & $3.4^{+0.8}_{-0.6}$ & $3.3^{+0.8}_{-0.6}$ \\\\\n\t\tHIRES post-upgrade RV jitter $\\sigma_{\\text{H,post}}$ [\\ms{}] & $3.3 \\pm 0.4$ & $3.3 \\pm 0.4$ \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table*}\n\n\\begin{figure*}\n\t\\includegraphics[width=\\columnwidth]{HR5183_pmRA.pdf}\n\t\\includegraphics[width=\\columnwidth]{HR5183_pmDec.pdf}\n\t\\caption{Our model for the proper motion of HR~5183, normalised to the star-planet barycentre, in right ascension (left) and declination (right). The two filled points in the main panels are the Hipparcos and Gaia proper motions while the unfilled points in the side panels represent the Hip-Gaia proper motions, which are averaged over the interval between the Hipparcos and Gaia observations. Due to the relatively large uncertainty on the Hipparcos proper motion almost all of our astrometric constraints are derived from the Gaia and Hip-Gaia measurements, which together imply an edge-on orbital inclination of $89.9^{+13.3}_{-13.5}\\degree$ for HR~5183~b.}\n\t\\label{figure:planet_PM}\n\\end{figure*}\n\nThe results of our orbital model for HR~5183~b are \\boldnew{provided} in Table \\ref{table:planet_orbit}. We measure a planetary orbital inclination of $89.9^{+13.3}_{-13.5}\\degree$ and a longitude of node of $224.0^{+18.2}_{-20.3}\\degree$. Our value for the orbital inclination is consistent with an edge-on orbit for the planet, and as a result our true mass of $3.31^{+0.18}_{-0.14}$~$M_J$ for HR~5183~b is almost identical to its minimum mass.\n\nOur fit to the RV data is shown in Figure~\\ref{figure:planet_RV}, and it is visually very similar to the RV model of \\citet{Blunt19}. In comparison to that work we find a slightly longer orbital period and higher eccentricity and semi-major axis, although these are consistent within the uncertainties; all other parameters that can be compared are essentially identical. To test whether these differences are a result of the inclusion of astrometric data we additionally ran a RV-only version of our model, and found that this model indeed resulted in identical results as \\citet{Blunt19}. Why exactly the inclusion of astrometric data leads to a preference for slightly longer orbital periods for HR~5183~b is not immediately clear, but one possibility is that the shortest orbital periods allowed by the radial velocities would result in a proper motion anomaly between the Gaia and Hip-Gaia measurements that is too large to match the observed signal.\n\nIn Figure~\\ref{figure:planet_PM} we display our model for the proper motion data. This demonstrates that the Hipparcos measurement is insensitive to the planetary signal, hence our constraints on the orbital inclination and longitude of node are largely derived from the difference between the Gaia and Hip-Gaia proper motions. It is also evident that the Gaia measurement is temporally fortuitously close to the maximum velocity displacement of HR~5183, allowing us to confidently detect the proper motion signal generated by HR~5183~b.\n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{binary_hists.pdf}\n\t\\caption{Posterior distributions of parameters from our LOFTI model of the HR~5183-HIP~67291 orbit. From left to right, the parameters are the semi-major axis, eccentricity, orbital inclination, argument of periastron, longitude of ascending node, time of periastron passage, and periastron distance. The long tails in semi-major axis and time of periastron have been truncated for clarity. Note that the longitude of node has been limited to the range (0, 180)$\\degree$ since this parameter is degenerate over the full 360$\\degree$ range.}\n\t\\label{figure:binary_orbit_posteriors}\n\\end{figure*}\n\n\\begin{table*}\n\t\\centering\n\t\\caption{Orbital parameters for the HR~5183-HIP~67291 binary.}\n\t\\label{table:binary_orbit}\n\t\\begin{tabular}{lcccc}\n\t\t\\hline\n\t\tParameter & Median & Mode & 68\\% CI & 95\\% CI \\\\\n\t\t\\hline\n\t\tRelative semi-major axis $a$ [AU] & \\boldnew{$16400$} & \\boldnew{$9000$} & \\boldnew{$\\left(8600, 22700\\right)$} & \\boldnew{$\\left(8500, 67800\\right)$} \\\\\n\t\t$\\log P$ [years] & $6.20$ & \\boldnew{$5.82$} & \\boldnew{$\\left(5.78, 6.42\\right)$} & \\boldnew{$\\left(5.77, 7.13\\right)$} \\\\\n\t\tEccentricity $e$ & \\boldnew{$0.59$} & \\boldnew{$0.96$} & \\boldnew{$\\left(0.35, 0.98\\right)$} & \\boldnew{$\\left(0.07, 0.98\\right)$} \\\\\n\t\tOrbital inclination $i$ [degrees] & \\boldnew{$79$} & \\boldnew{$81$} & \\boldnew{$\\left(72, 86\\right)$} & \\boldnew{$\\left(35, 87\\right)$} \\\\\n\t\tArgument of periastron $\\omega$ [degrees] & \\boldnew{$148$} & \\boldnew{$137$} & \\boldnew{$\\left(110, 217\\right)$} & \\boldnew{$\\left(8, 288\\right)$} \\\\\n\t\tLongitude of node $\\Omega$ [degrees] & \\boldnew{$118$} & \\boldnew{$122$} & \\boldnew{$\\left(107, 128\\right)$} & \\boldnew{$\\left(68, 165\\right)$} \\\\\n\t\tTime of periastron $T_0$ [$10^{5}$ years \\bmaroon{CE}] & \\boldnew{$-4.83$} & \\boldnew{$-2.00$} & \\boldnew{$\\left(-11.06, -1.72\\right)$} & \\boldnew{$\\left(-63.82, 0.02\\right)$} \\\\\n\t\tPeriastron distance $a(1-e)$ [AU] & $6800$ & \\boldnew{$270$} & \\boldnew{$\\left(130, 14900\\right)$} & \\boldnew{$\\left(100, 54500\\right)$} \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table*}\n\n\n\\subsection{Binary orbit} \\label{subsec:binary_results}\n\nOur parameter posteriors for the HR~5183-HIP~67291 binary orbit are shown in Figure \\ref{figure:binary_orbit_posteriors} and listed in Table \\ref{table:binary_orbit}. \\boldnew{Uncertainties on the orbital parameters are generally large}, an explicable circumstance considering the large scale of the binary orbit, but we are \\boldnew{nevertheless} able to derive meaningful constraints on all parameters. The semi-major axis \\boldnew{posterior} displays a sharp peak at \\boldnew{9000}~AU, with a long and essentially interminate tail towards larger values; \\boldnew{54\\%} of the posteriors have a semi-major axis \\boldnew{higher} than the 15400~AU projected separation. The orbital eccentricity \\boldnew{shows} a \\boldnew{sharp peak at $e=0.96$ with a tail spanning all the way to to $e=0$, with a modest secondary peak at $e\\approx0.25$}. As a result the periastron distance is less constrained than the semimajor axis, with a generally rising probability towards lower separations. \\boldnew{The orbital period of} the binary is reasonably well-constrained \\boldnew{in our model} with a modal value of \\boldnew{$\\sim7\\times10^{5}$} years, approximately four orders of magnitude larger than the planetary orbital period.\n\nRegarding the orbital inclination, our posterior \\boldnew{peaks sharply at $i=81\\degree$, followed by} a significant tail stretching all the way to $i=0\\degree$. The longitude of node \\boldnew{has a strong peak at $\\Omega=122\\degree$ along with a secondary peak at $\\Omega=110\\degree$, however, the entire} allowed parameter space is sampled in the posterior \\boldnew{at low probability;} furthermore, since the radial velocities of the two stars are statistically indistinguishable after `de-projection' (see Table \\ref{table:binary_astrometry}), we are unable to break the $180\\degree$ degeneracy in the longitude of node. Thus, the posteriors in longitude of node are mirrored around $180\\degree$, and the distribution given in Table \\ref{table:binary_orbit} and Figure \\ref{figure:binary_orbit_posteriors} are folded for clarity. \\boldnew{This degeneracy likewise affects the argument of periastron, which shows a strong maximum at $\\omega=137\\degree$ for the $\\Omega<180\\degree$ orbital solution.}\n\nThe significant probability \\boldnew{density for orbital eccentricities approaching $e=1$ in our posterior} suggests that hyperbolic orbits are consistent with the available data \\boldnew{for the binary}. We are unable to directly test this possibility in our model since LOFTI imposes an $e<1$ constraint on trial orbits, so we instead use the methodology outlined in Section \\ref{subsubsec:bound_prob} to determine the fraction of \\boldnew{non-hyperbolic orbits. From this model} we find a bound probability of \\boldnew{73\\%} for the binary, \\boldnew{higher than the 44\\% found by \\citet{Blunt19} using a similar method}. \\boldnew{Though this result is already encouraging, in Section \\ref{subsec:binary_discussion} we further argue that this estimate is likely to be pessimistic and that the binary is almost certainly gravitationally bound.}\n\n\\subsection{Planet-binary mutual inclination}\n\n\\begin{figure*}\n\t\\includegraphics[width=\\columnwidth]{HR5183_inc_node.pdf}\n\t\\includegraphics[width=\\columnwidth]{HR5183_delta_inc.pdf}\n\t\\caption{(Left) Distribution of inclinations and longitudes of node for HR~5183~b (blue-green) and HIP~67291 (purple-yellow). The planetary and stellar parameters appear to differ in their distributions \\boldnew{for the longitude of node}, suggesting misalignment between the two orbits. (Right) Distribution of planet-binary mutual inclinations \\boldnew{$\\Delta i$}. The blue histogram reflects our model distribution while the green histogram represents \\bmaroon{the prior, which is uniform in $\\cos\\Delta i$}. \\boldnew{We} observe a preference for misaligned orbits \\boldnew{($\\Delta i\\approx90\\degree$), which remains even after acknowledging the bias towards misaligned orbits resulting from the non-uniform prior.}}\n\\label{figure:inclinations}\n\\end{figure*}\n\nHaving now constrained the planetary and stellar orbits, we plot their orbital inclinations and longitudes of node in \\boldnew{the left panel of} Figure \\ref{figure:inclinations}. \\boldnew{The precision of the orbital parameters are generally similar in both cases, although the inclination distribution for HIP~67291 is significantly more asymmetrical than for HR~5183~b and there is a clear} $180\\degree$ degeneracy in the longitude of node \\boldnew{for the binary orbit.} The distributions of these parameters are suggestive of misalignment between the two companions; in particular, \\boldnew{while the two orbital inclinations are consistent with alignment at $i\\approx80\\degree$, the location of the $225\\degree$ peak in the planetary longitude differs significantly from the maxima at $120\\degree$ and $300\\degree$ for the binary orbit.} However, the parameter distributions for both companions are too broad to confidently infer orbital misalignment from this figure \\boldnew{alone}, so it is necessary to quantitatively assess the mutual inclination between the two orbits.\n\nFor any two companions $b$ and $c$ orbiting a star, the mutual inclination \\boldnew{$\\Delta i$} between their two orbits can be defined as\n\n\\begin{equation}\n\\label{equation:mutual_inclination}\n\\cos\\Delta i=\\cos i_b \\cos i_c + \\sin i_b \\sin i_c \\cos(\\Omega_b - \\Omega_c)\n\\:,\n\\end{equation}\n\n\\citep{DeRosa20, Xuan20}. Since all of these parameters have been determined to some extent in our results, we are able to directly constrain the mutual inclination between HR~5183~b and \\boldnew{HIP~67291}. \\boldnew{However}, it is important to note that the resulting prior on the mutual inclination is uniform in $\\cos\\Delta i$ rather than in $\\Delta i$, meaning that a randomly sampled posterior can have the appearance of supporting a misalignment.\n\nOur resulting mutual inclination distribution is shown in the right panel of Figure \\ref{figure:inclinations}. We compare our \\bmaroon{model} distribution with \\boldnew{an empirical} prior drawn from uniform sampling of the orbital inclinations and longitudes of node. Our distribution of mutual inclinations is \\boldnew{indicative} of planet-binary orbital misalignment, \\boldnew{with a peak in the mutual inclination of $\\Delta i=90\\degree$. This is primarily driven by the $\\approx75\\degree$ or $\\approx105\\degree$ disagreement in the longitudes of node, whereas the contribution from the orbital inclinations is generally small. However it is also evident that the prior is biased towards misalignment, with the peak in the uniformly sampled distribution likewise lying at $\\Delta i=90\\degree$. This means that the significance of our mutual inclination measurement must be overestimated. Nevertheless, the posterior distribution clearly favours mutual inclinations around $\\approx90\\degree$ more strongly than can be explained by prior bias alone; we find 3$\\sigma$ lower limit of $\\Delta i>13\\degree$ for the mutual inclination, whereas the equivalent for the uniformly sampled distribution is just $>4\\degree$. Arbitrarily setting up $\\Delta i<10\\degree$ as a definition for aligned orbits, we find that $1.33\\%$ of the prior qualifies as aligned whereas only $0.16\\%$ of the posterior distribution does the same, resulting in an odds ratio of $8.4$. These statistics strongly suggest that the misalignment between the planetary and stellar orbits is significant, even \\boldagain{when considering the} bias of the prior towards such a result.}\n\n\n\n\\section{Discussion} \\label{sec:discussion}\n\n\\subsection{HR 5183 b -- an eccentric giant planet with an edge-on orbit} \\label{subsec:planet_discussion}\n\nOur joint RV-astrometric model for HR~5183~b confirms the extremely high-eccentricity and long-period orbit for the planet found by \\citet{Blunt19}. With its orbital inclination now measured as $89.9^{+13.3}_{-13.5}\\degree$, we are able to unambiguously confirm that HR~5183~b is a planet with a true mass of $3.31^{+0.18}_{-0.14}$~$M_J$. The near-equivalence of the minimum and true planetary masses is relevant for our understanding of the dynamics of the HR~5183 system, since the dynamical studies of \\citet{Kane19} and \\citet{Mustill22} have both implicitly assumed that the planetary minimum mass of $3.23^{+0.15}_{-0.14}$~$M_J$ measured by \\citet{Blunt19} is a good approximation of the true mass; our results demonstrate that this assumption is valid.\n\nIt is notable that in the Gaia DR2 version of the HGCA \\citep{Brandt18}, astrometric acceleration of HR~5183 was not significantly detected. By contrast, in the Gaia EDR3 HGCA \\citep{Brandt21} acceleration of HR~5183 was detected at $>$3$\\sigma$ significance, allowing us to measure the orbital inclination of HR~5183~b. This rise in statistical significance can be attributed to two main factors. The first cause, more dominant of the two, is the substantial increase in astrometric precision between the two catalogues; \\citet{Brandt21} estimates that the precision of the EDR3 HGCA reflects an improvement by a factor of $\\sim$3 over the DR2 version, which \\bmaroon{can itself} largely be ascribed to the significant improvement in astrometric precision of Gaia EDR3 over DR2 \\citep{GaiaEDR3astrometry}. The second cause is the extended time coverage of Gaia EDR3. The timespan of astrometric observations used in the Gaia DR2 solution is 2014~Aug~22 -- 2016~May~23, while in Gaia EDR3 this has been extended to 2017~May~28 \\citep{GaiaDR2astrometry, GaiaEDR3astrometry}. This time extension is highly relevant because, as is evident from Figure \\ref{figure:planet_PM}, the average epoch of Gaia observations is \\boldnew{only} $\\sim$500 days prior to the maximum of the proper motion anomaly $\\Delta\\mu$ generated by HR~5183~b; as a result, the time extension of Gaia EDR3 has brought the average epoch of the Gaia astrometry closer to the epoch of maximum astrometric signal, increasing the detectability of HR~5183~b.\n\nThe nature of the available astrometric data does however constrain our ability to detect the reflex signal of HR~5183~b. The Hipparcos-Gaia proper motions used in this work are tangential velocities that are themselves time-averaged from position measurements taken over spans of several years. In contrast, the periastron passage of HR~5183~b is a rapid event, with a complete reversal in the sign of $\\Delta\\mu$ taking place in no more than a few hundred days; as a result the time-averaging of the proper motion data leads to a considerable loss of resolution around periastron passage, which necessarily limits our ability to precisely constrain the orbital inclination and longitude of node of HR~5183~b. We anticipate that once the full Gaia astrometric data is released with Gaia DR4 it will be possible to detect the planetary reflex signal with significantly finer time resolution, allowing for significantly reduced uncertainties for the orbital inclination and longitude of node for HR~5183~b compared to those measured here.\n\n\n\\subsubsection{Sky orbit and direct imaging} \\label{subsec:direct_imaging}\n\n\\begin{figure}\n\t\\includegraphics[width=0.85\\columnwidth]{HR5183_pos.pdf}\n\t\\caption{Our projected sky orbit for HR~5183~b. The star symbol at (0, 0) denotes the position of HR~5183. As in Figure~\\ref{figure:planet_RV}, the black line corresponds to the best-fitting model while the grey lines are drawn randomly from the posteriors; the axes scales have been limited for clarity. The broad range of allowed orbits reflects the significant uncertainties in the orbital period, inclination, and longitude of node. The white arrowhead indicates the direction of motion for the best-fitting orbit.}\n\t\\label{figure:planet_pos}\n\\end{figure}\n\nOwing to our astrometric constraints on the orbital inclination and longitude of node of HR~5183~b we can predict the relative orbit of the planet around HR~5183, which we plot in Figure~\\ref{figure:planet_pos}. The significant uncertainties in these parameters and the planetary orbital period manifest in the relatively wide range of sky orbits that are allowed by the available data, in particular the very broad distribution of possible separations at apoastron. Since the planetary orbit is observed edge-on the position angle of a given orbit is governed primarily by the longitude of node, hence the spread of orbits across a quarter of the figure is caused by the substantial $\\sim$20$\\degree$ uncertainty on this parameter.\n\nRestricting ourselves to the planetary sky separation we find considerably tighter constraints than for the full sky orbit, as shown in Figure~\\ref{figure:planet_sep}. Beginning with a wide spread of possible separations at earlier epochs, the planetary separation is well-constrained from BJD$\\approx$2455000 up until inferior conjunction at BJD 2457800. Periastron counter-intuitively occurs at the secondary peak in separation at BJD 2458100, after which the planet passes through superior conjunction and then begins to separate from HR~5183. Subsequent to superior conjunction the planet-star separation remains well-constrained \\bmaroon{with} the small degree of uncertainty primarily related to the orbital inclination\\bmaroon{,} inclinations closer to 90$\\degree$ corresponding to smaller separations.\n\nBased on these results we can improve upon the observations made by \\citet{Blunt19} regarding the prospects for direct detection of HR~5183~b. Assuming a random distribution of $i$ and $\\Omega$, \\citet[][\\bmaroon{figures 7 -- 9}]{Blunt19} predicted the separation and contrast of HR~5183~b at a series of epochs in the range of 2020-2025. The separation distributions found by those authors are generally composed of a sharp peak, corresponding to edge-on orbital inclinations, followed by an extended tail towards wider separations caused by near-polar orbital inclinations. As we have measured an edge-on inclination for HR~5183~b we can exclude these low-probability tails, so our separations as plotted in Figure \\ref{figure:planet_sep} correspond to the low-separation peaks in \\citet{Blunt19}.\n\nBecause the detectability of the planet through direct imaging increases with increasing separation, and we have excluded the tails towards larger separations found in \\citet{Blunt19}, the prospects for directly imaging HR~5183~b are restricted to the more challenging end of those presented in that work. Nevertheless, following the predicted contrasts given by those authors, the planet will reach the $300-500$ mas separation required for detection in the infrared by the second half of this decade for the entire range of plausible orbital inclinations \\bmaroon{--} even despite the relatively old age and low mass of the planet. Thus, HR~5183~b is likely to be \\bmaroon{detectable} with high-contrast imaging in the relatively near future. Direct \\bmaroon{observation} of the planet will allow for study of its atmospheric properties, and additionally the precise astrometry provided by its detection would allow for further improvements in the measurement of the orbital parameters of HR~5183~b.\n\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{HR5183_sep.pdf}\n\t\\caption{Planet-star sky separation over time for HR~5183~b. Minima in separation occur at inferior conjunction (BJD = 2457800) and superior conjunction (BJD$\\approx$2459000); note that periastron counter-intuitively occurs close to the secondary peak in separation between these epochs.}\n\t\\label{figure:planet_sep}\n\\end{figure}\n\n\n\\subsubsection{Transit probability}\n\nOur measurement of the planetary orbital inclination of $89.9^{+13.3}_{-13.5}\\degree$ motivates us to revisit the possibility that HR~5183~b may, despite its long orbital period, be a transiting planet. As is well-known, a planet may transit its parent star if its orbit is observed sufficiently near to edge-on from our line of sight. The occurrence of transits can be parametrised by $b<1$, where $b$ is the impact parameter, a term defined as\n\n\\begin{equation}\n\\label{equation:impact_parameter}\nb=\\frac{a\\cos i}{R_*} \\left(\\frac{1-e^2}{1-e\\sin\\omega}\\right)\n\\:,\n\\end{equation}\n\nWhere $R_*$ is the stellar radius and all remaining parameters are as in Section \\ref{subsec:planet_method} \\citep[][\\bmaroon{equation~7}]{Winn10}. Thus, the probability of observing transits is inversely proportional to the star-planet separation at inferior conjunction. Without knowledge of the planetary orbital inclination, \\citet{Blunt19} inferred a transit probability $p_{\\text{tra}}$ of $0.00185 \\pm 0.00010$ for HR~5183~b assuming a uniform inclination distribution. Since our measurement of the planetary orbital inclination is fully consistent with an edge-on orbit, it is worth revisiting the transit probability based on the planetary orbital parameters measured in this work.