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+{"text":"\\section{Introduction}\n\nFuture high-energy lepton colliders will measure the top quark\nproperties like its mass, width and couplings with an unprecedented\naccuracy and use the top quark as a means to search for new physics\nbeyond the Standard Model. Here, we discuss the matching of\nfixed-order QCD next-to-leading order (NLO-QCD calculations for\nexclusive $e^+e^- \\to W^+ b W^- \\bar{b}$ final states in the\ncontinuum, based on~\\cite{Nejad:2016bci}, with a resummed calculation\nin the threshold region, where fixed-order perturbation theory in the\nstrong coupling $\\alpha_s$ is not a good approximation anymore, but\nthe top velocity $v$ is an additional expansion parameter and\nCoulomb-singular terms $\\sim (\\alpha_s\/v)^n$ and (ideally also) large\nlogarithms $\\sim(\\alpha_s \\log v)^n$ have to be resummed. In\nSec.~\\ref{sec:matching}, after reviewing our calculational framework\nand the details of the continuum calculation for completeness, we\ndiscuss, based on~\\cite{Bach:2017ggt}, a previously known\nnon-relativistic QCD (NRQCD) effective field theory setup to compute a\nform factor accounting for the resummation of the threshold-singular\nterms at NLL accuracy, implemented it in the fixed-order calculation\nand matched the result to the QCD-NLO cross section in the transition\nregion between threshold and continuum. We thus obtained a\nfully-differential cross section, which gives reliable predictions for\nall center-of-mass energies. Depending on how inclusive the process\nis, we achieve LL + QCD-NLO (for very exclusive processes) or NLL +\nQCD-NLO precision (for inclusive processes) in the threshold\nregion. Finally, we conclude in Sec.~\\ref{sec:conclusions}. \n\n\n\\section{QCD-NLO (fixed-order) \\& Threshold Matching}\n\\label{sec:matching}\n\nIn the continuum, i.e. away from the threshold, QCD corrections are\nproperly described by fixed-order relativistic QCD-NLO perturbation\ntheory for the off-shell top pair production. For that purpose, we\nstudy either the process $e^+e^- \\to W^+ b W^- \\bar{b}$ or $e^+e^- \\to\n\\ell^+ e^- \\bar{\\nu}_e \\mu^+\\nu_\\mu b \\bar{b}$ including leptonic $W$\ndecays. Within the full four- or six-particle final state, there\nare double-resonant diagrams included (involving a top and an anti-top\npropagator), single-resonant diagrams and non-resonant irreducible\nbackground processes. To calculate total and fully differential\nQCD-NLO corrections for the top production processes, we take the\n\\texttt{WHIZARD} framework for (QCD-)NLO\nprocesses. \\texttt{WHIZARD}~\\cite{Kilian:2007gr} is a \nmulti-purpose event generator with its own matrix-element\ngenerator for tree-level amplitudes,\n\\texttt{O'Mega}~\\cite{Moretti:2001zz,Nejad:2014sqa} with support\nfor a plethora of models like\ne.g. supersymmetry~\\cite{Ohl:2002jp}. Users can use external models by\nthe interface to\n\\texttt{FeynRules}~\\cite{Christensen:2010wz}. \\texttt{WHIZARD} uses\nthe color-flow formalism~\\cite{Kilian:2012pz}, and it comes with its\nown parton shower implementation~\\cite{Kilian:2011ka}. QCD-NLO\napplications within \\texttt{WHIZARD} started with a hard-coded\nimplementation for the production of $b$ jets at\nLHC~\\cite{Binoth:2009rv,Greiner:2011mp}, while matching \nbetween resummed terms and fixed-order calculations have been tackled\nby combining fixed-order electroweak corrections to chargino\nproduction at the ILC with an all-order QED initial-state structure\nfunction~\\cite{Kilian:2006cj,Robens:2008sa}. \\texttt{WHIZARD} is also\nable to do automatic POWHEG matching for $e^+e^-$ \nprocesses~\\cite{Reuter:2016qbi}.\n\n\\texttt{WHIZARD} uses FKS subtraction~\\cite{Frixione:1995ms} and\ngenerates the automatically generates the phase space for all singular\nemission regions. Virtual matrix elements, color-correlated and\nspin-correlated matrix elements for the collinear and soft splittings\nare taken from the one-loop provider (OLP) program\n\\texttt{OpenLoops}~\\cite{Openloops}. The complex mass scheme is used,\nleading to a complex weak mixing angles. The input values are as follows: $m_W =\n80.385$ GeV, $m_Z = 91.1876$ GeV, $m_t = 173.2$ GeV, $m_H = 125$\nGeV. We use massive $b$-quarks of mass $m_b = 4.2$ GeV. Widths need to\nbe calculated at the same order and in the same scheme than the\nscattering process in order to guarantee properly normalized branching\nratios: $\\Gamma_Z^{\\text{LO}} = 2.4409$ GeV,  $\\Gamma_Z^{\\text{NLO}} =\n2.5060$ GeV, $\\Gamma_W^{\\text{LO}} = 2.0454$ GeV,\n$\\Gamma_W^{\\text{NLO}} = 2.0978$ GeV,  $\\Gamma_{t\\to Wb}^{\\text{LO}} =\n1.4986$ GeV, $\\Gamma_{t\\to Wb}^{\\text{LO}} = 1.3681$ GeV. As the\nmatrix elements for the full off-shell processes contain narrow\nresonances, particularly the $H\\to bb$ resonance, we use a\nresonance-aware version of the FKS subtraction formalism to make sure\nthat cancellations between real emissions and subtraction terms do\ncancel though the real emission could shift the kinematics on or off\nthe resonance compared to Born kinematics. This resonance-aware\ntreatment is automatically done in \\texttt{WHIZARD}. As we are using\nmassive $b$-quarks, no cuts are necessary for the  process $e^+e^- \\to\nW^+W^- b \\bar{b}$. The integrations for the full QCD-NLO are\nvery stable. We did two independent own integrations with the serial\nand the non-blocking MPI-parallelizable version~\\cite{MPI} of\n\\texttt{VAMP}~\\cite{Ohl:1998jn} inside \\texttt{WHIZARD}. \n\nFor the QCD-NLO corrections, we take the top mass as renormalization\nscale. The scale variations for the process $e^+e^- \\to W^+ b W^-\n\\bar{b}$ is very small, at the level of two per cent. After one has\nreplaced the top width in the matrix elements by a running top width\n$\\Gamma_t(\\mu_R)$ , the scale variations for the on-shell process\n$e^+e^- \\to t\\bar{t}$ behave the same way as for the off-shell\nprocess. \nThe \\texttt{WHIZARD} infrastructure immediately enables QCD-NLO\ncalculations\/simulations for polarized beams, to include QED\ninitial-state photon radiation as well as collider-specific \nbeamspectra.\n\nA kinematic fit to the shape of the rising of the cross section at the\ntop threshold is believed to be the most precise method to measure the\ntop quark mass with an ultimate precision of 30-80 MeV. For this the\nsystematic uncertainties of the experimental measurement -- especially\nthe details of the beam spectrum -- as well as the theoretical\nuncertainties have to be well under control. As shown above, close to\nthe kinematical threshold for the on-shell production of a $t\\bar{t}$ \n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=.44\\textwidth]\n                    {matched_nlofull_nlodecay_newscale_symm_comb}\n                    \\quad\n    \\includegraphics[width=.44\\textwidth]\n                    {matched_nlofull_nlodecay_newscale_symm_comb_isr}    \n  \\end{center}\n  \\caption{Matched NRQCD-NLL + QCD-NLO \n    calculation without (left) and with (right) QED ISR. The dashed\n    vertical line is the value of twice $M^{1\\text{S}}$. Blue is the\n    fixed QCD-NLO calculation, red is the fully matched\n    calculation. The matched calculation has a full envelope over\n    (symmetrized) scale uncertainties as well as variations over\n    switch-off functions. }\n  \\label{fig:threshold_full}  \n\\end{figure}\npair, fixed-order perturbation theory is not a good\napproximation. Very close to threshold, the effective field theory of\n(v\/p)NRQCD separates the hard scale $m_t$, the soft scale given by the\ntop momentum of the non-relativistic top quark with velocity $v$, $m_t\nv$ and the ultrasoft scale, given by the kinetic energy of the top\nquark, $m_t v^2$ and allows to resum large logarithms of $v$\nwith $\\alpha_s \\sim v \\sim 0.1$ close to\nthreshold. \"Fixed-order\" calculations resumming only Coulomb\nsingularities, but no velocity logarithms, for the totally inclusive \n$t\\bar{t}$ production have been carried out in NRQCD to\nNNNLO~\\cite{Beneke:2015kwa}. The large velocity logarithms have been\nresummed to next-to-next-leading logarithmic\n(NNLL)~\\cite{Hoang:2013uda} order\n(cf. also~\\cite{Hoang:2001mm,Pineda:2006ri} for predictions not\ncontaining the full set of NNLL ultrasoft logarithms). These NRQCD \ncalculations, based on the optical theorem, hold only for the\ntotal inclusive cross section in a narrow window around the\n$t\\bar{t}$ threshold. Here, we combine and match the NLL\nNRQCD-resummed process close to the top threshold with \nthe fixed-order (relativistic) QCD-NLO process in the continuum. By a\ncarefully performed matching procedure, our approach smoothly\ninterpolates between threshold region and continuum, and\nallows to study all kinds of differential distributions. \n\nThe matching is embedded into the \\texttt{WHIZARD}-\\texttt{OpenLoops}\nQCD-NLO fixed-order framework discussed above. The NLL resummed NRQCD\ncontributions are included in terms of (S-\/P-wave) form factors to the\n(vector\/axial vector) $\\gamma\/Z-t-\\bar{t}$ vertex. These form factors\nare obtained from the numerical solution of Schr\\\"odinger-type\nequations for the NLL Green functions computed by the \n\\texttt{Toppik}~\\cite{Jezabek:1992np,Harlander:1994ac,Hoang:1999zc}\ncode, which is included in \\texttt{WHIZARD},  for technical details\ncf.~\\cite{Bach:2017ggt}. In order to avoid double-counting between \nthe fixed-order QCD-NLO part and the resummed NLL-NRQCD part, one has\nto expand the form factors to first order in $\\alpha_s$ and subtract\nthose pieces. As the NRQCD resummed calculations are only available\nfor the top-vector and axial-vector currents, this removal of\ndouble-counting has to be done in a factorized approach within a\ndouble-pole approximation. In order to maintain gauge-invariance of\nthe factorized amplitudes, an on-shell projection of the exclusive\nfinal states to the top mass shell is performed, for details\ncf.~\\cite{Bach:2017ggt}. The implementation inside \\texttt{WHIZARD}\nhas been \n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=.40\\textwidth]\n                    {matched_nlofull_nlodecay_newscale_scalevars_symm}\n                    \\quad\n    \\includegraphics[width=.44\\textwidth]\n                    {BWp-inv}    \n  \\end{center}\n  \\caption{Left panel: Matched NRQCD-NLL + QCD-NLO total cross section\n    as on the left of Fig.~\\ref{fig:threshold_full}, but for a single\n    choice of switch off-function. $h$ and $f$ are renormalization\n    scale parameters as defined in\n    \\cite{Bach:2017ggt,Hoang:2013uda}. The grey bands display the\n    corresponding scale \n    variations with and without symmetrization. Right panel: $Wb$\n    invariant mass distribution at threshold ($\\sqrt{s}=344$ GeV) as\n    obtained with \\texttt{WHIZARD}. The red line represents the full\n    NRQCD-NLL + QCD-NLO matched, and the blue line the pure QCD-NLO\n    result. The associated bands are generated by the same scale\n    variations as in the left panel, here without symmetrization.}\n  \\label{fig:match_diff}\n\\end{figure}\nvalidated with analytical calculations for different invariant mass\ncuts on the reconstructed top quarks from Ref.~\\cite{Hoang:2010gu}.\n\nFor larger top velocity ($v \\gtrsim 0.4$) only the relativistic\nQCD-NLO result is valid. We define a switch-off function that smoothly\ninterpolates between the two regions. The possibility to vary this\narbitrary function and its parameters adds another theory uncertainty\nto the different scale variations~\\cite{Bach:2017ggt}. The\nresults of our matching procedure are displayed in\nFig.~\\ref{fig:threshold_full}. These plots \nshow the total inclusive cross section for the process $e^+e^- \\to W^+ b W^-\n\\bar{b}$, in the left panel without and in the right panel with QED\ninitial-state radiation (ISR). The dashed vertical line gives the\nvalue for $2 M^{1\\text{S}}$. The 1S mass $M^{1\\text{S}}$ is defined as\nhalf of the perturbative mass of a would-be 1S toponium state and\nrepresents a renormalon free short-distance mass, which we treat as an\ninput parameter in \\texttt{WHIZARD}. The blue\nline shows the QCD-NLO cross section including scale variations in the\nblue shaded areas. The red curve shows the NRQCD-NLL + QCD-NLO\nresult, while the shaded band contains all (symmetrized) scale\nvariations of the hard, soft and ultrasoft\nfactorization\/renormalization scales according\nto~\\cite{Hoang:2013uda} as well as variations of the switch-off\nfunction to a reasonable extent~\\cite{Bach:2017ggt}. The dotted black\nline shows the matched results without applying a switch-off function\nto the factorized NRQCD terms which deviates above threshold from\nthe relativistic QCD-NLO result. In Fig.~\\ref{fig:match_diff}, left\npanel, we see the matched result in the threshold region for a\nsingle choice  of switch-off parameters, but scale variations over the\nfull two-dimensional renormalization parameter range defined\nin~\\cite{Hoang:2013uda}. This shows that the \nscale variation bands for the resummed NLL result in the threshold\nregion are highly asymmetric with respect to the central value which\nmotivates to apply a symmetrization of the error bands around the\ncentral value. This symmetrization is also shown in\nFig.~\\ref{fig:threshold_full}. In the right panel of\nFig.~\\ref{fig:match_diff} we show as an example for a differential\ndistribution the invariant mass of the $W-b$ jet system. Blue is the\nfixed-order QCD-NLO distribution, while red is the fully matched\ndistribution including scale variations, here un-symmetrized. The\nratio plot in the bottom does not show a K factor, but the ratio of\nthe matched result to the QCD-NLO fixed order result. It shows an\nenhancement in the top mass peak due to threshold resummation by a\nfactor of 10-12.\n\n \n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn order to be able to study experimental event selections as well as\ndifferential distributions, we presented a matched threshold\ncalculation that smoothly interpolates the threshold region\ndescribed by non-relativistic QCD to the relativistic QCD-NLO\ncalculation. It constitutes the highest precision available at the\nlevel of the completely exclusive final state. Any of the presented\ndifferential distributions depending on the top mass may serve as a\ndifferent means to determine the top mass. We were not accomplishing\nthis task here, but rather showed a framework as a proof-of-principle\nof the matching procedure between threshold and continuum. For the\nproper matching to the continuum the fixed-order QCD-NLO calculations\nfor top-quark pair production including top and (leptonic) $W$ decays\nhave been done. All of this has been done in the  QCD-NLO framework of\nthe \\texttt{WHIZARD} event generator which allows to include all\nimportant physics of a lepton collider like polarization, QED ISR\nradiation and non-trivial beam spectra. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\\constraint{Paying for something purchased online cannot happen after receiving\nit}, \\constraint{The average time for a package to be delivered after purchase\nis between two and five days}, and \\constraint{The same shipping car can be\nused for delivering packages at most seven times per day} are various examples\nof constraints that are posed over business processes.\nThese constraints can be very general and can refer to a variety of requirements~\\cite{LyMMRA15}.  Non-compliance of certain constraints can be very costly\nand risky, so compliance checking\\footnote{One should differentiate between the\nproblems of \\emph{verification} and \\emph{compliance checking}.  Our focus is\non compliance checking: checking properties of execution logs.\nOn the other hand, in verification, one seeks to determine whether all possible\nexecutions of some given process model satisfy some property.  The kind of\nconstraints we are dealing with in this paper are typically quite expressive,\nso that verification would be undecidable and one needs to resort to compliance\nchecking.\nThere is also a third problem, \\emph{conformance checking}~\\cite{Aalst12},\nwhere we check that a given execution follows a given process model.  This\nproblem is outside the scope of this paper, although, formally speaking,\nconformance checking could be viewed as a kind of compliance checking.}\nand monitoring are of utmost importance to the enterprise~\\cite{WinterSR20}.\n\n\nConstraints can be very simple in terms of their scope, i.e., the process\ninstances they involve, and the conditions they impose such as\n\\constraint{Conducting a patient's surgery must be preceded by examining the\npatient} or \\constraint{Paying for something purchased online cannot happen\nafter receiving it}.  Those are examples of constraints to be enforced on activity instances belonging to\nthe same process instance.  This type of constraint is often referred to as\n\\emph{intra-instance}~\\cite{WarnerA06,WinterSR20}.  On the other\nhand, there are constraints that can be much more complex, both in their scope\nand in the conditions they impose.\nSpecifically, constraints where the scope spans multiple process\ninstances, or combinations of entities involved in multiple process instance,\nhave been referred to as \\emph{inter-instance}~\\cite{WarnerA06,MontaliMCMA13}, or, more\nrecently, \\emph{instance-spanning} constraint\n(ISC)~\\cite{FdhilaGRMI16,Rinderle-MaGFMI16}.  \\constraint{The same shipping\ncar can be used for delivering packages at most seven times per day} and\n\\constraint{Packages that are delivered to the same neighbourhood on the same\nday must be delivered by the same shipping car} are examples of ISC.\n\n\nIt should be noted, however, that whether a constraint is intra-instance or\ninstance-spanning is a relative matter; it depends on the design of the process\nmodel.  Indeed, in general, a single process may require sophisticated control-flow structures involving iterations and  multi-instance activities.  Figures~\\ref{fig:p1-models} and~\\ref{fig:p2-models} give\nsimple illustrations of the relative nature of ``intra'' versus ``inter''\ninstance.  Thus, while our focus is on studying ISC, similar features would be required when checking intra-instance constraints on such complex processes.  In what\nfollows, we will hence just talk about (process) \\emph{constraints} in general.\n\\begin{figure}\n     \\centering\n     \\begin{subfigure}[t]{0.34\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/p1model3.png}\n         \\caption{}\n         \\label{fig:p1model1}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[t]{0.64\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/p1STmodel3.png}\n         \\caption{}\n         \\label{fig:p1model2}\n     \\end{subfigure}\n\\caption{Consider the process $P_{AB}$ in Figure~\\ref{fig:p1model1}.\n\\constraint{There can be at most three orders per customer} is an example of\nan ISC~when posed against multiple instances of $P_{AB}$.  On the other hand,\nwhen the same constraint is posed against the iterative model in\nFigure~\\ref{fig:p1model2}, then it would be an intra-instance constraint.\nNote that $R$ added in Figure~\\ref{fig:p1model2} would resemble a\nreceive order activity.}\n\\label{fig:p1-models}\n\\end{figure}\n\\begin{figure}\n     \\centering\n     \\begin{subfigure}[t]{0.35\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/p2model2.png}\n         \\caption{}\n         \\label{fig:p2model1}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[t]{0.6\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/p2STmodel2.png}\n         \\caption{}\n         \\label{fig:p2model2}\n     \\end{subfigure}\n\\caption{Consider the two separate processes $P_{AB}$ and $P_{C}$ in\nFigure~\\ref{fig:p2model1}.  \\constraint{For every instance of $P_{AB}$, an\ninstance of $P_{C}$ must be instantiated for the same customer} is an example\nof ISC~that relates instances of the two processes based on a\ncommon attribute.  On the other hand, when the two processes are subprocesses\nof a single process as in Figure~\\ref{fig:p2model2}, the constraint would be\nan intra-instance constraint.}\n\\label{fig:p2-models}\n\\end{figure}\n\n\nConstraints must be checked against execution logs, which are files or\ndatabases holding data about past and current executions of all process\ninstances in the enterprise.  Two types of compliance checking are commonly\ndistinguished:\n\\begin{description}\n  \\item[Post-mortem checking] targets only full (completed) executions on a\n  historical log.\n  \\item[Compliance monitoring] checks the execution of the currently running\n  process instances, for a live log.\\footnote{Of course, in principle, post\n  mortem checking can also be performed within a live log.}\n\\end{description}\n\n\nThere is a striking similarity between the problem of compliance monitoring\nand the problem of \\emph{incremental view maintenance}, a well-researched\nproblem in databases~\\cite{GuptaMS93,GuptaM95,GuptaM99Book,ChirkovaY12Book,KennedyAK11,KochAKNNLS14}.\nThere, a \\emph{view} is the materialized result of a (possibly complex) query\nposed against a database.  The problem of view maintenance is then to keep the\nview consistent with its  definition under changes to the database.  In\ngeneral, these changes may be CRUD operations such as in particular insertions, deletions, or updates. This is perfectly in line with the execution of a process, where events witness the execution of tasks that, in turn, are typically associated to CRUD operations used to persist relevant event data in an underlying storage.