\n\nWe adopt a stellar radius $R_*=1.53^{+0.06}_{-0.05}$ $R_\\odot$ as in \\citet{Blunt19}. First assuming a uniform distribution of orbital inclinations, we measure $p_{\\text{tra}}=0.0010$. This is lower than found by \\citet{Blunt19}, a difference which we ascribe to our preference for slightly longer orbital periods (see Section \\ref{subsec:planet_results}). Secondly, when incorporating our constraints on the planetary orbital inclination we find $p_{\\text{tra}}=0.0033$. The factor of $\\sim$3 improvement in the transit probability appears to be quite modest considering that our inclination distribution peaks almost exactly at 90$\\degree$, but seeing as the extremely narrow range of orbital inclinations where transits occur (approximately $\\pm0.1\\degree$) is two orders of magnitude smaller than our inclination uncertainty, the relatively small increase in transit probability is understandable. Nevertheless, it should be remembered that \\bmaroon{if the measured} planetary orbital inclination \\bmaroon{did differ} significantly from 90$\\degree$\\bmaroon{, it} would obviously greatly reduce the transit probability.\n\nThe previous inferior conjunction of HR~5183~b occurred in early 2017, and the next such event will only occur after another orbital period of the planet has elapsed. Due to the large uncertainties on this parameter we cannot provide a precise prediction of the next epoch of inferior conjunction, but our median orbital period of 102~years is sufficient to conclude that detection of planetary transits -- should they occur -- will not be possible in the near future. Nevertheless, future improvements on the planetary orbital parameters from radial velocity, astrometric, and direct imaging observations will allow for refined measurements of the probability and time of transit, and the conclusions made here will undoubtedly be revised long before the next inferior conjunction.\n\n\\subsection{HIP 67291 -- the wide stellar companion to HR 5183} \\label{subsec:binary_discussion}\n\nIn this work we have used LOFTI \\citep{Pearce20} to constrain the orbital parameters of the wide HR~5183-HIP~67291 binary. With a sky separation of 490 arcseconds and a corresponding projected separation of 15400 AU, this system has one of the largest separations of any binary that has been modelled using this technique, and we were forced to account for perspective effects arising due to the non-linear nature of the celestial sphere in our model. We have found that the most likely binary orbits have semi-major axes of \\boldnew{9000~AU, orbital inclinations of $81\\degree$, and longitudes of node of $122\\degree$ (with a $\\pm180\\degree$ degeneracy for the latter)}. However, all of these parameters have substantial uncertainties in our posteriors, and orbits with very different parameters should be considered plausible. \\boldnew{We plot the sky orbit of the binary in Figure~\\ref{figure:binary_orbit}, which shows the strong preference for highly inclined, near edge-on orbits for HR~5183-HIP~67291.}\n\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{orbit_plot.pdf}\n\t\\caption{\\boldnew{Our projected sky orbit for the stellar binary. The star symbols mark the positions of HR~5183 and HIP~67291 respectively, while the orbits are drawn randomly from the posteriors of the LOFTI fit. The direction of motion is clockwise for all orbits in the figure. There is a clear preference for near-edge on orbits ($i\\approx80\\degree$) for the binary.}}\n\t\\label{figure:binary_orbit}\n\\end{figure}\n\n\\citet{Blunt19} performed a similar fit to the binary orbit as we have done, and it is informative to compare their model posteriors with ours. Our results differ in significant ways; \\boldnew{for example,} their most probable bound orbits show a preference for semi-major axes around $\\sim$25000~AU with very few orbits close to the peak in our results at \\boldnew{9000~AU. Likewise,} their \\boldnew{orbital inclination distribution is strongly peaked} at $\\sim$100$\\degree$ with a tail towards higher inclinations and seemingly no orbits below $i<90\\degree$, \\boldnew{while our inclination distribution peaks at $i=81\\degree$ with none of our posteriors reaching above $i>90\\degree$.} The clearest area of agreement between our results is in the \\boldnew{orbital eccentricity,} where we both \\boldnew{observe high values ($e>0.8$), and correspondingly in the periastron distance, where we find preferences for relatively small minimum separations ($a(1-e)<2000$~AU).}\n\nSince the data underlying our fits are similar -- The difference between the Gaia DR2 solutions used by \\citet{Blunt19} and the EDR3 solutions used in this work are generally insignificant -- we anticipate that these disparities are a result of methodological differences. First, we note that the astrometric acceleration generated by HR~5183~b substantially displaces the Gaia proper motion from the star-planet barycentre, and accounting for the planetary signal results in significant change to the stellar velocity. Secondly, our handling of perspective effects in the LOFTI model affects our velocities for HR~5183 as well, although the increased proper motion uncertainties arising from this procedure reduces the significance of these effects. Lastly, \\citet{Blunt19} incorporated Gaia parallaxes of the two stars as positional information in their orbital model, which is not the case with LOFTI. While the Gaia DR2 parallaxes of HR~5183 and HIP~67291 imply a difference in stellar distances from the solar system of $-0.157 \\pm 0.059$~pc (the negative sign meaning that HR~5183 is more distant), the Gaia EDR3 parallaxes for the pair result in a smaller difference of $-0.061 \\pm 0.029$~pc. As a smaller distance between the two stars implies a larger escape velocity, it follows that the more similar stellar distances would entail lower orbital eccentricities given identical velocities. We anticipate that these differences can explain the dissimilarity between our binary orbit and that of \\citet{Blunt19}.\n\nTo improve upon our constraints on the binary orbit, we suggest that stronger constraints on the stellar velocities are of importance. In particular, the radial velocity measurements for the two stars provide the largest portion of the uncertainties in the input data, and bringing these uncertainties below the 100 \\ms{} level will be important for improving constraints on the binary orbit. We furthermore note that HIP~67291 has not previously been targeted for precise radial velocities to our knowledge; therefore, as well as contributing to improved precision on the binary orbital parameters, high-precision radial velocities of this star could reveal the presence of planets orbiting the secondary component of this wide pair. \\bmaroon{One possible source of bias in the available data is that the Gaia~EDR3 astrometric solution for HR~5183 assumes a linear proper motion, which is demonstrably incorrect due to the periastron passage of HR~5183~b which occurs during the course of Gaia observations. Although we have corrected for the displacement in proper motion caused by HR~5183~b in our binary orbit model, it is possible that the underlying Gaia proper motion may be slightly biased due to the single-star nature of the EDR3 astrometric solution. Future Gaia data releases will account for non-linear proper motions in the astrometric model, thus accounting for this possible source of systematic error. Finally}, utilising the Gaia parallaxes for the binary orbit model may help to improve the constraints on some of the parameters.\n\n\n\\subsubsection{Appraising the binary bound orbit probability} \\label{subsubsec:bound_prob_discussion}\n\nIn Section \\ref{subsec:binary_results}, we found that a test particle model suggests that \\boldnew{73\\%} of possible orbits for HIP~67291 are bound (non-hyperbolic) based on the available data. \\boldnew{This is a substantial increase from \\citet{Blunt19}, who found a bound orbit fraction of 44\\% based on a similar model. However, our result implies a considerable 27\\% probability that HIP~67291 is gravitationally unbound, which justifies further consideration of the bound probability for the pair. In the following section we expand on the reasoning outlined by \\citet{Blunt19} regarding the nature of HR~5183-HIP~67291 to argue that the pair is almost certainly a gravitationally bound binary.}\n\nThere are three possible hypotheses for the relationship between HR~5183 and HIP~67291:\n\n\\begin{enumerate}\n\t\\item The two stars are unrelated, and their close physical separation is a coincidence.\n\t\\item The two stars are physically related, and form a binary with a gravitationally bound orbit.\n\t\\item The two stars are physically related, but are not presently on a bound orbit; they may have been bound in the past, but are now undergoing breakup.\n\\end{enumerate}\n\nHypothesis I was considered in Section \\ref{subsec:binary_method}, and can be firmly rejected as improbable; \\bmaroon{through a query of Gaia~EDR3 for stars with parallaxes matching within $\\pm~0.5$~mas and proper motions within $\\pm~1$~\\masyr{} of the corresponding values for HR~5183,} we \\bmaroon{empirically} estimate the probability of such a chance alignment in parallax and proper motion to be $2.0\\times10^{-5}$. Hypothesis II corresponds to the \\boldnew{73\\%} fraction of bound orbits, while Hypothesis III reflects the remaining \\boldnew{27\\%} of unbound orbits. However, while Hypothesis II is a physically plausible arrangement for the two stars, Hypothesis III is not. HR~5183 and HIP~67291 would have to have become unbound relatively recently to conceivably be observed at their present separation; we can make a first-order estimate of this time-scale by taking the observed relative tangential velocity between the stars ($\\approx$275 \\ms{} from the proper motions in Table \\ref{table:binary_astrometry}) and their projected separation of 15400 AU, which when put together result in a time-scale of separation of $\\sim$2.6 $\\times10^{5}$ years (disregarding gravitational effects, which would undoubtedly decrease this time-scale). Assuming that the probability of dissociation between the stars is uniform over the age of the system, and given an age for HR~5183 of $7.7^{+1.4}_{-1.2}$~Gyr \\citep{Blunt19}, the probability of observing this pair in such a recent state of dissociation would be\n\n\\begin{equation}\n\\frac{\\sim2.6\\times 10^{5}}{\\sim7.7\\times 10^{9}}\\approx 3.4 \\times 10^{-5}\n\\:.\n\\end{equation}\n\nThis indicates that the probability of observing this system in a recently-unbound state is very low. With this informed prior in mind, it follows that Hypothesis III is far less probable than the naive \\boldnew{27\\%} estimate provided by the test particle model, and conversely the probability that the two stars are on a bound orbit must be much greater than \\boldnew{73\\%}. We thus conclude that HR~5183 and HIP~67291 form a physical, gravitationally bound system. Our investigation into the binary orbital parameters, and the planet-binary mutual inclination, can therefore be justified.\n\nWe anticipate that the prevalence of hyperbolic orbits in our test particle model reflect the proportionally large uncertainties of the stellar velocities as compared to the escape velocity of the system. If this is the case, future improvements in the observational measurement uncertainties may alter the estimation of the bound probability.\n\n\n\\subsubsection{Considerations on biases in the binary orbit}\n\nRecently, \\citet{FerrerChavez21} have explored biases in the fitting of orbits using the OFTI algorithm of \\citet{Blunt17}; this forms the basis of the LOFTI model used to model the HR~5183-HIP~67291 orbit in this work, so their observations are likely to be relevant to our results. A key result of that work is that there is a strong degeneracy between orbital inclination and eccentricity, with edge-on circular orbits being difficult to distinguish from face-on eccentric orbits. This can perhaps be seen in our results since we find very broad posteriors for both of these parameters, but the peaks in these distributions suggest that orbits with near-edge on inclinations \\textit{and} \\bmaroon{high} eccentricities are generally preferred. However, those authors also observe that higher eccentricities leads to a bias towards edge-on inclinations, and it is quite possible that this bias is reflected in our results. Due to these biases, \\citet{FerrerChavez21} find that in the case of near edge-on orbital inclinations the eccentricity will be largely unconstrained (which is presumably what can be seen in \\boldnew{the breadth of our} eccentricity posterior) and that, qualitatively, the 68\\% confidence interval for the eccentricity will recover the true eccentricity less than 68\\% of the time.\n\nBased on these observations it appears that we should avoid drawing much of significance from our binary eccentricity posterior, \\boldnew{which shows a sharp peak at high values with a mode of $e=0.96$, as} it is highly likely that \\boldnew{this} is statistically biased due to the various inherent degeneracies with the orbital inclination. It is less clear to what extent our binary orbital inclination should be trusted, but considering the magnitude of biases found for the eccentricity it should be considered distinctly possible that our inclination posteriors are biased away from the true value. If this is the case, it would have strong implications for the measurement of a mutual inclination between the planetary and binary orbits in this system\\boldnew{; as our measurements of the orbital inclinations are both consistent with $i\\approx90\\degree$, any bias towards edge-on binary orbits would result in us \\textit{underestimating} the mutual inclination in this system.}\n\n\n\\subsection{\\boldnew{Planet-binary mutual inclination}}\n\nIn this study we have measured the orbital inclination and longitude of node of HR~5183~b for the first time, and we have combined these with the corresponding parameters for the orbit of HIP~67291 in an attempt to measure the planet-binary mutual inclination. Our results \\boldnew{suggest a significant} preference for misaligned orbits, with \\boldnew{a best-fit mutual inclination of $\\Delta i\\approx90\\degree$ and a 3$\\sigma$ lower limit of $\\Delta i>13\\degree$. This is primarily driven by disagreement between the longitudes of node of the planetary and binary orbits, which can be visually identified from the $\\approx90\\degree$ disagreement in the position angle of their sky orbits in Figure~\\ref{figure:planet_pos} and Figure~\\ref{figure:binary_orbit}. However we recognise that the prior for the mutual inclination is uniform in $\\cos\\Delta i$ rather than $\\Delta i$, which means our results are biased towards the $\\Delta i\\approx90\\degree$ that we observe in our posterior. Still, this bias is insufficient to explain our results alone, as the 3$\\sigma$ lower limit in mutual inclination for a uniformly sampled distribution is just $\\Delta i>4\\degree$. Only $0.16\\%$ of our posterior has a mutual inclination below $\\Delta i<10\\degree$ whereas the same statistic is $8.4$ times higher for the uniformly sampled distribution, a discrepancy which indicates that our preference for misaligned orbits is stronger than would be expected from chance. We therefore conclude that the orbits of the planetary and binary orbits are misaligned.}\n\n\\boldnew{In their recent study of the dynamical formation history of HR~5183~b, \\citet{Mustill22} predicted that the mutual inclination between the planetary orbit and the orbit of the wide stellar companion HIP~67291 could potentially be used to distinguish between formation mechanisms for HR~5183~b. Their results suggest that very low or very high mutual inclinations ($\\Delta i\\approx0\\degree$ or $\\approx180\\degree$) cannot be reproduced without invoking planet-planet scattering, while planet-binary interactions alone produce mutual inclinations of $\\approx45\\degree$ or $\\approx135\\degree$. Unfortunately, our measurement of the planet-binary mutual inclination is insufficiently precise} to test the predictions of \\citet{Mustill22}\\boldnew{; our empirical measurement of the planet-binary mutual inclination is consistent with a broad range of values, and is therefore consistent with formation of HR~5183~b with or without planet-planet scattering.}\n\nWhile we are therefore unable to \\boldnew{make inferences on the formation history of HR~5183~b} in this work, future studies may be able to do so if the uncertainties on the orbital inclinations and longitudes of node can be substantially reduced. For the planetary orbit, as noted in Section \\ref{subsec:planet_discussion}, we anticipate that the full release of the Gaia astrometric data will allow for significantly more precise constraints on these parameters than we have been able to reach here; further in the future, detection of HR~5183~b with direct imaging will undoubtedly lead to even greater improvements in the measurement of the astrometric orbital parameters. However, \\boldnew{for the HR~5183-HIP~67291 binary,} opportunities for improvement on the \\boldnew{orbital} parameter precision are slimmer. As discussed in Section \\ref{subsec:binary_discussion}, increased precision on the velocity measurements for the two stars is one of the most important areas for improvement for constraining the binary orbit, and while future Gaia releases will likely lead to more precise proper motion measurements, we suggest that observations with high-precision ground-based spectrographs will be important for improvement on the absolute radial velocities of HR~5183 and HIP~67291. Additionally, we have not used the individual parallaxes of the two stars to constrain the difference in their radial distances in our LOFTI model, and making use of this data may improve some of the constraints on the binary orbital parameters in future studies.\n\nAs discussed in the preceding section, \\citet{FerrerChavez21} have observed significant biases in parameter determination from orbit fitting using the OFTI algorithm. In particular, the biases in inclination observed in that work may pose a significant challenge for measuring planet-binary mutual inclinations. Further investigation of these parameter biases is of importance before mutual inclinations in systems similar to the one studied in this work can be securely measured.\n\n\n\\subsubsection{Comparison with other studies}\n\nIn this section we highlight some studies that have explored mutual inclinations between planetary and stellar orbits in a similar way to our work, and comment on the implications of these results.\n\n\\citet{Li21} have recently used Hipparcos-Gaia astrometry to measure true masses for nine planets discovered through radial velocities using the \\texttt{orvara} code \\citep{orvara}. Two of these planets reside in binary systems (HD~106515~Ab and HD~196067~b), and like our work the authors used Gaia EDR3 astrometry to constrain the binary orbits. Unlike our results for HR~5183~b the planetary orbital inclinations are found to be bimodal, complicating the interpretation of mutual inclination in these systems. \\bmaroon{For the HD~106515 system the results of \\citet{Li21} would be consistent with alignment for the prograde orbital solution for HD~106515~Ab, but would suggest strong misalignment for their retrograde solution. In the HD~196067-196068 system, strong disagreement between the planetary and stellar orbital inclinations suggests misalignment.} It is notable that both of these planets have relatively large eccentricities (HD~106515~Ab $=0.571 \\pm 0.012$, HD~196067~b $=0.70^{+0.14}_{-0.12}$); the possibility that the wide stellar companions in these systems have influenced the formation of these eccentric planets through dynamically interactions should be further investigated.\n\n\n\\citet{Newton19} presented the discovery of a transiting planet orbiting the primary component of the young visual binary DS Tucanae by the TESS mission \\citep{TESS}, and as well as characterising the planetary companion the authors used LOFTI to constrain the orbit of the binary and determined the rotational inclination of DS Tucanae A. All three inclinations were found to be close to alignment in inclination, with a planetary orbital inclination of $89.5^{+0.34}_{-0.41}\\degree$ (or $90.5^{+0.41}_{-0.34}\\degree$), a stellar orbital inclination of $96.9 \\pm 0.9\\degree$, and a stellar rotational inclination of $82-98\\degree$. Subsequent observations of the Rossiter-McLaughlin effect during transit have confirmed that the obliquity of the planetary orbit relative to the rotational axis of DS Tucanae A is low \\citep{Montet20, Zhou20, Benatti21}. The general alignment of orbital planes and rotational axes in this system is suggestive of primordial coplanarity \\citep{Montet20}.\n\n\\citet{Xuan20.inclinations} used Hipparcos-Gaia astrometry to measure the orbital inclinations of giant planets orbiting HD~113337 and HD~38529. Both systems contain two known planets, a debris disc, and widely separated M-type companions. The authors focused on measuring mutual inclinations between the planetary orbits and those of the debris discs, but also attempted to use LOFTI to constrain the orbit of HD~38529~B. While this fit provided only loose constraints on the stellar orbital parameters, this was sufficient for the authors to measure a planet-binary mutual inclination \\boldnew{$\\Delta i > 20\\degree$} at 3$\\sigma$ confidence. They further report a minimum planet-disc mutual inclination \\boldnew{$\\Delta i = 21-45\\degree$} (1$\\sigma$), which combine to suggest that the orbital planes in the HD~38529 system are generally misaligned.\n\n\n\\bmaroon{From the foregoing results, it is notable that the examples of systems with smaller binary separations are more consistent with alignment (DS~Tucanae, HD~106515), while wider binaries show evidence for misalignment (HD~38529, HD~196067-196068, and perhaps HR~5183-HIP~67291 from this work).} In this context, the recent work of \\bmaroon{\\citet{Christian22}} lends significant support to the interpretation that binary semi-major axis plays an important role in planet-binary orbital alignment. In that work the authors used LOFTI to measure the orbital inclinations of a sample of visual binaries containing transiting planets and planet candidates discovered by the TESS mission and found a statistically significant excess of systems consistent with alignment between the planetary and stellar orbits, \\bmaroon{further observing} that this overabundance of aligned systems is strongest for binary semi-major axes below <700~AU. Beyond >700~AU the distribution of binary orbital inclinations is more random, indicating that there is a significant fraction of systems with misaligned orbits. These observations satisfactorily match the properties of the systems discussed above. \\bmaroon{\\citet{Christian22}} suggest that a plausible explanation for this phenomenon is that during the protoplanetary disk phase, relatively close stellar companions on primordially misaligned orbits are capable of torquing the planet-forming disk into alignment prior to its dissipation. However, they note that for systems with semi-major axes below $\\lesssim200$~AU it is possible that the binaries formed in primordial state of alignment through disk fragmentation, a hypothesis which would provide a particularly attractive explanation for the alignment of not only orbital axes but also the rotational axis in the DS~Tucanae system.