\n\n\\emph{In this paper, we put forward the idea that incremental view maintenance is\napplicable to do compliance monitoring.}  To do so, we need to answer three\nquestions:\n\\begin{inparaenum}[(1)]\n\\item what is the database?\n\\item What are the updates?\n\\item What is the query?\n\\end{inparaenum}\n\nThe first two questions are easily answered: the log is the database, and events trigger insertions to the log to leave a trace about their occurrence. In this context, only insertion operations are thus used, to append the occurrence to an event to those occurred before. Every insertion, triggered by the execution of some activity instance, stores the corresponding event data in the database, including the timestamp of the event and which data payload it carries.\n\nWhat is then the query? To answer this question, we first need to indicate which dimensions we want to tackle when expressing constraints. Given the nature of ISC, we want to comprehensively tackle multi-perspective constraints dealing with several cases and their control-flow, time, and data dimensions. Instead of defining a specific constraint language that can accommodate such different perspectives, we directly employ full-fledged SQL for the purpose. Hence, a constraint is expressed as a query or, more precisely, an ensemble of queries, the number of which depends on whether compliance has to be assessed post-mortem or at runtime. In post-mortem checking, a constraint is expressed as a pair $(Q_{\\mathrm{case}}, Q_{\\mathrm{viol}})$ of two queries:\n\\begin{compactitem}\n      \\item $Q_{\\mathrm{case}}$ defines the ``scope\" of the constraint -- it returns the\n      set of cases to which the constraint applies;\n      \\item $Q_{\\mathrm{viol}}$ returns the subset of cases that violate the constraint.\n\\end{compactitem}\nAt runtime, we take inspiration from previous works in monitoring processes and temporal logic specifications \\cite{BaLS11,MMWA11,DDGM14}, and consider that each constraint may be, in principle, in one of four possible states: currently satisfied (resp., currently violated), that is, satisfied (resp., violated) by the current event data, but with a possible evolution of the system that will lead to violation (resp., satisfaction); permanently satisfied (resp., permanently violated), that is, satisfied (resp., violated) by the current event data, and staying in that state no matter which further events will occur in the future. For well-studied languages only tackling the control-flow dimension, such as variants of linear temporal logics over finite traces, such states can all be automatically characterized starting from a single formula formalizing the constraint of interest \\cite{DDMM22}. This is not the case for richer languages tackling also the data dimension, as in this setting reasoning on future continuations is in general undecidable \\cite{Del09,CDMP22}. We therefore opt for a pragmatic approach where constraint states are manually identified by the user through dedicated queries, as in  \\cite{MontaliMCMA13,CaMC19}. In particular, a monitored constraint comes with an ensemble of four queries:\n$(Q_{\\mathrm{case}}, Q_{\\mathrm{viol-perm}}, Q_{\\mathrm{viol-pending}}, Q_{\\mathrm{sat-pending}})$\n, where:\n   \\begin{compactitem}\n     \\item $Q_{\\mathrm{case}}$ is as before;\n     \\item $Q_{\\mathrm{viol-pending}}$ and $Q_{\\mathrm{sat-pending}}$ return the ``pending\" cases that, respectively, violate and satisfy the constraint at present, but for which upon acquisition of new events, their status may change.\n     \\item $Q_{\\mathrm{viol-perm}}$ returns permanent violations, i.e., those cases that irrevocably violate the constraint, that is, for which the constraint is currently violated and will stay so no matter which further events are collected.\n   \\end{compactitem}\n\n\nTo monitor constraints, we have used the system DBToaster for incremental query\nprocessing~\\cite{KennedyAK11,KochAKNNLS14,DBToasterWeb} in a\nproof-of-concept experiment.  We monitor a number of realistic constraints on\nexperimental data taken from the work by Winter et al.\\ \\cite{WinterSR20}.\nWe will present multiple examples demonstrating our approach in\nSections~\\ref{sec:examples} and~\\ref{sec:monitor} of the paper.\n\n\nImportantly, while we employ here the de-facto standard query language in databases,\nSQL, any other general data model (capable of suitably representing\nexecution logs) with a sufficiently expressive declarative query language would\ndo as well.  Examples are the RDF data model with SPARQL, or graph databases\nwith Cypher.  It should be noted, however, that incremental query processing is\nthe most advanced for SQL. Indeed, relational database management systems are\nstill the most mature database technology in development since the 1970s.\n\n\n\n\n\nThe rest of the paper is organized as follows.  In Section~\\ref{sec:examples},\nwe formalize our approach, discuss some examples of constraints and express\nthem as SQL~queries.  In Section~\\ref{sec:monitor}, we elaborate on the\nproblem of compliance monitoring.  In Section~\\ref{sec:exp}, we present the\nexperimental results.  In Section~\\ref{sec:seqdlog}, we discuss query language\nextensions for sequences that can be useful for an approach.  We conclude in\nSection~\\ref{sec:conc}.\n\n\n\\section{Post-mortem Analysis by Queries}\n\\label{sec:examples}\nWe capture a constraint as a query that returns the set of cases incurring in a violation.\n\n\\begin{definition}[Constraint, Post-mortem Variant]\nA \\emph{constraint} $C$ is a pair $(Q_{\\mathrm{case}}, Q_{\\mathrm{viol}})$ of queries\nwhere $Q_{\\mathrm{case}}$ is a \\emph{scoping query} that returns all the cases subject to the constraint $C$, while\n$Q_{\\mathrm{viol}}$ is a \\emph{violation detection query} that returns the violating cases such that $Q_{\\mathrm{viol}}$ is always a subset of $Q_{\\mathrm{case}}$.\n\\end{definition}\n\nThis definition settles our approach for post-mortem checking.  It is simply an\napplication of query answering, where the queries are asked against a\ndatabase instance (representing the execution log) that consists only of\ncompleted process instances.\nIn that case, when a tuple $t \\in Q_{\\mathrm{case}} \\setminus Q_{\\mathrm{viol}}$, then $t$ represents\na case that satisfies the constraint (i.e., $t \\in Q_{\\mathrm{sat}}$).\n\n\\begin{remark}\nNote that an equivalent approach is to represent the constraint as the pair of\nqueries $(Q_{\\mathrm{case}}, Q_{\\mathrm{sat}})$ instead.  The two approaches are interchangeable since\n$Q_{\\mathrm{sat}}$ can be defined in SQL~as follows (assuming that both $Q_{\\mathrm{case}}$ and $Q_{\\mathrm{viol}}$\nare materialized):\n\\lstinputlisting[language=SQL,style=mystyle]{SQL\/qsat.sql}\n\\end{remark}\n\n\\begin{example}\nFor an example of a constraint that its $Q_{\\mathrm{sat}}$ query is defined easier than its\n$Q_{\\mathrm{viol}}$ query, consider the constraint \\constraint{Activity B must be executed\nat least once in any process instance.} that is imposed over the process model\ngiven in Figure~\\ref{fig:pABAC}.  In this example, defining $Q_{\\mathrm{viol}}$ is more\ncomplicated as it requires negation.  On the other hand, $Q_{\\mathrm{sat}}$ is a simple\nexistentially quantified statement.\n\\end{example}\n\\begin{figure}\n  \\centering\n\\includegraphics[scale=0.32]{figures\/processAABBSTAC.png}\n\\caption{\\label{fig:pABAC} A process model of an example process.}\n\\end{figure}\n\nGuaranteeing that, for a constraint $(Q_{\\mathrm{case}}, Q_{\\mathrm{viol}})$, query $Q_{\\mathrm{viol}}$ always returns a subset of $Q_{\\mathrm{case}}$ is under the responsibility of the modeler. One way to ensure this is to write $Q_{\\mathrm{viol}}$ as a query that takes $Q_{\\mathrm{case}}$ and extends it with a filter to identify violations; however, alternative formulations may be preferred for readability and\/or performance needs.\n\n\\subsection{Database Schema}\n\nWe note that the structure of the database schema representing the data of the\nexecution log and how to get a database instance with the data are not issues\nthat we address in this paper.  These problems are orthogonal to what we\ndiscuss in this paper.  In the work by de Murillas et al. \\cite{MurillasRA19},\nthey showed how to \\emph{automatically} extract, transform, and load the log's\ndata from scattered sources into a database instance.  In the same work, they\ndevised a meta model that structures the database into a specific schema that\nis easily queried.\n\nThus, in our work, we assume that we can have a suitable database schema to\nwork with.  However, we will not be assuming the schema suggested by de\nMurillas et al. as it is very comprehensive, also integrating issues such as\nversioning and provenance.  For our purposes of giving illustrating examples,\nwe will assume the following two relations in our database:\n\\begin{itemize}\n  \\item A main $\\mathtt{Log}$ relation that has the following schema\n\n  $$\\mathtt{(CaseId, EventId, ActivityLabel, Timestamp, Lifecycle)}$$\n\n  The $\\mathtt{ActivityLabel}$ and $\\mathtt{Timestamp}$ attributes are mandatory\n  when working with (instance-spanning) constraints~\\cite{WinterSR20}.  The\n  $\\mathtt{Lifecycle}$ attribute describes the \\emph{lifecycle transition} of\n  an event.  This is useful when the events can span a time interval which is\n  typical in the constraints checking \\emph{concurrent} execution of\n  activities.  All of those attributes are parts of the XES standard\n  extensions~\\cite{XESLink}.\n\n\n  \\item An auxiliary $\\mathtt{EventData}$ relation that contains the extra\n  information of the logged events.  The attributes of this relation are not\n  fixed and they (depend on the application) change depending on the data,\n  however, the key of this relation is the pair $\\mathtt{(EventId, Lifecycle)}$.\n\\end{itemize}\n\n\\begin{remark}\nAn alternative approach to define the schema of $\\mathtt{EventData}$ relation\nis by following a semi-structured approach.  In that approach, the schema is\nfixed to be $\\mathtt{(EventId, Lifecycle, Attribute, Value)}$, where\n$\\mathtt{Attribute}$ could be the name of the attribute, while\n$\\mathtt{Value}$ is its value for that event.\n\\end{remark}\n\n\\subsection{Examples}\n\nIn the following examples, we assume that the relation $\\mathtt{EventData}$\nhas the following schema $\\mathtt{(EventId, Lifecycle, PackageId, CarId)}$.\nWe also assume that in our processes, we have two activities with the labels\n``purchase package'' and ``deliver package''.\n\n\n\\begin{example}[Same Shipping Car Constraint]\\label{ex:same_car1}\nConsider the constraint \\constraint{The same shipping car can be used for\ndelivering packages at most seven times per day}.  As we have mentioned before,\nwe have a great flexibility in defining what a violation is (in other words,\nwhat is the scope of the constraint).  One possibility is to define the cases\nto be tuples $\\mathtt{(CarId, Day)}$.  Following this view, the constraint can\nbe represented by the following pair of queries:\n\n\\lstinputlisting[language=SQL,style=mystyle]{SQL\/e1.sql}\n\nA less fine-grained scope: only having $\\mathtt{CarId}$.  An even more\nfine-grained scope: having tuples of $\\mathtt{(CarId, Day, CountOfDeliveries)}$\nas our cases.\n\n\\lstinputlisting[language=SQL,style=mystyle]{SQL\/e2.sql}\n\\end{example}\n\nExample~\\ref{ex:same_car1} demonstrates possible queries that define an\ninstance-spanning constraint.  To show the uniformity of our approach, the\nfollowing is an example of an intra-instance constraint.\n\n\\begin{example}[Average Shipping Time Constraint]\\label{ex:avg_deliver}\nConsider the constraint \\constraint{The average time for a package to be\ndelivered after purchase is between two and five days}.  In what follows,\nwe consider a case to be a package identifier.\n\n\\lstinputlisting[language=SQL,style=mystyle]{SQL\/e3.sql}\n\\end{example}\n\n\n\\section{Compliance Monitoring as Incremental View Maintenance}\n\\label{sec:monitor}\n\nNow, if we want to monitor a constraint \\emph{dynamically}, we will have to\nrefine our definition.  The reason is that the database instance representing\nthe execution log is continuously progressing.  Thus, the database instance\nwill contain the data of running (non-completed) process instances along with\nthe completed process instances.  Hence, at any moment, any case that is\nsubjected to some constraint will be in one of four different states \\cite{BaLS11,MMWA11,DDGM14}:\n\\begin{inparaenum}[1)]\n  \\item a \\emph{permanently} violating state;\n  \\item a \\emph{permanently} satisfying state;\n  \\item a \\emph{currently} violating state that may later be in a satisfying\n  state as a result of the occurrences of new events; and\n  \\item similarly, a \\emph{currently} satisfying state that may later be in a\n  violating state.\n\\end{inparaenum} We will refer to the last two states as \\emph{pending}\nstates.  Figure~\\ref{fig:states} shows the different states and how a case\ncould change its state upon the occurrence of new events.  Notice that it depends on the constraint under study whether all such four states have to be actually considered, or whether instead\nthe constraint only requires a subset thereof.\nExample~\\ref{ex:simple} discusses a\nsimple constraint such that we can have its cases belonging to the different\nstates.\n\n\nRegardless of the formal tools, languages, approaches, there is always a\n``methodology\" to go from informal specifications to formal realization.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.28]{figures\/states-colored3.png}\n\\caption{\\label{fig:states} A transition diagram of the different states that\na case could be in with respect to some constraint.  The diagram shows the\npossible ways the state of a case can change as time progresses.  Not shown\nin the diagram is that a case can also simply cease to be a case; furthermore,\nnew cases appear.}\n\\end{figure}\n\n\n\\begin{example}[Monitoring ``Followed-By\" Constraint]\\label{ex:simple}\n  Consider a process that comprises three activities with the labels $A, B$,\n  and $C$ whose process model is shown in Figure~\\ref{fig:pABC}. Let the\n  constraint that is imposed on this process be \\constraint{Every instance of\n  activity $A$ must be directly followed by an instance of activity $B$ within\n  20 hours}.  In Figure~\\ref{fig:traces}, we show five traces of that process\n  which correspond to five cases as the constraint is an intra-instance one.\n  The states of those five traces are distributed among the four different\n  states.\n\\end{example}\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.32]{figures\/processAABBSTC3.png}\n\\caption{\\label{fig:pABC} A process model of an example process.}\n\\end{figure}\n\\begin{figure}\n  \\centering\n  \\includegraphics[scale=0.32]{figures\/traces-colored.png}\n  \\caption{\\label{fig:traces} The plot contains five different traces of\n  the process whose model is shown in Figure~\\ref{fig:pABC}.  The x-axis\n  represents 12-hour intervals.  In a trace, double arrows ($\\Rightarrow$)\n  (respectively, single arrows ($\\rightarrow$)) denote time intervals that\n  are longer (respectively, equal or shorter) than 20 hours.  Each of the\n  five traces is coloured based on its state at the ``now'' point with regard\n  to the constraint \\constraint{Every instance of activity $A$ must be directly\n  followed by an instance of activity $B$ within 20 hours}.  For an example,\n  the fifth trace is in a \\emph{violating-permanent} state as the time span\n  between the second execution of activities $A$ and $B$ is greater than the\n  20-hour interval.}\n\\end{figure}\n\n\\begin{definition}[Constraint, Compliance monitoring Variant]\n\\label{def:monitoring}\nA \\emph{constraint} $C$ is represented by four queries $(Q_{\\mathrm{case}}, Q_{\\mathrm{viol-perm}},$\n$Q_{\\mathrm{viol-pending}}, Q_{\\mathrm{sat-pending}})$, where $Q_{\\mathrm{case}}$ returns all the cases subjected to the\nconstraint $C$, $Q_{\\mathrm{viol-perm}}$ returns the \\emph{permanently} violating cases,\n$Q_{\\mathrm{viol-pending}}$ returns the violating cases that later could be changed to\nnon-violating cases, while $Q_{\\mathrm{sat-pending}}$ returns the satisfying cases that later\ncould be changed to violating cases. (The cases not returned by none of these\nthree queries, are then the ones defined by $Q_{\\mathrm{sat-perm}}$.)\nOn any database instance, $Q_{\\mathrm{viol-perm}}$, $Q_{\\mathrm{viol-pending}}$, and $Q_{\\mathrm{sat-pending}}$ always return\nthree mutually exclusive subsets of $Q_{\\mathrm{case}}$.\n\\end{definition}\n\n\\begin{remark}\n  Typically the query $Q_{\\mathrm{viol}}$ in the post-mortem checking variant corresponds\n  to the union of the pair $Q_{\\mathrm{viol-perm}}$ and $Q_{\\mathrm{viol-pending}}$ in the compliance monitoring\n  variant.  Similarly, the query $Q_{\\mathrm{sat}}$ corresponds to the pair $Q_{\\mathrm{sat-perm}}$ and\n  $Q_{\\mathrm{sat-pending}}$.\n\\end{remark}\n\n\\begin{example}[Monitoring Same Shipping Car Constraint]\\label{ex:same_car2}\nConsider the same constraint as in Example~\\ref{ex:same_car1}.  The queries\nrepresenting this constraint can be defined as follows (where, $Q_{\\mathrm{case}}$ and\n$Q_{\\mathrm{viol-perm}}$ are defined as $Q_{\\mathrm{case}}$ and $Q_{\\mathrm{viol}}$ of Example~\\ref{ex:same_car1};\nrespectively.):\n\n\\lstinputlisting[language=SQL,style=mystyle]{SQL\/e4.sql}\nNote that $Q_{\\mathrm{viol-pending}}$  will always be empty for this constraint.\n\\end{example}\n\n\n\\section{Experiments}\n\\label{sec:exp}\n\nDBToaster is a state-of-the-art incremental query\nprocessor~\\cite{KennedyAK11,KochAKNNLS14,DBToasterWeb}.  As a proof-of-concept\nof our approach, we tested DBToaster on some of the constraints from the work\nof Winter et al.\\ on automatic discovery of ISC~\\cite{WinterSR20}.\nSpecifically, we worked with the constraints ISC1, ISC2a, ISC2b, ISC3, and ISC4\nfrom the paper.  We have also used the execution logs provided by these authors\nas sample input data~\\cite{ExecutionLogData}.\nTo manage our experiments, we performed some preprocessing steps that are\nmentioned in the Appendix~\\ref{app:expPrep}.  To assess the feasibility and\nusability of our approach, we have designed some experiments that ran over the\nmentioned five constraints.  The results of these experiments are discussed\nin Sections~\\ref{sub:rt},~\\ref{sub:qs}, and~\\ref{sub:tc}.\nAt the beginning, we give a brief demonstration on the processes and the\nconstraints used in the experiments in Section~\\ref{sub:expModel}.\n\n\n\\subsection{Experiment Data}\\label{sub:expModel}\n\nThe constraints used in the experiments are expressed over the three processes\nwhose models are shown in Figure~\\ref{fig:expProcesses}.  In the Figure, we\nhave ``Flyer Order\", ``Poster Order\", and ``Bill\" processes that are labelled\nas $P1$, $P3$, and $P2$, respectively.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{figures\/printingProcesses.png}\n  \\caption{\\label{fig:expProcesses} A figure showing the three process models\n  whose executions are used in the experiments~\\cite{WinterSR20}.}\n\\end{figure}\n\nThe ``Flyer Order\" process and ``Poster Order\" process are quite similar.\nBoth processes begin by the activity of receiving the order.  This is\nfollowed by designing the order activity, that is later followed by printing\nthe order.  In the end, the printed order is delivered.  The only difference\nbetween the two processes is the extra activity of sending the design to the\ncustomer for confirmation before the printing proceeds.  This is only part of\nthe ``Flyer Order\" process.  The customer either accepts the design, then the\nprocess proceeds as already mentioned.  Otherwise, if the customer rejects the\ndesign then the order is redesigned and the same happens until the customer is\nsatisfied with the flyer design.  That explains the loop appearing in $P1$.\nAny order whether it is for a flyer or a poster, has a corresponding initiated\n``Bill\" process.  This process is quite simple, it begins by the activity\nof writing the bill, then the bill is printed and later delivered.  Moreover,\nas you can see from the Figure, the printers are considered a shared resource\nbetween all the processes.\n\n\nThe constraints used in the experiments are the following:\n\\begin{description}\n   \\item{ISC1} There is exactly one delivery activity per day in which all the\n   finished orders\/bills of that day so far are delivered to the post office\n   simultaneously.\n\n   \\item{ISC2a} All print jobs must be completed within 10 minutes in at least\n   95\\% of all cases per month.\n\n   \\item{ISC2b} Printer 1 may only print 10 times per day.\n\n   \\item{ISC3} If a flyer or poster order is received $P2$ (i.e., bill process)\n   is started afterwards.  Moreover, the corresponding bill process must be\n   started before the order is delivered to the post office.\n\n   \\item{ISC4} Printing jobs that require different paper formats (i.e., A4\n   and Poster formats) cannot be printed concurrently on one printer where\n   concurrently means that one job starts, and before it finishes, the other\n   starts.\n\n\\end{description}\nWe slightly modified the original constraints~\\cite{WinterSR20} to better\nmatch with the log data~\\cite{ExecutionLogData}.\n\n\n\\subsection{Running Time}\\label{sub:rt}\n\nThe running time of three of the five monitored constraints is reported in\nFigure~\\ref{fig:runTime}, which shows averages over 10 runs.  The time is\nreported for every 300 insertions with total insertions 30636 (the number of\nevents in the dataset).  This experiment was performed on a personal laptop\nrunning macOS 12.2.1 with RAM of 16 GB and processor speed of 2.6Hz.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{figures\/exp\/LastRunningTimePlot-eps-converted-to.pdf}\n  \\caption{\\label{fig:runTime} A plot of the running time (in milliseconds)\n  taken to monitor the constraints ISC1, ISC3 and ISC4.  The running time of the\n  constraints ISC2a and ISC2b are omitted since they are quite similar to ISC4.}\n\\end{figure}\n\nThe slope of each curve is indicative of the average time needed, per event, to\nmaintain the queries defining the constraint.  We can see that this line is\nsignificantly higher for the first constraint; indeed, this constraint requires\nrather complex SQL~queries (shown in Appendix~\\ref{app:expSQL}).\nFor tested constraints ISC1 and ISC3, the slopes of these lines are less than\nhalf a millisecond, respectively less than 1\/6th of a millisecond.  For tested\nconstraints ISC2a, ISCb and ISC4, the slopes are less than 1\\% of a millisecond.\n\n\n\\subsection{Sizes of Queries}\\label{sub:qs}\n\nThe size (i.e., the number of cases) of each of the queries defining four of\nthe five monitored constraints is reported and plotted relative to time (i.e.,\nthe number of insertions).  This can show us how the cases are changing\ntheir status (pending or permanent, violating or satisfying).  A plot\nfor each of the four constraints is provided by Figure~\\ref{fig:ISCQS}.\nThe query size is reported every 500 insertions except for ISC2a\nwhich is done every 100 insertions instead, as it displays a more\nfine-grained behavior.\n\\begin{figure}\n     \\centering\n     \\begin{subfigure}[t]{0.49\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/exp\/ISC1QSZoomIntegrated-eps-converted-to.pdf}\n         \\caption{Plot of ISC1}\n         \\label{fig:ISC1QS}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[t]{0.49\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/exp\/ISC2aQSZoomIntegrated-eps-converted-to.pdf}\n         \\caption{Plot of ISC2a}\n         \\label{fig:ISC2aQS}\n     \\end{subfigure}\n     \\qquad\n     \\begin{subfigure}[b]{0.49\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/exp\/ISC3QSZoomIntegrated-eps-converted-to.pdf}\n         \\caption{Plot of ISC3}\n         \\label{fig:ISC3QS}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.49\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/exp\/ISC4QuerySize-eps-converted-to.pdf}\n         \\caption{Plot of ISC4}\n         \\label{fig:ISC4QS}\n     \\end{subfigure}\n    \\caption{Plots of the size of each of the queries of the tested constraints.\n    ISC2b is not shown as it has the same cases as ISC1 and has similar behavior\n    to ISC3, which are shown.  Since our measurements consist of 600 data points\n    (even 3000 for ISC2a), the plots are at rather high scale.  To show more\n    detail, we provide insets that zoom in on selected regions (orange rectangles).}\n    \\label{fig:ISCQS}\n\\end{figure}\n\n\n\\subsection{Tracing Cases}\\label{sub:tc}\nFor ISC2a and ISC1, we show in Figures~\\ref{fig:c2SC} and~\\ref{fig:c1SC} the\nevolution in status of all the individual cases over time.  This illustrates\nthat our approach is compatible with monitoring on a very detailed level.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{figures\/exp\/ISC2aCases.png}\n  \\caption{\\label{fig:c2SC} A plot showing the different cases of ISC2a and\n  how each of the cases is changing its status through time.  From the\n  previous plots, we see that in total we have six cases for this constraint.\n  The cases according to this constraint are months.}\n\\end{figure}\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{figures\/exp\/ISC1CasesZoomIntegrated.pdf}\n  \\caption{\\label{fig:c1SC} A plot showing the 101 different cases (days) of\n  ISC1 and how each of the cases is changing its status through time.  Here the\n  measurement consists of 30600 data points per case, so the plot is at a very\n  high scale.  The inset shows more detail by zooming on the selected region\n  (orange rectangle).}\n\\end{figure}\n\n\n\\section{Sequence Data Extensions of Query Languages}\n\\label{sec:seqdlog}\n\nWe have mentioned before that any data model with a sufficiently expressive\nquery language can be used to express the constraints.  Although, we chose\nto work with the relational data model with SQL~for the reasons we mentioned,\nit is interesting to briefly discuss query languages for the relational data\nmodel extended with sequences~\\cite{AnglesABBFGLPPS18,ShenDNR07}.  Indeed, a\ntrace is a sequence of events.  Hence, representing the relative order of the\nevents is quite natural in a sequence data model.  This level of abstraction,\nof viewing traces as sequences of abstract events, is often assumed when\nworking with temporal and dynamic\nlogics~\\cite{GiacomoFMP21,PesicSA07,GiacomoMGMM14}.\n\nSequence Datalog~\\cite{SeqDL,bonnermecca_sequences,meccabonner_termination} is\nan extension of the query language Datalog, to work with sequences as first\nclass citizens.   We will briefly showcase this language by considering a\ntypical example of a constraint that is handled using the temporal logic.\n\n\\begin{example}[Strict Sequencing~\\cite{GiacomoFMP21}]\n  Let $\\mathtt a$ and $\\mathtt b$ be two activities.  Consider that we want to\n  verify that the two activities are restricted by a \\emph{strict sequencing}\n  relation, which is one of the standard \\emph{ordering}\n  relations~\\cite{vanderAalstWM2004}.  There is a strict sequencing relation\n  between $\\mathtt a$ and $\\mathtt b$ if the log satisfies the following:\n  \\begin{itemize}\n    \\item there exists a trace where $\\mathtt a$ is immediately followed by\n    $\\mathtt b$; and\n    \\item there are not any traces where $\\mathtt b$ is immediately followed\n    by $\\mathtt a$.\n  \\end{itemize}\nThere are two possible violations of this constraint.  The first is not having\na trace with $\\mathtt b$ directly following $\\mathtt a$.  The other is having\na trace with $\\mathtt a$ directly following $\\mathtt b$.\n\nFor the purpose of expressing this constraint, assume we have the following\nschema for the $\\mathtt{Log}$ relation: $\\mathtt{(TraceId,Events)}$, where\n$\\mathtt{Events}$ are just a sequence of labels of activities.  Then, this\nconstraint can be expressed by the following Sequence Datalog program.\n{\\small\n\\begin{lstlisting}\na_before_b():- Log(@traceId, $pre.a.b.$post).\n\nviolation():- +a_before_b().\nviolation():- Log(@traceId, $pre.b.a.$post).\n\\end{lstlisting}}\nThis program illustrates a number of Sequence Datalog features:\n\\begin{itemize}\n  \\item the dot is the concatenation operator.\n  \\item \\verb|@traceId| is an \\emph{atomic} variable (indicated by the\n  \\verb|@| symbol) representing atomic values (in this case, trace identifiers).\n  \\item \\verb|$pre| and \\verb|$post| are \\emph{sequence} variables\n  (indicated by the \\verb|$| symbol) representing (possibly empty) sequences of\n  atomic values.\n\\end{itemize}\n\\end{example}\nThe utility of using Sequence Datalog can be appreciated if we compare the\nabove program with the same query expressed in SQL.\n\n\\lstinputlisting[language=SQL,style=mystyle]{SQL\/ordering.sql}\n\n\n\\section{Discussion}\n\\label{sec:conc}\n\nIn this paper, we have looked into the problems of post-mortem checking and\ncompliance monitoring of constraints over business processes.  Specifically,\nwe focused on ISC~as recently introduced in the process mining field, and\ncaught attention since it refers to complex constraints that span multiple\nprocess instances.  Although there have been extensive works on inventorying\nand categorizing ISC s~\\cite{Rinderle-MaGFMI16,WinterSR20}, a crisp definition\nof what is or is not an ISC, however, seems to be elusive.  Indeed, the notion\nof constraint is so broad that we propose to \\emph{define} any constraint as\ntwo or four queries posed against the database instance that represents a\n(partial) execution log.  This approach gives us huge flexibility, moreover,\nwe gain a lot from advances in database technology as demonstrated in the\nExperiments Section.\n\n\nIn using the DBToaster system for our experiments, we faced a few technical\nissues.  The main challenge was that the Scala version of DBToaster gets\nstuck when retrieving snapshots over the course of the insertions.  To\novercome this issue, to perform our measurements of counting cases over time,\nand how they evolve their constraint satisfaction status, we only retrieved\na snapshot after an initial sequence of insertions.  We then restart the\nmeasurement for one batch of insertions longer.  Another limitation is that\nSQL~is not yet fully supported, although complex queries can be expressed.\nThis required us to sometimes rewrite queries in equivalent form.  Finally,\nsome built-in functions (e.g., on strings or dates) are missing from the Scala\nversion.  Thus, those experiments should be seen more of a proof-of-concept of\nthe feasibility of our approach.\n\n\nIn this discussion, we briefly touch upon the main difference between our\napproach and the main approach that is used to monitor ISC.  This approach\nis based on the Event Calculus (EC)~\\cite{LyMMRA15,MontaliMCMA13,ma2016}.\nMost monitoring systems that are based on EC are implemented using Prolog.\nUsing EC to express a constraint seems to be very \\emph{procedural} albeit\nbeing defined in logical programming language.  For example, to monitor a\nconstraint such as ISC2b, in EC one would define a rule that increments a\ncounter every time a printing event occurs.  At the end, that counter value\nshould be at most 10 as per the constraint.  Since this is done in Prolog,\nthis will be asking the SAT solver if there exists an extension of the given\nsequence of events satisfying the specification of this counting process.\nA similar approach was followed in the paper by Montali et al.\\\n\\cite{MontaliMCMA13} to monitor business (intra-instance) constraint with\nthe EC\\@.  Events come in time, and Prolog rules that fire every\nnew time instant, are used to check various constraints dynamically.\nHowever, these incremental rules are manually implemented.  On the\ncontrary, using an incremental query processor shifts the focus on what the\nqueries (or constraints) themselves are rather than what the rules are that\nare responsible for this incremental maintenance.  Hence, our approach is\nmore declarative.\n\n\nAt the end of this discussion, we mention a few points for further research.\nSince there are some algorithms that are used to discover ISC~from execution\nlogs~\\cite{WinterSR20}, and these algorithms search for explicit patterns,\none could define a common language to report the results of those algorithms\nand use those results to automatically write the SQL~queries monitoring each\nof the reported constraints. Thus, the whole process could be automated.  Also,\none could try to rewrite the same queries differently and evaluate how the\ndifferent formulations affect the running times to incrementally maintain them.\n\n\\subsection*{Acknowledgments}\nWe thank Stefanie Rinderle-Ma and J\\\"{u}rgen Mangler for initial discussions.\n\n\\bibliographystyle{splncs04}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe structure of Hom-Lie algebra appeared first as a generalization of Lie algebra by Hartwig, Larsson and Silvestrov in \\cite{J D S}. In 1994, the concept of  $\\rho$-Lie algebra or Lie color algebra introduced by Bongaarts \\cite{BP1} and then in 1998, Scheunert and Zhang introduced the cohomology theory of Lie\ncolor algebras in \\cite{MR}. Also, in 2012, Yuan \\cite{LY} introduced the notion of a hom-Lie color algebra which can be viewed as an extension of Hom-Lie superalgebras to $G$-graded algebras, where $G$ is any abelian group. In 2015, Abdaoui, Ammarto and Makhlouf defined representations\nand a cohomology of the Hom-Lie color algebra in \\cite{AAM}. After two years, in 2017, $T^*$-extensions and abelian extensions of the Hom-Lie color algebras are studied by Bing Sun, Liangyun Chen and Yan Liu in \\cite{BLY}.\n\nFilippov, in 1985 introduced a concept that is called $n$-Lie algebra. These Lie algebras are represented with various names such as Filippov algebra, Nambu-Lie algebra, Lie $n$-algebra. The notion of $n$-Lie algebra has close relationships with many fields in mathematics\nand mathematical physics, for their applications refer to \\cite{TL, AG, HH, JN, JN1}. The cohomology theory and deformation theory for $n$-Lie algebras was introduced respectively by Takhtajan and Gautheron in \\cite{TL1, D1, PG}. H. Ataguema, A. Makhlouf and S. Silvestrov in \\cite{HAS} have introduced the notion of 3-hom-Lie\nalgebras and representations and module-extensions of 3-hom-Lie algebras have investigated by Y. Liu, L. Chena and Y. Ma in \\cite{LCM}. The notion of 3-Lie colour algebras have introduced by T. Zhang and have studied Cohomology and deformations of 3-Lie colour algebras by him (see \\cite{T}, for more details).\n\nIn this paper, we introduce the notion of 3-Hom-Lie colour algebras or 3-Hom-$\\rho$-Lie algebras and study the representation and deformation theory of this kind of Hom-Lie algebras.\n\nThis paper is arranged as follows: In Section 2, we recall some necessary background knowledge including $\\rho$-commutative and\nHom-$\\rho$-Lie algebras. In the next, we discuss about the 3-Hom-$\\rho$-Lie algebras and define representations, modules, $\\phi^k$derivations of it and show that representations and modules of 3-Hom-$\\rho$-Lie algebras are equivalent. This section also contains the $T^{\\star}$-extension of 3-Hom-$\\rho$-Lie algebras. Section 3 is contained abelian extensions of 3-Hom-$\\rho$-Lie algebras and the reader will get some results in this case. In this section we show that associated to any abelian extension, there is a representation and a 2-cocycle.  Section 4 is devoted to discuss about deformations and the Hom Nijenhuis operator of 3-Hom-$\\rho$-Lie algebras. Furthermore, we show that $\\omega$ generates a $t$-parameter infinitesimal deformation of the 3-Hom-$\\rho$-Lie algebra $A$ is equivalent to 3-Hom-$\\rho$-Lie algebra which is 1-cocycle of $A$ with coefficients in the adjoint representation.\n\n\n\n\\section{3-hom-$\\rho$-lie algebras}\nIn this section, we summarize some definitions concerning $\\rho$-commutative algebras and Hom-$\\rho$-Lie algebras. We also introduce the notion of 3-Hom-$\\rho$-Lie algebras. Representations, modules, $\\phi^k$-derivations and some results about them are studied in this section.\n\nLet $A$ be an associative and unital algebra over a field $k$ ($k = \\mathbb{R}$ or $k = \\mathbb{C}$), grading by an abelian group $(G, +)$ that is  the vector space $A$ has a $G$-grading $A = \\oplus_{a\\in G}A_a$\nsuch that $A_aA_b\\subset A_{a+b}$. A map\n$\\rho:G\\times G\\longrightarrow k^{\\star}$ is called a two-cycle if the following conditions hold\n\\begin{align}\n&\\rho(a,b) =\\rho(b,a)^{-1},\\quad a,b\\in G,\\\\\n&\\rho(a+b,c) =\\rho(a,c)\\rho(b,c),\\quad a,b,c\\in G.\n\\end{align}\nThe above conditions say that \n$\\rho(a,b)\\neq 0$, $\\rho(0,b)=1$ and $\\rho(c,c)=\\pm 1$ for all $a,b,c\\in A$ , $c\\neq 0$.\n\nLet us denote by $Hg(A)$ the set of homogeneous elements in $A$. The $\\rho$-commutator of two homogeneous elements $f,g$ is\n\\begin{equation}\\label{111}\n[f, g]_{\\rho} = fg-\\rho(|f|, |g|)gf,\n\\end{equation}\nwhere $|f|$ denotes the $G$-degree of a (non-zero)  homogeneous element $f\\in A$.\\\\\nA $\\rho$-commutative algebra is a $G$-graded algebra $A$ with a given two-cycle\n$\\rho$ such that $f g =\\rho(| f |, |g|)g f $ for all homogeneous elements $ f$ and\n$g$ in $A$ (i.e., $[f,g]_{\\rho}=0$).\n\\begin{definition}\nA 2-Hom-$\\rho$-Lie algebra or for simply a Hom-$\\rho$-Lie algebra is a $G$-graded vector space $A$ together with a bilinear map $[.,.]_{\\rho}:A\\times A\\longrightarrow A$, a two-cycle $\\rho$ and a linear map $\\phi:A\\longrightarrow A$ satisfying the following conditions\n\\begin{align*}\n&\\bullet |[f,g]_{\\rho}|= | f|+| g|,\\\\\n&\\bullet [f, g]_{\\rho} = -\\rho(f,g)[g, f]_{\\rho},\\\\\n&\\bullet \\rho(h,f)[\\phi(f), [g, h]_{\\rho}]_{\\rho}+\\rho(g,h)[\\phi(h), [f, g]_{\\rho}]_{\\rho}+\\rho(f,g)[\\phi(g), [h, f]_{\\rho}]_{\\rho} = 0.\n\\end{align*} \nThe third condition is equivalent to \n$$[\\phi(f), [g, h]_{\\rho}]_{\\rho}=[[f, g]_{\\rho},\\phi(h)]_{\\rho}+\\rho(f,g)[\\phi(g), [f, h]_{\\rho}]_{\\rho}.$$\n\\end{definition}\n\\begin{definition}\nA quadruple\n$ (A, [.,.,.], \\rho,\\phi)$ \nconsisting of a  $G$-graded vector space\n $A =\\bigoplus_{a\\in G} A_a$,\n a trilinear map\n $[.,.,.]: A \\times A\\times A \\longrightarrow A$,\na two-cycle\n $\\rho : G\\times G\\longrightarrow k^{\\ast}$\n and an even linear map\n$\\phi : A\\longrightarrow A$ is called a 3-Hom-$\\rho$-Lie algebra if the following condition are satisfied\n\\begin{align*} \n&(1)~~|[f_1,f_2,f_3]|=|f_1|+ |f_2| +|f_3|,\\\\\n&(2)~~[\\phi(f_1),\\phi(f_2),[g_1,g_2,g_3]]=[[f_1,f_2,g_1],\\phi(g_2),\\phi(g_3)]-\\rho(f_1+f_2,g_1)[\\phi(g_1),[f_1,f_2,g_2],\\phi(g_3)]\\\\\n&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ +\\rho(f_1+f_2, g_1+g_2)[\\phi(g_1),\\phi(g_2),[f_1,f_2,g_3]].\n\\end{align*}\nThe second property is called $\\rho$-fundamental identity.\n\\end{definition}\nNote that, the bracket introduced in the above definition has the $\\rho$-skew symmetry property with respect to the displacement of every two elements of itself.\n\\begin{definition}\nA 3-Hom-$\\rho$-Lie algebra  $(A, [.,.,.],\\rho,\\phi)$ is said to be multiplicative if $\\phi$ is a Lie algebra morphism, i.e. for $f,g,h\\in A$, $\\phi[f,g,h]=[\\phi(f),\\phi(g),\\phi(h)]$, regular if $\\phi$ is an automorphism for $[.,.,.]$, and involutive if $\\phi^2 = Id_A$.\n\\end{definition}\n\\begin{definition}\nLet $(A,[.,.,.]_A,\\phi)$ and $(B,[.,.,.]_B,\\psi)$ be two 3-Hom-$\\rho$-Lie algebra. A linear map $\\alpha:A\\longrightarrow B$ is said to be a morphism of 3-Hom-$\\rho$-Lie algebras if\n$$\\alpha[f,g,h]_A=[\\alpha(f),\\alpha(g),\\alpha(h)]_B,$$\nfor all $f,g,h\\in A$ and \n$$\\alpha\\circ\\phi=\\psi\\circ\\alpha.$$\n\\end{definition}\nLet us denote by $\\vartheta=:\\{(f,\\alpha(f))| f\\in A\\}\\subseteq A\\oplus B$ the group of linear maps $\\alpha:A\\longrightarrow B$.\n\nLet $(A,[.,.,.],\\rho,\\phi)$ be a multiplicative 3-Hom-$\\rho$-Lie algebra. We define the following operation on the fundamental set $\\mathcal{L}=\\wedge^2A$ by\n\\begin{equation}\\label{123}\n[(f_1,f_2), (g_1,g_2)]_{\\mathcal{L}}=([f_1, f_2, g_1],\\phi(g_2))+\\rho(f_1+f_2,g_1)(\\phi(g_1),[f_1,f_2,g_2]).\n\\end{equation}\nIf we define the even linear map $\\phi_1:\\mathcal{L}\\longrightarrow \\mathcal{L}$ by $\\phi_1(f_1,f_2)=(\\phi(f_1),\\phi(f_2))$, then we have the multiplicative Hom-$\\rho$-Lie algebra \n$(\\mathcal{L},[.,.]_{\\mathcal{L}},\\rho,\\phi_1)$.\n\nLet $A$ be a 3-Hom-$\\rho$-Lie algebra and $V$ be a $G$-graded vector space. Recall that ${\\rm End_G(V)} = {\\rm Hom_G(V, V )}$ and ${\\rm Hom_G(A, V)}$ are $G$-graded vector spaces.\n\\begin{definition}\nLet $A$ be a 3-Hom-$\\rho$-Lie algebra, $V$ be a $G$-graded vector space and $\\mu$ be a linear map from \n$\\mathcal{L}=\\wedge^2A$ to ${\\rm End_G(V)}$. Then $(V,\\mu)$ is called a representation of $A$ with respect to $\\beta\\in {\\rm End_G(V)}$ if the following conditions are satisfied\t\n\\begin{align}\n\\mu[(f_1,f_2),(g_1,g_2)]_{_\\mathcal{L}}\\circ \\beta&=\\mu(\\phi_1(f_1,f_2))\\mu(g_1,g_2)-\\rho(f_1+f_2,g_1+g_2)\\mu(\\phi_1(g_1,g_2))\\mu(f_1,f_2),\\label{1}\\\\\n\\mu([g_1,g_2,g_3],\\phi(f))\\circ \\beta&=\\mu(\\phi_1(g_1,g_2))\\mu(g_3,f)+\\rho(g_1,g_2+g_3)\\mu(\\phi_1(g_2,g_3))\\mu(g_1,f)\\label{2}\\\\\n&\\ \\ \\  +\\rho(g_1+g_2,g_3)\\mu(\\phi_1(g_3,g_1))\\mu(g_2,f),\\nonumber\\\\\n\\mu(\\phi(g),[f_1,f_2,f_3])\\circ \\beta&=\\rho(g,f_1+f_2)\\mu(\\phi_1(f_1,f_2))\\mu(g,f_3)+\\rho(g,f_2+f_3)\\rho(f_1,f_2+f_3)\\mu(\\phi_1(f_2,f_3))\\mu(g,f_1)\\\\\n&\\ \\ \\  +\\rho(g,f_1+f_3)\\rho(f_1+f_2,f_3)\\mu(\\phi_1(f_3,f_1))\\mu(g,f_2),\\nonumber\\\\\n\\mu(\\phi_1(f_1,f_2))\\mu(g_1,g_2)&=\\rho(f_1+f_2,g_1+g_2)\\mu(\\phi_1(g_1,g_2))\\mu(f_1,f_2)\\label{7.1}\\\\\n&\\ \\ \\ +\\rho(f_1+f_2,g_1)\\mu(\\phi(g_1),[f_1,f_2,g_2])\\circ\\beta +\\mu([f_1,f_2,g_1],\\phi(g_2))\\circ\\beta.\\nonumber\n\\end{align}\n\\end{definition}\n\\begin{example}\nLet $A$ be a 3-Hom-$\\rho$-Lie algebra, $V=A$ and $\\phi=\\beta\\in {\\rm End_G(A)}$. Then the linear map $ad:A\\times A\\longrightarrow {\\rm End_G(A)}$ defined by $ad(f_1,f_2)(f_3)=[f_1,f_2,f_3]$\nis a representation of $A$ with respect to $\\beta=\\phi$.\\\\\nIt is enough to check the conditions \\eqref{1} and \\eqref{2}. For the condition \\eqref{1}, by \\eqref{123} and the \n$\\rho$-fundamental identity of 3-Hom-$\\rho$-Lie algebra $A$, we have \n\\begin{align*}\nad[(f_1,f_2),(g_1,g_2)]_{_\\mathcal{L}}\\phi(g_3) &=ad([f_1,f_2,g_1],\\phi(g_2))\\phi(g_3)+\\rho(f_1+f_2,g_1) ad(\\phi(g_1),[f_1,f_2,g_2])\\phi(g_3)\\\\\n&=[[f_1,f_2,g_1],\\phi(g_2),\\phi(g_3)]+\\rho(f_1+f_2,g_1)[\\phi(g_1),[f_1,f_2,g_2],\\phi(g_3)]\\\\\n&=[\\phi(f_1),\\phi(f_2),[g_1,g_2,g_3]]-\\rho(f_1+f_2,g_1+g_2)[\\phi(g_1),\\phi(g_2),[f_1,f_2,g_3]]\\\\\n&=ad(\\phi(f_1),\\phi(f_2))ad(g_1,g_2)g_3-\\rho(f_1+f_2,g_1+g_2)ad(\\phi(g_1),\\phi(g_2))ad(f_1,f_2)g_3.\n\\end{align*}\nThe other conditions prove similarly.