\n\n\nA notable limitation of the method of \\bmaroon{\\citet{Christian22}} is that the longitude of node of planetary orbits cannot be measured with the transit method, which thus limits the measurement of planet-binary mutual inclinations to a minimum value for each \\boldnew{system}. Indeed, the only widely applicable exoplanet detection technique that can be used to measure the longitude of node is astrometry. As relatively few exoplanets have been detected \\boldnew{with this technique} (both relative, i.e. direct imaging, and absolute \\boldnew{astrometry}), and fewer still belong to multiple star systems, it is clearly not possible to conduct a study of planet-binary mutual inclinations based on astrometric data matching the scope of \\bmaroon{\\citet{Christian22}} at the present time. However, the Gaia mission is expected to detect many thousands of planets with astrometry by the conclusion of its nominal mission \\citep{Perryman14}, and this will undoubtedly provide a large sample of planets in binary star systems where it would be possible to measure the planet-binary mutual inclination. We therefore reason that once this sample of astrometrically-detected planets becomes available with future Gaia data releases, it would be highly interesting to investigate the distribution of planet-binary mutual inclinations for a large sample of systems, using similar techniques as applied in this work to the HR~5183-HIP~67291 system. The distribution of planet-binary mutual inclinations afforded by this would allow for indirect constraints on the processes of planet formation that occur in multiple star systems.\n\n\n\\subsection{Stellar multiplicity and the formation of highly eccentric planets} \\label{subsec:eccentricity_multiplicity}\n\n\\begin{table*}\n\t\\centering\n\t\\caption{Planets with $e\\geq 0.8$ in systems without known stellar companions. Masses are minimum masses ($m\\sin i$) unless otherwise noted.}\n\t\\label{table:eccentric_planets_nonbinary}\n\t\\begin{tabular}{lcccc}\n\t\t\\hline\n\t\tPlanet & Eccentricity & Mass [$M_J$] & Semi-major axis [AU] & Reference \\\\\n\t\t\\hline\n\t\tHD 219828 c & $0.8102 \\pm 0.0051$ & $14.6 \\pm 2.3$ & $5.79 \\pm 0.41$ & \\citet{Ment18} \\\\\n\t\tTOI-3362 b & $0.815^{+0.023}_{-0.032}$ & $5.029^{+0.668}_{-0.646}$ $^{{a}}$ & $0.153^{+0.002}_{-0.003}$ & \\citet{Dong21} \\\\\n\t\tHD 22781 b & $0.8191 \\pm 0.0023$ & $13.65 \\pm 0.97$ & $1.167 \\pm 0.039$ & \\citet{Diaz12} \\\\\n\t\tHD 43197 b & $0.83^{+0.05}_{-0.01}$ & $0.60^{+0.12}_{-0.04}$ & $0.92^{+0.01}_{-0.02}$ & \\citet{Naef10} \\\\\n\t\tKepler-419 b & $0.833 \\pm 0.013$ & $2.5 \\pm 0.3$ $^{{a}}$ & $0.370^{+0.007}_{-0.006}$ & \\citet{Dawson14} \\\\\n\t\tKepler-1656 b & $0.836^{+0.013}_{-0.012}$ & $0.153^{+0.013}_{-0.012}$ $^{{a}}$ & $0.197 \\pm 0.021$ & \\citet{Brady18} \\\\\n\t\tWASP-53 c & $0.8369^{+0.0069}_{-0.0070}$ & $>16.35^{+0.86}_{-0.82}$ & $>3.73^{+0.16}_{-0.14}$ & \\citet{Triaud17} \\\\\n\t\tHD 98649 b & $0.852^{+0.033}_{-0.022}$ & $9.7^{+2.3}_{-1.9}$ $^{{a}}$ & $5.97^{+0.24}_{-0.21}$ & \\citet{Li21} \\\\\n\t\tHD 76920 b & $0.8782 \\pm 0.0025$ & $3.13^{+0.41}_{-0.43}$ & $1.090^{+0.068}_{-0.077}$ & \\citet{Bergmann21} \\\\\n\t\tKepler-1704 b & $0.921^{+0.010}_{-0.015}$ & $4.15 \\pm 0.29$ $^{{a}}$ & $2.026^{+0.024}_{-0.031}$ & \\citet{Dalba21} \\\\\n\t\t\\hline\n\t\t\\multicolumn{4}{l}{$^{{a}}$ True mass $m$ rather than $m\\sin i$.} \\\\\n\t\\end{tabular}\n\\end{table*}\n\n\\begin{table*}\n\t\\centering\n\t\\caption{Planets with $e\\geq 0.8$ that are in known multi-star systems. Masses are minimum masses unless otherwise noted; binary separations are projected values except when otherwise indicated.}\n\t\\label{table:eccentric_planets_binary}\n\t\\begin{tabular}{lcccccc}\n\t\t\\hline\n\t\tPlanet & Eccentricity & Mass [$M_J$] & Semi-major axis [AU] & Planet Reference & Binary separation [AU] & Binary Reference \\\\\n\t\t\\hline\n\t\tHD 7449 b & $0.80^{+0.08}_{-0.06}$ & $1.09^{+0.52}_{-0.19}$ & $2.33^{+0.01}_{-0.02}$ & \\citet{Rodigas16} & 21 & \\citet{Rodigas16} \\\\\n\t\tHD 28254 b & $0.81^{+0.05}_{-0.02}$ & $2.15^{+0.04}_{-0.05}$ & $2.15^{+0.04}_{-0.05}$ & \\citet{Naef10} & 269 & \\citet{ElBadry21} \\\\\n\t\tHD 26161 b & $0.820^{+0.061}_{-0.050}$ & $13.5^{+8.5}_{-3.7}$ & $20.4^{+7.9}_{-4.9}$ & \\citet{Rosenthal21} & 561 & \\citet{ElBadry21} \\\\\n\t\tHD 108341 b & $0.85^{+0.09}_{-0.08}$ & $3.5^{+3.4}_{-1.2}$ & $2.00 \\pm 0.04$ & \\citet{Moutou15} & 382 & \\citet{ElBadry21} \\\\\n\t\tHD 156846 b & $0.84785 \\pm 0.00050$ & $10.57 \\pm 0.29$ & $1.096 \\pm 0.021$ & \\citet{Kane11} & 250 & \\citet{Tamuz08} \\\\\n\t\tHD 80869 b & $0.862^{+0.028}_{-0.018}$ & $4.86^{+0.65}_{-0.29}$ & $2.878^{+0.045}_{-0.046}$ & \\citet{Demangeon21} & 250 & \\citet{ElBadry21} \\\\\n\t\t\\textbf{HR 5183 b} & $0.87 \\pm 0.04$ & $3.31^{+0.18}_{-0.14}$ $^{{a}}$ & $22.3^{+11.0}_{-5.3}$ & This Work & 15400 & This Work \\\\\n\t\tBD+63 1405 b & $0.88 \\pm 0.02$ & $3.96 \\pm 0.31$ & $2.06 \\pm 0.14$ & \\citet{Dalal21} & 97 & \\citet{ElBadry21} \\\\\n\t\tHD 4113 b & $0.8999^{+0.0020}_{-0.0016}$ & $1.602^{+0.076}_{-0.075}$ & $1.298 \\pm 0.030$ & \\citet{Cheetham18} & $23.0^{+4.0}_{-2.7}$ $^{{b}}$ & \\citet{Cheetham18} \\\\\n\t\tHD 80606 b & $0.93226^{+0.00064}_{-0.00069}$ & $4.116^{+0.097}_{-0.100}$ $^{{a}}$ & $0.4565^{+0.0051}_{-0.0053}$ & \\citet{Bonomo17} & 1355 & \\citet{ElBadry21} \\\\\n\t\tHD 20782 b & $0.950 \\pm 0.001$ & $1.488^{+0.105}_{-0.107}$ & $1.365^{+0.047}_{-0.050}$ & \\citet{Udry19} & 9075 & \\citet{ElBadry21} \\\\\n\t\t\\hline\n\t\t\\multicolumn{7}{l}{$^{{a}}$ True mass rather than $m\\sin i$ $^{{b}}$ Semi-major axis based on an orbital model. As well as a brown dwarf-mass companion, HD 4113 also has a M0-1 stellar} \\\\\n\t\t\\multicolumn{7}{l}{companion at a projected separation of 2157 AU \\citep{Mugrauer14}.} \\\\\n\t\\end{tabular}\n\\end{table*}\n\nWhile HR~5183~b has one of the most eccentric orbits among known planets, it is hardly alone in this area of the parameter space. At the time of writing, a query to the NASA Exoplanet Archive\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}, accessed 2021-10-25.} for planets with $e\\geq 0.8$ returns 21 results, of which nine were discovered subsequent to 2015 and five were added to this group during 2021 alone. These planets are found at a wide range of orbital periods, from the 18.1-day period of TOI-3362~b \\citep{Dong21} to the $\\sim$100-year period of HR~5183~b. Most of these planets are of Jovian mass or above, with the least massive exemplar known being Kepler-1656~b with a mass of $0.153^{+0.013}_{-0.012}$ $M_J$ \\citep{Brady18}.\n\nAlthough previous studies have explored the evolution of single members of this high-eccentricity planet sample \\citep[e.g.][]{Dawson14, Santos16, Mustill22, Dong21}, few have previously considered this population as a whole. Considering that a substantial number of planets with $e\\geq 0.8$ are now known, it seems possible to begin to consider the origins of these planets and their extreme orbits.\n\nOne factor that is particularly worth considering in this context is the role of stellar multiplicity. It has been amply theoretically demonstrated that gravitational interactions with a wide companion can dynamically drive a planet towards high orbital eccentricities \\citep{Holman97, Mazeh97, Wu03, Takeda05}. This occurs as a result of the Kozai-Lidov effect \\citep{Kozai62, Lidov62}, which causes the eccentricity and mutual inclination of the planet to oscillate on long time-scales. \\citet{Blunt19} justifiably expressed scepticism that dynamical interactions with HIP~67291 could have played a role in the origin of HR~5183~b's high orbital eccentricity, owing to the wide projected separation of the binary (15400 AU). However, the results of \\citet{Mustill22} suggest that this is indeed plausible, in part due to the fact that the periastron distance of the binary is likely to be significantly smaller than the current projected separation (see Section \\ref{subsec:binary_results}). To this point, it is important to note that the orbits of wide binaries vary over time due to stellar flybys and interactions with the Galactic tide \\citep{Kaib13, CorreaOtto17.perturbations, Pearce21, Mustill22}, such that even binaries with large semi-major axes ($\\sim$10$^4$~AU) may undergo periods where the periastron distance reaches as low as $\\sim$100~AU, where dynamical interactions with inner planetary systems are greatly amplified \\citep{Kaib13, CorreaOtto17.potential}. Thus, the importance of the Kozai-Lidov effect in exciting planetary eccentricities cannot be discounted, even for very widely separated multi-star systems.\n\nHowever, \\citet{Carrera19} have found that planet-planet scattering can also produce planets with such high eccentricities, and have furthermore cast doubt on the role of the Kozai-Lidov effect in the formation of high-eccentricity planets by pointing out that such planets observed in wide binaries could be the result of planet-planet scattering, this itself possibly being initiated by the Kozai-Lidov effect \\citep[e.g.][]{Malmberg07, Mustill17}. Indeed, while the results of \\citet{Mustill22} indicate that Kozai-Lidov oscillations caused by HIP~67291 can reproduce the high eccentricity of HR~5183~b, it is substantially more probable that the Kozai-Lidov effect can generate the observed eccentricity if planet-planet scattering occurs first than if it acts on a lone planet. While these results may reduce the importance of the Kozai-Lidov effect in the formation of highly eccentric planets in favour of planet-planet scattering, it is important to distinguish this from the role of stellar multiplicity itself; it remains possible that the presence of a stellar companion can cause a planet-planet scattering event that in turn results in the formation of a high-eccentricity planet.\n\nRegardless of whether the Kozai-Lidov effect plays a direct or indirect role, if the presence of a stellar companion significantly contributes to the formation of high-eccentricity exoplanets then these planets would be expected to be more commonly found in multi-stellar systems. \\citet{Kaib13} have observed this effect among the general population of exoplanets, finding that the eccentricity distribution of planets in binary systems is skewed towards higher values. However the inverse hypothesis, where high-eccentricity planets are preferentially found in multi-star systems, has not yet been been demonstrated. \\citet{Mustill22} observed some evidence for this in systems containing exoplanets with $e\\geq 0.8$, but did not go so far as to evaluate the statistical significance of this result. We therefore aim to fully evaluate this hypothesis in this study.\n\nStarting with the 21 known systems containing planets with eccentricities above $e\\geq 0.8$, we have searched the astronomical literature for evidence of stellar companions. We have made particular use of the binary catalogue of \\citet{ElBadry21}, which is based on Gaia EDR3 \\citep{GaiaEDR3} and contains many binaries that have not been previously recognised elsewhere. We present our results in Table \\ref{table:eccentric_planets_nonbinary} (for single-star systems) and Table \\ref{table:eccentric_planets_binary} (for multi-star systems), with both tables reporting the planetary eccentricities, masses, and semi-major axes, and \\bmaroon{the latter} additionally reporting the binary projected separations.\n\nWe identify a total of 11 multiple star systems in our sample. All of these systems \\bmaroon{are binaries}, although as well as a stellar companion at a projected separation of 2157~AU HD~4113 contains a brown dwarf companion with a semi-major axis of $23.0^{+4.0}_{-2.7}$~AU \\citep{Cheetham18, Mugrauer14}, the closer brown dwarf companion being more relevant dynamically to \\bmaroon{the interior planet}. The stellar companions in these systems are found at a wide range of separations, from the 21~AU projected separation of HD~7449~B \\citep{Rodigas16} to the 15400~AU projected separation of HIP~67291 from HR~5183.\n\nThe 11 binary systems in our sample of 21 targets results in a multiplicity rate of $52 \\pm 16\\%$ (where the reported uncertainty is Poisson noise). To assess whether this multiplicity fraction differs significantly from the overall exoplanet population, we must compare our result with those of a larger sample. For this purpose we make use of the recent results of \\citet{Fontanive21}, who conducted a census of multiplicity of exoplanet host stars within 200 pc. The authors measured an overall raw multiplicity rate of $23.2 \\pm 1.6\\%$, and have also provided counts arranged by certain parameters, one of which is mass; these have been split between <0.1~$M_J$, $0.1-7$~$M_J$, and >7~$M_J$. Of the 21 planets in our $e\\geq 0.8$ sample all have masses above >0.1~$M_J$, so to derive a multiplicity rate more comparable to our sample we combine the system counts belonging to the higher-mass bins of \\citet{Fontanive21}, resulting in an adopted multiplicity fraction of $25.5 \\pm 1.8\\%$ for systems including giant planets (>0.1~$M_J$).\n\nWhile this appears to suggest that our high-eccentricity sample has a higher rate of stellar multiplicity by a factor of $\\sim$2, that this is a causal relationship should not yet be considered demonstrated because it cannot be taken for granted that the sample of $e\\geq 0.8$ planets is drawn from the same distribution as that of the overall giant planet population. To confirm that the planetary eccentricity does play a causative role, we aim to construct a control sample of planets with similar overall parameters to those in our target group, except in having lower orbital eccentricities. To accomplish this we use a methodology adapted from \\bmaroon{\\citet{Christian22}}.\n\nTo assemble our control sample, we first downloaded the confirmed planet table from the NASA Exoplanet Archive and removed all planets with $e\\geq 0.5$. From this table, we matched the three planets that provide the lowest values for the following metric:\n\n\\begin{equation}\n\\left(\\frac{a_c-a_p}{a_p}\\right)^2 + \\left(\\frac{m_c-m_p}{m_p}\\right)^2 + \\left(\\frac{M_c-M_p}{M_p}\\right)^2 + \\log_{10}\\left(\\frac{\\text{distance}_c}{\\text{distance}_p}\\right)\n\\:,\n\\end{equation}\n\nWhere $a$ is the planetary semi-major axis in AU, $m$ is the planetary (minimum) mass in $M_J$, $M$ is the stellar mass in $M_\\odot$, and distances are in parsecs; terms in $X_p$ reflect the parameters of our 21 target systems and planets, while terms in $X_c$ are as so for the control sample. In this way, we construct a control sample of planets with similar masses, semi-major axes, parent star masses, and distances, but crucially with lower eccentricities than our target planets. Since the observational biases that underlie exoplanet discovery differ significantly between detection methods, we match the discovery techniques used to detect our target planets with those of the control planets; thus, the four planets discovered through the transit method in our sample (TOI-3362~b and Kepler-419~b, -1656~b, -1704~b) were matched only with planets discovered via the transit method, and the remaining target planets were matched with planets discovered through the radial velocity method.\n\nWith three control matches for every target planets, we assemble a list of 63 planet matches in the control sample. However, a total of 8 planets are duplicated matches among different target planets, leaving us with a final count of 55 control planets in the same number of systems. We then searched for stellar companions to these systems in the same manner as conducted for the target sample. Our results for the control sample are provided in Appendix \\ref{appendix:data}. 13 control systems are identified as stellar multiples, resulting in a multiplicity fraction of $24 \\pm 7\\%$ for the control sample. This is entirely consistent with our adopted $25.5 \\pm 1.8\\%$ multiplicity fraction for systems with giant planets. This allows us to securely establish that the high $52 \\pm 16\\%$ multiplicity rate for our target sample is causally related to the high planetary eccentricities used to select that sample.\n\nWe assess the statistical significance of this overabundance of multi-star systems by conducting a Monte Carlo simulation where we assess the probability that a 21-system sample will contain $\\geq$11 multi-star systems given an underlying multiplicity fraction of $25.5 \\pm 1.8\\%$. As a sanity check we also employ a binomial probability test with the same parameters except for a fixed multiplicity fraction of $25.5\\%$. Both of these tests result in a probability of $p=0.0075$, suggesting that the rate of stellar multiplicity for our target sample differs from that of the overall giant planet population at a moderate $\\sim$2.4$\\sigma$ confidence level. We thus conclude that the factor of $\\sim$2 overabundance of stellar multiples among high-eccentricity planet hosts is likely to be a real effect, although a larger sample will be required to demonstrate this beyond doubt. \n\nWhile this result suggests that the Kozai-Lidov effect plays an important role in the formation of planets with strongly excited orbital eccentricities, it should be remembered that this does not necessarily imply that the Kozai-Lidov effect is the sole mechanism that leads to the formation of highly eccentric planets, as the role of planet-planet scattering cannot be discounted even in binary systems where the Kozai-Lidov effect is likely to occur \\citep[hence][]{Carrera19}. However, even if one does not allow a commanding role to the Kozai-Lidov effect, our result indicates that its role in the formation of high-eccentricity planets should not be lightly disregarded. Indeed, it appears likely that the Kozai-Lidov effect plays a significant role in the excitation of planetary eccentricities even in binary systems with very wide separations. In addition to the HR~5183-HIP~67291 system studied in this work with a 15400~AU binary projected separation, it is remarkable that the most eccentric planet currently known, HD~20782~b \\citep[$e=0.950 \\pm 0.001$,][]{Udry19}, is also found in a very wide binary with a projected separation of 9075~AU. While it is not unlikely that the stellar companions in these systems may approach to significantly smaller separations during periastron, thus producing stronger dynamical perturbations than at their current separations, it nevertheless remains the case that the dynamical influence from a stellar companion should not be discounted in its role in exciting planetary orbits due to a large binary separation alone.\n\n\n\\section{Conclusions} \\label{sec:conclusions}\n\nIn this work we have performed a new analysis of the HR~5183 system. We have used Hipparcos-Gaia astrometry \\citep{Brandt18, Brandt21} to measure the orbital inclination of the highly eccentric planet HR~5183~b discovered by \\citet{Blunt19}, and have found that the planet is observed at $i=89.9^{+13.3}_{-13.5}\\degree$, consistent with an edge-on orbit. We confirm the previously reported long orbital period and high eccentricity of HR~5183~b, finding $P=102^{+84}_{-34}$~years and $e=0.87 \\pm 0.04$. We have furthermore found that HIP~67291 almost certainly forms a physical binary with HR~5183 with a projected separation of 15400~AU, and have used LOFTI \\citep{Pearce20} to explore the orbit of this pair. While the uncertainties on the orbital parameters are significant, we find a most probable binary semi-major axis of \\boldnew{$\\sim$9000~AU} and orbital period of \\boldnew{$\\sim7\\times10^5$~years, with a strong preference for highly inclined orbits ($i\\approx80\\degree$)}. Combining our constraints on the orbital inclinations and longitudes of node for the planetary and binary orbits, we attempt the mutual inclination between the two orbits in the system \\boldnew{for the first time;} we observe a preference for planet-binary misalignment\\boldnew{ ($\\Delta i>13\\degree$ at 3$\\sigma$ confidence), which remains significant even after acknowledging that the mutual inclination prior is biased towards misaligned orbits. However, our measurement of the misalignment of orbits in the system is insufficiently precise to make use of the hypothesis of \\citet{Mustill22} that the planet-binary mutual inclination reflects the formation history or HR~5183~b.} Future observations will allow for the mutual inclination in this system to be more precisely constrained, possibly allowing for direct constraints on the formation of HR~5183~b.\n\nOf the 21 systems containing planets with eccentricities above $e\\geq 0.8$ currently known, 11 are found in multi-stellar systems. This rate of multiplicity exceeds that of the overall planet-host population by a factor of $\\sim$2, a result that we demonstrate to be moderately significant ($p=0.0075$). This provides observational support for the hypothesis that dynamical interactions with exterior stellar companions through the Kozai-Lidov effect plays a major role in the formation of highly eccentric exoplanets, although planet-planet scattering is likely to be an important factor as well. A larger sample of systems with highly eccentric exoplanets will be required before this overabundance of stellar multiples can be demonstrated beyond doubt.\n\nA growing number of long-period exoplanets have been detected with Hipparcos-Gaia astrometry to date \\citep[e.g.][]{Feng19, Xuan20, Li21}, providing new insights into the population of giant planets discovered by the radial velocity method. This group of detections will however undoubtedly be dwarfed by the number of planets that will be found in the Gaia DR4 astrometric solution \\citep{Perryman14}. Of the science discoveries that could be accomplished with this sample, we propose that the planet population revealed by Gaia will allow for the study of mutual inclinations for planets in wide binary systems to an extent that has not previously been possible, through the use of techniques similar to those utilised in this study. The distribution of planet-binary mutual inclinations afforded by this could be used to constrain the planet formation processes that occur in multiple star systems.