\n\\end{example}\n\\begin{definition}\nLet $(A,[.,.,.],\\rho,\\phi)$ be a 3-Hom-$\\rho$-Lie algebra. Consider the triple $(V,\\beta,\\cdot)$ consisting of a $G$-graded vector space $V$, an even homomorphism $\\beta$ of vector spaces and a linear operation $\\cdot:\\mathcal{L}\\times V\\longrightarrow V$ such that $\\mathcal{L}_{g^{\\prime}}\\cdot V_h\\subseteq V_{g^{\\prime}+h}$ for all $g^{\\prime},h\\in G$. Then $(V,\\beta,\\cdot)$ is called  an $A$-module if\n\\begin{align}\n[(f_1,f_2),(g_1,g_2)]_{_\\mathcal{L}}\\beta(m)&=\\phi_1(f_1,f_2)\\cdot((g_1,g_2)\\cdot m) -\\rho(f_1+f_2,g_1+g_2)\\phi_1(g_1,g_2)\\cdot((f_1,f_2)\\cdot m),\\label{3}\\\\\n([f_1,f_2,f_3],\\phi(g))\\cdot \\beta(m)&=\\phi_1(f_1,f_2)\\cdot ((f_3,g)\\cdot m)+\\rho(f_1,f_2+f_3)\\phi_1(f_2,f_3)\\cdot ((f_1,g)\\cdot m)\\label{4}\\\\\n&\\ \\ \\ +\\rho(f_1+f_2,f_3)\\phi_1(f_3,f_1)\\cdot((f_1,g)\\cdot m),\\nonumber\\\\\n(\\phi(f),[g_1,g_2,g_3])\\cdot \\beta(m)&=\\rho(f,g_1+g_2)\\phi_1(g_1,g_2)\\cdot ((f_2,g_3)\\cdot m)\\label{44}\\\\\n&\\ \\ \\ +\\rho(f,g_2+g_3)\\rho(g_1,g_2+g_3)\\phi_1(g_2,g_3)\\cdot ((f,g_1)\\cdot m)\\nonumber\\\\\n&\\ \\ \\  +\\rho(f,g_1+g_3)\\rho(g_1+g_2,g_3)\\phi_1(g_3,g_1)\\cdot((f,g_2)\\cdot m),\\nonumber\\\\\n\\phi_1(f_1,f_2)((g_1,g_2)\\cdot m)&=\\rho(f_1+f_2,g_1)(\\phi(g_1),[f_1,f_2,g_1])\\cdot\\beta(m)+([f_1,f_2,g_1],\\phi(g_2))\\cdot\\beta(m)\\\\\n&\\ \\ \\ +\\rho(f_1+f_2,g_1+g_2)\\phi_1(g_1,g_2)\\cdot((f_1,f_2)\\cdot m).\\nonumber\n\\end{align}\n\\end{definition} \n\\begin{lemma}\nLet $(A,[.,.,.],\\rho,\\phi)$ be a 3-Hom-$\\rho$-Lie algebra. The modules and the representations of $A$ are equivalent.\n\\end{lemma}\n\\begin{proof}\nLet $(V,\\mu,\\beta)$ be a representation of $A$. We define the linear operation $.:\\mathcal{L}\\times V\\longrightarrow V$ by $(f,g,m)\\mapsto(f,g)\\cdot m=\\mu(f,g)(m)$ and show that $(V,\\beta,\\cdot)$ is an $A$-module. For this purpose, it is sufficient to check two relations \\eqref{3} and \\eqref{4}. Then we have\n\\begin{align*}\n[(f_1,f_2),(g_1,g_2)]_{\\mathcal{L}}\\cdot \\beta(m)&=\\mu([(f_1,f_2),(g_1,g_2)]_{\\mathcal{L}})\\beta(m)=\\mu(\\phi_1(f_1,f_2))\\mu(g_1,g_2)m\\\\\n&\\ \\ \\ -\\rho(f_1+f_2,g_1+g_2)\\mu(\\phi_1(g_1,g_2))\\mu(f_1,f_2)m\\\\\n&=\\phi_1(f_1,f_2)\\cdot((g_1,g_2)\\cdot m) -\\rho(f_1+f_2,g_1+g_2)\\phi_1(g_1,g_2)\\cdot((f_1,f_2)\\cdot m).\n\\end{align*} \nFor the next condition, we have\n\\begin{align*}\n([f_1,f_2,f_3],\\phi(g))\\cdot \\beta(m)&=\\mu([f_1,f_2,f_3],\\phi(g)) \\beta(m)=\\mu(\\phi_1(f_1,f_2))\\mu(f_3,g)m\\\\\n&\\ \\ \\ +\\rho(f_1,f_2+f_3)\\mu(\\phi_1(f_2,f_3))\\mu(f_1,g)m+\\rho(f_1+f_2,f_3)\\mu(\\phi_1(f_3,f_1))\\mu(f_2,g)m\\\\\n&=\\phi_1(f_1,f_2)\\cdot ((f_3,g)\\cdot m)+\\rho(f_1,f_2+f_3)\\phi_1(f_2,f_3)\\cdot ((f_1,g)\\cdot m)\\\\\n&\\ \\ \\ +\\rho(f_1+f_2,f_3)\\phi_1(f_3,f_1)\\cdot((f_1,g)\\cdot m).\n\\end{align*}\nIn the same method, we can check the last two conditions.\\\\\nFor the converse, consider the linear map $\\mu:A\\times A\\longrightarrow {\\rm End_G(V)}$ by $\\mu(f,g)m=(f,g)\\cdot m$. It is easy to see that for an even homomorphism $\\beta:V\\longrightarrow V$, the triple $(V,\\beta,\\cdot)$ is an $A$-module.\n\\end{proof}\n\\begin{definition}\nLet $A$ be a 3-Hom-$\\rho$-Lie algebra and $(V,\\mu)$ be an $A$-module. An $n$-cochain on $A$ is a $\\rho$-skew symmetric morphism $\\omega$ from $\\wedge^{2n+1}(A)$ into $V$ of degree $|\\omega|$. Let us denote by $C^n(A,V)$ the set of all $n$-cochains on $A$, in the sense of \n$$C^n(A,V)=Hom(\\wedge^{2n+1}(A), V), \\quad \\omega(f_1,\\cdots, f_{2n+1})\\in V_{|f_1|+\\cdots |f_{2n+1}|+|\\omega|},$$\nwhere $f_1,\\cdots,f_{2n+1} \\in Hg(A)$.\n$\\omega\\in C^n(A,V)$ is called an $n$-Hom-cochain on $A$ if for $f_1,\\cdots,f_{2n+1} \\in Hg(A)$ and $\\beta\\in {\\rm End_G(V)}$, the following relation holds\n$$\\beta(\\omega(f_1,\\cdots,f_{2n+1})) = \\omega(\\phi(f_1),\\cdots, \\phi(f_{2n+1})).$$\nWe denote by $C^n_{\\phi}(A,V)$ the set of all $n$-Hom-cochains on $A$.\\\\\nLet $A$ be a multiplication 3-Hom-$\\rho$-Lie algebra and $\\beta=Id_V$. Define the coboundary operator $d_{n-1}:C^{n-1}_{\\phi}(A,V)\\longrightarrow C^n_{\\phi}(A,V)$ by\n\\begin{align*}\nd_{n-1}\\omega(f_1,\\cdots,f_{2n+1})&=(-1)^{n+1}\\rho(f_1+\\cdots+f_{2n-2}, f_{2n-1}+f_{2n+1})\\rho(f_{2n-1}+f_{2n},f_{2n+1})\\\\\n&\\ \\ \\ \\times\\mu(\\phi^{n-1}(f_{2n+1}), \\phi^{n-1}(f_{2n-1}))\\omega(f_1,\\cdots,f_{2n-2},f_{2n})\\\\\n&\\ \\ \\ +(-1)^{n+1}\\rho(f_1+\\cdots+f_{2n-2}, f_{2n}+f_{2n+1})\\rho(f_{2n-1},f_{2n}+f_{2n+1})\\\\\n&\\ \\ \\ \\times\\mu(\\phi^{n-1}(f_{2n}), \\phi^{n-1}(f_{2n+1}))\\omega(f_1,\\cdots,f_{2n-1})\\\\\n&\\ \\ \\ +\\sum_{k=1}^n (-1)^{k+1}\\rho(f_1+\\cdots+f_{2k-2}, f_{2k-1}+f_{2k})\\\\\n&\\ \\ \\ \\times\\mu(\\phi^{n-1}(f_{2k-1}), \\phi^{n-1}(f_{2k}))\\omega(f_1,\\cdots,\\widehat{f_{2k-1}},\\widehat{f_{2k}},\\cdots,f_{2n+1})\\\\\n&\\ \\ \\ +\\sum_{k=1}^n\\sum_{j=2k+1}^{2n+1}(-1)^k\\rho(f_1+\\cdots+f_{2k}, f_{2k+1}+\\cdots+f_{j-1})\\\\\n&\\ \\ \\ \\times\\omega(\\phi(f_1),\\cdots,\\widehat{\\phi(f_{2k-1})},\\widehat{\\phi(f_{2k})},\\cdots,[f_{2k-1},f_{2k},f_j],\\cdots, \\phi(f_{2n+1})),\n\\end{align*}\nfor $n\\geq 1$, and $\\omega\\in C^n_{\\phi}(A,V)$, where $\\widehat{\\phi(f_{2k-1})}$ means that $\\phi(f_{2k-1})$ is omitted. Note that $| d\\omega|=|\\omega|$ and $d_{n}\\circ d_{n-1}=0$ (the condition $d_{n}\\circ d_{n-1}=0$ does note follow if the condition $\\omega\\circ\\phi=\\omega$ is omitted, so it is necessary to define the differential operator on $n$-Hom-cochains).\\\\\nFor $n=1$:\n\\begin{align*}\nd_0\\omega(f_1,f_2,f_3)&=\\mu(f_1,f_2)\\omega(f_3)+\\rho(f_1+f_2,f_3)\\mu(f_3,f_1)\\omega(f_2)\\\\\n&\\ \\ \\ +\\rho(f_1,f_2+f_3)\\mu(f_2,f_3)\\omega(f_1)-\\omega([f_1,f_2,f_3]).\n\\end{align*}\nFor $n=2$:\n\\begin{align*}\nd_1\\omega(f_1,f_2,g_1,g_2,g_3)&=\\omega(\\phi(f_1),\\phi(f_2), [g_1,g_2,g_3])+\\mu(\\phi(f_1),\\phi(f_2))\\omega(g_1,g_2,g_3)\\\\\n&\\ \\ \\ -\\omega([f_1,f_2,g_1],\\phi(g_2),\\phi(g_3))+\\rho(f_1+f_2,g_1)\\omega(\\phi(g_1),[f_1,f_2,g_2],\\phi(g_3))\\\\\n&\\ \\ \\ -\\rho(f_1+f_2,g_1+g_2)\\omega(\\phi(g_1),\\phi(g_2),[f_1,f_2,g_3])\\\\\n&\\ \\ \\ -\\rho(f_1+f_2,g_2+g_3)\\rho(g_1,g_2+g_3)\\mu(\\phi(g_2),\\phi(g_3))\\omega(f_1,f_2,g_1)\\\\\n&\\ \\ \\ -\\rho(f_1+f_2,g_1+g_3)\\rho(g_1+g_2,g_3)\\mu(\\phi(g_3),\\phi(g_1))\\omega(f_1,f_2,g_2)\\\\\n&\\ \\ \\ -\\rho(f_1+f_2,g_1+g_2)\\mu(\\phi(g_1),\\phi(g_2))\\omega(f_1,f_2,g_3).\n\\end{align*}\n\\end{definition}\n\\begin{definition}\\label{1234}\nLet $A$ be a 3-Hom-$\\rho$-Lie algebra and $(V,\\mu)$ be an $A$-module. Then a morphism $\\nu\\in {\\rm Hom_G(A,V)}$ is called 0-Hom-cocycle if and only if $d_0\\nu=0$, in the other word\n\\begin{align*}\n\\mu(f_1,f_2)\\nu(f_3)&+\\rho(f_1+f_2,f_3)\\mu(f_3,f_1)\\nu(f_2)\\\\\n&+\\rho(f_1,f_2+f_3)\\mu(f_2,f_3)\\nu(f_1)=\\nu([f_1,f_2,f_3]).\n\\end{align*}\nAlso, $\\omega\\in {\\rm Hom(\\wedge^3A,V)}$ is called 1-Hom-cocycle with respect to $\\mu$ if and only if $d_1\\omega=0$.\n\\end{definition}\n\\begin{definition}\nLet $(A,[.,.,.],\\rho,\\phi)$ be a multiplicative 3-Hom-$\\rho$-Lie algebra. For any non-negative integer $k$, denote the $k$ times composition of $\\phi$ by $\\phi^k$ ($\\phi^k=\\phi\\circ\\cdots\\circ\\phi$ ($k$ times)) such that $\\phi^0=id$ and $\\phi^1=\\phi$. A $\\phi^k$-$\\rho$-derivation of degree $|X|$ on $A$ is a linear map $X : A \\longrightarrow A$ \nsuch that\\\\\n$$X\\circ\\phi=\\phi\\circ X\\quad i.e.,\\quad [X,\\phi]_{\\rho}=0,$$\nand\n\\begin{equation}\\label{5}\nX[f,g,h]= [X(f),\\phi^k(g),\\phi^k(h)]+ \\rho(X,f)[\\phi^k(f),X(g),\\phi^k(h)]+\\rho(X,f+g)[\\phi^k(f),\\phi^k(g),X(h)],\n\\end{equation}\nfor all $f,g,h\\in A$. Let us denote by $\\rho\\text{-}Der_{\\phi^k} A$ the set of all $\\phi^k$-$\\rho$-derivations of $A$. \n\\end{definition}\n\\begin{example}\nWe define the even linear map $ad_k(f_1,f_2):A\\longrightarrow A$ by $ad_k(f_1,f_2)(g)=[f_1,f_2,\\phi^k(g)]$ for $g\\in A$. If we assume that for any $f_1,f_2\\in A$,\n$\\phi(f_1)=f_1$ and $\\phi(f_2)=f_2$, then $ad_k(f_1,f_2)$ is a $\\phi^{k+1}$-derivation.\\\\\nIf we check the accuracy of the equality \\eqref{5}, the assertion follows. Thus, we have\n\\begin{align*}\nad_k(f_1,f_2)[f,g,h]&=[f_1,f_2,\\phi^k[f,g,h]]=[\\phi(f_1),\\phi(f_2),\\phi^k[f,g,h]]\\\\\n&=[\\phi(f_1),\\phi(f_2),[\\phi^k(f),\\phi^k(g),\\phi^k(h)]]\\\\\n&=[[f_1,f_2,\\phi^k(f)],\\phi^{k+1}(g),\\phi^{k+1}(h)]\\\\\n&\\ \\ \\ +\\rho(f_1+f_2,f)[\\phi^{k+1}(f),[f_1,f_2,\\phi^k(g)],\\phi^{k+1}(h)]\\\\\n&\\ \\ \\ +\\rho(f_1+f_2,f+g)[\\phi^{k+1}(f),\\phi^{k+1}(g),[f_1,f_2,\\phi^k(h)]]\\\\\n&=[ad_k(f_1,f_2)(f),\\phi^{k+1}(g),\\phi^{k+1}(h)]\\\\\n&\\ \\ \\ +\\rho(f_1+f_2,f)[\\phi^{k+1}(f),ad_k(f_1,f_2(g)),\\phi^{k+1}(h)]\\\\\n&\\ \\ \\ +\\rho(f_1+f_2,f+g)[\\phi^{k+1}(f),\\phi^{k+1}(g),ad_k(f_1,f_2)(h)].\n\\end{align*}\n$ad_k(f_1,f_2)$  is called an inner $\\phi^{k+1}$-derivation. Denote by ${\\rm Inn_{\\phi^k}(A)}$ the set of inner $\\phi^{k}$-derivation, i.e.,\n$${\\rm Inn_{\\phi^k}(A)}=\\{[f_1,f_2,\\phi^{k-1}(\\cdot)]|~~ f_1,f_2\\in A,~~ \\phi(f_i)=f_i~~ i=1,2\\}.$$\n\\end{example}\n\\begin{definition}\nLet $A$ be a 3-Hom-$\\rho$-Lie algebra.\\\\\n1: A linear map $X:A\\longrightarrow A$ is said to be a homogeneous generalized $\\phi^k$-derivation of degree $|X|$ of $A$, if there exist three linear maps $Y,Z,W:A\\longrightarrow A$ such that \n$$[X,\\phi]=0,~~[Y,\\phi]=0,~~[Z,\\phi]=0,~~[W,\\phi]=0,$$\nand\n\\begin{align*}\nW[f,g,h]&=[X(f),\\phi^k(g),\\phi^k(h)]+\\rho(X,f)[\\phi^k(f), Y(g),\\phi^k(h)]\\\\\n&\\ \\ \\ +\\rho(X,f+g)[\\phi^k(f),\\phi^k(g),Z(h)],\n\\end{align*}\nfor all $f,g,h\\in A$. We denote the set of all homogeneous generalized $\\phi^k$-derivation of degree \n$| X|$ of $A$ by $GDer_{\\phi^k}(A)$.\\\\\n2: We call $X:A\\longrightarrow A$ a homogeneous $\\phi^k$-quasi derivation of degree $|X|$ of $A$, if there exists a linear map $Y:A\\longrightarrow A$ such that\n$$[X,\\phi]=0,~~[Y,\\phi]=0,$$\nand\n\\begin{align*}\nY[f,g,h]&=[X(f),\\phi^k(g),\\phi^k(h)]+\\rho(X,f)[\\phi^k(f), X(g),\\phi^k(h)]\\\\\n&\\ \\ \\ +\\rho(X,f+g)[\\phi^k(f),\\phi^k(g),X(h)],\n\\end{align*}\nfor all $f,g,h\\in A$. We denote the set of all homogeneous $\\phi^k$-quasi derivation of degree \n$| X|$ of $A$ by $QDer_{\\phi^k}(A)$.\\\\\n3: We call $X:A\\longrightarrow A$ a homogeneous $\\phi^k$-centroid element of degree $|X|$ of $A$, if it satisfies for all $f,g,h\\in Hg(A)$\n\\begin{align*}\nX[f,g,h]&=[X(f),\\phi^k(g),\\phi^k(h)]=\\rho(X,f)[\\phi^k(f), X(g),\\phi^k(h)]\\\\\n&=\\rho(X,f+g)[\\phi^k(f),\\phi^k(g),X(h)].\n\\end{align*}\nWe denote the set of all homogeneous $\\phi^k$-centroid elements of degree \n$| X|$ of $A$ by $C_{\\phi^k}(A)$.\\\\\n4: $X:A\\longrightarrow A$ is said to be a homogeneous $\\phi^k$-quasi centroid element of degree $|X|$ of $A$, if it satisfies for all $f,g,h\\in Hg(A)$\n\\begin{align*}\nX[f,g,h]&=[X(f),\\phi^k(g),\\phi^k(h)]=\\rho(X,f)[\\phi^k(f), X(g),\\phi^k(h)]\\\\\n&=\\rho(X,f+g)[\\phi^k(f),\\phi^k(g),X(h)].\n\\end{align*}\nWe denote the set of all homogeneous $\\phi^k$-quasi centroid elements of degree \n$| X|$ of $A$ by $QC_{\\phi^k}(A)$.\n\\end{definition}\n\\begin{proposition}\nLet $(A,[.,.,.],\\rho,\\phi)$ be a multiplication 3-Hom-$\\rho$-Lie algebra. If $X\\in GDer_{\\phi^k}(A)$ and $X^{\\prime}\\in C_{\\phi^s}(A)$, then $X^{\\prime}X\\in GDer_{\\phi^{k+s}}(A)$.\n\\end{proposition}\n\\begin{proof}\nSince $X\\in GDer_{\\phi^k}(A)$, then there exist $Y,Z,W:A\\longrightarrow A$ such that \n\\begin{align*}\nW[f,g,h]&=[X(f),\\phi^k(g),\\phi^k(h)]+\\rho(X,f)[\\phi^k(f), Y(g),\\phi^k(h)]\\\\\n&\\ \\ \\ +\\rho(X,f+g)[\\phi^k(f),\\phi^k(g),Z(h)].\n\\end{align*}\nOn the other hand, since $X^{\\prime}\\in C_{\\phi^s}(A)$ we have\n\\begin{align*}\nX^{\\prime}W[f,g,h]&=X^{\\prime}[X(f),\\phi^k(g),\\phi^k(h)]+\\rho(X,f)X^{\\prime}[\\phi^k(f), Y(g),\\phi^k(h)]\\\\\n&\\ \\ \\ +\\rho(X,f+g)X^{\\prime}[\\phi^{k+s}(f),\\phi^{k+s}(g),Z(h)]\\\\\n&=[X^{\\prime}X(f),\\phi^{k+s}(g),\\phi^{k+s}(h)]+\\rho(X,f)[\\phi^{k+s}(f), X^{\\prime}Y(g),\\phi^{k+s}(h)]\\\\\n&\\ \\ \\ +\\rho(X,f+g)[\\phi^{k+s}(f),\\phi^{k+s}(g),X^{\\prime}Z(h)].\n\\end{align*}\nTherefore, $X^{\\prime}X\\in GDer_{\\phi^{k+s}}(A)$.\n\\end{proof}\n\\begin{proposition}\nLet $(A,[.,.,.],\\rho,\\phi)$ be a multiplication 3-Hom-$\\rho$-Lie algebra and $X\\in C_{\\phi^{k}}(A)$. Then $X$ is a $\\phi^k$-quasi derivation of $A$. \n\\end{proposition}\n\\begin{proof}\nAssuming that $f,g,h \\in Hg(A)$. So, we have\n\\begin{align*}\n[X(f),\\phi^k(g),\\phi^k(&h)]+\\rho(X,f)[\\phi^k(f), X(g),\\phi^k(h)]\\\\\n&\\ \\ \\ +\\rho(X,f+g)[\\phi^k(f),\\phi^k(g),X(h)]\\\\\n&=3[X(f),\\phi^k(g),\\phi^k(h)]=3X[f,g,h]\\\\\n&=X^{\\prime}[f,g,h].\n\\end{align*}\n\\end{proof}\n\\subsection{The Coadjoint Representation}\nWe consider $A^{\\star}$ as a dual space of $A$, $A^{\\star}$ is a $G$-graded space, where $A^{\\star}_a=\\{\\alpha\\in A^{\\star}| \\alpha(f)=0,~~\\forall~~f:~~|f|\\neq -a\\}$. Moreover, $A^{\\star}$ is a graded $A$-module. Also, since $A=\\oplus_{a\\in G}A_a$ and $A^{\\star}=\\oplus_{a\\in G}A^{\\star}_a$ are $G$-graded spaces, the direct sum \n$$A\\oplus A^{\\star}=\\oplus_{a\\in G}(A\\oplus A^{\\star})_a=\\oplus_{a\\in G}(A_a\\oplus A^{\\star}_a),$$\nis $G$-graded. Consider a homogeneous element of $A\\oplus A^{\\star}$ as $f+\\alpha$ such that $f\\in A$ and $\\alpha\\in A^{\\star}$, with $|f+\\alpha|=|f|=|\\alpha|$.\n\nLet $(A,[.,.,.]_A,\\rho,\\phi)$ be a 3-Hom-$\\rho$-Lie algebra and $(V,\\mu,\\beta)$ be a representation of $A$. Let $V^{\\star}$ be the dual vector space of $V$. We define the linear map $\\widetilde{\\mu}:A\\times A\\longrightarrow End(V^{\\star})$ by $\\widetilde{\\mu}(f_1,f_2)(\\varrho)=-\\rho(f_1+f_2,\\varrho)\\varrho\\circ\\mu(f_1,f_2)$, where $f_1,f_2\\in A,~~\\varrho\\in V^{\\star}$ and set $\\widetilde{\\beta}(\\varrho)=\\varrho\\circ\\beta$.\n\\begin{proposition}\nLet $(A,[.,.,.]_A,\\rho,\\phi)$ be a 3-Hom-$\\rho$-Lie algebra and $(V,\\mu,\\beta)$ be a representation of $A$. Then the triple $(V^{\\star},\\widetilde{\\mu},\\widetilde{\\beta})$ defines a representation of $A$ if and only if \n\\begin{align*}\n&(1).~~\\beta(\\mu[(f_1,f_2),(g_1,g_2)]_{\\mathcal{L}})=\\mu(f_1,f_2)\\mu(\\phi(g_1),\\phi(g_2))-\\rho(f_1+f_2,g_1+g_2)\\mu(g_1,g_2)\\mu(\\phi(f_1),\\phi(f_2)),\\\\\n&(2).~~ \\beta\\mu([g_1,g_2,g_3],\\phi(f))=-\\rho(g_1+g_2,g_3+f)\\mu(g_3,f)\\mu(\\phi(g_1),\\phi(g_2))\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad -\\rho(g_2+g_3,g_1+f)\\mu(g_1,f)\\mu(\\phi(g_2),\\phi(g_3))\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad-\\rho(g_3+g_1,g_2+f)\\mu(g_2,f)\\mu(\\phi(g_3),\\phi(g_1)),\\\\\n&(3).~~ \\beta\\mu(\\phi(g),[f_1,f_2,f_3])=-\\rho(f_1+f_2,f_3)\\mu(g,f_3)\\mu(\\phi(f_1),\\phi(f_2))-\\mu(g,f_1)\\mu(\\phi(f_2),\\phi(f_3))\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad-\\rho(f_1+f_2,f_3)\\rho(f_1+f_3,f_2)\\mu(g,f_2)\\mu(\\phi(f_3),\\phi(f_1)),\\\\\n&(4).~~  \\rho(f_1+f_2,g_1)\\beta\\mu(\\phi(g_1),[f_1,f_2,g_2])+\\beta\\mu([f_1,f_2,g_1],\\phi(g_2))=\\mu(f_1,f_2)\\mu(\\phi(g_1),\\phi(g_2))\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad-\\rho(f_1+f_2,g_1+g_2)\\mu(g_1,g_2)\\mu(\\phi(f_1),\\phi(f_2)).\n\\end{align*} \n\\end{proposition}\n\\begin{proof}\nAssuming that the conditions (1)-(4) hold. \nWe prove that $\\widetilde{\\mu}$ is a representation of $A$, thus we must check the properties \\eqref{1}-\\eqref{7.1} for $(\\widetilde{\\mu},\\widetilde{\\beta},V^{\\star})$. For this, we check the property \\eqref{1} and the others will prove similarly. Using (1), we start with the computation of the left-hand side of \\eqref{1}:\n\\begin{align*}\n\\widetilde{\\mu}[(f_1,f_2), (g_1,g_2)]_{\\mathcal{L}}\\widetilde{\\beta}(\\varrho)(m)&=-\\rho(f_1+f_2+g_1+g_2,\\varrho)(\\varrho\\circ\\beta)(\\mu[(f_1,f_2), (g_1,g_2)]_{\\mathcal{L}})\\\\\n&=\\rho(f_1+f_2+g_1+g_2,\\varrho)\\rho(f_1+f_2,g_1+g_2)\\varrho\\mu(g_1,g_2)\\mu(\\phi(f_1),\\phi(f_2))(m)\\\\\n&\\ \\ \\ -\\rho(f_1+f_2+g_1+g_2,\\varrho)\\varrho\\mu(f_1,f_2)\\mu(\\phi(g_1),\\phi(g_2))(m)\\\\\n&=\\widetilde{\\mu}(\\phi(f_1),\\phi(f_2))\\widetilde{\\mu}(g_1,g_2)\\varrho(m)\\\\\n&\\ \\ \\ -\\rho(f_1+f_2,g_1+g_2)\\widetilde{\\mu}(\\phi(g_1),\\phi(g_2))\\widetilde{\\mu}(f_1,f_2)\\varrho(m).\n\\end{align*}\nTherefore\n\\begin{align*}\n\\widetilde{\\mu}[(f_1,f_2), (g_1,g_2)]_{\\mathcal{L}}\\circ\\widetilde{\\beta}&=\\widetilde{\\mu}(\\phi(f_1),\\phi(f_2))\\widetilde{\\mu}(g_1,g_2)\\\\\n&\\ \\ \\ -\\rho(f_1+f_2,g_1+g_2)\\widetilde{\\mu}(\\phi(g_1),\\phi(g_2))\\widetilde{\\mu}(f_1,f_2).\n\\end{align*}\nThe proof of the converse is also straightforward.\n\\end{proof}\n\\begin{corollary}\nLet $(A,[.,.,.]_A,\\rho,\\phi)$ be a 3-Hom-$\\rho$-Lie algebra with the adjoint representation and $A^{\\star}$ be the dual of $A$. The linear map $ad^{\\star}:A\\times A\\longrightarrow {\\rm End(A^{\\star})}$ defined by $ad^{\\star}(f_1,f_2)(\\varrho)(f_3)=-\\rho(f_1+f_2,\\varrho)\\varrho[f_1,f_2,f_3]_A=-\\rho(f_1+f_2,\\varrho)\\varrho(ad(f_1,f_2)(f_3)$ for $f_1,f_2,f_3\\in A$ and $\\varrho\\in A^{\\star}$ is a representation of $A$ that is called coadjoint representation.\n\\end{corollary}\n\\begin{theorem}\nLet $(A,[.,.,.]_A,\\rho,\\phi)$ be a 3-Hom-$\\rho$-Lie algebra with the adjoint representation and $A^{\\star}$ be the dual of $A$. Consider the linear map $\\omega:A\\times A\\times A\\longrightarrow A^{\\star}$ and let $\\widetilde{\\mu}=ad^{\\star}$. The $G$-graded space $A\\oplus A^{\\star}$ together with the bracket and linear map \n\\begin{align*}\n[f_1+\\alpha_1,f_2+\\alpha_2,f_3+\\alpha_3]_{A\\oplus A^{\\star}}&=[f_1,f_2,f_3]_A+\\omega(f_1,f_2,f_3)+\\widetilde{\\mu}(f_1,f_2)(\\alpha_3)\\\\\n&\\ \\  \\ +\\rho(f_1+f_2,f_3)\\widetilde{\\mu}(f_3,f_2)(\\alpha_2)+\\rho(f_1,f_2+f_3)\\widetilde{\\mu}(f_2,f_3)(\\alpha_1),\\\\\n\\phi^{\\prime}(f+\\alpha)=\\phi(f)+\\alpha\\circ\\phi,\\ \\ \\ \\ \\ \\ \\ &\n\\end{align*}\nfor all $f_i\\in A$ and $\\alpha_i\\in A^{\\star},~~~ i=1,2,3$, is a 3-Hom-$\\rho$-Lie algebra if and only if $\\omega$ is a 1-Hom-cocycle with respect to $\\widetilde{\\mu}$.\n\\end{theorem}\n\\begin{proof}\nUsing \\eqref{2}-\\eqref{7.1} and Definition \\ref{1234}, the result easily follows.\n\\begin{definition}\nThe 3-hom-$\\rho$-Lie algebra $(A\\oplus A^{\\star}, [.,.,.]_{A\\oplus A^{\\star}} , \\phi^{\\prime})$ is called the $T^{\\star}_{\\omega}$-extension of $(A, [.,.,.]_A, \\phi)$.\n\\end{definition}\n\\end{proof}\n\\section{abelian extension of 3-Hom-$\\rho$-lie algebra}\nIn this section, we discuss about extensions and abelian extensions of 3-Hom-$\\rho$-Lie algebra $A$ and show that associated to any abelian extension, there is a\nrepresentation and a 2-cocycle. We assume that the 3-Hom-$\\rho$-Lie algebra $A$ is multiplicative.\n\\begin{definition}\nA sub-vector space $I\\subseteq A$ is called a Hom subalgebra of $(A,[.,.,.]_A,\\phi)$ if $[I,I,I]_A\\subseteq I$ and $\\phi(I)\\subseteq I$, $I$ also is called a Hom ideal of $A$ if $\\phi(I)\\subseteq I$ and $[I,A,A]_A\\subseteq I$. $I$ is said to be a Hom abelian ideal of $A$ if $[A,I,I]_A=0$.\n\\end{definition}\n\\begin{definition}\nLet $(A,[.,.,]_A,\\rho,\\phi)$, $(V,[.,.,.]_V,\\phi_V)$ and $(B,[.,.,.]_B,\\psi)$ be three 3-Hom-$\\rho$-Lie algebras and $i:V\\longrightarrow B$, $p:B\\longrightarrow A$ be homomorphisms. The sequence \n\\begin{equation*}\n\\begin{tikzcd}\n0\\arrow{r} & V \\arrow{r}{i} & B\\arrow{r}{p} & A \\arrow{r} & 0,\n\\end{tikzcd}\n\\end{equation*}\nof 3-Hom-$\\rho$-Lie algebras is a short exact sequence if ${\\rm Im(i)}={\\rm Ker(p)}$, ${\\rm Ker(i)=0}$, ${\\rm Im(p)}=A$ and $\\phi_V(V)=\\psi(V)$. In this case, $B$ is called an extension of $A$ by $V$ and denote it by $E_B$. Also, we call $B$ an abelian extension of $A$ if $V$ is a Hom abelian ideal of $B$ i.e., $[.,u,v]_B=0$\nfor all $u,v\\in V$. A linear map $\\delta:A\\longrightarrow B$ is called a section of $p:B\\longrightarrow A$ if $p\\circ\\delta=id_{A}$ and $\\delta\\circ\\phi=\\psi\\circ\\delta$.\n\\end{definition}\n\\begin{definition}\nTwo extensions $0\\xrightarrow{} V\\xrightarrow{i} B\\xrightarrow{p} A\\xrightarrow{} 0$ and \n$0\\xrightarrow{} V\\xrightarrow{j} B\\xrightarrow{q} A\\xrightarrow{} 0$ of 3-Hom-$\\rho$-Lie algebra $A$ are equivalent if there exists a morphism $F:B\\longrightarrow\\tilde{B}$ of 3-Hom-$\\rho$-Lie algebras such that the following diagram commutes: \n\\begin{equation*}\n\\begin{tikzcd}\n0\\arrow{r} & V \\arrow{r}{i}\\arrow{d}{id}  & B\\arrow{r}{p}\\arrow{d}{F} & A \\arrow{r}\\arrow{d}{id} & 0\\\\\n0\\arrow{r} & V \\arrow{r}{j} & \\tilde{B}\\arrow{r}{q}& A \\arrow{r} & 0.\n\\end{tikzcd}\n\\end{equation*}\n\\end{definition}\n\\vspace{1 cm}\nLet $B$ be an abelian extension of $A$ by $V$ and $\\delta:A\\longrightarrow B$ be a section. Define $\\mu:\\wedge^2A\\longrightarrow{\\rm End(V)}$ by \n\\begin{equation}\\label{t22}\n\\mu(f)(u)=\\mu(f_1,f_2)(u)=[\\delta(f_1),\\delta(f_2),u]_B=ad(\\delta(f))u,\n\\end{equation}\nfor all $f=(f_1,f_2)\\in\\wedge^2A, u\\in V$.\n\\begin{proposition}\nLet $(V,\\phi_V)$, $(A,\\phi)$ and $(B,\\psi)$ be multiplication 3-Hom-$\\rho$-Lie algebras and $B$ be an abelian extension of $A$ by $V$. Cosider $\\mu$ with \\eqref{t22}. Then, $(V,\\phi_V,\\mu)$ is a representation of $(A,\\phi)$ and does not depend on the choice of the section $\\delta$. Moreover, equivalent abelian extensions give the same representation.\n\\end{proposition}\n\\begin{proof}\nAt first, we show that $\\mu$ is independent of the choice of $\\delta$. If $\\delta^{\\prime}:A\\longrightarrow B$ is another section, then \n$$p(\\delta(f_i)-\\delta^{\\prime}(f_i))=f_i-f_i=0,$$\nthus, $\\delta(f_i)-\\delta^{\\prime}(f_i)\\in V$. So $\\delta^{\\prime}(f_i)=\\delta(f_i)+u$ for some $u\\in V$. Since $[.,u,v]_B=0$ for all $u,v\\in V$, we deduce that\n\\begin{align*}\n[\\delta^{\\prime}(f_1),\\delta^{\\prime}(f_2),w]_B&=[\\delta(f_1)+u,\\delta(f_2)+v,w]_B\\\\\n&=[\\delta(f_1),\\delta(f_2)+v,w]_B+[u,\\delta(f_2)+v,w]_B\\\\\n&=[\\delta(f_1),\\delta(f_2),w]_B+[\\delta(f_1),v,w]_B+[u,\\delta(f_2),w]_B+[u,v,w]_B\\\\\n&=[\\delta(f_1),\\delta(f_2),w]_B.\n\\end{align*}\nSo, $\\mu$ is independent of the choice of $\\delta$. In the next, we show that $(V,\\phi_V,\\mu)$ is a representation of $(A,\\phi)$. For this it is enough to check the conditions \\eqref{1} and \\eqref{2}. Let $\\beta=\\phi_V\\in {\\rm End(V)}$. By the third property of 3-Hom-$\\rho$-Lie algebras, we have \n\\begin{align*}\n[\\psi(\\delta(f_1)),\\psi(\\delta(f_2)),[\\delta(g_1),\\delta(g_2),u]_B]_B&=[[\\delta(f_1),\\delta(f_2),\\delta(g_1)]_B,\\psi(\\delta(g_2)),\\psi(u)]_B\\\\\n&\\ \\ \\ +\\rho(f_1+f_2,g_1)[\\psi(\\delta(g_1)),[\\delta(f_1),\\delta(f_2),\\delta(g_2)]_B,\\psi(u)]_B\\\\\n&\\ \\ \\ +\\rho(f_1+f_2,g_1+g_2)[\\psi(\\delta(g_1)),\\psi(\\delta(g_2)),[\\delta(f_1),\\delta(f_2),u)]_B]_B.\n\\end{align*} \nUsing \\eqref{t22} and this fact that $\\psi\\circ\\delta=\\delta\\circ\\phi$, we have\n\\begin{align*}\n\\mu(\\phi(f_1),\\phi(f_2))\\mu(g_1,g_2)u&=\\mu[(f_1,f_2),(g_1,g_2)]_A\\circ\\phi_V(u)\\\\\n&\\ \\ \\ +\\rho(f_1+f_2,g_1+g_2)\\mu(\\phi(g_1),\\phi(g_2))\\mu(f_1,f_2)u.\n\\end{align*}\nTherefore\n\\begin{align*}\n\\mu[(f_1,f_2),(g_1,g_2)]_A\\circ\\phi_V(u)&=\\mu(\\phi(f_1),\\phi(f_2))\\mu(g_1,g_2)u\\\\\n&\\ \\ \\ -\\rho(f_1+f_2,g_1+g_2)\\mu(\\phi(g_1),\\phi(g_2))\\mu(f_1,f_2)u.\n\\end{align*}\nThis gives us the condition \\eqref{1}. In the continues, we try to prove the correctness of the condition \\eqref{2}.\\\\\nSince $\\phi_V(V)=\\psi(V)$, $\\delta\\circ\\phi=\\psi\\circ\\delta$, $[\\delta(f_1),\\delta(f_2),\\delta(g_1)]_B-\\delta[f_1,f_2,g_1]_A\\in V$ and $V$ is abelian ideal then \n\\begin{align}\n\\mu([f_1,f_2,g_1],\\phi(g_2))\\phi_V(u)&=[\\delta[f_1,f_2,g_1],\\delta(\\phi(g_2)),\\phi_V(u)]\\nonumber\\\\\n&=[[\\delta(f_1),\\delta(f_2),\\delta(g_1)],\\psi\\circ\\delta(g_2),\\psi(u)].\\label{231}\n\\end{align}\nOn the other hand, we have\n\\begin{align*}\n[\\psi(\\delta(f_1)),\\psi(u),[\\delta(g_1),\\delta(g_2),\\delta(g_3)]_B]_B&=[[\\delta(f_1),u,\\delta(g_1)]_B,\\psi(\\delta(g_2)),\\psi(\\delta(g_3))]_B\\\\\n&\\ \\ \\ +\\rho(f_1+u,g_1)[\\psi(\\delta(g_1)),[\\delta(f_1),u,\\delta(g_2)]_B,\\psi(\\delta(g_3))]_B\\\\\n&\\ \\ \\ +\\rho(f_1+u,g_1+g_2)[\\psi(\\delta(g_1)),\\psi(\\delta(g_2)),[\\delta(f_1),u,\\delta(g_3)]_B]_B.\n\\end{align*}\nBy invoking \\eqref{231} and this fact that $\\delta\\circ\\phi=\\psi\\circ\\delta$ and $\\beta=\\phi_V$, we conclude that\n\\begin{align*}\n\\rho(f_1+u,g_1+g_2+g_3)\\mu(&[g_1,g_2,g_3],\\phi(f_1))\\phi_V(u)=\\rho(f_1+u,g_1+g_2)\\rho(f_1+u,g_3)\\mu(\\phi(g_1),\\phi(g_2))\\mu(g_3,f_1)u\\\\\n& \\ \\ \\ +\\rho(f_1+u+g_1,g_2+g_3)\\rho(f_1+u,g_1)\\mu(\\phi(g_2),\\phi(g_3))\\mu(g_1,f_1)u\\\\\n&\\ \\ \\ + \\rho(f_1+u+g_2,g_3)\\rho(g_1,g_3)\\rho(f_1+u,g_1+g_2)\\mu(\\phi(g_3),\\phi(g_1))\\mu(g_2,f_1)u.\n\\end{align*}\nSo, this statements lead us to\n\\begin{align*}\n\\mu([g_1,g_2,g_3],\\phi(f_1))\\circ\\phi_V&=\\mu(\\phi(g_1),\\phi(g_2))\\mu(g_3,f_1)\\\\\n& \\ \\ \\ +\\rho(g_1,g_2+g_3)\\mu(\\phi(g_2),\\phi(g_3))\\mu(g_1,f_1)\\\\\n&\\ \\ \\ + \\rho(g_1+g_2,g_3)\\mu(\\phi(g_3),\\phi(g_1))\\mu(g_2,f_1).\n\\end{align*}\nTherefore, the result holds. At last, we investigate that equivalent abelian extension give the same representation. For this, suppose that $E_B$ and $E_{\\tilde{B}}$ are equivalent abelian extensions presented by \n\\begin{equation*}\n\\begin{tikzcd}\n\t0\\arrow{r} & V \\arrow{r}{i} & B\\arrow{r}{p}& A \\arrow{r} & 0\\\\\n\t0\\arrow{r} & V \\arrow{r}{j} & \\tilde{B}\\arrow{r}{q}& A \\arrow{r} & 0,\n\\end{tikzcd}\n\\end{equation*}\n and $F:B\\longrightarrow \\tilde{B}$ is the 3-Hom-$\\rho$-Lie algebra homomorphism, satisfying $F\\circ i=j$, $q\\circ F=p$. Choose linear sections $\\delta$ and $\\delta^{\\prime}$ of $p$ and $q$. So, we obtain $q\\circ F(\\delta(f_i))=p\\circ\\delta(f_i)=f_i=q\\circ\\delta^{\\prime}(f_i)$. Then, $F\\circ\\delta(f_i)-\\delta^{\\prime}(f_i)\\in{\\rm Ker(q)}\\cong V$. Thus, we have\n $$[\\delta(f_1), \\delta(f_2),u]_B=[F\\circ\\delta(f_1),F\\circ\\delta(f_2),u]_{\\tilde{B}}=[\\delta^{\\prime}(f_1),\\delta^{\\prime}(f_2),u]_{\\tilde{B}}.\n $$\n Therefore, we get the result.\n\\end{proof}\n\\begin{proposition}\nLet $\\delta:A\\longrightarrow B$ be a section of the abelian extension of $A$ by $V$. Define the map \n$$\\omega(f_1,f_2,f_3)=[\\delta(f_1),\\delta(f_2),\\delta(f_3)]_B-\\delta([f_1,f_2,f_3]_A),$$\nfor all $f_1, f_2, f_3\\in A$. Then $\\omega$ is a 1-cocycle, where the representation $\\mu$ is given by \\eqref{t22}.\n\\end{proposition}\n\\begin{proof}\nSince $B$ is a 3-Hom-$\\rho$-Lie algebra, we have\n\\begin{align}\\label{t25}\n[\\psi(\\delta(f_1)), \\psi(\\delta(f_2)), [\\delta(g_1), \\delta(g_2), \\delta(g_3)&]_B]_B=[[\\delta(f_1), \\delta(f_2), \\delta(g_1)]_B, \\psi(\\delta(g_2)), \\psi(\\delta(g_3))]_B\\nonumber\\\\\n&\\ \\ \\  +\\rho(f_1+f_2,g_1)[\\psi(\\delta(g_1)), [\\delta(f_1), \\delta(f_2), \\delta(g_2)]_B, \\psi(\\delta(g_3))]_B\\nonumber\\\\\n&\\ \\ \\  +\\rho(f_1+f_2+g_1+g_2)[\\psi(\\delta(g_3)), \\psi(\\delta(g_3)), [\\delta(f_1), \\delta(f_2), \\delta(g_3)]_B ]_B.\n\\end{align}\nOn the other hand, we have\n\\begin{align*}\n[\\psi(\\delta(f_1)), \\psi(\\delta(f_2)), [\\delta(g_1), \\delta(g_2), \\delta(g_3)&]_B]_B=[\\psi(\\delta(f_1)), \\psi(\\delta(f_2)), \\omega(g_1, g_2, g_3)+\\delta[g_1, g_2, g_3]_A]_B\\\\\n&= \\mu(\\phi(f_1), \\phi(f_2))\\omega(g_1, g_2, g_3) +[\\psi(\\delta(f_1)), \\psi(\\delta(f_2)),\\delta[g_1, g_2, g_3]_A]_B\\\\\n&= \\mu(\\phi(f_1), \\phi(f_2))\\omega(g_1, g_2, g_3) +\\omega(\\phi(f_1), \\phi(f_2), [g_1, g_2, g_3]_A)\\\\\n&\\ \\ \\ +\\delta[\\psi(\\delta(f_1)), \\psi(\\delta(f_2)),[g_1, g_2, g_3]_A]_B.\n\\end{align*}\nSimilarly, the right hand side of \\eqref{t25} is equal to\n\\begin{align*}\n&\\rho(f_1+f_2+g_1, g_2+g_3)\\mu(\\phi(g_2), \\phi(g_3))\\omega(f_1, f_2, g_1)+\\omega([f_1, f_2, g_1]_A,\\phi(g_2), \\phi(g_3))\\\\\n&\\ \\ \\ +\\delta[[f_1, f_2, g_1]_A, \\phi(g_2), \\phi(g_3)]_A+\\rho(f_1+f_2, g_1)\\rho(f_1+f_2+g_2, g_3)\\rho(g_1, g_2)\\mu(\\phi(g_3), \\phi(g_1))\\omega(f_1, f_2, g_2)\\\\\n&\\ \\ \\ +\\rho(f_1+f_2, g_1)\\delta[\\phi(g_1), [f_1, f_2, g_2]_A, \\phi(g_3)]_A+\\rho(f_1+f_2, g_1+g_2)\n\\mu(\\phi(g_1), \\phi(g_2))\\omega(f_1, f_2, g_3)\\\\\n&\\ \\ \\ +\\rho(f_1+f_2, g_1+g_2)\\omega(\\phi(g_1), \\phi(g_2), [f_1, f_2, g_3]_A) +\\rho(f_1+f_2, g_1+g_2)\\delta[\\phi(g_1), \\phi(g_2), [f_1, f_2, g_3]_A]_A.\n\\end{align*}\nSo, we get\n\\begin{align*}\n\\omega(\\phi(f_1), \\phi(f_2), [g_1, g_2, g_3]_A) &+\\mu(\\phi(f_1), \\phi(f_2))\\omega(g_1, g_2, g_3)=\\omega([f_1, f_2, g_1]_A,\\phi(g_2), \\phi(g_3))\\\\\n&\\ \\ \\ +\\rho(f_1+f_2, g_1)\\omega(\\phi(g_1),[f_1, f_2, g_2]_A,\\phi(g_3))\\\\\n&\\ \\ \\ +\\rho(f_1+f_2, g_1+g_2)\\omega(\\phi(g_1),\\phi(g_3), [f_1, f_2, g_3]_A)\\\\\n&\\ \\ \\ +\\rho(f_1+f_2, g_1)\\rho(f_1+f_2+g_2, g_3)\\rho(g_1, g_3)\n\\mu(\\phi(g_3), \\phi(g_1))\\omega(f_1, f_2, g_2)\\\\\n&\\ \\ \\ +\\rho(f_1+f_2, g_1+g_2)\n\\mu(\\phi(g_1), \\phi(g_2))\\omega(f_1, f_2, g_3)\\\\\n&\\ \\ \\ +\\rho(f_1+f_2+g_1, g_2+g_3)\\mu(\\phi(g_2), \\phi(g_3))\\omega(f_1, f_2, g_1).\n\\end{align*}\nTherefore, $\\omega$ is a 1-cocycle.\n\\end{proof}\n\\section{Infinitesimal deformations of 3-Hom-$\\rho$-Lie algebras}\nIn this section, we introduce infinitesimal deformations of 3-Hom-$\\rho$-Lie algebras and define Hom-Nijenhuis operator of it.\n\nLet $A$ be a 3-Hom-$\\rho$-Lie algebra and $\\omega:\\wedge^3A\\longrightarrow A$ be a morphism. Consider a $t$-parametrized family of linear operations\n$$[f,g,h]_t=[f,g,h]_A+t\\omega(f,g,h).$$\nIf $A$ with all the brackets $[.,.,.]_t$ endow regular 3-Hom-$\\rho$-Lie algebra structure $(A, [.,.,.]_t, \\rho,\\phi)$ which is denoted by $A_t$, we\nsay that $\\omega$ generates a $t$-parameter infinitesimal deformation of the 3-Lie colour algebra $A$.\n\\begin{theorem}\n$\\omega$ generates a $t$-parameter infinitesimal deformation of the 3-Hom-$\\rho$-Lie algebra $A$ is equivalent to \n\\begin{enumerate}\n\\item \n$\\omega$ itself defines a 3-Hom-$\\rho$-Lie algebra structure on $A$.\n\\item\n$\\omega$ is a 1-cocycle of $A$ with coefficients in the adjoint representation.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nSince $[.,.,.]_t$ endow $(A, [.,.,.]_t, \\rho,\\phi)$ a regular 3-Hom-$\\rho$-Lie algebra structure, then, \n\\begin{align*} \n[\\phi(f_1),\\phi(f_2),[g_1,g_2,g_3]_t]_t&=[[f_1,f_2,g_1]_t,\\phi(g_2),\\phi(g_3)]_t-\\rho(f_1+f_2,g_1)[\\phi(g_1),[f_1,f_2,g_2]_t,\\phi(g_3)]_t\\\\\n&\\ \\ \\ +\\rho(f_1+f_2, g_1+g_2)[\\phi(g_1),\\phi(g_2),[f_1,f_2,g_3]_t]_t.\n\\end{align*}\nThe left-hand side is equal to\n\\begin{align*}\n[\\phi(f_1),\\phi(f_2),[g_1,g_2,g_3]_A]_A &+t\\{\\omega(\\phi(f_1), \\phi(f_2), [g_1, g_2, g_3]_A)+[\\phi(f_1),\\phi(f_2), \\omega(g_1, g_2, g_3)]_A\\}\\\\\n& +t^2\\omega(\\phi(f_1), \\phi(f_2), \\omega(g_1, g_2, g_3)).\n\\end{align*}\nThe right-hand side too is equal to\n\\begin{align*}\n&[[f_1, f_2, g_1]_A, \\phi(g_1), \\phi(g_3)]+t\\omega([f_1, f_2, g_1]_A, \\phi(g_2), \\phi(g_3))+[t\\omega(f_1,f_2,g_1),\\phi(g_2),\\phi(g_3)]_A\\\\\n&+t^2\\omega(\\omega(f_1, f_2,g_1),\\phi(g_2),\\phi(g_3))+\\rho(f_1+f_2, g_1)[\\phi(g_1), [f_1, f_2,g_2]_A,\\phi(g_3)]_A\\\\\n&+\\rho(f_1+f_2, g_1)t\\omega(\\phi(g_1),[f_1,f_2,g_2]_A,\\phi(g_3))\\\\\n&+\\rho(f_1+f_2, g_1)[\\phi(g_1), t\\omega(f_1,f_2,g_2),\\phi(g_3)]_A+\\rho(f_1+f_2, g_1)t^2\\omega(\\phi(g_1),\\omega(f_1,f_2,g_2),\\phi(g_3))\\\\\n&+\\rho(f_1+f_2,g_1+g_2)[\\phi(g_1),\\phi(g_2),[f_1,f_2,g_3]_A]_A+\\rho(f_1+f_2, g_1+g_2)t\\omega(\\phi(g_1),\\phi(g_2),[f_1,f_2,g_3]_A)\\\\\n&+\\rho(f_1+f_2, g_1+g_2)[\\phi(g_1),\\phi(g_2),t\\omega(f_1,f_2,f_3)]_A+\\rho(f_1+f_2, g_1+g_2)t^2\\omega(\\phi(g_1),\\phi(g_2),\\omega(f_1,f_2,g_3)).\n\\end{align*}\nThus, we have\n\\begin{align*}\n\\omega(\\phi(f_1), \\phi(f_2), [g_1, g_2, g_3]_A) +[f_1,f_2,\\omega(g_1,g_2,g_3)]_A&=\\omega([f_1,f_2,g_1]_A, \\phi(g_2), \\phi(g_3))\\\\\n&+\\rho(f_1+f_2, g_1)\\omega(\\phi(g_1), [f_1,f_2,g_2]_A, \\phi(g_3))\\\\\n&+\\rho(f_1+f_2, g_1+g_2)\\omega(\\phi(g_1),\\phi(g_2), [f_1,f_2,g_3]_A)\\\\\n&+[\\omega(f_1,f_2,g_1),\\phi(g_2),\\phi(g_3)]_A\\\\\n&+\\rho(f_1+f_2,g_1)[\\phi(g_1),\\omega(f_1,f_2,g_2),\\phi(g_3)]_A\\\\\n&+\\rho(f_1+f_2,g_1+g_2)[\\phi(g_1), \\phi(g_2), \\omega(f_1, f_2, g_3)]_A,\n\\end{align*}\nand \n\\begin{align*}\n\\omega(\\phi(f_1),\\phi(f_2),\\omega(g_1,g_2,g_3))&=\\omega(\\omega(f_1,f_2,g_1),\\phi(g_2),\\phi(g_3))+\\rho(f_1+f_2,g_1)\\omega(\\phi(g_1),\\omega(f_1,f_2,g_2),\\phi(g_3))\\\\ &\\ \\ \\ +\\rho(f_1+f_2,g_1+g_2)\\omega(\\phi(g_1),\\phi(g_3),\\omega(f_1,f_2,g_3)).\n\\end{align*}\nTherefore, $\\omega$ defines a 3-Hom-$\\rho$-Lie algebra structure on $A$ and $\\omega$ is a 1-cocycle of $A$ with the coefficient in the adjoint representation.\n\\end{proof}\nAn infinitesimal deformation is said to be trivial if there exists a grade-preserving map $N:A\\longrightarrow A$\nsuch that for $T_t = id + tN: A_t \\longrightarrow A$ the following relation holds\n$$T_t[f_1, f_2, f_3]_t = [T_tf_1, T_tf_2, T_tf_3].$$\n\\begin{definition}\nA linear operator $N:A\\longrightarrow A$ is called a Hom Nijenhuis operator if \n\\begin{align}\\label{t123}\nN^2[f_1, f_2, f_3] &= N[Nf_1, f_2, f_3] + N[f_1, Nf_2, f_3] + N[f_1, f_2, Nf_3]\\nonumber\\\\\n&\\ \\ \\ -([Nf_1, Nf_2, f_3] + [Nf_1, f_2, Nf_3] + [f_1, Nf_2, Nf_3]).\n\\end{align}\nIf we define $[.,.,.]_N$ by \n$$[f_1,f_2,f_3]_N= [Nf_1, f_2, f_3] + [f_1, Nf_2, f_3] + [f_1, f_2, Nf_3] -N[f_1, f_2, f_3],$$\nthen \\eqref{t123} is equivalent to\n$$N[f_1, f_2, f_3]_N = [Nf_1, Nf_2, f_3] + [Nf_1, f_2, Nf_3] + [f_1, Nf_2, Nf_3].$$\n\\end{definition}\n\\begin{theorem}\nLet $N$ be a Nijenhuis operator for $A$. Setting \n$$\\omega(f_1, f_2, f_3) = [f_1,f_2,f_3]_N,$$\nthen $\\omega$ is an infinitesimal\ndeformation of $A$. Furthermore, this deformation is a trivial one.\n\\end{theorem}\n\\begin{proof}\nBy a direct calculation, we can see $d_1\\omega=0$, therefore $\\omega$ is a 1-cocycle of\n$A$ with the coefficients in the adjoint representation. We must check the $\\rho$-fundamental identity for $\\omega$, that is \n\\begin{align*}\n\\omega(\\phi(f_1), \\phi(f_2), \\omega(g_1, g_2, g_3)) &=\\omega(\\omega(f_1, f_2, g_1), \\phi(g_2), \\phi(g_3))\n+\\rho(f_1 + f_2, g_1)\\omega(\\phi(g_1), \\omega(f_1, f_2, g_2), \\phi(g_3))\\\\\n&\\ \\ \\ +\\rho(f_1 + f_2, g_1 + g_2)\\omega(\\phi(g_1), \\phi(g_2), \\omega(f_1, f_2, g_3)).\n\\end{align*}\nThis identity follows easily by a direct calculation, using the $\\rho$-fundamental identity for $A$ and this fact that $N$ is a Hom Nijenhuis operator for $A$. Suppose that $T_t=id + tN$, then\n\\begin{align*}\nT_t[f_1, f_2, f_3]_t &=id + tN([f_1, f_2, f_3]_t)=id + tN([f_1,f_2,f_3]_A+t\\omega(f_1,f_2,f_3))\\\\ &=[f_1, f_2, f_3] + t\\omega(f_1, f_2, f_3) + tN([f_1, f_2, f_3] + t\\omega(f_1, f_2, f_3))\\\\\n&= [f_1, f_2, f_3] + t(\\omega(f_1, f_2, f_3) + N[f_1, f_2, f_3]) + t2N\\omega(f_1, f_2, f_3),\n\\end{align*}\nand\n\\begin{align*}\n[T_tf_1, T_tf_2, T_tf_3] &= [f_1 + tNf_1, f_2 + tNf_2, f_3 + tNf_3]\\\\\n&= [f_1, f_2, f_3] + t([Nf_1, f_2, f_3] + [f_1, Nf_2, f_3] + [f_1, f_2, Nf_3])\\\\\n&\\ \\ \\ +t2([Nf_1, Nf_2, f_3] + [Nf_1, f_2, Nf_3] + [f_1, Nf_2, Nf_3])\n+t3[Nf_1, Nf_2, Nf_3].\n\\end{align*}\nThen, we have\n$$T_t[f_1, f_2, f_3]_t =[T_tf_1, T_tf_2, T_tf_3],$$\nwhich implies that the infinitesimal deformation is trivial.\n\\end{proof}\n\n\\bigskip \\addcontentsline{toc}{section}{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe problem of structure-preservation in numerical discretizations of partial differential equations has primarily been studied in two disjoint stages, the first involving the semi-discretization of the spatial degrees of freedom, and the second having to do with the time-integration of the resulting coupled system of ordinary differential equations. Implicit in such an approach is the use of tensor product meshes in spacetime. In the context of spatial semi-discretization, the notion of structure-preservation is focused on compatible discretizations~(see~\\citet{Ar2018}, and references therein), that preserve in some manner the functional and geometric relationships between the different function spaces that arise in the partial differential equation, and in the context of time-integration, geometric numerical integrators~(see~\\citet{HaLuWa2006}, and references therein) aim to preserve geometric invariants like the symplectic or Poisson structure, energy, momentum, and the nonlinear manifold structure of the configuration spaces, like its Lie group, homogeneous space, or Riemannian structure.\n\nLagrangian partial differential equations are an important class of partial differential equations that exhibit geometric structure that can benefit from numerical discretizations that preserve such structure. This can either be viewed as an infinite-dimensional Lagrangian system with time as the independent variable, or a finite-dimensional Lagrangian multisymplectic field theory~\\cite{MaPeShWe2001} with space and time as independent variables. Lagrangian variational integrators~\\cite{MaWe2001, MaPaSh1998} are a popular method for systematically constructing symplectic integrators of arbitrarily high-order, and satisfy a discrete Noether's theorem that relates group-invariance with momentum conservation. A group-invariant (and hence momentum-preserving) variational integrator can be constructed from group-equivariant interpolation spaces~\\cite{GaLe2018}.\n\nIn this paper, we will demonstrate how compatible discretization, multisymplectic variational integrators, and group-equivariant interpolation spaces can be combined to yield a natural geometric structure-preserving discretization framework for Lagrangian field theories. \n\n\\paragraph{\\bf Multisymplectic Formulation of Classical Field Theories} The variational principle for Lagrangian PDEs involve a multisymplectic formulation \\cite{MaPaSh1998, MaPeShWe2001}. The base space $X$ consists of independent variables, denoted by $(x^0,\\ldots,x^n)\\equiv(t,x)$, where $x^0\\equiv t$ is time, and $(x^1,\\ldots,x^n)\\equiv x$ are space variables. The dependent field variables, $(y^1,\\ldots, y^m)\\equiv y$, form a fiber over each spacetime basepoint. The independent and field variables form the configuration bundle, $\\pi:Y\\rightarrow X$. The configuration of the system is specified by a section of $Y$ over $X$, which is a continuous map $\\phi:X\\rightarrow Y$, such that $\\pi\\circ\\phi=1_{X}$. This means that for every $(t,x)\\in X$, $\\phi((t,x))$ is in the fiber over $(t,x)$, which is $\\pi^{-1}((t,x))$.\n\n\\begin{figure}[h]\n\\centerline{\n\\begin{overpic}\n[scale=0.925]\n{Figures\/avi5}\n\\put(10,5){$X$}\n\\put(95,30){$t$}\n\\put(10,70){$x$}\n\\put(85,60){$\\phi$}\n\\end{overpic}}\n\\vspace*{-0.15in}\n  \\caption{\\footnotesize A section of the configuration bundle: the horizontal axes represent spacetime, and the vertical axis represent dependent field variables. The section $\\phi$ gives the value of the field variables at every point of spacetime.}\n  \\label{fig:section_configuration_bundle}\n\\end{figure}\n\nFor ODEs, the Lagrangian depends on position and its time derivative, which is an element of the tangent bundle $TQ$, and the action is obtained by integrating the Lagrangian in time. In the multisymplectic case, the Lagrangian density is dependent on the field variables and the partial derivatives of the field variables with respect to the spacetime variables, and the action integral is obtained by integrating the Lagrangian density over a region of spacetime. The multisymplectic analogue of the tangent bundle is the first jet bundle $J^1 Y$, consisting of the configuration bundle $Y$, and the first partial derivatives of the field variables with respect to the independent variables. In coordinates, we have $\\phi(x^0,\\ldots, x^n)=(x^0,\\ldots x^n, y^1,\\ldots y^m)$, which allows us to denote the partial derivatives by\n$ v^a_\\mu={y^a}_{,\\mu}=  \\partial y^a \/ \\partial x^\\mu$.\nWe can think of $J^1 Y$ as a fiber bundle over $X$. Given a section $\\phi:\nX\\rightarrow Y$, we obtain its first jet extension, $j^1 \\phi:X\\rightarrow J^1 Y$, that is given by\n\\[ j^1 \\phi (x^0,\\ldots, x^n) = \\left( x^0,\\ldots, x^n, y^1,\\ldots, y^m, {y^1}_{,0},\\ldots, {y^m}_{,n} \\right),\\]\nwhich is a section of the fiber bundle $J^1 Y$ over $X$. We refer to sections of $J^1Y$ of the form $j^1\\phi$, where $\\phi$ is a section of $Y$, as holonomic.\nThe Lagrangian density is a bundle map $\\mathcal{L}:J^1 Y\\rightarrow \\Omega^{n+1}(X)$. Given the action functional,\n$S[\\phi] = \\int_{X} \\mathcal{L}(j^1 \\phi), $\nHamilton's principle states that $\\delta S = 0$ (subject to compactly supported variations). As we will see, this is the basis of Lagrangian multisymplectic variational integrators~\\cite{MaPaSh1998}.\n\nThe variational structure of a Lagrangian field theory is given by the Cartan form, which in coordinates has the expression\n\\begin{equation}\\label{Cartan form}\n\\Theta_{\\mathcal{L}} = \\frac{\\partial L}{\\partial v^a_\\mu} dy^a \\wedge d^nx_\\mu + \\left(L - \\frac{\\partial L}{\\partial v^a_\\mu}v^a_\\mu\\right) d^{n+1}x.\n\\end{equation}\nThis can be defined intrinsically as the pullback of the canonical $(n+1)$-form on the dual jet bundle by the covariant Legendre transform $\\mathbb{F}\\mathcal{L}: J^1Y \\rightarrow J^1Y^*$. Then, the action can be expressed as $S[\\phi] = \\int_X\\mathcal{L}(j^1\\phi) = \\int_X (j^1\\phi)^*\\Theta_{\\mathcal{L}}.$ The variation of the action is then expressed as\n$$ dS[\\phi]\\cdot V = -\\int_X (j^1\\phi)^*(j^1V \\lrcorner\\ \\Omega_{\\mathcal{L}}) + \\int_{\\partial X} (j^1\\phi)^* (j^1V \\lrcorner\\ \\Theta_{\\mathcal{L}}), $$\nwhere $\\Omega_{\\mathcal{L}} = -d\\Theta_{\\mathcal{L}}$ defines the multisymplectic form and $j^1V$ denotes the jet prolongation of the vector field $V$ (for details, see \\citet{GoIsMaMo1998}). Hence, the variation of the action is completely specified by the Cartan form; we will show that a finite element discretization of the variational principle gives rise to a discrete form and subsequently we will express variational properties of the discrete system in terms of the discrete Cartan form.\n\nIn this paper, we will take the configuration bundle $Y = H\\Lambda^k(X)$, the space of square integrable $k-$forms on $X$ with square integrable exterior derivative. In this setting, the appropriate analogue of the jet bundle is $J^1_{H\\Lambda^k} := H\\Lambda^k \\times dH\\Lambda^k$, where the jet extension of a field $\\phi \\in H\\Lambda^k$ is $j^1\\phi = (x,\\phi,d\\phi)$ (i.e., we take the Lagrangian theory to depend on the exterior derivative of the field and not depending more generally on all first-order derivatives; for scalar fields, $k=0$, these are equivalent).\n\n\\paragraph{\\bf Finite Element Exterior Calculus} The notion of compatible discretization is a research area that has garnered significant interest and activity in the finite element community, motivated by the seminal work of \\citet{ArFaWi2006} on finite element exterior calculus that provides a broad generalization of Hiptmair's work on mixed finite elements for electromagnetism \\citep{Hi2002}. This arises from the fundamental role that the de~Rham complex of exterior differential forms plays in mixed formulations of elliptic partial differential equations, and the realization that many of the most successful mixed finite element spaces, such as Raviart--Thomas and N\\'ed\\'elec elements, can be viewed as finite element subspaces of the de~Rham complex that satisfy a bounded cochain projection property, so that the set of mixed finite elements form a subcomplex that provides stable approximations of the original problem.