\n\n\n\\section*{Acknowledgements}\n\n\\bmaroon{We thank the referee for many helpful suggestions that have improved this manuscript. We thank Alex Mustill \\boldnew{and Timothy D. Brandt} for helpful comments.} The authors would like to thank Sarah Blunt, Jerry W. Xuan, and Judah van Zandt for their courtesy and co-operation during the inception of this project. This research has made use of the SIMBAD database and VizieR catalogue access tool, operated at CDS, Strasbourg, France. This research has made use of NASA's Astrophysics Data System. \\boldagain{This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.} This work has made use of data from the European Space Agency (ESA) mission {\\it Gaia} (\\url{https:\/\/www.cosmos.esa.int\/gaia}), processed by the {\\it Gaia} Data Processing and Analysis Consortium (DPAC, \\url{https:\/\/www.cosmos.esa.int\/web\/gaia\/dpac\/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\\it Gaia} Multilateral Agreement.\n\n\n\n\\section*{Data Availability}\n\nThe radial velocity data used in this article are found in \\citet{Blunt19} (\\url{https:\/\/vizier.cds.unistra.fr\/viz-bin\/VizieR?-source=J\/AJ\/158\/181}). The astrometric data used here are found in \\citet{GaiaEDR3} (\\url{https:\/\/vizier.cds.unistra.fr\/viz-bin\/VizieR?-source=I\/350}) and \\citet{Brandt21} (\\url{https:\/\/doi.org\/10.3847\/1538-4365\/abf93c}).\n \n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and summary}\n\nIn 1977 Jaffe predicted a deeply bound 6-quark state with $I=0$, $J=0$, ${S}=-2$\nfrom the bag-model, called the $H$-dibaryon \\cite{Jaffe:1976yi}. \nSubsequently, many experimental searches for the $H$-dibaryon were carried out,\nbut so far no convincing signal was found \\cite{Yoon}.\nHowever, recently evidence for a bound $H$-dibaryon was claimed based on\nlattice QCD calculations \\cite{Beane,Inoue,Inoue11a,Beane11a}. Extrapolations of the\nsimulations, performed for pion masses $M_\\pi \\gtrsim 400$ MeV, to the physical mass\nsuggest that the $H$-dibaryon could be either loosely bound or move into\nthe continuum \\cite{Beane11,Shanahan11}.\n\nIn this paper, we analyze various issues related to these lattice results\nin the framework of chiral effective field theory (EFT) for the baryon-baryon ($BB$) \ninteraction at leading order (LO) in the Weinberg counting. This scheme\nhas proven successful for the few data on hyperon-nucleon scattering \\cite{Polinder06}\nand also for the bounds that exist on the $BB$ interactions with\nstrangeness $S=-2$ \\cite{Polinder07}. There is one low-energy constant (LEC)\nin the $S=-2$ sector, corresponding to the SU(3) flavor-singlet channel, \nthat can not be fitted by hyperon-nucleon data and can be\nfine-tuned to produce a bound $H$ with a given binding energy. This framework\nalso allows us to study the quark mass dependence\\footnote{Because of the\nGell-Mann-Oakes-Renner relation, the pion mass squared is proportional\nto the average light quark mass. Therefore, the notions ``quark mass dependence''\nand ``pion mass dependence'' can be used synonymously.} of the binding energy of the\n$H$, see the calculations of the quark mass dependence of the deuteron\nbinding energy in \\cite{Beane:2002vs,Beane03,Epe02,Epe02a}. \nAnother important issue to be addressed here is how this quark mass \ndependence is affected when the SU(3) breaking manifested in the masses\nof the octet baryons is accounted for.\nIn the real world the $BB$ thresholds are not degenerate but \nthey all differ for the relevant $\\Lambda\\Lambda$, $\\Sigma\\Sigma$ \nand $\\Xi N$ systems and that has very definite dynamical consequences, \nas we will show in what follows.\n\nThe pertinent results of our study can be summarized as follows:\n\\begin{itemize}\n\\item[(i)] We have analyzed the effective range expansion for the \n$\\Lambda\\Lambda$ $^1S_0$ channel, assuming a loosely bound $H$ dibaryon.\nIt shows a very different behaviour to the case of the deuteron in\nthe neutron-proton $^3S_1$ channel. In fact, any attraction supplied \nby the flavor-singlet channel contributes with a much larger weight \nto the $\\Xi N$ interaction than to $\\Lambda\\Lambda$, according to SU(3)\nflavor symmetry, so that the $H$-dibaryon should predominantly be a \n$\\Xi N$ bound state.\n\\item[(ii)] \nWe observe that, for pion masses below 400~MeV, the dependence of the binding\nenergy of the $H$ is linearly decreasing with decreasing pion mass, in\nagreement with the findings in Ref.~\\cite{Beane11}. In particular, a $H$\nbinding energy (BE) adjusted to the value found by NPLQCD \\cite{Beane} \nat $M_\\pi = 389\\,$MeV, is reduced by 7~MeV at the physical pion mass. \nFor larger pion masses, this dependence is weakened. Note, however, that\nfor such large pion masses this should only be considered a trend as \nthe chiral EFT is constructed for masses\/momenta well below the chiral\nsymmetry breaking scale.\n\\item[(iii)] We find a much more drastic effect caused by\nthe SU(3) breaking related to the\nvalues of the three thresholds $\\Lambda\\Lambda$, $\\Sigma\\Sigma$ and $\\Xi N$.\nFor physical values the BE of the $H$ is reduced by as much as 60~MeV as \ncompared to a calculation based on degenerate (i.e. SU(3) symmetric) $BB$ \nthresholds. \nTranslating this observation to the situation in the HAL QCD \\cite{Inoue} \ncalculation, we see that the bound state has disappeared at the physical point. \nFor the case of the NPLQCD \ncalculation, a resonance in the $\\Lambda\\Lambda$ system might survive.\n\\end{itemize}\n\nOur manuscript is organized as follows: In Sec.~\\ref{sec:2}, we recall the\nbasic formalism of the $BB$ interaction in the framework of chiral\nEFT. Sec.~\\ref{sec:3} contains a detailed discussion of the quark mass\ndependence of the BE of the $H$ and the influence of the SU(3) breaking\nthrough the various two-baryon thresholds. In Sec.~\\ref{sec:4} we try to \nmake direct contact to the results published by the NPLQCD and HAL QCD\ncollaborations.\n\n\\section{The baryon-baryon interaction to leading order}\n\\label{sec:2}\n\nFor details on the derivation of the chiral $BB$ potentials for the strangeness sector \nat LO using the Weinberg power counting, we refer the reader to Refs.~\\cite{Polinder06,Polinder07},\nsee also Refs.~\\cite{Savage1,Korpa,Savage2}.\nHere, we just briefly summarize the pertinent ingredients of the chiral EFT for $BB$ interactions. \n\\begin{table*}[t]\n\\renewcommand{\\arraystretch}{1.2}\n\\centering\n\\begin{tabular}{|l|c|c|l|c|l|}\n\\hline\n&Channel &Isospin &$C_{1S0}$ &Isospin &$C_{3S1}$\\\\\n\\hline\n$S=0$&$NN\\rightarrow NN$ &$1$ & $C^{27}$ &$0$ &$C^{10^*}$\\\\\n\\hline\n$S=-1$&$\\Lambda N \\rightarrow \\Lambda N$ &$\\frac{1}{2}$ &$\\frac{1}{10}\\left(9C^{27}+C^{8_s}\\right)$\n&$\\frac{1}{2}$ &$\\frac{1}{2}\\left(C^{8_a}+C^{10^*}\\right)$\\\\\n&$\\Lambda N \\rightarrow \\Sigma N$ &$\\frac{1}{2}$ &$\\frac{3}{10}\\left(-C^{27}+C^{8_s}\\right)$\n&$\\frac{1}{2}$ &$\\frac{1}{2}\\left(-C^{8_a}+C^{10^*}\\right)$\\\\\n&$\\Sigma N \\rightarrow \\Sigma N$ &$\\frac{1}{2}$ &$\\frac{1}{10}\\left(C^{27}+9C^{8_s}\\right)$\n&$\\frac{1}{2}$ &$\\frac{1}{2}\\left(C^{8_a}+C^{10^*}\\right)$\\\\\n&$\\Sigma N \\rightarrow \\Sigma N$ &$\\frac{3}{2}$ &$C^{27}$\n&$\\frac{3}{2}$ &$C^{10}$\\\\\n\\hline\n$S=-2$&$\\Lambda\\Lambda \\rightarrow \\Lambda\\Lambda$ &$0$ & $\\frac{1}{40}\\left(27C^{27}+8C^{8_s}+5C^{1}\\right)$\n & & \\\\\n&$\\Lambda\\Lambda \\rightarrow \\Xi N$ &$0$ &$\\frac{-1}{40}\\left(18C^{27}-8C^{8_s}-10\\,C^{1}\\right)$\n & & \\\\\n&$\\Lambda\\Lambda \\rightarrow \\Sigma\\Sigma$ &$0$ &$\\frac{\\sqrt{3}}{40}\\left(-3C^{27}+8C^{8_s}-5C^{1}\\right)$\n & & \\\\\n&$\\Xi N \\rightarrow \\Xi N$ &$0$ &$\\frac{1}{40}\\left(12C^{27}+8C^{8_s}+20\\,C^{1}\\right)$\n &$0$ &$C^{8_a}$\\\\\n&$\\Xi N \\rightarrow \\Sigma\\Sigma$ &$0$ &$\\frac{\\sqrt{3}}{40}\\left(2C^{27}+8C^{8_s}-10\\,C^{1}\\right)$\n &$1$ &$\\frac{\\sqrt{2}}{6}\\left(C^{10}+C^{10^*}-2C^{8_a}\\right)$\\\\\n&$\\Sigma\\Sigma \\rightarrow \\Sigma\\Sigma$ &$0$ &$\\frac{1}{40}\\left(C^{27}+24C^{8_s}+15C^{1}\\right)$\n &$1$ &$\\frac{1}{6}\\left(C^{10}+C^{10^*}+4C^{8_a}\\right)$\\\\\n&$\\Xi N \\rightarrow \\Xi N$ &$1$ &$\\frac{1}{5}\\left(2C^{27}+3C^{8_s}\\right)$\n &$1$ &$\\frac{1}{3}\\left(C^{10}+C^{10^*}+C^{8_a}\\right)$\\\\\n&$\\Xi N \\rightarrow \\Sigma\\Lambda$ &$1$ &$\\frac{\\sqrt{6}}{5}\\left(C^{27}-C^{8_s}\\right)$\n &$1$ &$\\frac{\\sqrt{6}}{6}\\left(C^{10}-C^{10^*}\\right)$\\\\\n&$\\Sigma\\Lambda \\rightarrow \\Sigma\\Lambda$ &$1$ &$\\frac{1}{5}\\left(3C^{27}+2C^{8_s}\\right)$\n &$1$ &$\\frac{1}{2}\\left(C^{10}+C^{10^*}\\right)$\\\\\n&$\\Sigma\\Lambda \\rightarrow \\Sigma\\Sigma$ & & &$1$ &$\\frac{\\sqrt{3}}{6}\\left(C^{10}-C^{10^*}\\right)$\\\\\n&$\\Sigma\\Sigma \\rightarrow \\Sigma\\Sigma$ &$2$ &$C^{27}$\n & & \\\\\n\\hline\n\\end{tabular}\n\\caption{Various LO baryon-baryon contact potentials for the ${}^1S_0$ and ${}^3S_1$ partial\nwaves in the isospin basis. $C^{27}$ etc. refers to the corresponding ${\\rm SU(3)_f}$\nirreducible representation.}\n\\label{tab:1}\n\\end{table*}\n\\renewcommand{\\arraystretch}{1.0}\n\n\nThe LO potential consists of four-baryon contact terms without derivatives and of \none-pseudoscalar-meson exchanges. \n The LO ${\\rm SU(3)}_{\\rm f}$ invariant contact terms for the octet $BB$\ninteractions that are Hermitian \nand invariant under Lorentz transformations follow from the Lagrangians\n\\begin{eqnarray}\n{\\mathcal L}^1 &=& C^1_i \\left<\\bar{B}_a\\bar{B}_b\\left(\\Gamma_i B\\right)_b\\left(\\Gamma_i B\\right)_a\\right>\\ , \\quad\n{\\mathcal L}^2 = C^2_i \\left<\\bar{B}_a\\left(\\Gamma_i B\\right)_a\\bar{B}_b\\left(\\Gamma_i B\\right)_b\\right>\\ , \\nonumber \\\\\n{\\mathcal L}^3 &=& C^3_i \\left<\\bar{B}_a\\left(\\Gamma_i B\\right)_a\\right>\\left<\\bar{B}_b\\left(\\Gamma_i B\\right)_b\\right>\\ .\n\\label{eq:2.1}\n\\end{eqnarray}\nHere $a$ and $b$ denote the Dirac indices of the particles, $B$ is the irreducible octet (matrix) \nrepresentation of ${\\rm SU(3)}_{\\rm f}$, and the $\\Gamma_i$ are the usual elements of the \nClifford algebra \\cite{Polinder06}. As described in Ref.~\\cite{Polinder06}, \nto LO the Lagrangians in Eq.~(\\ref{eq:2.1}) give rise to only six independent \nlow-energy coefficients (LECs), the $C_i^j$s in Eq.~(\\ref{eq:2.1}), due to \n${\\rm SU(3)}_{\\rm f}$ constraints. They need to be determined by a fit to experimental data. \nIt is convenient to re-express the $BB$ potentials in terms of the ${\\rm SU(3)_f}$ \nirreducible representations, see e.g. Refs.~\\cite{Swart,Dover}.\nThen the contact interaction is given by\n\\begin{equation}\nV=\n\\frac{1}{4}(1-\\mbox{\\boldmath $\\sigma$}_1\\cdot \\mbox{\\boldmath $\\sigma$}_2) \\, C_{1S0}\n+ \\frac{1}{4}(3+\\mbox{\\boldmath $\\sigma$}_1 \\cdot\\mbox{\\boldmath $\\sigma$}_2) \\, C_{3S1} \\ ,\n\\label{contact}\n\\end{equation}\nand the constraints imposed by the assumed ${\\rm SU(3)}_{\\rm f}$ symmetry on the interactions\nin the various $BB$ channels for the $^1S_0$ and $^3S_1$ partial waves can be\nreadily read off from Table~\\ref{tab:1}.\n \nThe lowest order ${\\rm SU(3)}_{\\rm f}$ invariant pseudoscalar-meson--baryon\ninteraction Lagrangian embodying the appropriate symmetries was also discussed in \\cite{Polinder06}. \nThe invariance under ${\\rm SU(3)}_{\\rm f}$ \ntransformations implies specific relations between the various coupling constants, namely\n\\begin{equation}\n\\begin{array}{rlrlrl}\nf_{NN\\pi} = & f, & f_{NN\\eta_8} = & \\frac{1}{\\sqrt{3}}(4\\alpha -1)f, & f_{\\Lambda NK} = & -\\frac{1}{\\sqrt{3}}(1+2\\alpha)f, \\\\\nf_{\\Xi\\Xi\\pi} = & -(1-2\\alpha)f, & f_{\\Xi\\Xi\\eta_8} = & -\\frac{1}{\\sqrt{3}}(1+2\\alpha )f, & f_{\\Xi\\Lambda K} = & \\frac{1}{\\sqrt{3}}(4\\alpha-1)f, \\\\\nf_{\\Lambda\\Sigma\\pi} = & \\frac{2}{\\sqrt{3}}(1-\\alpha)f, & f_{\\Sigma\\Sigma\\eta_8} = & \\frac{2}{\\sqrt{3}}(1-\\alpha )f, & f_{\\Sigma NK} = & (1-2\\alpha)f, \\\\\nf_{\\Sigma\\Sigma\\pi} = & 2\\alpha f, & f_{\\Lambda\\Lambda\\eta_8} = & -\\frac{2}{\\sqrt{3}}(1-\\alpha )f, & f_{\\Xi\\Sigma K} = & -f.\n\\end{array}\n\\label{su3}\n\\end{equation}\nHere $f\\equiv g_A\/2F_\\pi$, where $g_A$ is the nucleon axial-vector strength\nand $F_\\pi$ is the weak pion \ndecay constant. We use the values $g_A= 1.26$ and $F_\\pi = 92.4$~MeV.\nFor $\\alpha$, the $F\/(F+D)$-ratio \\cite{Polinder06}, we adopt \nthe SU(6) value: $\\alpha=0.4$, which is consistent with recent determinations\nof the axial-vector coupling constants \\cite{Ratcliffe}.\n\nThe spin-space part of the LO one-pseudoscalar-meson-exchange potential is similar to the \nstatic one-pion-exchange potential in chiral EFT for nucleon-nucleon\ninteractions, see e.g. \\cite{Epe98} (recoil and relativistic corrections give \nhigher order contributions),\n\\begin{eqnarray}\nV^{B_1B_2\\to B_1'B_2'}&=&-f_{B_1B_1'P}f_{B_2B_2'P}\\frac{\\left(\\mbox{\\boldmath $\\sigma$}_1\\cdot{\\bf q}\\right)\\left(\\mbox{\\boldmath $\\sigma$}_2\\cdot{\\bf q}\\right)}{{\\bf q}^2+M^2_P}\\ ,\n\\label{eq:14}\n\\end{eqnarray}\nwhere $M_P$ is the mass of the exchanged pseudoscalar meson. The transferred \nand average momentum, ${\\bf q}$ and ${\\bf k}$, are defined in terms of the final and initial \ncenter-of-mass (c.m.) momenta of the baryons, ${\\bf p}'$ and ${\\bf p}$, as \n${\\bf q}={\\bf p}'-{\\bf p}$ and ${\\bf k}=({\\bf p}'+{\\bf p})\/2$. \nIn the calculation we use the physical masses of the exchanged pseudoscalar mesons. \nThe explicit ${\\rm SU(3)}$ breaking reflected in the mass splitting between the \npseudoscalar mesons and, in particular, the small mass of the pion relative to\nthe other members of the octet leads to sizeable differences in the range of\nthe interactions in the different channels and, thus, induces an essential dynamical \nbreaking of ${\\rm SU(3)}$ symmetry in the $BB$ interactions.\nThe $\\eta$ meson was identified with the octet $\\eta$ ($\\eta_8$) and its physical \nmass was used.\n\nThe reaction amplitudes are obtained from the solution of a coupled-channels \nLippmann-Schwinger (LS) equation for the interaction potentials: \n\\begin{eqnarray}\n&&T_{\\rho''\\rho'}^{\\nu''\\nu',J}(p'',p';\\sqrt{s})=V_{\\rho''\\rho'}^{\\nu''\\nu',J}(p'',p')+\n\\sum_{\\rho,\\nu}\\int_0^\\infty \\frac{dpp^2}{(2\\pi)^3} \\, V_{\\rho''\\rho}^{\\nu''\\nu,J}(p'',p)\n\\frac{2\\mu_{\\nu}}{q_{\\nu}^2-p^2+i\\eta}T_{\\rho\\rho'}^{\\nu\\nu',J}(p,p';\\sqrt{s})\\ .\n\\label{LS} \n\\end{eqnarray}\nThe label $\\nu$ indicates the particle channels and the label $\\rho$ the partial wave. \n$\\mu_\\nu$ is the pertinent reduced mass. The on-shell momentum in the intermediate state, \n$q_{\\nu}$, is defined by $\\sqrt{s}=\\sqrt{m^2_{B_{1,\\nu}}+q_{\\nu}^2}+\\sqrt{m^2_{B_{2,\\nu}}+q_{\\nu}^2}$. \nRelativistic kinematics is used for relating the laboratory energy $T_{{\\rm lab}}$ of the hyperons \nto the c.m. momentum.\n\nIn \\cite{Polinder06,Polinder07} \nthe LS equation was solved in the particle basis, in order to incorporate the correct physical\nthresholds. Since here we are only interested in the $H$ dibaryon we work in the isospin\nbasis. Then for $J=0$, $I=0$, and $S=-2$ we have to consider the three coupled channels\n${\\Lambda}\\La$, $\\Xi N$ and ${\\Sigma}\\Si$. We use the following (isospin averaged) masses \n$m_{\\Lambda}=1115.6$ MeV, $m_{\\Sigma}=1192.5$ MeV, $m_\\Xi=1318.1$ MeV, and $m_N=939.6$ MeV so\nthat the ${\\Lambda}\\La$, $\\Xi N$, and ${\\Sigma}\\Si$ thresholds are at\n2231.2, 2257.7, and 2385.0~MeV, respectively. Furthermore, the potentials in the LS \nequation are cut off with a regulator function, $\\exp\\left[-\\left(p'^4+p^4\\right)\/\\Lambda^4\\right]$, \nin order to remove high-energy components of the baryon and pseudoscalar meson fields \\cite{Epe05}.\nWe consider cut-off values in the range 550, ..., 700 MeV, similar to what was used for \nchiral $NN$ potentials \\cite{Epe05}.\n\nThe imposed ${\\rm SU(3)}$ flavor symmetry implies that only five of the six LECs \ncontribute to the $YN$ interaction, namely $C^{27}$, $C^{10}$, $C^{10^*}$, $C^{8_s}$, \nand $C^{8_a}$, cf. Table~\\ref{tab:1}. \nThese five contact terms were determined in \n\\cite{Polinder06} by a fit to the $YN$ scattering data. Since the $NN$ data\ncannot be described with a LO EFT, ${\\rm SU(3)}$ constraints from the $NN$ interaction \nwere not implemented explicitly. As shown in Ref.~\\cite{Polinder06}, a good\ndescription of the 35 low-energy $YN$ scattering can be \nobtained for cutoff values $\\Lambda=550,...,700$ MeV and for natural values of the LECs. \nThe sixth LEC ($C^{1}$) is only present in the $S=-2$ channels with isospin zero,\ncf. Table~\\ref{tab:1}. There is scarce experimental information on these \nchannels that could be used to fix this LEC, but it turned out that the quality \nof the existing data do not really allow to constrain its value reliably \n\\cite{Polinder07}. Even with the value of the sixth LEC chosen so that \n$C^{{\\Lambda}\\La\\to {\\Lambda}\\La}_{1S0} = 0$, agreement with those data can be achieved. In this case \na scattering length of $a_{^1S_0}^{\\Lambda\\Lambda} = -1.52$~fm \\cite{Polinder07} \nis obtained. \nAnalyses of the measured binding energy of the double-strange hypernucleus\n${}^{\\;\\;\\;6}_{\\Lambda\\Lambda}{\\rm He}$ \\cite{Takahashi:2001nm} suggest that\nthe ${\\Lambda}\\La$ scattering length could be in the range of \n$-1.3$ to $-0.7$ fm \\cite{Gal,Rijken,Fujiwara}.\n \n\\section{Quark mass dependence of the binding energy and SU(3) breaking effects}\n\\label{sec:3}\n\nChiral effective field theory itself does not allow one to make any\npredictions with regard to the existence of the $H$ dibaryon because, as stated,\none of the contact terms (${C^1}$) occurs just in the channel in question, cf. \nTable~\\ref{tab:1}, and has to be determined from there. \nIf we take over the values for $C^{27}$ and $C^{8_s}$ as fixed from the $Y N$ data \nand assume that ${C^1} = 0$ then we find no bound state for $\\Lambda\\Lambda$ - \nneither in LO \\cite{Polinder06} nor based on the preliminary NLO results \\cite{Hai10}. \nHowever, if we assume that ${C^1} \\neq 0$ and vary its value then \nin both cases a near-threshold bound state can be produced for values of natural size.\n\nAt the same time, the framework of chiral effective field theory in which our \n$\\Lambda\\Lambda$ interaction is derived is very well suited to shed light \non the general characteristics of a $H$-dibaryon, should it indeed exist.\nIn particular, it allows us study the implications of the imposed \n(approximate) SU(3) flavor symmetry and to explore the dependence of \nthe properties of an assumed $H$-dibaryon on\nthe masses of the relevant mesons and baryons. The latter aspect is a \nrather crucial issue in view of the fact that the available lattice QCD \ncalculations were not performed at the physical masses of the involved \nparticles. \n\nTo start our discussion, let us assume that the $H$-dibaryon is a \n(loosely) bound $BB$ state \nand that its binding energy $E_H$ is similar to that of\nthe deuteron $D$. For pedagogical purposes we fix the value of the flavor-singlet LEC \n${C^1}$ in such a way that\n$\\gamma_{H} = \\gamma_{D} = 0.23161\\,$fm\n($E = - \\gamma^2 \/ m_{B}$, where $m_{B}$ is either $m_N$ or $m_\\Lambda$),\nbecause of the well-known relation between the binding energy and the effective\nrange parameters \\cite{Schwinger47,Bethe} \n\\begin{equation}\n\\frac{1}{a} \\simeq {\\gamma} - \\frac{1}{2}{r} {\\gamma}^2 .\n\\nonumber\n\\end{equation}\nThis relation is very well fulfilled for the deuteron and the corresponding\nneutron-proton $^3S_1$ scattering length ($a=5.43\\,$fm) and effective range\n($r=1.76\\,$fm). One would naively expect that the same should happen for the\n$H$-dibaryon. However, it turns out that the corresponding results for\n${\\Lambda}\\La$ in the $^1S_0$ partial wave are quite different, namely\n$a=3.00\\,$fm and $r=-4.98\\,$fm. Specifically, the effective range\nis much larger and, moreover, negative. Clearly, the\nproperties of the $H$-dibaryon are not comparable to those of the\ndeuteron, despite the fact that both bound states are close to the\nelastic threshold.\nIndeed, if one recalls the expressions for the relevant potentials\nas given in Table \\ref{tab:1},\n\\begin{equation}\nV^{{\\Lambda}\\La \\rightarrow {\\Lambda}\\La} = \\frac{1}{40}\\left(27C^{27}+8C^{8_s}+{ 5}{\n C^{1}}\\right) \\, , \\ \\ \\ \nV^{\\Xi N \\rightarrow \\Xi N} = \\frac{1}{40}\\left(12C^{27}+8C^{8_s}+{ 20}{\n C^{1}}\\right) \\, ,\n\\nonumber \n\\end{equation}\none can see that the attraction supplied by the SU(3) flavor-singlet state ($C^{1}$)\ncontributes with a much larger weight to the $\\Xi N$ channel than to ${\\Lambda}\\La$.\nThis indicates that the presumed $H$-dibaryon could be predominantly a\n$\\Xi N$ bound state. We have confirmed this conjecture by evaluating explicitly\nthe phase shifts in the ${\\Lambda}\\La$ and $\\Xi N$ channels, cf. the discussion in the\nnext section. Indeed, one finds that the phase shift for the $\\Xi N$ channel\nis rather similar to the $NN$ $^3S_1$ case. Specifically, the $\\Xi N$ ($^1S_0$) phase shift \n$\\delta(q_{\\Xi N})$ fulfills $\\delta(0)-\\delta(\\infty) = 180^\\circ$ in \nagreement with the Levinson theorem. \nThe ${\\Lambda}\\La$ ($^1S_0$) phase behaves rather differently and \nsatisfies $\\delta(q_{{\\Lambda}\\La}=0) - \\delta(\\infty) = 0$.\nNote that there have been earlier discussions on this issue in\nthe context of $S=-2$ baryon-baryon interactions derived within the\nquark model \\cite{Oka,Nakamoto}. \n\nLet us now consider variations of the masses of the involved particles. \nThe dependence of the $H$ binding energy on the pion mass $M_\\pi$ is \ndisplayed in Fig.~\\ref{fig:mpi} (left). For the case considered above, \nenlarging the pion mass to around 400 MeV \n(i.e. a value corresponding to the NPLQCD calculation \\cite{Beane}) increases the\nbinding energy to around 8 MeV and a further change of $M_\\pi$ to 700 MeV\n(corresponding roughly to the HAL QCD calculation \\cite{Inoue}) yields then 13 MeV, \ncf. the solid line. \nReadjusting $C^1$ so that we predict a $H$ binding energy of 13.2 MeV for\n$M_\\pi=389$ MeV, corresponding to the latest result published by \nNPLQCD~\\cite{Beane11a}, yields the dashed curve. \nFig.~\\ref{fig:mpi} includes also results of a calculation where $C^1$ was \nfixed in order to reproduce their earlier value of around 16 MeV \\cite{Beane} \n(dash-dotted curve), to facilitate a more direct comparison with\nchiral extrapolations~\\cite{Beane11,Shanahan11} based on that value.\nIt is obvious that the dependence on $M_\\pi$ we get agrees well -- at least on a \nqualitative level -- with that presented in Ref.