\n\n\\paragraph{\\bf Group-equivariant interpolation}\n\nThe study of group-equivariant approximation spaces~\\cite{GaLe2018} for functions taking values on manifolds is motivated by the applications to geometric structure-preserving discretization of Lagrangian and Hamiltonian PDEs with symmetries. In particular, when the Lagrangian density for a Lagrangian PDE with symmetry is discretized using a Lagrangian multisymplectic variational integrator constructed from an approximation space that is equivariant with respect to the symmetry group, the resulting numerical method automatically preserves the momentum map associated with the symmetry of the PDE. In essence, such variational discretizations exhibit a discrete analogue of Noether's theorem, which connects symmetries of the Lagrangian with momentum conservation laws.\n\nMany intrinsic geometric flows such as the Ricci flow and the Einstein equations involves computing the evolution of a Riemannian or pseudo-Riemannian metric on spacetime. Additionally, these intrinsic geometric flows can often be formulated variationally, so it is natural to consider group-equivariant approximation spaces taking values on Riemannian or pseudo-Riemannian metrics with a view towards constructing variational discretizations that preserve the associated momentum maps.\n\nA now standard approach to constructing an approximation space for functions taking values on a Riemannian manifold that is equivariant with respect to Riemannian isometries is the method of geodesic finite elements introduced independently by~\\citet{Sa2012} and~\\citet{Gr2013}. Given a Riemannian manifold $(M,g)$, the geodesic finite element $\\varphi:\\Delta^n\\rightarrow M$ associated with a set of linear space finite elements $\\{ v_i:\\Delta^n\\rightarrow\\mathbb{R} \\}_{i=0}^n$ is given by the Fr\\'echet (or Karcher) mean,\n\\[ \\varphi(x)=\\arg \\min_{p\\in M} \\sum\\nolimits_{i=0}^n v_i(x) (\\operatorname{dist} (p,m_i)) ^2,\\]\nwhere the optimization problem involved can be solved using optimization algorithms developed for matrix manifolds~(see~\\citet{AbMaSe2008}, and references therein). The spatial derivatives of the geodesic finite element can be computed in terms of an associated optimization problem. The advantage of the geodesic finite element approach is that it inherits the approximation properties of the underlying linear space finite element, but it can be expensive to compute, since it entails solving an optimization problem on a manifold.\n\nAn alternative approach to group-equivariant interpolation for functions taking values on symmetric spaces was introduced in \\citet{GaLe2018}, which, in particular, is applicable to the interpolation of Riemannian and pseudo-Riemannian metrics. It uses the generalized polar decomposition~\\cite{munthe2001generalized} to construct a local diffeomorphism between a symmetric space and a Lie triple system, which lifts a scalar-valued interpolant to a symmetric space-valued interpolant.\n\n\n\\paragraph{\\bf Lagrangian Variational Integrators}\n\nVariational integrators~(see \\cite{MaWe2001}, and references therein) are a class of geometric structure-preserving numerical integrators that are based on a discretization of Hamilton's principle. They are particularly appropriate for the simulation of Lagrangian and Hamiltonian ODEs and PDEs, as they automatically preserve many geometric invariants, including the symplectic structure, momentum maps associated with symmetries of the system, and exhibit bounded energy errors for exponentially long times.\n\nIn the case of Lagrangian ODEs, variational integrators are based on constructing computable approximations $L_d:Q\\times Q\\rightarrow \\mathbb{R}$ of the exact discrete Lagrangian,\n\\[ L_d^E(q_0,q_1,h)=\\ext_{\\substack{q\\in C^2([0,h],Q) \\\\ q(0)=q_0, q(h)=q_1}} \\int_0^h L(q(t), \\dot q(t)) dt,\\]\nwhich can be viewed as Jacobi's solution of the Hamilton--Jacobi equation. Given a discrete Lagrangian $L_d$, one introduces the discrete action sum $\\mathbb{S}_d=\\sum_{k=0}^{n-1} L_d(q_k, q_{k+1})$, and then the discrete Hamilton's principle states that $\\delta \\mathbb{S}_d=0$, for fixed boundary conditions $q_0$ and $q_n$. This leads to the discrete Euler--Lagrange equations, \n\\[D_2 L_d(q_{k-1},q_k)+D_1 L_d(q_k,q_{k+1})=0,\n\\]\nwhere $D_i$ denotes the partial derivative with respect to the $i$-th argument. This implicitly defines the discrete Lagrangian map $F_{L_d}:(q_{k-1},q_k)\\mapsto(q_k,q_{k+1})$ for initial conditions $(q_{k-1},q_k)$ that are sufficiently close to the diagonal of $Q\\times Q$.\nIt is also equivalent to the implicit discrete Euler--Lagrange equations,\n\\[\np_k=-D_1 L_d(q_k, q_{k+1}),\\qquad p_{k+1}=D_2 L_d(q_k, q_{k+1}),\\]\nwhich implicitly defines the discrete Hamiltonian map $\\tilde{F}_{L_d}:(q_k,p_k)\\mapsto(q_{k+1},p_{k+1})$, which is automatically symplectic. This clearly follows from the fact that these equations are precisely the characterization of a symplectic map in terms of a Type I generating function. The two equations in the implicit discrete Euler--Lagrange equations can be used to define the discrete Legendre transforms, \\(\\mathbb{F}^{\\pm}L_{d}: Q\\times Q \\rightarrow T^{*}Q\\):\n\\begin{align*}\n\\mathbb{F}^{+}L_{d}&:(q_{0},q_{1}) \\rightarrow (q_{1},p_{1}) = (q_{1},D_{2}L_{d}(q_{0},q_{1})), \\\\\n\\mathbb{F}^{-}L_{d}&:(q_{0},q_{1}) \\rightarrow (q_{0},p_{0}) = (q_{0},-D_{1}L_{d}(q_{0},q_{1})).\n\\end{align*}\nThe following commutative diagram illustrates the relationship between the discrete Hamiltonian flow map, discrete Lagrangian flow map, and the discrete Legendre transforms,%\n\\begin{align*}\n\\xymatrix{\n& (q_{k},p_{k})  \\ar[rr]^{\\tilde{F}_{L_{d}}} & & (q_{k+1},p_{k+1}) & \\\\\n& & & & \\\\\n(q_{k-1},q_{k}) \\ar[uur]^{\\mathbb{F}^{+}L_{d}} \\ar[rr]_{F_{L_{d}}} & & (q_{k},q_{k+1}) \\ar[rr]_{F_{L_{d}}} \\ar[uur]^{\\mathbb{F}^{+}L_{d}} \\ar[uul]_{\\mathbb{F}^{-}L_{d}}& &(q_{k+1},q_{k+2}) \\ar[uul]_{\\mathbb{F}^{-}L_{d}}\n}\n\\end{align*}\nIf the discrete Lagrangian is invariant under the diagonal action of a Lie group $G$, i.e., $L_d(q_0, q_1)=L_d(gq_0, gq_1)$, for all $g\\in G$, then the discrete Noether's theorem states that there is a discrete momentum map that is automatically preserved by the variational integrator. The bounded energy error of variational integrators can be understood by performing backward error analysis~\\cite{Ha1994,BeGi1994}, which then shows that the discrete flow map is approximated with exponential accuracy by the exact flow map of the Hamiltonian vector field of a modified Hamiltonian.\n\n\\paragraph{\\bf Multisymplectic Hamiltonian Variational Integrators.}\nFor Hamiltonian PDEs (see, for example, \\citet{MaSh1999}) the action is a functional on the field and multimomenta values (more precisely, sections of the restricted dual jet bundle),\n$$ S[\\phi,p] = \\int [p^\\mu \\partial_\\mu \\phi - H(\\phi,p) ]d^{n+1}x, $$\nwhere the integration is over some $(n+1)$-dimensional region. The variational principle gives the De Donder--Weyl equations $\\partial_\\mu p^\\mu = -\\partial H\/\\partial \\phi$, $\\partial_\\mu\\phi = \\partial H\/\\partial p^\\mu$. Defining $z = (\\phi, p^0, \\dots, p^n)$ and $K^{\\mu}$ as the $(n+2)\\times (n+2)$ skew-symmetric matrix with value $-1$ in the $(0,\\mu+1)$, $1$ in the $(\\mu+1,0)$ entry, and $0$ in every other entry (with indexing from $0$ to $n+1$), the De Donder--Weyl equations can be written in the form\n$$ K^0\\partial_0 z + \\dots + K^n\\partial_n z = \\nabla_z H. $$\nThis formulation of Hamiltonian PDEs was studied in \\citet{Br1997}; in particular, it was shown that such a system admits a multisymplectic conservation law of the form $\\partial_\\mu \\omega^\\mu(V,W) = 0$, where the $\\omega^\\mu$ are two-forms corresponding to $K^\\mu$ and the conservation law holds when evaluated on first variations $V, W$. For discretizing such equations, multisymplectic integrators have been developed which admit a discrete analogue of this multisymplectic conservation law (see, for example, \\citet{BrRe2006}). Such multisymplectic integrators have traditionally not been approached from a variational perspective. \n\nHowever, in \\citet{TrLe2021}, we developed a systematic method for constructing variational integrators for multisymplectic Hamiltonian PDEs which automatically admit a discrete multisymplectic conservation law and a discrete Noether's theorem by virtue of the discrete variational principle. The construction is based on a discrete approximation of the boundary Hamiltonian that was introduced in~\\citet{VaLiLe2011}, \n$$ H_{\\partial U}(\\varphi_A,\\pi_B) = \\ext\\Big[ \\int_B p^\\mu \\phi d^nx_\\mu - \\int_U (p^\\mu \\partial_\\mu\\phi - H(\\phi,p) ) d^{n+1}x \\Big],$$\nwhere $\\partial U = A \\sqcup B$, boundary conditions are placed on the field value $\\phi$ on $A$ and normal momenta value on $B$, and one extremizes over the sections $(\\phi,p)$ over $U$ satisfying the specified boundary conditions. The boundary Hamiltonian is a generating functional in the sense that the Type II variational principle generates the normal momenta value along $A$ and the field value along $B$,\n$$ \\frac{\\delta H_{\\partial U}}{\\delta \\varphi_A} = -p^n|_A, \\quad \\frac{\\delta H_{\\partial U}}{\\delta \\pi_B} = \\phi|_B. $$\nA variational integrator is then constructed by first approximating the boundary Hamiltonian using a finite-dimensional function space and quadrature, and subsequently enforcing the Type II variational principle. For example, with particular choices of function spaces and quadrature, \\citet{TrLe2021} recover the class of multisymplectic partitioned Runge--Kutta methods.\n\nIn this paper, we take a different approach in several regards. First, we focus on Lagrangian field theories as opposed to Hamiltonian field theories. For Hamiltonian field theories, the momenta are related to the field and its derivative by the Legendre transform; this falls out from the variational principle so one does not need to enforce it beforehand. Thus, in this sense, the momenta and field values can be considered as independent before enforcing the variational principle. On the other hand, for Lagrangian field theories, the Lagrangian depends on both the field value and its first derivative, so one cannot na\\\"\\i vely treat the two as independent; that is, the Lagrangian depends on holonomic sections of the jet bundle. As we will see, this will mean that we need to pay particular attention to the holonomic condition when discretizing via a finite element projection. Furthermore, as opposed to constructing variational integrators from a generating functional (the analogue in the Lagrangian framework would be the boundary Lagrangian, see \\citet{VaLiLe2011}), in this paper, we instead investigate directly discretizing the variational principle $\\delta S = 0$ utilizing projections into finite-dimensional subspaces. Finally, for simplicity, we do not utilize any quadrature approximations of the various integrals which we encounter; for strong nonlinearities in the Lagrangian, one generally has to utilize quadrature to construct an efficient discretization. However, the theory that we outline is also applicable to the case of applying a quadrature rule, given that one applies the quadrature rule to the action before enforcing the variational principle, so that the resulting discretization is still variational; we will elaborate on this in Remark \\ref{Remark on Quadrature}. For this reason, we will assume exact integration in order to keep the exposition simple. \n\n\\paragraph{\\bf Main Contributions.} This paper studies the variational finite element discretization of Lagrangian field theories from two perspectives; we begin by investigating directly discretizing the full variational principle over the full spacetime domain, which we refer to as the ``covariant\" approach, and subsequently study semi-discretization of the instantaneous variational principle on a globally hyperbolic spacetime, which we refer to as the ``canonical\" approach. This paper can be considered a discrete analogue to the program initiated in \\citet{GoIsMaMo1998, GoIsMaMo2004}, which lays the foundation for relating the covariant and canonical formulations of Lagrangian field theories through their (multi)symplectic structures and momentum maps. One of the goals of understanding the relation between these two different formulations is to systematically relate the covariant gauge symmetries of a gauge field theory to its initial value constraints. This is seen, for example, in general relativity, where the diffeomorphism gauge invariance gives rise to the Einstein constraint equations over the initial data hypersurface (see, for example, \\citet{Go2012}). When one semi-discretizes such gauge field theories, the discrete initial data must satisfy an associated discrete constraint. We aim to make sense of the discrete geometric structures in the covariant and canonical discretization approaches as a foundation for understanding the discretization of gauge field theories.\n\nIn Section \\ref{Covariant Discretization Section}, we begin by formulating a discrete variational principle in the covariant approach, utilizing the finite element construction to appropriately restrict the variational principle. We show that a cochain projection from the underlying de Rham complex onto the finite element spaces yields a natural discrete variational principle that is compatible with the holonomic jet structure of a Lagrangian field theory. In Section \\ref{Variational Structure Section}, we then show that discretizing by cochain projections leads to a naturality relation between the continuous variational problem and the discrete variational problem; this naturality then implies that discretization and the variational principle commute and also, that discretizing at the level of the configuration bundle or at the level of the jet bundle are equivalent. Subsequently, by decomposing the finite element spaces into boundary and interior components, we define a discrete Cartan form in analogy with the continuum Cartan form which will, in a sense, encode the discrete variational structure. With particular choices of finite element spaces, this discrete Cartan form recovers the notion of the discrete Cartan form introduced by \\citet{MaPaSh1998}; however, we note that our notion of a discrete Cartan form is more general and furthermore, since our discrete variational problem is naturally related to the continuum variational problem, we are able to explicitly discuss in what sense the discrete Cartan form converges to the continuum Cartan form.  Using this discrete Cartan form, in Sections \\ref{Multisymplectic Section} and \\ref{Noether's Theorem Section}, we state and prove discrete analogues of the multisymplectic form formula and Noether's theorem. Finally, in Section \\ref{Variational Complex Section}, we reinterpret and concisely summarize the preceding sections by interpreting the discrete variational structures as elements of a discrete variational complex. \n\nIn Section \\ref{Semi-Discretization Section}, we study the semi-discretization of the canonical formulation of a Lagrangian field theory on a globally hyperbolic spacetime. In Section \\ref{Semi-Discrete EL Section}, we discretize the instantaneous variational principle utilizing cochain projections onto finite element spaces over a Cauchy surface, which gives rise to a semi-discrete Euler--Lagrange equation. In Section \\ref{Semi-discrete Symplectic Structure Section}, we relate this semi-discrete Euler--Lagrange equation to a Hamiltonian flow on a symplectic semi-discrete phase space. We will discuss in what sense the symplectic structure on the semi-discrete phase space arises from a symplectic structure on the continuum phase space. Subsequently, we will investigate the energy-momentum map structure associated to the semi-discrete phase space in Section \\ref{Energy-Momentum Section}, and discuss how, under appropriate equivariance conditions on the projection, the energy-momentum map structure on the semi-discrete phase space arises as the pullback of the energy-momentum map structure on the continuum phase space. This lays a foundation for understanding initial value constraints when discretizing field theories with gauge symmetries. Finally, in Section \\ref{Tensor Product Discretization Section}, we relate the covariant and canonical discretization approaches in the case of tensor product finite element spaces. \n\nThe underlying theme of this paper is that, when one discretizes the variational principle utilizing compatible discretization techniques, the associated (covariant or canonical) discretization inherits discrete variational structures which can be viewed as pullbacks or projections of the associated continuum variational structures. These discrete variational structures allow one to investigate structure-preservation under discretization of important physical properties, such as momentum conservation, symplecticity, and (gauge) symmetries. \n\n\\section{Covariant Discretization of Lagrangian Field Theories}\\label{Covariant Discretization Section}\nIn this section, we discretize the covariant Euler--Lagrange equations which arise from the variational principle $\\delta S[\\phi] = 0$ for the action $S: \\phi \\mapsto \\int_X \\mathcal{L}(j^1\\phi)$ where $\\phi \\in H\\Lambda^k$ is a section of the configuration bundle and $j^1\\phi = (x,\\phi,d\\phi)$. To utilize the finite element method, we take our base space $X$ to be a bounded $(n+1)-$dimensional polyhedral domain with boundary $\\partial X$, equipped with a finite element triangulation $\\mathcal{T}_h$. We will assume $X$ has a Riemannian or Lorentzian metric. For this discretization, we perform the variation over a finite element space, and subsequently study how the multisymplectic and covariant momentum map structures are affected by discretization. In particular, we show how these structures are preserved for particular choices of finite element spaces, namely spaces whose projections are cochain maps or group-equivariant interpolation spaces. \n\nFirst, we derive the Euler--Lagrange equations in the Hilbert space setting, where we take the first jet bundle to be $J^1_{H\\Lambda^k} = H\\Lambda^k \\times dH\\Lambda^k.$ Fixing the trace of $\\phi$ on $\\partial X$, the variational principle is to find $\\phi \\in H\\Lambda^k$ such that $\\delta S[\\phi]\\cdot v = 0$ for all $v \\in \\mathring{H}\\Lambda^k$. This yields\n\\begin{align*}\n0 &= \\delta S[\\phi]\\cdot v = \\int_X \\Big(\\delta_2\\mathcal{L}(j^1\\phi)\\cdot v + \\delta_3\\mathcal{L}(j^1\\phi)\\cdot dv\\Big)\\\\\n&= (\\partial_2\\mathcal{L}(j^1\\phi),v)_{L^2H\\Lambda^k} + (\\partial_3\\mathcal{L}(j^1\\phi),dv)_{L^2H\\Lambda^{k+1}}\\\\\n&= (\\partial_2\\mathcal{L}(j^1\\phi),v)_{L^2H\\Lambda^k} + (d^*\\partial_3\\mathcal{L}(j^1\\phi),dv)_{L^2H\\Lambda^k}\n\\end{align*}\nwhere $j^1\\phi = (x,\\phi,d\\phi)$, $\\delta_i$ denotes the variation with respect to the $i^{th}$ argument, and in the second line we apply the Riesz representation theorem to express $\\delta_i\\mathcal{L}(j^1\\phi)$ as an element $\\partial_i\\mathcal{L}(j^1\\phi)$ of $H\\Lambda^k$ (noting that for a general class of Lagrangians $\\mathcal{L}(j^1\\phi)$, e.g. polynomial in $\\phi$ and $d\\phi$, $\\delta_i\\mathcal{L}(j^1\\phi)$ is a bounded linear functional, since the domain is compact). Note that this can equivalently be stated as $dS[\\phi]\\cdot V = 0$ for all $V \\in \\mathfrak{X}(\\mathring{H}\\Lambda^k)$. \n\nThere is a slight subtlety in the Lagrangian formulation of the variation principle, in that the variation with respect to the derivative of the field is regarded as independent from the variation of the field. To be more precise, one should regard the third argument of $\\mathcal{L}(x,\\phi,\\psi)$, $\\psi \\in dH\\Lambda^k$, as independent of $\\phi$ and subsequently enforce that the argument is holonomic utilizing the variational principle, i.e., that $(x,\\phi,\\psi) = j^1\\phi = (x,\\phi,d\\phi)$. To do this, we use the Hamilton--Pontryagin principle, introducing a Lagrange multiplier to enforce the holonomic condition. The action reads\n$$ S[\\phi,\\psi,p] = \\int_X \\mathcal{L}(x,\\phi,\\psi) - (p,d\\phi - \\psi)_{L^2\\Lambda^{k+1}}  $$\nwhere $\\phi \\in H\\Lambda^k, \\psi \\in dH\\Lambda^k, p \\in L^2\\Lambda^{k+1}$. Since $\\psi \\in dH\\Lambda^k,$ we can express it as $\\psi \\equiv d\\chi$. We enforce that $S$ is stationary with respect to variations $\\tilde{\\phi} \\in \\mathring{H}\\Lambda^k$, $d\\tilde{\\chi} \\in d\\mathring{H}\\Lambda^k, \\tilde{p} \\in L^2\\Lambda^{k+1}$ (note there is no boundary condition assumed on the variation associated with $p$, since this is an auxilliary variable). This yields\n\\begin{align*}\n0 &= \\delta_1S[\\phi,d\\chi,p]\\cdot \\tilde{\\phi} = (\\partial_2\\mathcal{L}(x,\\phi,d\\chi),\\tilde{\\phi})_{L^2\\Lambda^k} - (p,d\\tilde{\\phi})_{L^2\\Lambda^{k+1}},\\\\\n0 &= \\delta_2S[\\phi,d\\chi,p]\\cdot d\\tilde{\\chi} = (\\partial_3\\mathcal{L}(x,\\phi,d\\chi), d\\tilde{\\chi})_{L^2\\Lambda^{k+1}} + (p, d\\tilde{\\chi})_{L^2\\Lambda^{k+1}},\\\\\n0 &= \\delta_3S[\\phi,d\\chi,p]\\cdot \\tilde{p} = (\\tilde{p}, d\\phi - d\\chi)_{L^2\\Lambda^{k+1}}.