~\\cite{Beane11}. Specifically, our \ncalculation exhibits the same trend (a decrease of the binding energy with \ndecreasing pion mass) and our binding energy of 9 MeV at the physical \npion mass is within the error bars of the results given in~\\cite{Beane11}. \nOn the other hand, we clearly observe a non-linear dependence of the binding\nenergy on the pion mass. As a consequence, scaling our results to the binding\nenergy reported by the HAL QCD collaboration \\cite{Inoue} (30-40 MeV for \n$M_\\pi \\approx 700-1000$~MeV) yields binding energies of more than 20 MeV\nat the physical point, which is certainly outside of the range suggested in\nRef.~\\cite{Beane11}. However, we note that for such large pion masses the LO\nchiral EFT can not be trusted quantitatively. \nWe remark that in our simulations the curves corresponding to different binding \nenergies remained roughly parallel even up to such large values as suggested \nby the HAL QCD collaboration. \n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[height=9.cm,keepaspectratio,angle=-90]{mpi.ps}\\includegraphics[height=9.cm,keepaspectratio,angle=-90]{msi.ps}\n\\caption{Dependence of the binding energy of the $H$-dibaryon on the pion mass \n$M_\\pi$ (left) and on the $\\Sigma$ mass $m_\\Sigma$ (right).\nThe solid curve correspond to the case where $C^1$ is fixed such that $E_H = -1.87\\,$MeV \nfor physical masses while for the dashed curve $C^1$ is fixed to yield $E_H = -13.2\\,$MeV for \n$M_\\pi = 389$ MeV. The pion mass dependence is also shown for $E_H = -16.6\\,$MeV \n(dash-dotted curve).\nThe asterisks and crosses represent results where, besides the \nvariation of $m_\\Sigma$, $m_\\Xi+m_N = 2 m_{\\Lambda}$ is assumed so that the $\\Xi N$ threshold \ncoincides with that of the ${\\Lambda}\\La$ channel. \nThe vertical (dotted) lines indicate the physical ${\\Lambda}\\La$ and ${\\Sigma}\\Si$ thresholds. \n}\n\\label{fig:mpi}\n\\end{figure}\n\nOur finding that any $H$-dibaryon is very likely a bound $\\Xi N$ state rather than a ${\\Lambda}\\La$\nstate, which follows from the assumed (approximate) SU(3) symmetry of the interaction, \nsuggests that not only the pion mass\nbut also the masses of the baryons play a significant role for the concrete\nvalue of binding energy. \nIndeed, the physical difference between the ${\\Lambda}\\La$ and $\\Xi N$ thresholds of around\n26~MeV implies that the $H$-dibaryon considered above is, in reality, bound by roughly\n28~MeV with respect to its ``proper'' threshold. Accordingly, one intuitively expects that \nin a fully SU(3) symmetric case, where the masses of all octet baryons coincide, the \nbound state would remain basically fixed to the $\\Xi N$ threshold and then would lie also \nabout 28 MeV below the ${\\Lambda}\\La$ threshold. In the concrete case of $J=0$, $I=0$, $S=-2$ we are \ndealing with three coupled channels, namely ${\\Lambda}\\La$, $\\Xi N$, and ${\\Sigma}\\Si$. \nSince we know from our experience with coupled-channel problems \n\\cite{Polinder06,Hai05,Hai10a,Hai11} that coupling effects are sizeable and \nthe actual separation of the various thresholds plays a crucial role we investigated also\nthe dependence of the $H$ binding energy on the thresholds (i.e. on the $\\Sigma$, and on \nthe $\\Xi$ and $N$ masses). Corresponding results are displayed in Fig.~\\ref{fig:mpi} on\nthe right side. The solid curve is again the result based on our reference case \nwith a binding energy of -1.87~MeV for physical masses of the pion and the baryons. \nWhen we now decrease the $\\Sigma$ mass so that its nominal threshold of 2385~MeV\nmoves downwards and finally coincides with the one of the ${\\Lambda}\\La$ channel (2231.2~MeV),\nwe observe a rather drastic change in the $H$ binding energy. Note that the direct\ninteraction in the $\\Sigma\\Sigma$ channel is actually repulsive for the low-energy\ncoefficients $C^{27}$ and $C^{8_s}$ fixed from the $YN$ data plus the pseudoscalar\nmeson exchange contributions with coupling constants determined from the SU(3)\nrelations Eq.~(\\ref{su3}). And it remains repulsive even for $C^{1}$ values that produce\na bound $H$-dibaryon. But the coupling between the channels generates a sizeable\neffective attraction which increases when the channel thresholds come closer. \nThe dashed curve is a calculation with the contact term $C^{1}$ fixed to simulate \nthe binding energy (-13.2~MeV) of the NPLQCD collaboration at $M_\\pi=389\\,$MeV.\nAs one can see, the dependence on the binding energy on the $\\Sigma$ mass is rather\nsimilar. The curve is simply shifted downwards by around 4.5~MeV, i.e. by the difference\nin the binding energy observed already at the physical masses. \nThe asterisks and crosses represent results where, besides the variation of the \n$\\Sigma\\Sigma$ threshold, the $\\Xi N$ threshold is shifted to coincide with that\nof the ${\\Lambda}\\La$ channel. This produces an additional increase of the $H$ binding \nenergy by 20~MeV at the physical $\\Sigma\\Sigma$ threshold and by 9~MeV \nfor that case where all three $BB$ threshold coincide. \nAltogether there is an increase in the binding energy of roughly 60~MeV \nwhen going from the physical point to the case of baryons with identical\nmasses. This is significantly larger than the variations due to the pion mass\nconsidered before. Note that we have kept the pion mass at its physical value\nwhile varying the $BB$ thresholds. \n\n\\section{Application to lattice QCD results}\n\\label{sec:4}\n \nAfter these exemplary studies let us now try to connect with the published \n$H$ binding energies from the lattice QCD calculations \\cite{Beane,Inoue}. \nThe results obtained by the HAL QCD collaboration are obviously for the SU(3)\nsymmetric case and the corresponding masses are given in Table I of \nRef.~\\cite{Inoue}. Thus, we can take those masses and then fix the LEC\n$C^1$ so that we reproduce their $H$ binding energy with those masses. To be\nconcrete, we use $m_{ps} = 673\\,$MeV and $m_{B} = 1485\\,$MeV, and fix \n$C^1$ so that $E_H = -35\\,$MeV. When we now let the masses of the baryons and\nmesons go to their physical values the bound state moves upwards, crosses\nthe ${\\Lambda}\\La$ threshold, crosses also the $\\Xi N$ threshold and then \ndisappears. In fact, qualitatively this outcome\ncan be already read off from the curves in Fig.~\\ref{fig:mpi} by combining the\neffects from the variations in the pion and the baryon masses. Based on those\nresults one would expect a shift of the $H$ binding energy in the order of\n60 to 70~MeV for the mass parameters of the HAL QCD calculation. \n\nIn case of the NPLQCD calculation we take the values provided in \nRef.~\\cite{Beane11b}, i.e. $m_N=1151.3\\,$MeV, $m_{\\Lambda}=1241.9\\,$MeV, $m_{\\Sigma}=1280.3\\,$MeV,\nand $m_\\Xi=1349.6\\,$MeV. Those yield\nthen 17~MeV for the $\\Xi N$-${\\Lambda}\\La$ threshold separation (to be compared \nwith the physical value of roughly 26~MeV) and \n77~MeV for the ${\\Sigma}\\Si$-${\\Lambda}\\La$ separation (physical value around 154~MeV). \nWith those baryon masses we fix again the LEC $C^1$ so that we \nreproduce the $H$ binding energy given by the NPLQCD collaboration, namely \n$E_H =-13.2$~MeV \\cite{Beane11a}. We use also $M_\\pi = 389\\,$MeV,\nbut we take the physical masses for the other pseudoscalar mesons ($K$ and $\\eta$).\nAgain we let the masses of the baryons and of the pion go to their \nphysical values. Also here the bound state moves upwards and crosses\nthe ${\\Lambda}\\La$ threshold. However, in the NPLQCD case the state survives\nand remains below the $\\Xi N$ threshold at the physical point. Specifically,\nwe observe a resonance at a kinetic energy of 21~MeV in the ${\\Lambda}\\La$\nsystem or, more precisely, a quasi-bound state in the $\\Xi N$ system around \n5~MeV below its threshold. \n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[height=9.cm,keepaspectratio,angle=-90]{phla.ps}\\includegraphics[height=9.cm,keepaspectratio,angle=-90]{phxi.ps}\n\\caption{Phase shifts for ${\\Lambda}\\La$ ($^1S_0$) (left) and $\\Xi N$ ($^1S_0$) (right)\nas a function of the pertinent laboratory energies. \nThe solid line is the result for our reference $BB$ interaction that produces a bound $H$ at\n$E_H=-1.87$ MeV. The other curves are results for interactions that are fine-tuned to the \n$H$ binding energies found in the lattice QCD calculations of the HAL QCD (dashed) and \nNPLQCD (dash-dotted) collaborations, respectively, for the pertinent meson (pion) and\nbaryon masses as described in the text. \n}\n\\label{fig:phases}\n\\end{figure}\n\nThe results reported so far were all obtained with the LECs as fixed from\nthe $YN$ data for the cutoff value $\\Lambda = 550$~MeV \\cite{Polinder06}.\nIn order to investigate the stability of our results we \nconsidered also variations of the cutoff mass $\\Lambda$ in the LS equation \nEq.~(\\ref{LS}) between 550 and 700~MeV, as in \\cite{Polinder07},\nand repeated the exercise described above. Those variations led to \nchanges of the predicted resonance energy in the order of 1~MeV only. \nEven an exploratory calculation that utilizes the low-energy coefficients \ndetermined in a preliminary NLO study of the $YN$ system \\cite{Hai10} \nyielded practically the same result. \n\nIt is interesting to observe that the chiral extrapolation of the lattice\nQCD results performed by Shanahan et al. \\cite{Shanahan11} yields results\nthat are similar to ours. In that reference the authors conclude \nthat the $H$-dibaryon is likely to be unbound by 13$\\pm$14 MeV at the physical point. \nLet us emphasize, however, that our values are not really comparable\nwith theirs. In our analysis we assume that the $H$-dibaryon is actually \na bound $BB$ state -- which seems to be the case also in the lattice QCD\nstudies \\cite{Beane,Inoue}. On the other hand, in Ref.~\\cite{Shanahan11} it is assumed that\nthe $H$ is a compact, multi-quark state rather than a loosely bound\nmolecular state. How such a genuine multi-quark state would be influenced\nby variations of the $BB$ thresholds is completely unclear. It depends,\namong other things, on whether and how strongly this state couples to the\n${\\Lambda}\\La$, $\\Xi N$, and ${\\Sigma}\\Si$ channels. So far there is no information\non this issue from lattice QCD calculations. Clearly, in case of a strong and \npredominant coupling to the ${\\Lambda}\\La$ alone, variations of the ${\\Sigma}\\Si$ and\n$\\Xi N$ would not influence the $H$ binding energy significantly. However,\nshould it couple primarily to the $\\Xi N$ and\/or ${\\Sigma}\\Si$ channels\nthen we expect a sensitivity to their thresholds comparable to what we found in\nour study for the case of a bound state. \n\nFinally, for illustrative purposes, let us show phase shifts for the\n$^1S_0$ partial wave of the ${\\Lambda}\\La$ and $\\Xi N$ channels. This is done in\nFig.~\\ref{fig:phases}. The solid line is the result for our reference $BB$ \ninteraction that produces a loosely bound $H$ dibaryon with $E_H=-1.87$ MeV.\nThe phase shift for the $\\Xi N$ channel (left side) is rather similar to the \none for the $^3S_1$ $NN$ partial wave where the deuteron resides, \nsee e.g. \\cite{Epe05}. \nSpecifically, it starts at $180^o$, decreases smoothly and eventually \napproaches zero for large energies, fulfilling the Levinson theorem. \nThe result for ${\\Lambda}\\La$ ($^1S_0$) (left side) behaves rather differently. \nThis phase commences at zero degrees, is first negative but becomes positive within 20 MeV \nand finally turns to zero again for large energies.\nThe dashed curve corresponds to the interaction that was fitted to the result\nof the HAL QCD collaboration and reproduces their bound $H$ dibaryon with\ntheir meson and baryon masses. \nThe phase shift of the $\\Xi N$ channel, calculated with physical masses, \nshows no trace of a bound state anymore. Still the phase shift \nrises up to around $60^o$ near threshold, a behavior quite similar to that of \nthe $^1S_0$ $NN$ partial wave where there is a virtual state (also called\nantibound state \\cite{Pearce}). Indeed, such a virtual state seems to \nbe present too in the $\\Xi N$ channel as a remnant of the original bound state.\nThe effect of this virtual state can be seen in the ${\\Lambda}\\La$ phase shift where it\nleads to an impressive cusp at the opening of the $\\Xi N$ channel, \ncf. the dashed line on the left side. \nIn the $\\Xi N$ phase shifts for the NPLQCD case (dash-dotted curve) the\npresence of a bound state is clearly visible. The corresponding ${\\Lambda}\\La$\nphase shift exhibits a resonance-like behavior at the energy where the\n(quasi-bound) $H$ dibaryon is located. \n\nGiven the present uncertainties of the lattice QCD results for the \n$H$ dibaryon binding energy we have focused here primarily on \nqualitative features of the $H$ that can be inferred from chiral \neffective field theory. Clearly, once more precise lattice data \nbecome available one should also perform a careful assessment of the\nuncertainties involved in the EFT calculation.\n\n\\acknowledgments{\n We would like to thank S.~R.~Beane, A.~Gal, and M.~J.~Savage for their\nconstructive comments. \n This work is supported by the Helmholtz Association by funds provided to\nthe virtual institute ``Spin and Strong QCD'' (VH-VI-231), by the EU-Research\nInfrastructure Integrating Activity ``Study of Strongly Interacting Matter''\n(HadronPhysics2, grant n. 227431) under the Seventh Framework Program of the EU,\nand by the DFG (SFB\/TR 16 ``Subnuclear Structure of Matter'').\n}\n\n\\bigskip \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section*{Reproducibility}\n\nThe code for our work is available at \\url{https:\/\/github.com\/privacytrustlab\/soteria_private_nn_inference}.\n\\section{Garbled Circuit for Efficient Neural-Networks}\\label{sec:gcbinary}\n\nWe investigate the problem of performing private inference on neural networks. Let $W$ be the model parameters stored on the server, $x$ be the client's input, $y$ be the expected output and $f$ is the inference function to be computed. Given this, we want to compute ${f(x;\\theta)\\rightarrow y}$. We aim for the following main goals:\n\\begin{itemize}\n\t\\item {\\em Confidentiality:}\n\tThe solution should preserve confidentiality of the model parameters $\\theta$ from the users and that of $x$ and $y$ from the server. We assume an honest-but-curious threat model. \n\n\t\\item {\\em Accuracy:}\n\tThe drop in accuracy of the privately computed inference function should be negligible as compared to the accuracy of the model on plaintext data.\n\n\t\\item {\\em Performance:}\n\tThe private computation should demonstrate acceptable performance (runtime and communication) overhead.\n\\end{itemize}\n\n\\paragraph{\\em Garbled circuits.} \nGC protocol allows construction of any function as a boolean circuit with a one time setup cost of data exchange~\\cite{YaoGC}. In our setting, the client is the {\\em garbler} and the server is the {\\em evaluator}. In the setup phase, the client first transforms the function into a boolean circuit with two-input gates. The function (model architecture) and the circuit are known to both the parties, but its parameters and input are private. The client then garbles the circuit. This process involves creating a garbled computation table (GCT), which is an encrypted version of the truth table for the boolean circuit. The entries for this table are randomly permuted, so that the order does not leak information. The client then shares the garbled circuit and its encrypted inputs to the circuit (binary values representing $x$) with the server. In the next phase, the parties perform an oblivious transfer (OT) protocol~\\cite{ObliviousTransfer_paper}, so the server obtains the encryption of its inputs to the circuit (binary values representing $W$), without leaking information about its parameters to the client. Then, the server evaluates the circuit, and obtains the output value which is encrypted. The server transfers the output to the client which can match the encrypted values to their plaintext and obtain $f(x;\\theta)$. \n\n\\paragraph{\\em Performance.} \nIn GC, the communication and computation overhead is directly dependent on the number of AND, OR gates in the boolean circuit. Prior research has proposed several techniques that make it free for the GC to execute XOR, XNOR and NOT gates~\\cite{kolesnikov2008improved}. Given this prior work, the communication overhead of the GC protocol for a given circuit is proportional to its security parameter and the number of non-XOR gates in the circuit. The total runtime for evaluating a circuit is the sum of the time required during the {\\bf online} (evaluation) and {\\bf offline} (garbling and oblivious transfer) computation.\n\n\\paragraph{\\em Efficient neural networks.} \nIn {\\scshape Soteria}{}, we leverage the above-mentioned properties of GC to design an optimized neural network algorithm. Neural network algorithms are shown to be flexible with respect to their architectures i.e., multiple models with different configuration can achieve a similar level of accuracy. We take advantage of this observation and propose designing neural network {\\bf architectures} that help optimize the performance of executing inference with garbled circuits. The number of gates in a circuit corresponding to a neural network depends on its activation functions and the size of its parameter vector. \n\nNeural networks have shown to exhibit relatively high accuracy for various tasks even with low precision parameters. {\\bf Binary neural networks}~\\cite{BNN} are designed with the lowest possible size for each parameter, i.e., one bit to represent $\\{-1, +1\\}$ values. Using BNNs naturally aligns with our selected cryptographic primitive because each wire in garbled circuits represents 1 bit value (representing $-1$ in the model with $0$ in the circuit). Binarizing the model parameters further allows us to heavily use the free XOR, XNOR and NOT gates in garbled circuits, thus minimizing the computation and communication overhead of private inference. This has recently been shown in the performance evaluation of garbled circuits on binary neural networks~\\cite{XONN}. \n\nIn neural networks, {\\bf linear functions} such as those used in the convolutional or fully connected layers form an important part of the network. These functions involve dot product vector multiplications. Instead of using multiplications, this can be computed very efficiently using XNOR-popcount: \n${\\mathbf{x} \\cdot \\mathbf{w} = 2\\times\\operatorname{bitcount}(\\operatorname{xnor}(\\mathbf{x}, \\mathbf{w})) - N}$, where $N = |\\mathbf{x}|$. In binary neural networks, the output of activation functions is also binary. But, the output of XNOR-popcount is not a binary number, thus, according to BNN algorithms one would need to compare it with $0$; positive numbers will be converted to $1$ and negative numbers will be converted to $0$. \n\nWe can compute some {\\bf non-linear functions} such as maxpool very efficiently in BNNs. Max-pooling is a simple operation which returns the maximum value from a vector, which in the case of neural networks is usually a one-dimensional representation of a 2D max-pooling window. In binary neural networks, maxpool need to simply return $1$ if there is a $1$ in the vector. This is achieved by a logical OR-operation over the elements of the vector. \n\nTo achieve a learning capacity for binary neural networks similar to full-precision models, we would need to scale up the the number of model parameters. We can increase the number of kernels in a convolution layer and the number of nodes in a fully connected layer, by a given {\\bf scaling factor}. This technique has been used in the prior work~\\cite{XONN}, and enables learning more accurate models, however at the cost of increasing the number of computations in the network.\n\n\\section{Conclusions}\n\nWe introduce {\\scshape Soteria}{}, a system that takes advantage of the power of neural architecture search algorithms in order to design model architectures which jointly optimize accuracy and efficiency for private inference. We use garbled circuits (GC) as our cryptographic primitive, due to its flexibility. However, other secure multi-party computation schemes can also be used to enrich the set of secure operations that could be chosen by the architecture search algorithm. .Instead of model post-processing, we also enable the stochastic gradient descent algorithm to train a sparse model (setting some parameters to 0), hence further improving the efficiency of the trained model. To this end, we train ternary neural networks which have shown to have a significant potential in learning reasonably accurate models. We construct optimal architectures that balance accuracy and inference efficiency on GC on MNIST and CIFAR10 datasets. As opposed to the prior work that build cryptographic schemes around given fixed models, {\\scshape Soteria}{} provides a flexible solution that can be adapted to the accuracy and performance requirements of any given system, and enables trading off between requirements.\n\n\\section{Empirical Evaluation} \\label{sec:evaluation}\n\nWe evaluate the efficiency of our method in two ways. We show how using ternary neural networks on fixed model architectures, as used in the prior work, can reduce the overhead of secure inference on neural networks. This is due to the sparsity of such models. We also present the performance of {\\scshape Soteria}{} architectures, in which model complexity is optimized along with the model accuracy. \n\n\\subsection{Experimental Setup}\n\nWe evaluate our work on MNIST and CIFAR10 image classification datasets, as they have been extensively used in the literature to evaluate the performance of cryptographically secure neural network schemes. We run our experiments on an AWS c5.2xlarge instance, running Ubuntu 18.04 LTS on an Intel Xeon 8124M at 3.0 GHz.\n\nWe use PyTorch 1.3~\\cite{Pytorch_web}, a python-based deep learning framework to implement our architecture search algorithm and train the ternary models. We use Synopsys Design Compiler~\\cite{SynopsysDC_web}, version L-2016.03-SP5-2, to synthesize SystemVerilog code into the gate-level netlist. Our synthesis runs the TinyGarble gate library infrastructure\\footnote{We use TinyGarble with 21ecca7cb75b33fd7508771fd35f03657dd44e5e gitid on \\url{https:\/\/github.com\/esonghori\/TinyGarble} master branch.}.\n\nWe execute the garbled circuit protocol on the boolean circuit generated as described in previous sections. We compute the number of non-XOR gates in the generated boolean circuit netlist as a measure of its complexity. We measure the exact performance of {\\scshape Soteria}{} as its runtime during the offline and online phases of the protocol, and its communication cost. \n\nWe present the experimental setup of prior work and {\\scshape Soteria}{}, including their CPU specification and link to available software codes, in Table~\\ref{table:summaryexistingwork}. We also present the details of all neural network architectures which we evaluate in this paper in Table~\\ref{tab:architectures} in Appendix~\\ref{Appendix:architectures}.