\n\\end{align*}\nThe third equation holds for all $\\tilde{p} \\in L^2\\Lambda^{k+1}$ and hence $d\\phi = d\\chi$ in $L^2\\Lambda^{k+1}$. Substituting this into the first two equations, setting $\\tilde{\\phi} = \\tilde{\\chi}$, and adding the resulting equations together gives the weak Euler--Lagrange equations that we derived formally from directly working with the action $S[\\phi] = \\int_X\\mathcal{L}(j^1\\phi)$. Thus, it is in this sense that the variational principle treats the derivative of the field as independent of the field. Although the resulting equations were equivalent in this case, we will see shortly that when we discretize the variational principle, one has to be careful about the holonomic condition. \n\nTo formulate a discrete variational principle, let $\\{\\Lambda^m_h\\}_{m=0}^{n+1}$ be a subcomplex of finite element spaces approximating $\\{H\\Lambda\\}$ with projections $\\pi^m_h: H\\Lambda^m \\rightarrow \\Lambda^m_h$. This provides an approximation of $J^1_{H\\Lambda^k} = H\\Lambda^k \\times dH\\Lambda^k$ by $\\pi^k_hH\\Lambda^k \\times \\pi^{k+1}_h(dH\\Lambda^k)$. Consider the associated  degenerate action $S_h: H\\Lambda^k \\rightarrow \\mathbb{R}$ defined by\n\\begin{equation}\\label{Degenerate Action}\nS_h[\\phi] = \\int_X \\mathcal{L}(x,\\pi^k_h\\phi,\\pi^{k+1}_hd\\phi);\n\\end{equation}\nwe refer to this as a degenerate action since the projections have nontrivial kernels, as projections from infinite-dimensional spaces to finite-dimensional subspaces. We immediately see that the formal variation of $S_h$ where one treats the derivative of $\\phi$ as independent from $\\phi$ will not work in this setting, since the argument of $\\mathcal{L}$ is not holonomic (i.e., of the form $(x,\\psi,d\\psi)$ for some $\\psi$). To resolve this issue, we utilize the Hamilton--Pontryagin approach, where we weakly enforce the condition that the argument is holonomic. The degenerate Hamilton--Pontryagin action is\n\\begin{align*}\nS_h[\\phi,d\\chi,p] = \\int_X \\mathcal{L}(x,\\pi^k_h\\phi,\\pi^{k+1}_h d\\chi) - (\\pi^{k+1}_hp, d\\pi^k_h\\phi - \\pi^{k+1}_h d\\chi)_{L^2\\Lambda^{k+1}}.\n\\end{align*}\nEnforcing stationarity with respect to variations $\\tilde{\\phi} \\in \\mathring{H}\\Lambda^k, d\\tilde{\\chi} \\in d\\mathring{H}\\Lambda^k, \\tilde{p} \\in L^2\\Lambda^{k+1}$ gives\n\\begin{align*}\n0 &= \\delta_1S_h[\\phi,d\\chi,p] = (\\partial_2\\mathcal{L}(x,\\pi^k_h\\phi,\\pi^{k+1}_h d\\chi),\\pi^k_h\\tilde{\\phi})_{L^2\\Lambda^k} - (\\pi^{k+1}_h p, d\\pi^k_h\\tilde{\\phi})_{L^2\\Lambda^{k+1}}, \\\\\n0 &= \\delta_2S_h[\\phi,d\\chi,p] = (\\partial_3\\mathcal{L}(x,\\pi^k_h\\phi,\\pi^{k+1}_h d\\chi),\\pi^{k+1}_h d\\tilde{\\chi})_{L^2\\Lambda^{k+1}} + (\\pi^{k+1}_h p, \\pi^{k+1}_h d\\tilde{\\chi})_{L^2\\Lambda^{k+1}}, \\\\\n0 &= \\delta_3S_h[\\phi,d\\chi,p] = (\\pi^{k+1}_h \\tilde{p}, d\\pi^k_h\\phi - \\pi^{k+1}_h d\\chi)_{L^2\\Lambda^{k+1}}.\n\\end{align*}\nEven ignoring issues of degeneracy of the Lagrangian itself (e.g., due to gauge freedom), these equations do not uniquely determine $(\\phi,d\\chi,p)$ due to the nontrivial kernel of the projections; however, they do determine $(\\phi,d\\chi,p)$ if we restrict to the images of the projections, i.e., that the fields are in the associated finite-dimensional subspaces. In this case, performing an analogous substitution to the case of the continuum Hamilton--Pontryagin principle gives\n\\begin{equation}\\label{DEL from HP Principle}\n(\\partial_2\\mathcal{L}(x,\\pi^k_h\\phi, d\\pi^k_h\\phi),\\pi^k_h\\tilde{\\phi})_{L^2\\Lambda^k} + (\\partial_3\\mathcal{L}(x,\\pi^k_h\\phi, d\\pi^k_h\\phi), \\pi^k_h\\tilde{\\phi})_{L^2\\Lambda^{k+1}} = 0.\n\\end{equation}\nOne could ask whether this equation arises from an action $S_h[\\phi]$ without utilizing the Hamilton--Pontryagin principle, so that the associated discrete variational structures are manifest and directly comparable to the continuum variational structures. The issue with the action (\\ref{Degenerate Action}) is that the argument of $\\mathcal{L}$ is not holonomic; however, if we assume that the projections are cochain projections, i.e., that $\\pi^{k+1}_hd = d\\pi^k_h$, then the argument of $\\mathcal{L}$ is holonomic. Assuming cochain projections, we can formally apply the variational principle to \n$$ S_h[\\phi]= \\int_X \\mathcal{L}(x,\\pi^k_h\\phi,\\pi^{k+1}_hd\\phi) = \\int_X \\mathcal{L}(x,\\pi^k_h\\phi,d\\pi^k_h\\phi) $$\nand recover equation (\\ref{DEL from HP Principle}). Hence, for the rest of the paper, we will assume that the projections are cochain projections (with the caveat that in Section \\ref{Semi-Discretization Section} regarding semi-discretization, we will assume that the projections are cochain projections with respect to the spatial exterior derivative).\n\\begin{assumption}[Cochain Projections]\\label{Cochain Projection Assumption}\nThe projections $\\pi^m_h: H\\Lambda^m \\rightarrow \\Lambda^m_h$ are cochain projections, i.e., that $\\pi^{k+1}_hd = d\\pi^k_h$.\n\\end{assumption}\nFurthermore, we will generally denote the projections as $\\pi_h$, where the degree of the differential forms that they act on are implicitly understood.\n\nThus, with the assumption of cochain projections, we can interpret the variational principle associated to the degenerate action as the variational principle applied to the original action $S$ restricted to $\\Lambda^k_h$; we will make this statement more precise in Section \\ref{Variational Structure Section}. Instead of enforcing the variational principle $\\delta S[\\phi]\\cdot v = 0$ for all compactly supported $v \\in H\\Lambda^k$, the finite-dimensional reduction to the problem is given by enforcing the variational principle $\\delta S[\\phi]\\cdot v = 0$ for all $v \\in \\Lambda^k_h$ with vanishing trace on the boundary, we denote the space of such $v$ by $\\mathring{\\Lambda}^k_h$. With the assumption of cochain projections, we can formally treat the variation in the derivative of $\\phi \\in \\Lambda^k_h$ as independent of $\\phi$, and hence we can compute the (discrete) Euler--Lagrange equations formally, as one would do in the continuum case. The variational principle thus yields a discrete weak form of the Euler--Lagrange equation: find $\\phi \\in \\Lambda^k_h$ such that\n\\begin{equation}\\label{DEL 1a}\n0 = \\delta S[\\phi]\\cdot v = \\big(\\partial_2\\mathcal{L}(j^1\\phi),v\\big)_{L^2\\Lambda^k} + \\big(\\partial_3\\mathcal{L}(j^1\\phi),dv\\big)_{L^2\\Lambda^{k+1}}, \\text{ for all } v \\in \\mathring{\\Lambda}^k_h.\n\\end{equation}\nIntegrating by parts, this gives\n\\begin{equation}\\label{DEL 1b}\n0 = \\big(\\partial_2\\mathcal{L}(j^1\\phi),v\\big)_{L^2\\Lambda^k} + \\big(d^*\\partial_3\\mathcal{L}(j^1\\phi),v\\big)_{L^2\\Lambda^{k}} + \\int_{\\partial X}v \\wedge * \\partial_3\\mathcal{L}(j^1\\phi), \\text{ for all } v \\in \\mathring{\\Lambda}^k_h,\n\\end{equation}\nwhere the codifferential $d^*$ is interpreted in the weak sense. Note the boundary term vanishes since $v \\in \\mathring{\\Lambda}^k_h$, but we include it explicitly since it will be necessary in the formulation of the multisymplectic form formula and Noether's theorem (where one generally has nonzero variations on the boundary).\n\n\nWe refer to these equivalent equations, (\\ref{DEL 1a}) and (\\ref{DEL 1b}), as the discrete Euler--Lagrange equations (DEL). Fixing a basis of shape functions $\\{v_i\\}$ for $\\mathring{\\Lambda}^k_h$, expressing $\\phi = \\phi^j v_j$, and choosing $v = v_i$, (\\ref{DEL 1a}) is equivalent to a (generally nonlinear) system of equations for the unknown components $\\phi^i$. Letting $[i]$ denote the set of indices $j$ such that $\\text{supp}(v_j) \\cap \\text{supp}(v_i)$ has positive measure, the system of equations can be written\n$$ \\big(\\partial_2\\mathcal{L}(j^1( \\sum_{j \\in [i]} \\phi^j v_j)),v_i\\big)_{L^2\\Lambda^k} + \\big(\\partial_3\\mathcal{L}(j^1(\\sum_{j \\in [i]}  \\phi^j v_j)),dv_i\\big)_{L^2\\Lambda^{k+1}} = 0,\\ i=1,\\dots,\\dim \\mathring{\\Lambda}^k_h. $$\n\n\\begin{example}[Nonlinear Poisson\/Wave Equation]\\label{Nonlinear Wave Eq Ex, 1}\nWe consider the nonlinear (scalar) Poisson\/wave equation in $1+1$ spacetime dimensions on a rectangular domain, $X = [a,b] \\times [c,d]$,\n$$ \\partial_t^2 \\phi + \\epsilon \\partial_x^2\\phi + N'(\\phi) = 0, $$\nwhere for the Poisson equation $\\epsilon = +1$ and for the wave equation $\\epsilon = - 1$. The Lagrangian is given by $L(\\phi, \\phi_t, \\phi_x) = \\frac{1}{2} (\\partial_t\\phi)^2 + \\epsilon \\frac{1}{2}(\\partial_x\\phi)^2 - N(\\phi)$, or equivalently,\n$$ \\mathcal{L}(\\phi,d\\phi) = \\frac{1}{2} d\\phi\\wedge\\star d\\phi - N(\\phi) dt \\wedge dx. $$\nwhere the metric corresponding to the Poisson equation is $g = \\text{diag}(1,1)$ and corresponding to the wave equation is $g = \\text{diag}(1,-1)$. Compute $\\partial_3\\mathcal{L} = d\\phi$, $\\partial_2\\mathcal{L} = -N'(\\phi)$, so the discrete Euler--Lagrange equation reads: find $\\phi \\in \\Lambda^0_h$ such that\n$$ (d\\phi, dv) - (N'(\\phi),v) = 0, \\text{ for all } v \\in \\mathring{\\Lambda}^0_h. $$\n(where the product $(\\cdot,\\cdot)$ is computed with the appropriate metric signature). We subdivide $X$ into a regular rectangular mesh and use a tensor-product basis of hat functions $\\psi_{ij}(t,x) = \\chi_i(t) \\xi_j(x) $ subordinate to this mesh.\n\nExpressing $\\phi = \\phi^{ij} \\psi_{ij}$ and taking $v = \\psi_{mn}$, the above equation reads\n$$ \\sum_{ij \\in [mn]} \\Big( \\phi^{ij} (\\chi_i'(t), \\chi_m'(t)) (\\xi_j(x),\\xi_n(x)) + \\epsilon \\phi^{ij}(\\chi_i(t), \\chi_m(t)) (\\xi_j'(x),\\xi_n'(x))\\Big) - (N'(\\phi), \\psi_{mn}) = 0. $$\nSince $[m] = \\{m-1,m,m+1\\},$ this gives a nine-point integrator on the interior elements of the mesh. Explicitly, we compute the stiffness and mass matrix elements\n\\begin{align*}\n\\{(\\chi_i'(t),\\chi_m'(t))\\}_{i \\in [m]} &= \\frac{1}{\\Delta t} \\{-1,2,-1\\}, \\\\\n\\{(\\chi_i(t),\\chi_m(t))\\}_{i \\in [m]} &= \\Delta t \\left\\{ \\frac{1}{6}, \\frac{2}{3}, \\frac{1}{6} \\right\\}\n\\end{align*}\n(and similarly for the $x$ direction). This gives \n$$ \\frac{ \\phi^{m+1 \\tilde{n}} - 2 \\phi^{m \\tilde{n}} + \\phi^{m-1 \\tilde{n}}}{\\Delta t^2} +\\epsilon \\frac{\\phi^{\\tilde{m} n+1} - 2 \\phi^{\\tilde{m} n} + \\phi^{\\tilde{m} n-1} }{\\Delta x^2} + \\frac{1}{\\Delta t \\Delta x} (N'(\\phi),\\psi_{mn}) = 0, $$\nwhere $\\phi^{m \\tilde{n}} = \\frac{1}{6}(\\phi^{m n+1} +4 \\phi^{mn} + \\phi^{mn-1})$ and $\\phi^{\\tilde{m} n} = \\frac{1}{6}(\\phi^{m+1 n} +4 \\phi^{mn} + \\phi^{m-1 n})$. Noting that $(N'(\\phi),\\psi_{mn}) = \\delta\\mathcal{N}\/\\delta \\phi^{mn}$, where $\\mathcal{N} = \\int N(\\phi) dt\\wedge dx$, this reproduces the nine-point variational integrator derived by \\citet{Ch2008}. As was shown in \\citet{Ch2008}, using mid-point quadrature, this method reduces to the multisymplectic integrator derived by \\citet{MaPaSh1998}. We will continue this example in the subsequent section to provide an example of the discrete Cartan form, see Example \\ref{Nonlinear Wave Eq Ex, 2}.\n\\end{example}\n\nIn order to provide local statements of the multisymplectic form formula and Noether's theorem, we now localize the DEL. For a region $U \\subset X$, we say that a node $i$ is an interior point of $U$ if $U$ contains all simplices touching $i$. Denote $\\bar{U}$ as the union of all simplices touching interior nodes $i$ of $U$; we say that $U$ is regular if $U = \\bar{U}$. We define the admissible variations with respect to a regular region $U$ as the space of all $v \\in \\mathring{\\Lambda}^k_h$ such that $v|_U \\in \\mathring{\\Lambda}^k_h(U)$. We define the localized action $S_U[\\phi] = \\int_U \\mathcal{L}(j^1\\phi)$ and the associated localized DEL,\n\\begin{align}\\label{local DEL} 0 = \\delta S_U[\\phi]\\cdot v &= \\big(\\partial_2\\mathcal{L}(j^1\\phi),v\\big)_{L^2\\Lambda^k(U)} + \\big(\\partial_3\\mathcal{L}(j^1\\phi),dv\\big)_{L^2\\Lambda^{k+1}(U)}  \\\\\n&= \\big(\\partial_2\\mathcal{L}(j^1\\phi),v\\big)_{L^2\\Lambda^k(U)} + \\big(d^*\\partial_3\\mathcal{L}(j^1\\phi),v\\big)_{L^2\\Lambda^{k}(U)} + \\int_{\\partial U}v \\wedge * \\partial_3\\mathcal{L}(j^1\\phi) \\nonumber\n\\end{align}\nwhich is enforced for all regular $U$ and admissible $v$ (as before, the boundary term vanishes for admissible $v$, but we write it explicitly as it will arise later). \n\\begin{prop}\nThe localized DEL (\\ref{local DEL}), ranging over all regular $U$ and admissible $v$, are equivalent to the DEL (\\ref{DEL 1b})\n\\begin{proof}\nTo see that the localized DEL imply the DEL, choose $U = X$ which is trivially regular; the space of admissible variations with respect to $X$ is then just $\\mathring{\\Lambda}^k_h$. To see that the DEL imply the localized DEL, let $U$ be regular and $v$ be admissible. Since $\\text{supp}(v) \\subset U$, the integrals over $X$ in the DEL can be replaced by integrals over $U$.\n\\end{proof}\n\\end{prop}\n\n\\subsection{Variational Structure of Discretization}\\label{Variational Structure Section}\\label{Variational Structure Section}\nIn this section, we aim to elucidate the variational structure that arises from discretizing the variational principle utilizing cochain projections. Recalling that the Cartan form (\\ref{Cartan form}) encodes the variational structure of a Lagrangian field theory, we will construct a discrete analogue of the Cartan form, which will naturally encode the variational structure of the discretized theory. \n\nWe first show that the restricted variational principle over the finite-dimensional subspace $\\mathring{\\Lambda}^k_h$ can be interpreted as a Galerkin variational integrator. Restricting the configuration space to $\\mathring{\\Lambda}^k_h$, we can view the action as a function of the components $\\phi^i$ in the expansion $\\phi = \\phi^i v_i$.\n$$ S[\\phi^i] = \\int \\mathcal{L}(x,\\phi^i v_i, \\phi^i dv_i). $$\nTaking the variation of $S$ with respect to $\\phi^j$, \n\\begin{align*}\n\\frac{\\delta S[\\phi^i]}{\\delta \\phi^j} &= \\int \\Big( \\frac{\\delta \\mathcal{L}}{\\delta \\phi} \\cdot \\frac{\\delta (\\phi^iv_i)}{\\delta \\phi^j} + \\frac{\\delta \\mathcal{L}}{\\delta (d\\phi)} \\cdot \\frac{\\delta (\\phi^idv_i)}{\\delta \\phi^j} \\Big) = \\int \\Big( \\frac{\\delta \\mathcal{L}}{\\delta \\phi} \\cdot v_j + \\frac{\\delta \\mathcal{L}}{\\delta (d\\phi)} \\cdot dv_j \\Big) \\\\\n&= (\\partial_2\\mathcal{L},v_j) + (\\partial_3\\mathcal{L},dv_j),\n\\end{align*}\nwhich shows that the conditions $\\delta S\/\\delta \\phi^j = 0$ is equivalent to the DEL (\\ref{DEL 1b}). Similarly, the localized DEL (\\ref{local DEL}) is equivalent to the conditions $\\delta S_U\/\\delta \\phi^j = 0$ for all interior nodes $j$. That is, the DEL can be interpreted as a Galerkin variational integrator. From this viewpoint of the DEL, we see that given appropriate choices of function spaces (and possibly a choice of quadrature rule), our discrete Euler--Lagrange equation reproduces multisymplectic variational integrators based on finite differences or nodal value finite element spaces (e.g., as discussed in \\citet{MaPaSh1998} and \\citet{Ch2008}). However, the discrete variational principle in the form $\\delta S[\\phi]\\cdot v = 0$, for $\\phi \\in \\Lambda^k_h$ and $v \\in \\mathring{\\Lambda}^k_h$, is expressed explicitly at the level of function spaces and hence, will allow us to examine the discrete variational structure more directly. Along with allowing more general approximating finite element spaces, this also has the advantage of stating properties of the discrete variational principle at the level of function spaces. Consequently, as we will see, properties such as multisymplecticity and Noether's theorem can be stated in a geometric way, which makes no explicit reference to finite differencing or quadrature. \n\nWhen the variational principle to the Lagrangian system defined by $\\mathcal{L}$ is enforced only over the subspace $\\mathring{\\Lambda}^k_h$ of the configuration bundle, the discrete dynamics (\\ref{DEL 1b}) are recovered. To recast (\\ref{DEL 1b}) in terms of the Cartan form, we note that variations $\\delta S[\\phi]\\cdot v$ is equivalently given by the differential of the action paired with a vertical vector field $dS_\\phi \\cdot V$. This follows since $T(H\\Lambda^k) \\cong H\\Lambda^k \\times H\\Lambda^k$, so $V(\\phi)$ can be identified with some $v \\in H\\Lambda^k$. We define a discrete vertical vector field as an element of $\\mathfrak{X}(\\Lambda^k_h)$. Then, in particular, a constant vertical vector field $V \\in \\mathfrak{X}(\\Lambda^k_h)$ can be identified with $v \\in \\Lambda^k_h$; i.e., viewing $\\mathfrak{X}(\\Lambda^k_h)$ as the space of (sufficient regularity) maps $\\Lambda^k_h \\rightarrow \\Lambda^k_h$, we have $V(\\phi) = v$ for all $\\phi$; in which case, we denote $V = V_v$ and the space of such vector fields as $V_h$ (note the time-$\\epsilon$ flow of $V_v$ on any section $\\phi \\in \\Lambda^k_h$ is given by $\\phi + \\epsilon v$). We denote the spaces $\\mathring{\\mathfrak{X}}(\\Lambda^k_h)$ and $\\mathring{V}_h$ as the subspaces of the above spaces which vanish on $\\partial X$. Then (\\ref{DEL 1b}) is equivalent to finding $\\phi \\in \\mathring{\\Lambda}^k_h$ such that\n\\begin{equation}\\label{DEL 2a}\n0 = dS[\\phi] \\cdot V = -\\int_X (j^1\\phi)^*(j^1V \\lrcorner\\, \\Omega_{\\mathcal{L}}), \\text{ for all } V \\in \\mathring{\\mathfrak{X}}(\\Lambda^k_h),\n\\end{equation}\nor equivalently using constant vector fields,\n\\begin{equation}\\label{DEL 2b}\n0 = \\delta S[\\phi]\\cdot v = dS[\\phi] \\cdot V_v = -\\int_X (j^1\\phi)^*(j^1V_v \\lrcorner\\, \\Omega_{\\mathcal{L}}), \\text{ for all } V_v \\in \\mathring{V}_h,\n\\end{equation}\nwhere $j^1V$ is the vector field jet prolongation of $V$. \n\nBy the above, we can view the Lagrangian structure associated to the equations (\\ref{DEL 2b}) as the restriction of the full Lagrangian structure to the discrete space. The next natural question to ask would be: is there some sense in which the discrete equations, which arises as a restriction of the variational principle, can instead be viewed as a variational principle on the full configuration bundle? Since we assume that the projection maps $\\pi_h: H\\Lambda^m \\rightarrow \\Lambda^m_h$ are cochain projections on the Hilbert de Rham complex, there is a natural relation between the dynamics of the restricted Lagrangian structure and variations on the full space of the degenerate Lagrangian. To see this, recall that the Lagrangian density $\\mathcal{L}: J^1_k \\rightarrow \\wedge^{n+1}(T^*X)$ is a bundle map, so it induces a map on the space of sections, $\\mathcal{L}: J^1_{H\\Lambda^k} \\rightarrow \\Lambda^{n+1}(X)$. In equation (\\ref{Degenerate Action}), we defined a degenerate Lagrangian density, $\\mathcal{L}_h: J^1_{H\\Lambda^k} \\rightarrow \\Lambda^{n+1}(X)$ given by $\\mathcal{L}_h(x,\\phi,d\\psi) = \\mathcal{L}(x,\\pi_h\\phi,\\pi_h d\\psi)$ with associated degenerate action $S_h[\\phi] = \\int_X \\mathcal{L}_h(j^1\\phi)$. In the case of a cochain projection, we can then view the variations of $S$ restricted to $\\Lambda^k_h$ as variations of $S_h$ on the full configuration bundle. \n\n\\begin{prop}{\\textbf{(Naturality of Discrete Variational Structure)}}\\label{Naturality}\n\\\\ The restricted variational structures are related to the degenerate variational structures by\n\\begin{align}\n\\mathcal{L}(j^1\\pi_h\\phi &) = \\mathcal{L}_h(j^1\\phi), \\label{Naturality Eqn 1} \\\\ \ndS[\\pi_h\\phi]\\cdot (T\\pi_h \\cdot &V_v) = dS_h[\\phi]\\cdot V_v, \\label{Naturality Eqn 2}\n\\end{align}\nfor $\\phi \\in H\\Lambda^k$ and $v \\in H\\Lambda^k$.\n\\begin{proof}\nFor (\\ref{Naturality Eqn 1}), since $\\pi_h$ is a cochain projection,\n$$ \\mathcal{L}(j^1\\pi_h\\phi) = \\mathcal{L}(x,\\pi_h\\phi, d\\pi_h\\phi) = \\mathcal{L}(x,\\pi_h\\phi, \\pi_h d\\phi) = \\mathcal{L}_h(x,\\phi,d\\phi) = \\mathcal{L}_h(j^1\\phi). $$\nThen, (\\ref{Naturality Eqn 2}) follows similarly, noting that since $V_v$ generates the flow $\\phi + \\epsilon v$ on $\\phi$, $T\\pi_h \\cdot V_v$ generates the flow $\\pi_h\\phi + \\epsilon \\pi_h v$ on $\\pi_h\\phi$, which gives\n\\begin{align*}\ndS[\\pi_h\\phi]\\cdot (T\\pi_h\\cdot V_v) &= \\delta S[\\pi_h\\phi]\\cdot \\pi_h v \n\\\\ &= \\frac{d}{d\\epsilon}\\Big|_{\\epsilon = 0} S[\\pi_h\\phi + \\epsilon \\pi_hv)] = \\frac{d}{d\\epsilon}\\Big|_{\\epsilon = 0} S[\\pi_h(\\phi + \\epsilon v)]\n\\\\ &= \\frac{d}{d\\epsilon}\\Big|_{\\epsilon = 0} S_h[\\phi + \\epsilon v] = \\delta S_h[\\phi]\\cdot v = dS_h[\\phi] \\cdot V_v,\n\\end{align*}\nwhere $S_h = S \\circ \\pi_h$ follows from the cochain map property. \n\\end{proof}\n\\end{prop}\n\nThe naturality equations (\\ref{Naturality Eqn 1}) and (\\ref{Naturality Eqn 2}) reveal that the process of discretization of the full field dynamics, in the case of using cochain projections for discretization, is itself associated to an action on the full field space; i.e., the discretization is compatible with the structure of a Lagrangian theory. A corollary is that the equation (\\ref{DEL 2b}) can be seen as either arising from the variation of the full action $S$ at a discrete field $\\pi_h\\phi$, or as from the variation of the discrete action $S_h$ at the full field $\\phi$. This shows that the variations associated to $S_h$ on the full field space are degenerate, since they are equivalently given by the variations of $S$ on the projected space. Thus, the finite-dimensionality of the restricted variational principle on $S$ can be interpreted as the degeneracy of the variational principle of $S_h$ on the full space, where two fields are equivalent if their difference is in $\\ker(\\pi_h)$ (and similarly for two vertical variations if their difference is in $\\ker(T\\pi_h)$). In other words, our finite-dimensional variational problem on the discrete space arises as a degenerate variational problem over the infinite-dimensional space, where the set of equivalence classes forms a finite-dimensional space, with the canonical representative $i_h\\pi_h\\phi$ for the equivalence class of $\\phi$, where $i_h:\\Lambda^k_h \\hookrightarrow H\\Lambda^k$ is the inclusion map. \n\nFurthermore, the above naturality relation shows that projecting the equations obtained from the variational principle applied to the continuum action is equivalent to first discretizing the action through the projection and subsequently applying the variational principle. Thus, when discretizing via cochain projections, the variational principle and discretization commute:\n\\[\\begin{tikzcd}\n\t{S: H\\Lambda^k \\rightarrow \\mathbb{R}} &&& {S_h: \\Lambda^k_h \\rightarrow \\mathbb{R}} \\\\\n\t\\\\\n\t\\\\\n\t{\\text{Weak EL}} &&& {\\text{Discrete EL .