\n\n\\subsection{Ternary Neural Networks}\n\nAs discussed in section~\\ref{section:tnn}, we use ternary neural networks (TNNs) instead of binary networks as it provides significant performance gains with GC without any post-processing (i.e., the model is {\\em trained} to be sparse). We perform two small experiments to illustrate the benefit of the sparsity (fraction of parameters with weight 0) of ternary models. Further, we analyze the effect of the scale of the network in the tradeoff between model accuracy and performance of private inference.\n\n\\paragraph{\\em Sparsity.} \nFigure~\\ref{fig:TNN-nonXOR} shows the number of non-XOR gates for a toy example: a 4-kernel $3\\times3$ convolution operation taking input of size $32\\times32\\times3$ with padding sized $1$ ternary neural network. We randomly set a fraction of parameters to zero to manually control the sparsity of the model. A BNN model is equivalent to the case where the sparsity is $0$. We observe that as the sparsity increases the number of non-XOR gates decrease with almost the same factor. This can result in reducing both the communication overhead and the inference runtime, as we will see in training large models.\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\input{plots\/0weight-nonxor.tex}\n\t\\caption{Sparsity of a model versus its circuit complexity. We measure sparsity as the fraction of model parameters with $0$ weight. We quantify circuit complexity as the number of non-XOR gates. The numbers are computed on a ternary neural network with a $3\\times3$ convolution operation with $4$ kernels and a $32\\times32\\times3$ input. For this experiment, we assign 0 weights to a random set of parameters, to get different levels of sparsity and corresponding number of non-XOR gates.}\n\t\\label{fig:TNN-nonXOR}\n\\end{figure}\n\n\\input{tables\/scaling-factor.tex}\n\n\\begin{figure}[t!]\n\t\\input{plots\/scaling-factor.tex}\n\t\\caption{Test accuracy versus inference runtime for a ternary neural network trained on a fixed MNIST (m1) architecture, in various scale. See Appendix~\\ref{Appendix:architectures} for description of the model.}\n\t\\label{fig:scaling-factor}\n\\end{figure}\n\n\\input{tables\/layer-comp.tex}\n\nNote that while training a TNN, we cannot control the sparsity of the network. In Table~\\ref{tab:component-costs}, we show the result of training binary and ternary models on MNIST dataset, model architecture m3. The table reports the GC costs of the components of the network for both BNNs and TNNs. We can observe that the ternary model has a significant sparsity (about $0.3$). This results in constructing smaller circuits for the model, which reduces the inference cost of using GC protocol over ternary neural networks. This is reflected in the smaller number of non-XOR gates in the ternary circuits constructed on convolution and fully connected operations. The costs for maxpool operation will remain the same, as it does not have any learnable parameter.\n\n\n\\paragraph{\\em Scale.}\nAs discussed in Section~\\ref{sec:gcbinary}, we need to scale the network operations to achieve a higher capacity for binary and ternary models and obtain better accuracies. Figure~\\ref{fig:scaling-factor} demonstrates the impact of scale of the network on accuracy of the model and its GC runtime for private inference. Although inference time increases linearly with the scaling factor, accuracy improves upto a certain extent with scaling (scaling factor of 1) and then becomes almost constant. This denotes that we can select a sweet spot for scaling factor thereby optimizing for both accuracy and performance.\n\n\nTable~\\ref{tab:scaling-factor} shows how scaling factor affects the number of parameters in the network. As the scaling factor in TNNs increases, the accuracy increases. However, with diminishing returns after a certain limit. The inference cost of the circuit also increases, as is evident from the growth of runtime with change in scaling factor. Note that the sparsity is about $0.24$ for scaling factor $3$ for the ternary neural network, which means that the effective size of the model (hence its performance cost) remains comparable to a binary neural network (without any scaling), albiet with better accuracy. As a reference, the test accuracy of a BNN model with the same architecture is $0.9514$. \n\n\n\\paragraph{\\em Comparison with prior work.}\nTable~\\ref{tab:comparison-results} reports the results of our experiments when compared to prior work. In this subsection, we present the outcome of basic {\\scshape Soteria}{} on {\\em fixed} model architectures which are used in the literature, thus only discussing the effect of sparsity of ternary neural networks on the tradeoff between accuracy and performance costs.\nWe use three different architectures for each of the two datasets, which have been used in existing work. m1-3 are used with the MNIST dataset, while m4-6 are used with the CIFAR10 dataset. See Appendix~\\ref{Appendix:architectures} for the descriptions of model architectures. We use the same scaling factors for our networks as used by XONN~\\cite{XONN}, which is the only other comparable work with quantized (binary) weights and inputs, for a fair comparison. \n\n\nWe observe that for MNIST datasets, the basic TNN {\\scshape Soteria}{} models (m1-m3) provide better runtime and communication performance on average than prior work with maximum drop in accuracy of only $0.016$ (for model m3). This shows that {\\scshape Soteria}{} is useful in designing custom models that provide optimal performance guarantees while retaining high prediction accuracy. For CIFAR10 datasets, we observe that for models used in prior work (m4 to m6), our basic TNN models exhibit a slightly higher drop in accuracy of $0.8$, but provide a computation and communication gain, on average, as compared to prior work. Overall, our results show that {\\scshape Soteria}{} provides a flexible approach of training private models given the constraint on performance and accuracy of the model.\n\n\n\\input{tables\/state-of-the-art-comp.tex}\n\n\\subsection{Architecture Search}\n\nWe present the details of our empirical analysis of {\\scshape Soteria}{}. In particular, we evaluate the cost function we used in the architecture search algorithm, the effect of the performance regularization during the search, the effect of the model size on the tradeoff between accuracy and inference runtime, and compare {\\scshape Soteria}{} with the prior work.\n\n\\paragraph{\\em Implementation.} \nTo handle conversion of the model into a digital circuit supported by TinyGarble, we first build a representation of the model and parameters in SystemVerilog, and then synthesize and optimize our circuit (using Synopsys Design Compiler) to use circuit elements supported by TinyGarble. In this first step, we designed a collection of parameterized components (notably dot product, and maxpool) to use as building blocks our architecture search algorithm. Each component is flexibly designed to efficiently accept arbitrary size input and output, and is composed to form the complete model. In a general setting, hardware-level code is typically straight-forward to generate. However, to enable TNNs with {\\scshape Soteria}{}, we have to dynamically define the sparsity of the modules depending on the result of model training and architecture search. Along with the parameter data, we define the sparsity information which is used during \\texttt{generate} phases in SystemVerilog to build the sparse network in hardware (taking advantage of the 0-valued parameters of the model). Altogether, this allows us to build and evaluate the models constructed by {\\scshape Soteria}{}.\n\n\\input{tables\/cost-archsearch.tex}\n\n\\paragraph{\\em Cost function for regularized architecture search.}\nAs presented in Section~\\ref{sec:method}, our algorithm searches for the models that are not only accurate but also are efficient with respect to the costs of using garbled circuits on the model architecture. For this, we modify the score value that the DARTS architecture search algorithm gives to each operation (e.g., maxpool, or convolution with different dimensions) with a regularized penalty factor proportional to the performance cost of the operation. Table~\\ref{table:operationcost} presents the communication and runtime cost of each operation that we use in our algorithm. The penalty factor is computed as the average of the relative communication cost and relative runtime cost of each operation, with respect to the most costly operation ({CONV$5\\times5$}). We use this penalty factor in the experiments.\n\n\\input{tables\/lambda-parameters.tex}\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\input{plots\/lambda-accuracy.tex}\n\t\\\\(a) CIFAR10\n\t\\\\[5pt]\n\t\\input{plots\/lambda-accuracy-MNIST.tex}\n\t\\\\(b) MNIST\n\t\\caption{Inference runtime versus test accuracy of a garbled ternary model, constructed with {\\scshape Soteria} architecture search algorithm, for various values of circuit cost regularization $\\lambda$. We obtain the architecture for a neural network with (a) $3$ cells for CIFAR10 dataset, and (b) $1$ cell for MNIST dataset. All experiments use a scaling factor of 3.}\n\t\\label{fig:lambda-vs-accuracy}\n\\end{figure}\n\n\\paragraph{\\em Balancing accuracy and inference costs.}\nFor the architecture search in {\\scshape Soteria}{}, we balance accuracy and inference cost over GC protocol, using a regularization factor $\\lambda$. With $\\lambda=1$ the importance of the penalty factor is maximum, and $\\lambda=0$ represents the case where we ignore the performance cost. We execute the search process with different values of lambda. We constrain the search to finding a 3-cell architecture for CIFAR10 dataset and 1-cell architecture for MNIST dataset, with each cell having 4 operations in sequence. We run the architecture search algorithm for 100 epochs, and subsequently train the obtained architectures for 200 epochs.\n\nFigure~\\ref{fig:lambda-vs-accuracy} presents trade-off between test accuracy of the optimal architectures and their inference runtime. Table~\\ref{tab:lambda-parameters} provides the statistics on the number of model parameters and the model sparsity for different values of $\\lambda$ for the MNIST dataset. As $\\lambda$ increases, cheaper operations, that have fewer trainable parameters are chosen by the search process, which improves the inference runtime at the expense of the model accuracy. As we observe, $\\lambda=0.6$ can provide a reasonable balance between both accuracy and the inference cost. This is where the search algorithm identifies cheaper operations that collectively can result in equivalent accuracy that can be achieved using more expensive operations. It is very important to note that selecting $\\lambda$ depends on how much cost or accuracy drop we can tolerate for a given setting. Thus, {\\scshape Soteria}{} enables adapting private inference to the specific requirements and limitations of a system.\n\n\\begin{figure}[t!]\n\t\\input{plots\/numcells-accuracy.tex}\n\t\\caption{Impact of the model's depth (as the number of cells in the {\\scshape Soteria}{} architecture search algorithm) on test accuracy and inference runtime of garbled ternary model on CIFAR10. Regularization term $\\lambda$ is $0.6$. Scaling factor is 3.}\n\t\\label{fig:numcells-vs-accuracy}\n\\end{figure}\n\n\\paragraph{\\em Finding the optimal depth for the model (number of cells).}\nThe number of cells that the final architecture will have is a manually-set hyperparameter, that practically defines the depth of the model architecture. We perform an experiment on the CIFAR10 dataset, using cells with 4 operations in sequence and $\\lambda=0.6$. We run the search process for 100 epochs, and train each resultant architecture for 200 epochs. Figure~\\ref{fig:numcells-vs-accuracy} presents the results of executing the search with different number of cells. It illustrates the model accuracies along with their inference runtime.\n\nWe can observe that for a single cell architecture, the accuracy levels are low, as the network could not process the features required to perform a generalizable classification. The test accuracy peaks for 3-cell architecture, suggesting that it has enough operations to process the required features of the inputs. In parallel, we can see that as the number of cells increases, the runtime also increases.\n\n\\paragraph{\\em Comparison with prior work.}\nTable~\\ref{tab:comparison-results} reports the results for {\\scshape Soteria}{} trained models that are optimized for both accuracy and efficiency. For the MNIST dataset, we observe that our model with $\\lambda =0$ gives the best model as compared to prior work while balancing the runtime performance ($0.034$) and accuracy of $0.9811$. Increasing $\\lambda =0.6$ increases the accuracy by a small value. Similarly, for CIFAR10 datasets, we observe that a {\\scshape Soteria}{}-trained model with $\\lambda =0.6$ gives better accuracy than prior work as compared to the numbers reported in brackets. In addition, {\\scshape Soteria}{} models provide acceptable runtime performance and communication overhead that outperforms the results from prior work. Our evaluation on MNIST and CIFAR10 datasets confirm that {\\scshape Soteria}{} is effective in training models that are customized to perform well for both performance and accuracy.\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nMachine learning models are susceptible to several security and privacy attacks throughout their training and inference pipelines. Defending each of these threats require different types of security mechanisms. The most important requirement is that the sensitive input data as well as the trained model parameters remains confidential at all times. In this paper, we focus on private computation of inference over deep neural networks, which is the setting of machine learning-as-a-service. Consider a server that provides a machine learning service (e.g., classification), and a client who needs to use the service for an inference on her data record. The server is not willing to share the proprietary machine learning model, underpinning her service, with any client. The clients are also unwilling to share their sensitive private data with the server. We consider an honest-but-curious threat model. In addition, we assume the two parties do not trust, nor include, any third entity in the protocol. In this setting, our first objective is to design a secure protocol that protects the confidentiality of client data as well as the prediction results against the server who runs the computation. The second objective is to preserve the confidentiality of the model parameters with respect to the client. We emphasize that the protection against the indirect inference attacks that aim at reconstructing model parameters~\\cite{tramer2016stealing} or its training data~\\cite{shokri2017membership}, by exploiting model predictions, is not our goal.\n\nA number of techniques provide data confidentiality while computing and thereby allow {\\em private computation}. The techniques include computation on trusted processors such as Intel SGX~\\cite{hunt2018chiron, ohrimenko2016oblivious}, and computation on encrypted data, using homomorphic encryption, garbled circuits, secret sharing, and hybrid cryptographic approaches that jointly optimize the efficiency of private inference on neural networks~\\cite{YaoGC,gentry,yao1982protocols,brakerski2011fully,beaver1991efficient}. To provide private inference with minimal performance overhead and accuracy loss, the dominant line of research involves adapting cryptographic functions to (an approximation of) a given {\\em fixed} model~\\cite{MiniONN, SecureML, Delphi, DeepSecure, XONN, EzPC, Gazelle, Chameleon}. \nHowever, the alternative approach of {\\em searching or designing} a network architecture for a given set of efficient and known cryptographic primitives is unexplored in the literature.\n\n{\\bf Our Contributions.} In this work, we approach the problem of privacy-preserving inference from a novel perspective. Instead of modifying cryptographic schemes to support neural network computations, we advocate modification of the training algorithms for efficient cryptographic primitives. Research has shown that training algorithms for deep learning are inherently flexible with respect to their neural network architecture. This means that different network configurations can achieve similar level of prediction accuracy. We exploit this fact about deep learning algorithms and investigate the problem of optimizing deep learning algorithms to ensure efficient private computation. \n\nTo this end, we present {\\scshape Soteria}{} --- an approach for constructing deep neural networks optimized for performance, accuracy and confidentiality. Among all the available cryptographic primitives, we use garbled circuits as our main building block to address the {\\em confidentiality} concern in the design of {\\scshape Soteria}{}. Garbled circuits are known to be efficient and allow generation of constant depth circuits even for non-linear function which is not possible with other primitives such as GMW or FHE. We show that neural network algorithms can be optimized to efficiently execute garbled circuits while acheiving high accuracy guarantees. We observe that the efficiency of evaluating an inference circuit depends on two key factors: the model parameters and the network structure. With this observation, we design a regularized architecture search algorithm to construct neural networks. {\\scshape Soteria}{} selects optimal parameter sparsity and network structure with the objective to guarantee acceptable {\\em performance} on garbled circuits and high model {\\em accuracy}. \n\n\n\\section{Specifications of the Model Architectures} \n\\label{Appendix:architectures}\n\n\\input{tables\/architectures-exp.tex}\n\n\\end{document}\n\n\\section{{\\scshape Soteria}{}} \\label{sec:method}\n\nAll the techniques which we discuss in Section~\\ref{sec:gcbinary}, can help in reducing the overhead of the garbled circuit protocol on a neural network. However, besides the size of model parameters, which is reduced in binary neural networks, the {\\em model size} and its {\\em structure} also play significant roles in determining the number of non-XOR gates in the garbled circuit of neural networks. For example, the configurations of the convolutional layers directly affects the overhead of garbled circuits on neural networks. Besides, not all model parameters are of the same value for the machine learning task, and models with the same structure but with larger sparsity can result in similar accuracy, but significantly lower overhead for private computation. \n\nIn this paper, we design {\\scshape Soteria}{} to automatically learn the model architecture and its connections so as to optimize the cost of private inference in addition to optimizing accuracy. This is a different approach than simply fine-tunning or compressing a model, as we aim at including the cost of private computation as part of the objective of {\\em architecture learning} and {\\em parameter learning} of the model. To this end, we build {\\scshape Soteria}{} on top of two well-established classes of machine learning algorithms to search for the models that balance accuracy and performance: neural architecture search algorithms, and ternary neural network algorithms.\n\n\\subsection{Neural architecture search for constructing efficient models for private inference}\n\nArchitecture search algorithms for neural networks are designed to replace the manual design of complex deep models. The objective is to learn the model structure for which we hope to obtain a high accuracy when trained on the training set. A number of such algorithms are designed recently. NAS~\\cite{NAS_survey} is one of the first neural architecture search algorithms. It comprises of three components --- a \\textit{search space} which is the domain of architectures over which the search will be executed, the \\textit{search strategy}, which defines how the search space has to be explored, and a \\textit{performance estimator}, to estimate the performance of a particular discovered architecture on unseen data. Multiple techniques have been proposed to minimize the computation cost of the search process, by tweaking the search strategy and the performance estimator such as ENAS and DARTS~\\cite{nas_rl, enas, DARTS}. DARTS is a differentiable neural architecture search algorithm, which is orders of magnitude faster than other search algorithms ~\\cite{DARTS}. DARTS automatically constructs the model architecture by stacking a number of {\\em cells}. Each cell is a directed acyclic graph, where each node is a neural {\\em operation} (e.g., convolution with different dimensions, maxpool, identity). The architecture search algorithm learns the optimal construction of cells that would maximize the accuracy of the model on some validation set. During the search algorithm, we use stochastic gradient descent to continuously update the probability of using different candidate operations for each connection in the internal graph of a cell. These probabilities reflect the usefulness of {\\em each} operation for different positions in the cell. Let $\\alpha_{o}^{(i,j)}$ be the fitting score associated with operation $o$ to connect nodes $i$ and $j$ in the directed acyclic graph inside the cell. The probability of choosing a particular operation to connect node $i$ to $j$ is computed as a softmax of the score $\\alpha_o^{(i,j)}$ over all possible operations.\n\nIn {\\scshape Soteria}{}, we modify the computation of the $\\alpha_o^{(i,j)}$ scores over candidate operations. In order to include the cost of private inference, we penalize each operation proportional to its computation and communication overhead. Let $\\gamma(o)$ be the penalty or the cost function for an operation $o$. The penalty factor could be the normalized runtime and communication cost of an operation, which can be computed empirically on garbled operations. In our experiments, we compute the penalty factor for different operations in Table~\\ref{table:operationcost}. We update the fitting score $\\alpha_o^{(i,j)}$ by replacing it with $\\alpha_o^{(i,j)} (1-\\lambda \\gamma(o))$, where $\\lambda$ is our regularization term. Larger values of $\\lambda$ would result in models that prefer training efficient models over accurate models. \n\nBy regularizing the architecture search algorithm, we effectively guide the algorithm to identify a configuration for cells which optimize both model accuracy and performance of private inference. This enables fine-tuning the model {\\em before} being trained to be efficient on our cryptographic primitives. As we balance the trade-off between accuracy and performance, {\\scshape Soteria}{} can construct models which {\\em by design} satisfy the requirements of our system. In our experiments, we evaluate the performance of models under different values of $\\lambda$, and how this factor can be used to balance different costs of confidentiality for neural networks. \n\n\\subsection{Ternary (Sparse Binary) Neural Network} \n\n\\label{section:tnn}\n\\label{sec:ternarization-of-weights}\n\nFor building a system that enables efficient private inference, we prefer to reduce the number of parameters in the network. One approach is to train a model and then compress the model, however, that might not result in the best construction of the model as far as the model accuracy is concerned. Besides, to be aligned with our approach of {\\em constructing} model architectures, we would prefer to learn model structures which are sparse. One well-established machine learning technique is to learn a model with ternary values ($-1, 0, +1$). This effectively means that some of the network connections (parameters) are removed (for parameters with value $0$). Ternary neural networks try to minimize the distance between the full precision model parameters and their ternary values~\\cite{twn}. \n\nIn building models for {\\scshape Soteria}{}, we train models with ternary parameters and binary activation functions. This would enable us to still use the techniques for binary neural networks, as discussed in Section~\\ref{sec:gcbinary}, however on a smaller circuit (due to the model's sparsity). We incorporate ternary neural networks into our regularized architecture search algorithm to find cells containing only ternary convolution and max-pooling layers that operate on binary inputs and ternary parameters.\n\nThe procedure for converting full precision model parameters, during training, to ternary involves comparing the model parameters with a threshold in the forward pass of the gradient descent algorithm. We follow the established algorithms in this domain~\\cite{twn}. For the parameters $W_i$ in an operation, we convert the parameter to $+1$ when it is larger than threshold $\\Delta$, we set it to $-1$ if it is smaller than $-\\Delta$, and we set it to $0$ otherwise. The threshold is computed as ${\\Delta = \\frac{0.7}{n} \\sum_{i=1}^{n}\\left|\\mathrm{W}_{i}\\right|}$. Output of the functions are also binarized similarly, by comparing them with $0$, where positive values are converted to $+1$, and the negative values are converted to~$-1$. All these transformations happen {\\em during} the model training, so the final model is optimal given the ternary restrictions. Besides, the training algorithm finds the optimal level of sparsity for the model which does not conflict with its accuracy. \n\n\n\n\n\n\n\n\\section{Related Work}\nThere has been several approaches that introduce new techniques for secure machine learning, or build up on existing techniques by trying to optimize bottlenecks.\n\n\\paragraph{\\em Homomorphic Encryption.} In CryptoNets~\\cite{Cryptonets}\\cite{Crypto-nets}, the authors modify the neural network operation by using square function as an activation and average pool instead of maxpool to reduce the non-linear functions to low degree polynomial to control the noise. Similar approaches of using homomorphic encryption on data and optimizing the machine learning operations to limit the noise have been explored extensively~\\cite{HomomorphicDiscreteNN, MLConfidential, BostClassificationEncrypted}. \n\nHesamifard et al.~\\cite{PrivacyML_Cloud} explore using homomorphic encrypted data for training the neural networks. CryptoDL~\\cite{CryptoDL} explores various activation functions with low polynomial degree that can work well with homomorphic encrypted data and proposed an activation using the derivative of ReLU function. However, using homomorphic encryption adds to an additional computational overhead and most of the non-linear activations cannot be effectively computed which results in a degradation in reliability of the deep learning systems.\n\n\\paragraph{\\em Secure Multiparty Computation.} Secure multiparty computation requires a very low computation overhead but requires extensive communication between the parties. It has been used for several machine learning operations. \nDeepSecure~\\cite{DeepSecure} only uses GC to compute all the operations in the neural network. They rely on pre-processing of the data by reducing the dimensions to improve the performance and is implemented on the TinyGarble~\\cite{TinyGarble_paper} library. Chameleon~\\cite{Chameleon} uses a combination of arithmetic sharing, garbled circuit and boolean sharing to compute the neural networks for secure inference. They rely on third party server to perform computation in the offline phase resulting in better performance than the previous work. \n\nXONN~\\cite{XONN} leverages Binary Neural Networks with GC. Binarization dramatically reduces the inference latency for the network compared to other frameworks that utilize full-precision weights and inputs, as it converts matrix multiplications into simple XNOR-popcounts. They use TinyGarble library as well to implement the Boolean circuits for GC. \nPrio~\\cite{Prio} uses a secret sharing~\\cite{ShamirSS} based protocol to compute aggregate statistics over private data from multiple sources. They deploy a secret-shared non-interactive zero-knowledge proof mechanism to verify whether data sent by clients is well-formed, and then decode summed encodings of clients' data to generate aggregate statistic. They extend the application of Prio to foundational machine learning techniques such as least squares regression.\n\n\\paragraph{\\em Hybrid Schemes.} A judicious combination of homomorphic encryption and multiparty computation protocol have shown to give some additional benefits in terms of runtime and communication costs. \nGazelle~\\cite{Gazelle} uses lattice based Packed Additive homomorphic encryption to compute dot product and convolution but relies on garbled circuits for implementing non-linear operations like Maxpool and ReLU. They reduce the overall bandwidth by packing ciphertexts and re-encryption to refresh the noise budget. Delphi~\\cite{Delphi} builds upon this work and uses Architecture Search to select optimal replacement positions for expensive ReLU activation function with a quadratic approximation with minimal loss in accuracy. \n\nMiniONN~\\cite{MiniONN} pre-computes multiplication triplets using homomorphic encryption for GMW protocol followed by SPDZ~\\cite{MPC_SomewhatHE, Breaking_SPDZ_Limits} protocol. The multiplication triplets are exchanged securely using additive homomorphic encryption like Paillier or DGK.\nSecureML~\\cite{SecureML} uses garbled circuits and additive homomorphic encryption to speed up some NN operations. However, the conversion costs between of homomorphic encryption and Yao's garbled circuits is expensive and the performance of homomorphic encryption scales poorly with increasing security parameter~\\cite{ABY}. \n\nHence, we rely on only garbled circuit protocol to efficiently compute neural network operations during inference with low communication bandwidth, low computation complexity and low memory footprint using binary neural networks while maintaining the accuracy. Most of the previous work have relied heavily on optimizing the complex cryptographic operations to work well with the neural networks. We show that it is possible to optimize the neural network to get an efficient privacy preserving neural network architectures.\n\n\\paragraph{\\em Trusted Computing.} Some research uses trusted processors where they assume that the underlying hardware is trustworthy and outsource all the machine learning computations to the trusted hardware. Chiron~\\cite{hunt2018chiron} is a training system for privacy-preserving machine learning as a service which conceals the training data from the operator. It uses Intel Software Guard Extensions (SGX) and runs the standard ML training in an enclave and confines it in a Ryoan sandbox~\\cite{Ryoan} to prevent it from leaking the training data outside the enclave.\n\nOhrimenko et al.~\\cite{ohrimenko2016oblivious} propose a solution for secure multiparty ML by using trusted Intel SGX-enabled processors and used oblivious protocols between client and server where the input and outputs are blinded. However, the memory of enclaves is limited and it is difficult to process memory and computationally intensive operations like matrix multiplication in the enclaves with paralellism. To address this, Slalom~\\cite{Slalom} provides a methodology to outsource the matrix multiplication to a faster untrusted processor and verify the computation.\n\n\n\\section{Selecting the Cryptographic Primitive} \\label{sec:selection}\n\n\\input{tables\/crypto-primitives.tex}\n\nIn designing {\\scshape Soteria}{}, we make several design choices with the goal of achieving efficiency. The most important among them is the selection of the underlying cryptographic primitive to ensure privacy of data. Several cryptographic primitives such as partially homomorphic encryption schemes (PHE) and fully homomorphic encryption schemes (FHE), Goldreich-Micali-Widgerson protocol (GMW), arithmetic secret sharing (SS), and Yao's garbled circuit (GC) have been proposed to enable two-party secure computation. Each of these primitives perform differently with respect to the factors such as efficiency, functionality, required resources and so on. PHE schemes allow either addition or multiplication operations but not both on encrypted data~\\cite{paillier1999public, elgamal1985public}. In contrast, FHE schemes enable both addition and multiplication on encrypted data~\\cite{gentry,FHE, van2010fully, brakerski2011fully} but incur huge performance overhead. SS involves distributing the secret shares among non-trusting parties such that any operation can be computed on encrypted data without revealing the individual inputs of each party~\\cite{beaver1991efficient}. GMW~\\cite{GMW} and GC~\\cite{yao1982protocols} allow designing boolean circuits and evaluating them between a client and a server. The differences between these schemes might make it difficult to decide which primitive is the best fit for designing a privacy-preserving system for a particular application. Therefore, we first outline the desirable properties specifically for private neural network inference and then compare these primitives with respect to these properties (see Table~\\ref{table:cryptoprimitives}). We select a cryptographic scheme that satisfies all our requirements.\n\n{\\bf Expressiveness.} This property ensures that the cryptographic primitive supports encrypted computation for a variety of operations. With the goal to enable private computation for neural networks, we examine the type of computations required in deep learning algorithms. Neural network algorithms are composed of linear and non-linear operations. Linear operations include computation required in the execution of fully-connected and convolution layers. Non-linear operations include activation functions such as Tanh, Sigmoid and ReLU. The research in deep learning is at its peak with a plethora of new models being proposed by the community to improve the accuracy of various tasks. Hence, we desire that the underlying primitive should be expressive with respect to any new operations used in the future as well. PHE schemes offer limited operations (either addition or multiplication) on encrypted data. This limits their usage in applications that demand expressive functionalities such as neural network algorithms. Alternative approaches such as FHE, SS, GMW and GC protocols allow arbitrary operations.\n\n{\\bf Computation Efficiency.} Efficiency is one of the key factors while designing a client-server application such as a neural network inference service on the cloud. FHE techniques have shown to incur orders of magnitude overhead for computation of higher-degree polynmials or non-linear functions. Existing approaches using FHE schemes have restricted its use to compute only linear functions. However, most of the neural network architectures such as CNNs have each linear layer followed by a non-linear layer. To handle non-linear operations, previous solutions either approximate them to linear functions or switch to cryptographic primitives that support non-linearlity~\\cite{Cryptonets, SecureML, Gazelle, Delphi}. Approximation of non-linear functions such as ReLU highly impacts the accuracy of the model. Switching between schemes introduces additional computation cost which is directly proportional to the network size. In comparison to FHE, research has shown that SS, GMW and GC schemes provide constructions with reasonable computation overhead for both linear and non-linear operations.\n\n{\\bf Communication Overhead.} The communication costs incurred for private computation contribute to the decision of selecting our cryptographic primitive, as the network should not become a bottleneck in the execution of the private machine learning as a service. We expect the client and server to interact only once during the setup phase and at the end of the execution to receive the output. We aim to remain backward compatible to the existing cloud service setting where the client does not need to be online at all time between the request and response. In contradiction to this property, the GMW scheme requires communication rounds proportional to the depth of the circuit. To evaluate every layer with an AND gate, the client and server have to exchange secrets among them forcing the client to be online throughout the execution. Similarly, construction of non-linear bitwise functions with arithmetic secret shares require communication rounds logarithmic to the number of bits in the input. This makes the use of these schemes almost infeasible in the cloud setting that have a high-latency network. Unlike these primitives, Yao's garbled circuits combined with recent optimizations require an exchange of data only once at the beginning of the protocol. \n\nWe select GC as our underlying cryptographic primitive in {\\scshape Soteria}{} as it satisfies all the desired properties for a designing private inference for cloud service applications. \n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nLet $X$ and $Y$ be sets and $F$, $G$, and $H$ be locally\nconvex spaces consisting of functions defined on $X$, $Y$,\nand $X\\times Y$ respectively. Suppose the function\n$(x,y)\\to f(x)g(y)$ belongs to $H$ for any $f\\in F$ and\n$g\\in G$. In applications, it is often important to find\nout whether the following statement holds:\n\\begin{itemize}\n \\item[(K)] For any separately continuous bilinear mapping\n $\\psi$ from $F\\times G$ to a Hausdorff complete space\n $\\tilde H$, there is a unique continuous linear mapping\n $\\tilde \\psi\\colon H\\to \\tilde H$ such that $\\psi=\\tilde\\psi\n \\circ\\Phi$, where the bilinear mapping $\\Phi\\colon\n F\\times G\\to H$ is defined by the relation\n $\\Phi(f,g)(x,y) = f(x)g(y)$.\n\\end{itemize}\nStatements of this type and their analogues for multilinear\nmappings are known as kernel theorems. In this paper, we\npropose a convenient technique for proving results of this\ntype in the case, where all considered spaces are\n(LF)-spaces (i.e., countable inductive limits of Fr\\'echet\nspaces).\n\nIn the language of topological tensor products~\\cite{Grot},\n(K) means that $H$ can be identified with a dense subspace\nof the completion $F\\hat\\otimes_i G$ of $F\\otimes G$ with\nrespect to the inductive\\footnote{Recall that the inductive\n(projective) topology on $F\\otimes G$ is the strongest\nlocally convex topology on $F\\otimes G$ such that the\ncanonical bilinear mapping $(f,g)\\to f\\otimes g$ is\nseparately continuous (resp., continuous).} topology. In\nparticular, to prove~(K), it suffices to show that $H$ can\nbe identified with $F\\hat\\otimes_i G$. However, this\nrequires proving the completeness of $H$, which may present\ndifficulty for concrete functional (LF)-spaces (and is\nactually unnecessary, as we shall see). To circumvent this\nproblem, we introduce a notion of the semi-completed tensor\nproduct of (LF)-spaces (see~Definition~\\ref{d1}). The\nsemi-completed tensor product $F\\tilde \\otimes \\,G$ of\n(LF)-spaces $F$ and $G$ is an (LF)-space which is\ncanonically identified with a dense subspace of\n$F\\hat\\otimes_i G$, and, therefore, we can prove~(K) by\nidentifying $H$ with $F\\tilde \\otimes\\, G$. Moreover,\nsemi-completed tensor products possess a natural\nassociativity property which turns out to be very useful\nfor the treatment of multilinear mappings.\n\nThe paper is organized as follows. In Sec.~\\ref{s2}, we\ngive the definition of semi-completed tensor products and\ndescribe their basic properties. In Sec.~\\ref{s3}, we find\nsimple conditions ensuring the coincidence of $F\\tilde\n\\otimes G$ with a given space $H$ in the case, where $F$,\n$G$, and $H$ are functional (LF)-spaces. In Sec.~\\ref{s4},\nwe apply the obtained results to proving kernel theorems\nfor some spaces of entire analytic functions arising in\nnonlocal quantum field theory (see~\\cite{SS2}).\n\nIn Secs.~\\ref{s2} and~\\ref{s3}, we assume that the\nconsidered vector spaces are either real or complex. The\nground field (${\\mathbb R}$ or ${\\mathbb C}$) is denoted by ${\\mathbb K}$.\n\n\\section{Semi-completed tensor products of (LF)-spaces}\n\\label{s2}\n\nA Hausdorff locally convex space $F$ is called an\n(LF)-space if there exist a sequence $(F_i)$ of Fr\\'echet\nspaces and a sequence $p_i$ of continuous mappings from\n$F_i$ to $F$ such that $F=\\bigcup_i \\mathrm{Im}\\,p_i$ and\nthe topology of $F$ coincides with the inductive topology\nwith respect to the mappings $p_i$. The following important\nresult was proved by Grothendieck (see~Th\\'eor\\`eme~A\nof~\\cite{Grot}).\n\n\\begin{lemma}\n \\label{l0}\nLet $F$ be a Hausdorff locally convex space, $(F_i)$ be a\nsequence of Fr\\'echet spaces, and $(p_i)$ be a sequence of\ncontinuous mappings from $F_i$ to $F$. Let $\\tilde F$ be a\nFr\\'echet space and $p\\colon \\tilde F\\to F$ be a continuous\nmapping such that $\\mathrm{Im}\\,p\\subset \\bigcup_i\n\\mathrm{Im}\\,p_i$. Then there is an index $i$ such that\n$\\mathrm{Im}\\,p\\subset \\mathrm{Im}\\,p_i$ and if $p_i$ is\ninjective, then there is a continuous mapping $\\tilde\np\\colon \\tilde F\\to F_i$ such that $p=p_i\\circ \\tilde p$.\n\\end{lemma}\n\nIt follows from Lemma~\\ref{l0} that the topology of an\n(LF)-space $F$ coincides with the inductive topology with\nrespect to an arbitrary countable family of continuous\nmappings of Fr\\'echet spaces into $F$ provided that the\nimages of these mappings cover $F$.\n\nLet $F$ be a Hausdorff locally convex space and $\\tilde F$\nbe a Fr\\'echet space. We say that $\\tilde F$ is a Fr\\'echet\nsubspace of $F$ (notation $\\tilde F\\prec F$) if $\\tilde\nF\\subset F$ and the inclusion mapping $\\tilde F\\to F$ is\ncontinuous. The relation $\\prec$ is a partial order on the\nset $\\mathfrak F_F$ of all Fr\\'echet subspaces of $F$. In\nfact, $\\mathfrak F_F$ is a lattice: the least upper bound\n$F_1\\vee F_2$ of $F_1,F_2\\in \\mathfrak F_F$ is the space\n$F_1+F_2$ endowed with the inductive topology with respect\nto the inclusion mappings $F_{1,2}\\to F_1+F_2$; the\ngreatest lower bound $F_1\\wedge F_2$ is the space $F_1\\cap\nF_2$ endowed with the projective topology with respect to\nthe inclusion mappings $F_1\\cap F_2\\to F_{1,2}$. In\nparticular, the set $\\mathfrak F_F$ is directed. By\nLemma~\\ref{l0}, a countable set $U\\subset \\mathfrak F_F$ is\ncofinal in $\\mathfrak F_F$ if and only if\n$F=\\bigcup_{\\tilde F\\in U} \\tilde F$. If $F$ is an\n(LF)-space, then $\\mathfrak F_F$ contains a countable\ncofinal subset. Indeed, let a sequence $(F_i)$ of Fr\\'echet\nspaces and a sequence $(p_i)$ of continuous linear mappings\nfrom $F_i$ to $F$ be such that $F=\\bigcup_i\n\\mathrm{Im}\\,p_i$. Let $\\tilde F_i$ be the space\n$\\mathrm{Im}\\,p_i$ endowed with the inductive topology with\nrespect to the mapping $p_i$. Then $\\tilde F_i\\in \\mathfrak\nF_F$ for all $i$ and, therefore, $\\tilde F_i$ form the\nrequired subset.\n\nFor a locally convex space $F$, we denote by $\\hat F$\nthe Hausdorff completion of $F$.\nGiven\nlocally convex spaces $F_1,\\ldots,F_n$, the tensor product\n$F_1\\otimes\\ldots\\otimes F_n$ endowed by the inductive\n(resp., projective) topology will be denoted by\n$F_1\\otimes_i\\ldots\\otimes_i F_n$ (resp., by\n$F_1\\otimes_\\pi\\ldots\\otimes_\\pi F_n$). The Hausdorff\ncompletions of $F_1\\otimes_i\\ldots\\otimes_i F_n$ and\n$F_1\\otimes_\\pi\\ldots\\otimes_\\pi F_n$ will be denoted by\n$F_1\\hat\\otimes_i\\ldots\\hat\\otimes_i F_n$ and\n$F_1\\hat\\otimes_\\pi\\ldots\\hat\\otimes_\\pi F_n$ respectively.\nIf $F_1,\\ldots,F_n$ are Fr\\'echet spaces, then the\ninductive and projective topologies on\n$F_1\\otimes\\ldots\\otimes F_n$ coincide. In this case, we\nshall omit the indices $i$ and $\\pi$ in the notation for\ntensor products.\n\n\\begin{definition}\n\\label{d1} Let $F_1,\\ldots,F_n$ be (LF)-spaces. The\nsemi-completed tensor product\n$F_1\\tilde\\otimes\\ldots\\tilde\\otimes F_n$ of\n$F_1,\\ldots,F_n$ is defined to be the Hausdorff space\nassociated with the inductive limit\n\\[\n\\varinjlim_{(\\tilde F_1,\\ldots,\\tilde F_n)\\in \\mathfrak\nF_{F_1}\\times\\ldots\\times\\mathfrak F_{F_n}}\\,\\tilde\nF_1\\hat\\otimes\\ldots\\hat\\otimes\\tilde F_n.\n\\]\n\\end{definition}\n\nEach of the sets $\\mathfrak F_{F_1},\\ldots,\\mathfrak\nF_{F_n}$ contains a countable cofinal subset. It hence\nfollows that $\\mathfrak F_{F_1}\\times\\ldots\\times\\mathfrak\nF_{F_n}$ also contains a countable cofinal subset and,\ntherefore, $F_1\\tilde\\otimes\\ldots\\tilde\\otimes F_n$ is an\n(LF)-space. For $\\tilde F=(\\tilde F_1,\\ldots,\\tilde F_n)\\in\n\\mathfrak F_{F_1}\\times\\ldots\\times\\mathfrak F_{F_n}$, let\n$\\iota_{\\tilde F}$ be the natural continuous linear mapping\nfrom $\\tilde F_1\\otimes\\ldots\\otimes \\tilde F_n$ to\n$F_1\\otimes_i\\ldots\\otimes_i F_n$ determined by the\ninclusion mappings $\\tilde F_i\\to F_i$, $i=1,\\ldots,n$, and\n$\\rho_{\\tilde F}$ be the canonical continuous linear\nmapping from $\\tilde F_1\\hat\\otimes\\ldots\\hat\\otimes\\tilde\nF_n$ to $F_1\\tilde\\otimes\\ldots\\tilde\\otimes F_n$.\n\n\\begin{lemma} \\label{l0a}\nLet $F_1,\\ldots,F_n$ be (LF)-spaces. There is a unique\ncontinuous linear mapping $j\\colon\nF_1\\otimes_i\\ldots\\otimes_i F_n\\to\nF_1\\tilde\\otimes\\ldots\\tilde\\otimes F_n$ such that\n\\[\nj\\iota_{\\tilde F}= \\rho_{\\tilde F} \\lambda_{\\tilde F},\n\\]\nfor any $\\tilde F=(\\tilde F_1,\\ldots,\\tilde F_n)\\in\n\\mathfrak F_{F_1}\\times\\ldots\\times\\mathfrak F_{F_n}$,\nwhere $\\lambda_{\\tilde F}$ is the canonical mapping from\n$\\tilde F_1\\otimes\\ldots\\otimes \\tilde F_n$ to $\\tilde\nF_1\\hat\\otimes\\ldots\\hat\\otimes \\tilde F_n$. The mapping\n$j$ is a topological isomorphism of\n$F_1\\otimes_i\\ldots\\otimes_i F_n$ onto a dense subspace of\n$F_1\\tilde\\otimes\\ldots\\tilde\\otimes F_n$. There is a\nunique continuous linear mapping $k\\colon\nF_1\\tilde\\otimes\\ldots\\tilde\\otimes F_n\\to\nF_1\\hat\\otimes_i\\ldots\\hat\\otimes_i F_n$ such that $k\\circ\nj$ is the canonical mapping $F_1\\otimes_i\\ldots\\otimes_i\nF_n\\to F_1\\hat\\otimes_i\\ldots\\hat\\otimes_i F_n$. The\nmapping $k$ is a topological isomorphism of\n$F_1\\tilde\\otimes\\ldots\\tilde\\otimes F_n$ onto a dense\nsubspace of $F_1\\hat\\otimes_i\\ldots\\hat\\otimes_i F_n$. For\nevery $f\\in F_1\\tilde\\otimes\\ldots\\tilde\\otimes F_n$, there\nis a bounded subset $B$ of $F_1\\otimes_i\\ldots\\otimes_i\nF_n$ such that $f\\in \\overline{j(B)}$, where the bar means\nclosure.\n\\end{lemma}\n\nThe mapping $j$ described in Lemma~\\ref{l0a} will be called\nthe canonical mapping from $F_1\\otimes_i\\ldots\\otimes_i\nF_n$ to $F_1\\tilde\\otimes\\ldots\\tilde\\otimes F_n$. For\n$f_1\\in F_1,\\ldots, f_n\\in F_n$, we set\n\\[\nf_1\\tilde\\otimes \\ldots \\tilde\\otimes f_n =\nj(f_1\\otimes\\ldots\\otimes f_n).\n\\]\nThe $n$-linear mapping $(f_1,\\ldots, f_n)\\to\nf_1\\tilde\\otimes \\ldots \\tilde\\otimes f_n$ will be called\nthe canonical $n$-linear mapping from\n$F_1\\times\\ldots\\times F_n$ to\n$F_1\\tilde\\otimes\\ldots\\tilde\\otimes F_n$.\n\n\\begin{definition}\\label{d2}\nLet $F_1,\\ldots,F_n$ be (LF)-spaces, $H$ be a locally\nconvex space, and $\\varphi$ be an $n$-linear mapping from\n$F_1\\times\\ldots\\times F_n$ to $H$. We say that $\\varphi$\nis (F)-continuous if for any $\\tilde F_1\\in\\mathfrak\nF_{F_1},\\ldots,\\tilde F_n\\in\\mathfrak F_{F_n}$, there are a\nFr\\'echet space $\\tilde H$, a continuous $n$-linear mapping\n$\\psi\\colon \\tilde F_1 \\times\\ldots\\times \\tilde\nF_n\\to\\tilde H$, and a continuous linear mapping $l\\colon\n\\tilde H\\to H$ such that $l\\circ\\psi$ coincides with the\nrestriction of $\\varphi$ to $\\tilde F_1 \\times\\ldots\\times\n\\tilde F_n$.\n\\end{definition}\n\nObviously, every (F)-continuous $n$-linear mapping is\nseparately continuous. If $H$ is Hausdorff and complete,\nthen every separately continuous $n$-linear mapping\n$\\varphi\\colon F_1\\times\\ldots\\times F_n\\to H$ is\n(F)-continuous. Indeed, for any $\\tilde F_1\\in\\mathfrak\nF_{F_1},\\ldots,\\tilde F_n\\in\\mathfrak F_{F_n}$, the\nrestriction of $\\varphi$ to $\\tilde F_1 \\times\\ldots\\times\n\\tilde F_n$ can be decomposed as $l\\circ\\psi$, where $\\psi$\nis the canonical $n$-linear mapping from $\\tilde F_1\n\\times\\ldots\\times \\tilde F_n$ to $\\tilde F_1\n\\hat\\otimes\\ldots\\hat\\otimes \\tilde F_n$ and $l\\colon\n\\tilde F_1 \\hat\\otimes\\ldots\\hat\\otimes \\tilde F_n\\to H$ is\na continuous linear mapping.\n\nIt easily follows from the above definitions that the\ncanonical $n$-linear mapping from $F_1\\times\\ldots\\times\nF_n$ to $F_1\\tilde\\otimes\\ldots\\tilde\\otimes F_n$ is\n(F)-continuous. Moreover, the space\n$F_1\\tilde\\otimes\\ldots\\tilde\\otimes F_n$ has the following\nuniversal property:\n\n\\begin{lemma}\\label{l0b}\nLet $F_1,\\ldots,F_n$ be (LF)-spaces, $H$ be a Hausdorff\nlocally convex space, and $\\varphi$ be an (F)-continuous\n$n$-linear mapping from $F_1\\times\\ldots\\times F_n$ to $H$.\nThen there is a unique continuous linear mapping $l\\colon\nF_1\\tilde\\otimes\\ldots\\tilde\\otimes F_n\\to H$ such that\n\\[\n\\varphi(f_1,\\ldots,f_n) =\nl(f_1\\tilde\\otimes\\ldots\\tilde\\otimes f_n),\\quad f_1\\in\nF_1,\\ldots, f_n\\in F_n.\n\\]\n\\end{lemma}\n\n\\begin{theorem}\\label{t0}\nLet $F_1,\\ldots,F_n$ be (LF)-spaces. For every $1\\leq m<\nn$, there is a unique topological isomorphism\n$F_1\\tilde\\otimes\\ldots\\tilde\\otimes F_n\\simeq\n(F_1\\tilde\\otimes\\ldots\\tilde\\otimes F_m)\\tilde\\otimes\\,\n(F_{m+1}\\tilde\\otimes\\ldots\\tilde\\otimes F_n)$ taking\n$f_1\\tilde\\otimes\\ldots\\tilde\\otimes f_n$ to\n$(f_1\\tilde\\otimes\\ldots\\tilde\\otimes f_m)\\tilde\\otimes\\,\n(f_{m+1}\\tilde\\otimes\\ldots\\tilde\\otimes f_n)$ for any\n$f_1\\in F_1,\\ldots, f_n\\in F_n$.\n\\end{theorem}\n\n\n\\section{Tensor products of functional (LF)-spaces}\n\\label{s3}\n\nGiven a locally convex space $F$, we denote by $F'$ and\n$\\dv{\\cdot}{\\cdot}$ the continuous dual of $F$ and the\ncanonical bilinear form on $F'\\times F$ respectively. We\ndenote by $F'_\\sigma$, $F'_\\tau$, and $F'_b$ the space $F'$\nendowed with its weak topology, Mackey topology, and strong\ntopology respectively.\n\nLet $X$ and $Y$ be sets and $F$ and $G$ be locally convex\nspaces consisting of functions defined on $X$ and $Y$\nrespectively. We denote by $\\Pi(F,G)$ the linear space\nconsisting of all functions $h(x,y)$ on $X\\times Y$ such\nthat $h(x,\\cdot)\\in G$ for every $x\\in X$ and the function\n$h_v(x)=\\dv{v}{h(x,\\cdot)}$ belongs to $F$ for every $v\\in\nG'$.\n\n\\begin{lemma}\n \\label{l1}\nLet $X$ and $Y$ be sets and $F$, $G$, and $H$ be Hausdorff\ncomplete locally convex spaces consisting of scalar functions\ndefined on $X$, $Y$, and $X\\times Y$ respectively. Let $F$ be\nB-complete, $G$ be nuclear, and the topologies of $F$, $G$, and\n$H$ be stronger than that of simple convergence. Suppose the\nfunction $(x,y)\\to f(x)g(y)$ on $X\\times Y$ belongs to $H$ for\nevery $f\\in F$ and $g\\in G$ and the bilinear mapping $\\Phi\\colon\nF\\times G\\to H$ taking $(f,g)$ to this function is continuous.\nThen $\\Phi$ induces an injective continuous linear mapping\n$F\\widehat{\\otimes}_\\pi G\\to H$ whose image coincides with\n$\\Pi(F,G)$.\n\\end{lemma}\n\n\\begin{proof}\nWithout loss of generality, we can assume that $H={\\mathbb K}^{X\\times Y}$.\nLet $\\Phi_*\\colon F\\widehat{\\otimes}_\\pi G \\to H$ be the\ncontinuous linear mapping determined by $\\Phi$. As usual, let\n$\\mathfrak B_e(F'_\\sigma, G'_\\sigma)$ denote the space of\nseparately continuous bilinear forms on $F'_\\sigma\\times\nG'_\\sigma$ equipped with the biequicontinuous convergence topology\n(i.e., the topology of the uniform convergence on the sets of the\nform $A\\times B$, where $A$ and $B$ are equicontinuous sets in\n$F'$ and $G'$ respectively). Let $S$ be the natural continuous\nlinear mapping $F\\widehat{\\otimes}_\\pi G \\to \\mathfrak\nB_e(F'_\\sigma, G'_\\sigma)$ which takes $f\\otimes g$ to the\nbilinear form $(u,v)\\to \\dv{u}{f}\\dv{v}{g}$. Since $G$ is nuclear,\n$S$ is a topological isomorphism (see~\\cite{Grot}, Chapitre~2,\nTh\\'eor\\`eme~6 or~\\cite{Schaefer}). Further, let $T$ be the linear\nmapping $\\mathfrak B_e(F'_\\sigma, G'_\\sigma)\\to {\\mathbb K}^{X\\times Y}$\ndefined by the relation $(Tb)(x,y)=b(\\delta_x,\\delta_y)$, $b\\in\n\\mathfrak B_e(F'_\\sigma, G'_\\sigma)$ (if $x\\in X$ and $y\\in Y$,\nthen $\\delta_x$ and $\\delta_y$ are the linear functionals on $F$\nand $G$ such that $\\dv{\\delta_x}{f}=f(x)$ and\n$\\dv{\\delta_y}{g}=g(y)$; they are continuous because the\ntopologies of $F$ and $G$ are stronger than the topology of simple\nconvergence). Obviously, $T$ is continuous and $TS$ coincides with\n$\\Phi_*$ on $F\\otimes G$. By continuity, we have $\\Phi_*=TS$\neverywhere on $F\\widehat{\\otimes}_\\pi G$. Moreover, $T$ is\ninjective because $\\delta$-functionals are weakly dense in $F'$\nand $G'$. To prove the statement, we therefore have to show that\n$\\mathrm{Im}\\,T=\\Pi(F,G)$.\n\nLet $b\\in \\mathfrak B_e(F'_\\sigma, G'_\\sigma)$ and $h=Tb$. The\nbilinear form $b$ determines two linear mappings $L_1\\colon F'\\to\nG$ and $L_2\\colon G'\\to F$ such that $b(u,v)=\\dv{v}{L_1\nu}=\\dv{u}{L_2 v}$ for any $u\\in F'$ and $v\\in G'$. For $x\\in X$\nand $y\\in Y$, we have\n$h(x,y)=\\dv{\\delta_y}{L_1\\delta_x}=(L_1\\delta_x)(y)$, i.e.,\n$h(x,\\cdot)=L_1\\delta_x$. Further, for $v\\in G'$ and $x\\in X$, we\nhave $h_v(x)=\\dv{v}{h(x,\\cdot)}=\n\\dv{v}{L_1\\delta_x}=\\dv{\\delta_x}{L_2v}=(L_2 v)(x)$. Hence\n$h_v=L_2 v$ belongs to $G$ and, therefore, $h\\in \\Pi(F,G)$. Thus,\nwe have the inclusion $\\mathrm{Im}\\,T\\subset\\Pi(F,G)$.\n\nWe now prove the converse inclusion. Let $h\\in \\Pi(F,G)$ and\n$L\\colon G'_\\tau\\to F$ be the linear mapping taking $v\\in G'$ to\n$h_v$. We claim that the graph $\\mathcal G$ of $L$ is closed. It\nsuffices to show that if an element of the form $(0,f)$ belongs to\nthe closure $\\bar{\\mathcal G}$ of $\\mathcal G$, then $f=0$.\nSuppose the contrary that there is $f_0\\in F$ such that $f_0\\ne 0$\nand $(0,f_0)\\in \\bar{\\mathcal G}$. Let $x_0\\in X$ be such that\n$f_0(x_0)\\ne 0$ and let the neighborhood $U$ of $f_0$ be defined\nby the relation $U=\\{f\\in F : |\\dv{\\delta_{x_0}}{f-f_0}|<\n|f_0(x_0)|\/2\\}$. Let $V=\\{v\\in G' :\n|\\dv{v}{h(x_0,\\cdot)}|<|f_0(x_0)|\/2\\}$. If $f\\in U$ and $v\\in V$,\nthen we have $|h_v(x_0)|<|f_0(x_0)|\/2<|f(x_0)|$. Hence the\nneighborhood $V\\times U$ of $(0,f_0)$ does not intersect $\\mathcal\nG$. This contradicts to the assumption that $(0,f_0)\\in\n\\bar{\\mathcal G}$, and our claim is proved. Being nuclear and\ncomplete, $G$ is semireflexive and hence $G'_\\tau$ is barrelled.\nWe can therefore apply the closed graph theorem (\\cite{Schaefer},\nTheorem~IV.8.5) and conclude that $L$ is continuous. Let the\nbilinear form $b$ on $F'\\times G'$ be defined by the relation\n$b(u,v)=\\dv{u}{h_v}$. The continuity of $L$ implies that $b\\in\n\\mathfrak B_e(F'_\\sigma, G'_\\sigma)$. Since $b(\\delta_x,\n\\delta_y)=h(x,y)$, we have $h=Tb$. Thus, $h\\in\\mathrm{Im}\\,T$ and\nthe lemma is proved.\n\\end{proof}\n\n\\begin{theorem}\\label{t1}\nLet $X$ and $Y$ be sets, $F$, $G$, and $H$ be (LF)-spaces\nconsisting of scalar functions on $X$, $Y$, and $X\\times Y$\nrespectively. Suppose the topologies of $F$, $G$, and $H$\nare stronger than that of simple convergence and $G$ can be\ncovered by a countable family of its nuclear Fr\\'echet\nsubspaces. Let the following conditions be satisfied:\n\\begin{itemize}\n\\item[($i$)] For every $f\\in F$ and $g\\in G$, the function\n$(x,y)\\to f(x)g(y)$ on $X\\times Y$ belongs to $H$ and the\nbilinear mapping $\\Phi\\colon F\\times G\\to H$ taking $(f,g)$\nto this function is (F)-continuous.\n\\item[($ii$)] For any $h\\in H$, one can find Fr\\'echet subspaces\n$\\tilde F\\subset F$ and $\\tilde G\\subset G$ such that\n$h(x,\\cdot)\\in \\tilde G$ for every $x\\in X$ and the\nfunction $h_v(x)=\\dv{v}{h(x,\\cdot)}$ belongs to $\\tilde F$\nfor every $v\\in \\tilde G'$.\n\\end{itemize}\nThen $\\Phi$ induces the topological isomorphism\n$F\\tilde\\otimes \\,G\\simeq H$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\Phi_*\\colon F\\tilde{\\otimes}\\, G\\to H$ be the\ncontinuous linear mapping determined by $\\Phi$. For $\\tilde\nF\\in\\mathfrak F_F$ and $\\tilde G\\in\\mathfrak F_G$, let\n$\\Phi_*^{\\tilde F,\\tilde G}$ be the continuous linear\nmapping $\\tilde F\\hat\\otimes\\, \\tilde G\\to H$ determined by\nthe restriction of $\\Phi$ to $\\tilde F\\times \\tilde G$. We\nhave\n\\begin{equation}\\label{eq}\n\\Phi_*^{\\tilde F,\\tilde G}= \\Phi_*\\,\\rho_{\\tilde F,\\tilde\nG},\n\\end{equation}\nwhere $\\rho_{\\tilde F,\\tilde G}$ is the canonical mapping\nfrom $\\tilde F\\hat\\otimes\\, \\tilde G$ to\n$F\\tilde{\\otimes}\\, G$. Let $\\mathcal N$ be the subset of\n$\\mathfrak F_G$ consisting of all nuclear Fr\\'echet\nsubspaces of $G$. By the assumption, $\\mathcal N$ is\ncofinal in $\\mathfrak F_G$. Let $h\\in H$. Condition~(ii)\nmeans that $h\\in\\Pi(\\tilde F,\\tilde G)$ for some $\\tilde\nF\\in\\mathfrak F_F$ and $\\tilde G\\in\\mathfrak F_G$. Let\n$\\tilde G_1\\in \\mathcal N$ be such that $\\tilde\nG\\prec\\tilde G_1$. Then we have $\\Pi(\\tilde F,\\tilde\nG)\\subset \\Pi(\\tilde F,\\tilde G_1)$ and, therefore, $h\\in\n\\Pi(\\tilde F,\\tilde G_1)$. By Lemma~\\ref{l1}, we have\n$\\Pi(\\tilde F,\\tilde G_1)=\\mathrm{Im}\\, \\Phi_*^{\\tilde\nF,\\tilde G_1}$ and in view of~(\\ref{eq}) we conclude that\n$h\\in \\mathrm{Im}\\,\\Phi_*$. Thus, $\\Phi_*$ is surjective.\nBy Lemma~\\ref{l1}, the mappings $\\Phi_*^{\\tilde F,\\tilde\nG}$ are injective for any $\\tilde F\\in\\mathfrak F_F$ and\n$\\tilde G\\in \\mathcal N$. Since $\\mathcal N$ is cofinal in\n$\\mathfrak F_G$, it follows from~(\\ref{eq}) that the\nrestriction of $\\Phi_*$ to the image of $\\rho_{\\tilde\nF,\\tilde G}$ is injective for any $\\tilde F\\in\\mathfrak\nF_F$ and $\\tilde G\\in\\mathfrak F_G$. This implies the\ninjectivity of $\\Phi_*$ because the spaces\n$\\mathrm{Im}\\,\\rho_{\\tilde F,\\tilde G}$ cover $F\\tilde\n\\otimes\\, G$. We have thus proved that $\\Phi_*$ is a\none-to-one mapping from $F\\tilde \\otimes\\, G$ onto $H$.\nSince both $F\\tilde \\otimes\\, G$ and $H$ are (LF)-spaces,\nwe can apply the open mapping theorem (\\cite{Grot},\nTh\\'eor\\`eme~B) and conclude that $\\Phi_*$ is a topological\nisomorphism. The theorem is proved.\n\\end{proof}\n\n\\section{Applications to spaces of analytic functions}\n\\label{s4}\n\nLet $U$ be a cone in ${\\mathbb R}^k$. We say that a cone $W$ is a\nconic neighborhood of $U$ if $W$ has an open\nprojection\\footnote{The projection $\\Pr W$ of a cone\n$W\\subset {\\mathbb R}^k$ is by definition the intersection of $W$\nwith the unit sphere in ${\\mathbb R}^k$; the projection of $W$ is\nmeant to be open in the topology of this sphere. } and\ncontains $U$.\n\n\\begin{definition} \\label{d3}\nLet $\\beta(s)$ be continuous monotone indefinitely\nincreasing convex function on the semi-axis $s\\geq 0$ and\n$U$ be a nonempty cone in ${\\mathbb R}^k$. For $B>0$, the Fr\\'echet\nspace $\\mathcal E^{\\beta, B}(U)$ consists of entire\nanalytic functions on ${\\mathbb C}^k$ having the finite norms\n\\[\n\\|f\\|_{U,N,B'}=\\sup_{z=x+iy\\in{\\mathbb C}^k}|f(z)|(1+|x|)^N\n\\exp\\left(-\\beta(B'|y|)-\\beta(B'\\delta_U(x))\\right)\n\\]\nfor any $B'>B$ and any nonnegative integer $N$, where\n$\\delta_U(x)=\\inf_{x'\\in U}|x-x'|$ is the distance from $x$\nto $U$. The space $\\mathcal E^{\\beta}(U)$ is defined by the\nrelation $\\mathcal E^{\\beta}(U)=\\bigcup_{B>0,\\,W\\supset U}\n\\mathcal E^{\\beta, B}(W)$, where $W$ runs over all conic\nneighborhoods of $U$ and the union is endowed with the\ninductive limit topology.\n\\end{definition}\n\nThe spaces $\\mathcal E^\\beta({\\mathbb R}^k)$ proved to be useful for\nthe analysis of infinite series in the Wick powers of free\nfields converging to nonlocal fields~\\cite{SS2}. If\n$\\beta(s)=s^{1\/(\\nu-1)}$, $\\nu<1$, then $\\mathcal\nE^\\beta({\\mathbb R}^k)$ coincides with the Gelfand-Shilov space\n$S^\\nu({\\mathbb R}^k)$ (see~\\cite{GS} for the definition and\nproperties of Gelfand-Shilov spaces; to avoid confusion, we\nuse the notation $S^\\nu$ instead of the standard\n$S^\\beta$). The spaces $\\mathcal E^{\\beta}(U)$ over cones\nare introduced to describe the localization properties of\nanalytic functionals belonging to $\\mathcal\nE^{\\prime\\beta}({\\mathbb R}^k)$. More precisely, a closed cone\n$K\\subset {\\mathbb R}^k$ is called a carrier cone of $u\\in \\mathcal\nE^{\\prime\\beta}({\\mathbb R}^k)$ if $u$ has a continuous extension to\n$\\mathcal E^{\\prime\\beta}(K)$. The notion of carrier cone\nreplaces the notion of support of a generalized function\nfor elements of $E^{\\prime\\beta}({\\mathbb R}^k)$. In particular,\nevery $u\\in \\mathcal E^{\\prime\\beta}({\\mathbb R}^k)$ has a uniquely\ndetermined minimal carrier cone (this was proved\nin~\\cite{Soloviev} for the case of the space $S^0$; the\ngeneral case can be treated in the same way using the\nestimates for plurisubharmonic functions obtained\nin~\\cite{Smirnov}).\n\nHere, we shall prove a kernel theorem for the spaces\n$\\mathcal E^{\\beta}(U)$. For this, we introduce, in\naddition to $\\mathcal E^{\\beta}(U)$, similar spaces\nassociated with finite families of cones.\n\n\\begin{definition} \\label{d4}\nLet $U_1,\\ldots,U_n$ be nonempty cones in\n${\\mathbb R}^{k_1},\\ldots,{\\mathbb R}^{k_n}$ respectively. We define the\nspace $\\mathcal E^\\beta(U_1,\\ldots,U_n)$ by the relation\n\\[\n\\mathcal E^\\beta(U_1,\\ldots,U_n) =\n\\bigcup_{B>0,\\,W_1\\supset U_1,\\ldots,W_n\\supset U_n}\n\\mathcal E^{\\beta,B}(W_1\\times\\ldots\\times W_n),\n\\]\nwhere the union is taken over all conic neighborhoods\n$W_1,\\ldots,W_n$ of $U_1,\\ldots,U_n$ and is endowed with\nthe inductive limit topology.\n\\end{definition}\n\n\nIn what follows, we assume for definiteness that the norm\n$|\\cdot|$ on ${\\mathbb R}^k$ is uniform: $|x|=\\max_{1\\leq j\\leq\nk}|x_j|$. For any cone $U$, the space\n$\\mathcal{E}^{\\beta,B}(U)$ belongs to the class of the\nspaces $\\mathcal H(M)$ introduced in~\\cite{SS1}. The space\n$\\mathcal H(M)$ is defined\\footnote{The definition of\n$\\mathcal H(M)$ given here is slightly less general than\nthat in~\\cite{SS1}, but it is sufficient for our purposes.}\nby a family $M=\\{M_\\gamma\\}_{\\gamma\\in\\Gamma}$ of strictly\npositive continuous functions on ${\\mathbb C}^k$ and consists of all\nentire analytic functions on ${\\mathbb C}^k$ with the finite norms\n\\[\n\\sup_{z\\in {\\mathbb C}^k} M_\\gamma(z)|f(z)|.\n\\]\nIt is supposed that (a) for every\n$\\gamma_1,\\gamma_2\\in\\Gamma$, one can find\n$\\gamma\\in\\Gamma$ and $C>0$ such that $M_\\gamma\\geq C\n(M_{\\gamma_1}+M_{\\gamma_2})$, and (b) there is a countable\nset $\\Gamma'\\subset \\Gamma$ with the property that for\nevery $\\gamma\\in \\Gamma$, one can find $\\gamma'\\in \\Gamma'$\nand $C>0$ such that $C M_\\gamma\\leq M_{\\gamma'}$ (a family\nof functions on ${\\mathbb C}^k$ satisfying~(a) and ~(b) will be\ncalled a defining family of functions). Given a cone\n$U\\subset {\\mathbb R}^k$ and $B>0$, we define the family $M^{U,B}$\nof functions on ${\\mathbb C}^k$ indexed by the set ${\\mathbb Z}_+\\times\n(B,\\infty)$ by the relation\n\\begin{equation}\\label{xxx}\nM^{U,B}_{N,B'}(x+iy)= (1+|x|)^{N}\n\\exp\\left(-\\beta(B'|y|)-\\beta(B'\\delta_U(x))\\right),\\quad\nN\\in{\\mathbb Z}_+,\\,B'>B.\n\\end{equation}\nThen all above conditions are satisfied and we have\n\\begin{equation}\\label{id}\n\\mathcal H(M^{U,B})= \\mathcal{E}^{\\beta,B}(U).\n\\end{equation}\nBy Lemma~12 of~\\cite{SS1}, $\\mathcal H(M)$ is a nuclear\nFr\\'echet space if the following conditions are satisfied:\n\\begin{itemize}\n\\item[(I)] For every $\\gamma\\in \\Gamma$, there\nis $\\gamma'\\in \\Gamma$ such that\n$M_\\gamma(z)\/M_{\\gamma'}(z)$ is integrable on ${\\mathbb C}^k$ and\ntends to zero as $|z|\\to\\infty$.\n\\item[(II)] For every $\\gamma\\in \\Gamma$, there are\n$\\gamma'\\in \\Gamma$, a neighborhood of the origin $\\mathcal\nB$ in ${\\mathbb C}^k$, and $C>0$ such that $M_\\gamma(z)\\leq\nCM_{\\gamma'}(z+\\zeta)$ for any $z\\in {\\mathbb C}^k$ and $\\zeta\\in\n\\mathcal B$.\n\\end{itemize}\nIt is straightforward to verify that the family $M^{U,B}$\nsatisfies~(I) and~(II). The space\n$\\mathcal{E}^{\\beta,B}(U)$ is therefore nuclear for any\n$B>0$ and cone $U\\subset{\\mathbb R}^k$.\n\nLet $M=\\{M_\\gamma\\}_{\\gamma\\in \\Gamma}$ and\n$N=\\{N_\\omega\\}_{\\omega\\in \\Omega}$ be defining families of\nfunctions on ${\\mathbb C}^{k_1}$ and ${\\mathbb C}^{k_2}$ respectively. We\ndenote by $M\\otimes N$ the family formed by the functions\n\\[\n(M\\otimes N)_{\\gamma\\omega}(z_1,z_2)=M_\\gamma(z_1)\nN_\\omega(z_2),\\quad (\\gamma,\\omega)\\in \\Gamma\\times \\Omega.\n\\]\nClearly, if $M\\otimes N$ is a defining family of functions.\nThe following result was proved in~\\cite{SS1}.\n\n\\begin{lemma}\n \\label{l3}\nLet $M=\\{M_\\gamma\\}_{\\gamma\\in \\Gamma}$ and\n$N=\\{N_\\omega\\}_{\\omega\\in \\Omega}$ be defining families of\nfunctions on ${\\mathbb C}^{k_1}$ and ${\\mathbb C}^{k_2}$ respectively and let\n$h\\in \\mathcal H(M\\otimes N)$. Suppose $N$ satisfies\nconditions~{\\rm (I)} and~{\\rm (II)}. Then $h(z,\\cdot)\\in\n\\mathcal H(N)$ for every $z\\in {\\mathbb C}^{k_1}$ and the function\n$h_v(z)=\\dv{v}{h(z,\\cdot)}$ belongs to $\\mathcal H(M)$ for\nall $v\\in \\mathcal H'(N)$.\n\\end{lemma}\n\nLet $V_1\\subset{\\mathbb R}^{k_1}$ and $V_1\\subset{\\mathbb R}^{k_1}$ be\nnonempty cones. It follows from~(\\ref{xxx}) and the\nmonotonicity of $\\beta$ that\n\\begin{equation}\\nonumber\nM^{V_1\\times V_2,2B}_{N,2B'}(z_1,z_2)\\leq\nM^{V_1,B}_{N,B'}(z_1)\\,M^{V_2,B}_{N,B'}(z_2) \\leq\nM^{V_1\\times V_2,B}_{2N,B'}(z_1,z_2),\\quad\nz_{1,2}\\in{\\mathbb C}^{k_{1,2}},\n\\end{equation}\nfor any $B'>B>0$ and $N\\in{\\mathbb Z}_+$. We hence have continuous\ninclusions\n\\begin{equation}\\label{zzz}\n\\mathcal E^{\\beta,B}(V_1\\times V_2)\\subset \\mathcal\nH(M^{V_1,B}\\otimes M^{V_2,B})\\subset \\mathcal\nE^{\\beta,2B}(V_1\\times V_2).\n\\end{equation}\n\n\\begin{lemma}\\label{l4}\nLet $U_1,\\ldots,U_n$ be nonempty cones in\n${\\mathbb R}^{k_1},\\ldots,{\\mathbb R}^{k_n}$ respectively. Let $1\\leq m0$ and $W_1,\\ldots,W_n$ be conic neighborhoods of\n$U_1,\\ldots,U_n$ respectively. Let $V_1 =\nW_1\\times\\ldots\\times W_m$ and $V_2 =\nW_{m+1}\\times\\ldots\\times W_n$. In view of~(\\ref{id}) it\nfollows from~(\\ref{zzz}) that $\\Phi$ induces a continuous\nbilinear mapping $\\mathcal E^{\\beta,B}(V_1)\\times \\mathcal\nE^{\\beta,B}(V_2)\\to \\mathcal E^{\\beta,B}(V_1\\times V_2)$.\nThis implies that $\\Phi$ is (F)-continuous and~(i) is\nfulfilled. Let $h\\in \\mathcal\nE^{\\beta,B}(W_1\\times\\ldots\\times W_n)$. It follows\nfrom~(\\ref{zzz}) and Lemma~\\ref{l3} that (ii) will be\nsatisfied if we set $\\tilde F = \\mathcal E^{\\beta,B}(V_1)$\nand $\\tilde G = \\mathcal E^{\\beta,B}(V_2)$. The lemma is\nproved.\n\\end{proof}\n\nThe next result follows from Lemma~\\ref{l4} and\nTheorem~\\ref{t0} by induction on $n$.\n\n\\begin{theorem}\\label{t2}\nLet $U_1,\\ldots,U_n$ be nonempty cones in\n${\\mathbb R}^{k_1},\\ldots,{\\mathbb R}^{k_n}$ respectively and $\\Phi\\colon\n\\mathcal E^\\beta(U_1)\\times\\ldots\\times \\mathcal\nE^\\beta(U_n)\\to \\mathcal E^\\beta(U_1,\\ldots,U_n)$ be the\nbilinear mapping defined by the relation\n\\[\n\\Phi(f_1,\\ldots,f_n)(z_1,\\ldots,z_n) = f_1(z_1)\\ldots\nf_n(z_n).\n\\]\nThen $\\Phi$ is (F)-continuous and induces the topological\nisomorphism\n\\[\n\\mathcal E^\\beta(U_1)\\tilde\\otimes\\ldots\\tilde\\otimes\n\\,\\mathcal E^\\beta(U_n)\\simeq \\mathcal\nE^\\beta(U_1,\\ldots,U_n).\n\\]\n\n\\end{theorem}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}