}}\n\t\\arrow[\"{\\substack{\\text{Variational} \\\\ \\text{Principle}}}\"', from=1-1, to=4-1]\n\t\\arrow[\"{\\text{Discretize}}\", from=1-1, to=1-4]\n\t\\arrow[\"{\\quad \\text{Discretize}}\", from=4-1, to=4-4]\n\t\\arrow[\"{\\substack{\\text{Variational} \\\\ \\text{Principle}}}\", from=1-4, to=4-4]\n\\end{tikzcd}\\]\nThis generalizes the result of \\citet{Le2004} where it was shown that discretization via discrete exterior calculus and the variational principle commute in the case of electromagnetism. In particular, the result of \\citet{Le2004} follows from the above, since one can view discrete exterior calculus in the framework of finite element exterior calculus as a particular low-order example; namely, through the use of Whitney forms. \n\nAs a final remark on the above naturality relation, a more fundamental issue for discretization is whether one should discretize at the level of the configuration bundle or the jet bundle. One can discretize sections of the configuration bundle, via $\\phi \\mapsto \\pi^k_h\\phi$ and subsequently work with the restricted Lagrangian $\\mathcal{L}(j^1(\\pi_h\\phi))$, or one can discretize sections of the jet bundle, via $j^1\\phi \\mapsto (\\pi^k_h \\times \\pi^{k+1}_h) j^1\\phi$; in general, these methods are not equivalent. However, in the case of cochain projections, these two discretization processes are equivalent; i.e., the following diagram commutes\n\n\\centerline{ \\xymatrix@C+5pc{ \\phi \\ar@{|->}[r]^{\\pi_h^k} \\ar@{|->}[d]^{j^1} & \\pi_h\\phi \\ar@{|->}[d]^{j^1} \\\\ j^1\\phi \\ar@{|->}[r]^{\\pi^k_h \\times \\pi^{k+1}_h} & j^1(\\pi_h\\phi),  } }\n\nso there is no ambiguity in which discretization procedure to use. Furthermore, regarding Assumption \\ref{Cochain Projection Assumption}, the above diagram shows that we only need the existence of the space $\\Lambda^{k+1}_h$ and the projection $\\pi^{k+1}_h$ such that the above diagram commutes and thus, one can perform the discretization solely using $\\Lambda^k_h$ and $\\pi^k_h$, without reference or implementation of $\\Lambda^{k+1}_h$ and $\\pi^{k+1}_h$. In particular, as discussed in, for example, \\citet{ArFaWi2006, ArFaWi2010} and \\citet{Ar2018}, there is a large class of classical finite element spaces for which such cochain projections exist, so our theory is broadly applicable.\n\nIn order to state discrete analogues of the multisymplectic form formula and Noether's theorem, we will have to consider variations which are nonzero on the boundary of a regular region $U$. To do this, consider the following decomposition. Let $U$ be a regular region and let $v \\in \\Lambda^k_h$. Consider $v$ restricted to $U$. In general, since we are not assuming $v$ be an admissible variation relative to $U$, $v$ may have nonzero trace along $\\partial U$. Decompose $v = v_\\partial + v_{in}$ where $v_\\partial$ denotes the boundary component of $v$ consisting of the expansion of $v$ with respect to all shape functions which have nonzero trace on $\\partial U$; while $v_{in} = v - v_\\partial$ corresponds to the expansion of $v$ into shape functions with vanishing trace on the boundary. Let $\\mathcal{T}[\\partial U]$ denote the set of all top-dimensional elements in $\\mathcal{T}_h$ on which shape functions with nonvanishing trace on $\\partial U$ are supported. We extend this decomposition to vector fields analogously $V = V_\\partial + V_{in}$, noting again by our previous discussion that for $V \\in \\mathfrak{X}(\\Lambda^k_h)$, $V(\\phi)$ can be identified with some $v \\in \\Lambda^k$ (and analogously for vector fields on the jet bundle). \n\n\\begin{remark}\nIf one considers Lagrange polynomial nodal shape functions (corresponding to point value degrees of freedom), then the shape functions which are nonzero on the boundary are those associated to the nodes on $\\partial U$. In this case, $\\mathcal{T}[\\partial U]$ consists of those top-dimensional elements touching the boundary, i.e., the one-ring of the boundary $\\partial U$. For general (local) shape functions, internal nodes may give rise to shape functions which are nonzero on the boundary, so $\\mathcal{T}[\\partial U]$ will generally consist of the elements touching $\\partial U$ and the elements touching those elements, i.e., the two-ring of the boundary $\\partial U$. In any case, we consider discretization by the finite element method due to the local support property of the shape functions, which will allow the discrete Cartan form defined below to be localized on $\\mathcal{T}[\\partial U]$.\n\\end{remark}\n\nWe can now consider variations which do not vanish on $\\partial U$. In particular, we compute for a solution $\\phi_h$ of the discrete Euler--Lagrange equation and $V \\in \\mathfrak{X}(\\Lambda^k_h),$\n\\begin{align*}\ndS_U[\\phi_h]\\cdot V &= -\\int_U (j^1\\phi_h)^*(j^1V\\, \\lrcorner\\, \\Omega_{\\mathcal{L}}) + \\int_{\\partial U}(j^1\\phi_h)^*(j^1V\\, \\lrcorner\\, \\Theta_{\\mathcal{L}})  \\\\\n&= -\\int_U (j^1\\phi_h)^*(j^1V_{in} \\, \\lrcorner\\, \\Omega_{\\mathcal{L}}) - \\int_U (j^1\\phi_h)^*(j^1V_{\\partial} \\, \\lrcorner\\, \\Omega_{\\mathcal{L}}) + \\int_{\\partial U}(j^1\\phi_h)^*(j^1V\\, \\lrcorner\\, \\Theta_{\\mathcal{L}}) \\\\ \n&= -\\sum_{T \\in \\mathcal{T}[\\partial U]} \\int_T (j^1\\phi_h)^*(j^1V_{\\partial} \\, \\lrcorner\\, \\Omega_{\\mathcal{L}}) + \\int_{\\partial U}(j^1\\phi_h)^*(j^1V\\, \\lrcorner\\, \\Theta_{\\mathcal{L}}),\n\\end{align*}\nwhere the term involving $V_{in}$ vanishes by the DEL. This boundary variation formula will be our candidate for a discrete Cartan form, as it encodes the contribution to the action from $V$ nonvanishing on and near the boundary, and will allow us to state discrete analogues of the multisymplectic form formula and Noether's theorem.\n\\begin{definition}[Discrete Cartan Form]\\label{Discrete Cartan Form}\nThe discrete Cartan one-form, evaluated at a solution $\\phi_h$ of the discrete Euler--Lagrange equation (\\ref{DEL 1b}), is defined by\n\\begin{subequations}\n\\begin{equation}\\label{Discrete Cartan Form 1}\n\\Theta^h_U(\\phi_h)\\cdot V \\equiv dS_U[\\phi_h]\\cdot V_{\\partial} = \\int_{\\partial U}(j^1\\phi_h)^*(j^1V\\, \\lrcorner\\, \\Theta_{\\mathcal{L}}) -\\sum_{T \\in \\mathcal{T}[\\partial U]} \\int_T (j^1\\phi_h)^*(j^1V_{\\partial} \\, \\lrcorner\\, \\Omega_{\\mathcal{L}}),\n\\end{equation}\nor in terms of the Lagrangian density,\n\\begin{equation}\\label{Discrete Cartan Form 2}\n\\Theta^h_U(\\phi_h)\\cdot V = \\int_{\\partial U}  \\Big(* \\partial_3\\mathcal{L}(j^1\\phi_h) \\Big) \\wedge V(\\phi_h)\\ + \\sum_{T \\in \\mathcal{T}[\\partial U]} \\int_T (\\partial_2\\mathcal{L}(j^1\\phi_h) + d^*\\partial_3\\mathcal{L}(j^1\\phi_h))\\wedge\\star V(\\phi_h)_{\\partial}.\n\\end{equation}\n\\end{subequations}\n\\end{definition}\nEven though the continuum Cartan form only involves integration on $\\partial U$, we will see why this is the appropriate definition in the discrete setting. In stating the discrete analogues of the multisymplectic form formula and Noether's theorem, we will contrast this to the (integrated) Cartan form of the continuum theory,\n\\begin{equation}\\label{Continuum Cartan Form}\n\\Theta_U(\\phi)\\cdot V \\equiv \\int_{\\partial U} (j^1\\phi)^* (j^1V \\lrcorner \\, \\Theta_{\\mathcal{L}}) = \\int_{\\partial U}  \\Big(* \\partial_3\\mathcal{L}(j^1\\phi) \\Big) \\wedge V(\\phi).\n\\end{equation}\n\n\\begin{remark}[Quadrature]\\label{Remark on Quadrature}\nAlthough, in our exposition, we have assumed that with the given Lagrangian and choice of finite element space one can evaluate the integrals involved exactly, one can more generally utilize quadrature to approximate the action before enforcing the variational principle. For a regular region $U$, let us consider quadrature nodes $\\{c_a \\in U\\}$ and associated quadrature weights $\\{b_a\\}$. With finite element shape functions $\\{v_j\\}$ and expressing the density as $\\mathcal{L} = L d^{n+1}x$, the associated discrete action is given by applying quadrature,\n\\begin{equation}\\label{Discrete Action with Quadrature}\n\\mathbb{S}_U[\\{\\phi^j\\}] = \\sum_a b_a L(j^1(\\phi^iv_i))|_{c_a}.\n\\end{equation}\nThe variation in the direction $w = w^kv_k$ is given by\n\\begin{equation}\\label{Discrete Variation with Quadrature}\n\\delta \\mathbb{S}_U[\\{\\phi^j\\}]\\cdot \\{w^k\\} = \\sum_a b_a \\frac{\\partial}{\\partial \\phi^k} \\Big[ L(j^1(\\phi^iv_i))\\big|_{c_a} \\Big] w^k.\n\\end{equation}\nThe associated discrete Euler--Lagrange equation is given by enforcing the variational principle for variations $w$ with vanishing trace on $\\partial U$. Then, the discrete Cartan form with quadrature (at a solution of the discrete Euler--Lagrange equation), is defined by taking an arbitrary variation and removing the term on the interior which vanishes by the discrete Euler--Lagrange equation. In particular, it is given by summing over all $a$ such that $c_a$ is contained in the support of some shape function with nonvanishing trace on the boundary; we denote the set of all such $a$ by $\\mathcal{I}[\\partial U]$. Hence, the discrete Cartan form with quadrature is given by \n$$ \\Uptheta^h_U(\\phi)\\cdot W = \\sum_{a \\in \\mathcal{I}[\\partial U]} b_a \\frac{\\partial}{\\partial \\phi^k} \\Big[ L(j^1(\\phi^iv_i))\\big|_{c_a} \\Big] w^k, $$\nwhere $W(\\phi) = w^kv_k$.\n\nUsing this discrete Cartan form, an analogous statement of discrete multisymplecticity that we state below holds in this setting, with the caveat that the first variations are defined relative to the discrete Euler--Lagrange equations with quadrature. Similarly, an analogous statement to the discrete Noether's theorem below also holds in this setting, with the caveat that the group action leaves the discrete action with quadrature, equation (\\ref{Discrete Action with Quadrature}), invariant. This is a direct consequence of the fact that the formulation with quadrature is still variational, since we applied the quadrature rule to the action, before enforcing the variational principle (see Section \\ref{Variational Complex Section}). In general, if one applies quadrature after enforcing the variational principle, i.e., to the equations of motion (\\ref{DEL 1b}), the system is not variational. To see this, we compute the variation of the action first,\n$$ \\delta S_U[\\phi^jv_j]\\cdot (w^kv_k) = \\int_X [\\partial_2\\mathcal{L}(j^1(\\phi^iv_i)) + d^*\\partial_3\\mathcal{L}(j^1(\\phi^iv_i))]\\wedge \\star w^kv_k, $$\n(for $w$ with vanishing trace on $\\partial U$) and subsequently apply quadrature, so that the above becomes\n$$ \\sum_a b_a\\Big[*\\Big( [\\partial_2\\mathcal{L}(j^1(\\phi^iv_i)) + d^*\\partial_3\\mathcal{L}(j^1(\\phi^iv_i))]\\wedge \\star w^kv_k \\Big) \\Big]\\Big|_{c_a}. $$\nIn general, this is not equal to (\\ref{Discrete Variation with Quadrature}), except when $\\phi$ a scalar field, using nodal interpolating shape functions and quadrature points at those nodes, in which case they are the same. Thus, for a variational formulation, one should generally apply quadrature before enforcing the variational principle. For the rest of the paper, we will return to the assumption that one can evaluate the various integrals exactly, but keeping in mind that similar results hold in the case of quadrature. \n\\end{remark}\n\nWe make several additional remarks regarding this candidate (\\ref{Discrete Cartan Form 1}) for a discrete Cartan form. We defined the discrete Cartan form as the variation of the action (which may be nonvanishing on the boundary) at a solution of the discrete Euler--Lagrange equations. Even though this functional involves integration over top-dimensional regions $T \\in \\mathcal{T}[\\partial U]$, it only depends on the degrees of freedom which contribute to the nonzero value of $V$ on $\\partial U$ and so makes sense as a candidate for a discrete Cartan form. In the continuum variational problem, boundary variations can be supported arbitrarily close to $\\partial U$, whereas in the finite element variational problem, this is not the case, so the discrete Cartan form (which encodes the contribution of the variation of the action by boundary variations) should indeed contain the additional terms involving integration over the elements of $\\mathcal{T}[\\partial U]$. These terms shrink relative to the integral over $\\partial U$ in the following heuristic sense. The terms involving $\\mathcal{T}[\\partial U]$ are $O(h)$ smaller than the term over $\\partial U$: the cardinality of $\\mathcal{T}[\\partial U]$ scales like the number of boundary faces in $\\partial U$, which is $O(h^{-n})$; on the other hand, the size of $T$ is $O(h^{n+1})$, so the terms in the discrete Cartan form involving the sum over $\\mathcal{T}[\\partial U]$ is $O(h)$, whereas the first term is $O(1)$ for a fixed region $U$. Thus, as $h \\rightarrow 0$, for a fixed region $U$, the Cartan form formally only involves the first contribution, as expected. In other words, as we refine the mesh, $\\partial U$ stays (roughly) the same, while the region containing only elements touching $\\partial U$ shrinks (and a similar remark applies to the discrete multisymplectic form formula and the additional terms involving the sum over $\\mathcal{T}[\\partial U]$, so that the multisymplectic form formula is formally recovered in the limit). This can be combined with bounds on the integrands to show convergence more rigorously, which we will sketch when discussing Noether's theorem. \n\nWe now show that Defintion \\ref{Discrete Cartan Form} recovers the notion of the discrete Cartan form introduced in \\citet{MaPaSh1998} and further examined in \\citet{Ch2008}, in the case that the degrees of freedom are the nodal values of the field with nodal interpolating shape functions. As previously remarked, in this case, the shape functions which are nonzero on $\\partial U$ are those associated to nodes on $\\partial U$. Consider a single node $i$ on $\\partial U$ and let $v_i$ be the shape function associated to the degree of freedom on the node; note that $v_i$ (restricted to $U$) is supported in some $T_i \\in \\mathcal{T}[\\partial U]$ and denote $F_i = \\partial T_i \\cap \\partial U$. Consider a variation by an amount $V v_i\\ (V \\in \\mathbb{R})$. \\citet{MaPaSh1998} and \\citet{Ch2008} define the discrete Cartan form associated to this node as $\\frac{\\delta S_U[\\phi^jv_j]}{\\delta \\phi^i} V$, viewing the action as a function of the components in the expansion of $\\phi = \\phi^jv_j$. Then, compute\n\\begin{align*}\n\\frac{\\delta S_U[\\phi^jv_j]}{\\delta \\phi^i} V &= \\int_U \\Big( \\frac{\\delta \\mathcal{L}}{\\delta \\phi} \\cdot \\frac{\\delta (\\phi^jv_j)}{\\delta \\phi^i} V + \\frac{\\delta \\mathcal{L}}{\\delta (d\\phi)} \\cdot \\frac{\\delta (\\phi^j dv_j)}{\\delta \\phi^i} V \\Big) \\\\ \n&= \\int_U \\Big( \\frac{\\delta \\mathcal{L}}{\\delta \\phi} \\cdot Vv_i + \\frac{\\delta \\mathcal{L}}{\\delta (d\\phi)} \\cdot V dv_i \\Big) = \\int_{U} \\Big( \\partial_2 \\mathcal{L} \\wedge \\star Vv_i + \\partial_3\\mathcal{L} \\wedge \\star Vdv_i \\Big) \\\\\n&= \\int_U (\\partial_2\\mathcal{L} + d^*\\partial_3\\mathcal{L}) \\wedge \\star V v_i + \\int_{\\partial U} (* \\partial_3\\mathcal{L})\\wedge Vv_i \\\\\n&= \\int_{T_i} (\\partial_2\\mathcal{L} + d^*\\partial_3\\mathcal{L}) \\wedge \\star V v_i + \\int_{F_i} (* \\partial_3\\mathcal{L})\\wedge Vv_i\n\\end{align*}\nThen, summing over all such variations on each node on $\\partial U$, one recovers our discrete Cartan form, equation (\\ref{Discrete Cartan Form 2}). There are several generalizations which our discrete Cartan form makes relative to the discrete Cartan form of \\citet{MaPaSh1998} and \\citet{Ch2008}. First, note that their Cartan form is defined in terms of the nodal values of the field, which implicitly suppresses the fact that the Cartan form involves integration over both $\\partial U$ and elements of $\\mathcal{T}[\\partial U]$. Our explicit formula for the discrete Cartan form lends itself more easily to showing convergence to the continuum Cartan form, as we sketched heuristically above and will discuss further when discussing Noether's theorem. That the discrete Cartan form involves integration over elements neighboring the boundary is inevitable, since a variation of the field value on the boundary induces changes to the field values on elements of $\\mathcal{T}[U]$. Furthermore, since we allow for general finite element spaces, we immediately get several generalizations. First, note that the dimension of the spacetime is arbitrary in our formulation, so this discrete Cartan form holds beyond the $1+1$ spacetime dimensions that they utilize explicitly in their framework (although this is not a fundamental restriction in their theory). Furthermore, our framework allows for arbitrary degree of differential forms, as opposed to just scalar fields. In particular, the degrees of freedom associated to the boundary variations need not be nodal values, but can be determined by more general degrees of freedom, such as moments or flux type degrees of freedom (e.g., when considering a theory involving vector fields, which we identify with $1-$forms via the metric). Furthermore, these degrees of freedom determining the boundary variations may be close to (i.e., in $\\mathcal{T}[\\partial U]$) but not necessarily on $\\partial U$. \n\n\\begin{example}[Discrete Cartan Form for the Nonlinear Poisson\/Wave Equation]\\label{Nonlinear Wave Eq Ex, 2}\nRecall the discretization of the nonlinear Poisson\/wave equation given in Example \\ref{Nonlinear Wave Eq Ex, 1}. Consider a regular region $U \\subset X$; for simplicity, we take $U$ to be a rectangular region $U = [t_0, t_M] \\times [x_0, x_N]$ (with no loss of generality, since any regular region is a union of such rectangular regular regions), where the vertices of $U$ are given by $\\{(t_i,x_j)\\}_{i,j=0}^{M,N}$ where $t_i = t_0 + i \\Delta t, x_j = x_0 + j \\Delta x$. We index the piecewise linear nodal interpolating shape functions $\\psi_{ij}(t,x) = \\chi_i(t)\\xi_j(x)$ by the node $(t_i,x_j)$ which it interpolates; i.e., $\\chi_i(t_k)\\xi_j(x_l) = \\delta_{ik}\\delta_{jl}$. Let \n$$ \\phi_h = \\sum_{i,j=0}^{M,N}\\phi^{ij}_h \\chi_i \\xi_j $$\nbe a solution of the associated discrete Euler--Lagrange equation, restricted to $U$.  \n\nRecall the definition of the discrete Cartan form as the variation of the action by $w \\in \\Lambda^0_h(U)$ (with generally nonvanishing trace on $\\partial U$). Letting $w = w_{in} + w_{\\partial} \\in \\Lambda^0_h(U)$ and $W \\in \\mathfrak{X}(\\Lambda^0_h)$ such that $W(\\phi_h) = w$, we have $\\delta S_U[\\phi_h]\\cdot w_{in} = 0$ and hence,\n\\begin{align}\\label{Discrete Cartan Form for Wave Eq}\n \\Theta^h_U(\\phi_h)\\cdot W &= \\delta S_U[\\phi_h]\\cdot w = \\delta S_U[\\phi_h]\\cdot (w - w_{in}) = \\delta S_U[\\phi_h] \\cdot w_{\\partial} \\\\\n &= \\sum_{T \\in \\mathcal{T}[\\partial U]} \\int_T \\left[ d\\phi_h \\wedge * dw_\\partial - N'(\\phi_h) \\wedge * w_\\partial \\right]. \\nonumber\n\\end{align}\nAs discussed above, the discrete Cartan form reproduces the discrete Cartan form in \\citet{MaPaSh1998} and \\citet{Ch2008}. However, we will now explicitly show this for this example. We express the action as a function of the components $\\phi_h^{ij}$:\n$$ S_U[\\{\\phi^{ij}_h\\}] = \\int_U [d\\phi_h \\wedge * d\\phi_h - N(\\phi_h) dt \\wedge dx] = \\int_U \\left[ \\frac{1}{2} \\sum_{i,j}\\sum_{k,l}\\phi^{ij}_h\\phi^{kl}_h d\\psi_{ij} \\wedge * d \\psi_{kl} - N(\\sum_{k,l}\\phi_h^{kl}\\psi_{kl})dt \\wedge dx \\right]. $$\nLet $ij \\in \\mathcal{I}[\\partial U]$, i.e., the index corresponds to a node on $\\partial U$ (consisting of indices $ij$ such that either $i = 0 \\text{ or } M$ or $j = 0 \\text{ or } N$). \\citet{MaPaSh1998} and \\citet{Ch2008} define the discrete Cartan form associated to this node as\n\\begin{equation} \\label{MPS Discrete Cartan for Wave Eq}\n\\frac{\\partial S_U[\\{\\phi_h^{kl}\\}]}{\\partial \\phi_h^{ij}} d\\phi^{ij}_h \\end{equation}\n(where here $d$ is the vertical exterior derivative along the fiber and not the exterior derivative on the base space). Compute\n$$ \\frac{\\partial S_U[\\{\\phi^{kl}_h\\}]}{\\partial \\phi_h^{ij}} = \\int_U \\left[ \\sum_{k,l}\\phi^{kl}_h d\\psi_{ij}\\wedge * d\\psi_{kl} - N'(\\sum_{k,l}\\phi^{kl}_h\\psi_{kl}) \\psi_{ij} dt\\wedge dx \\right]. $$\nWith coordinate $\\phi_h^{ij}$ on $\\Lambda^0_h$, we can express the vector field $W = \\sum_{k,l} W^{kl} \\partial\/\\partial \\phi_h^{kl}$ and hence $W^{kl}(\\phi_h) = w^{kl}$. Pairing (\\ref{MPS Discrete Cartan for Wave Eq}) with $W$ and summing over all $ij \\in \\mathcal{I}[\\partial U]$, we see that this gives (\\ref{Discrete Cartan Form for Wave Eq}), since $w_\\partial = \\sum_{ij \\in \\mathcal{I}[\\partial U]} w^{ij} \\psi_{ij}$ and $\\psi_{ij}$ for $ij \\in \\mathcal{I}[\\partial U]$ are supported on $\\cup_{ T \\in \\mathcal{T}[\\partial U]}T$.\n\nFinally, we now discuss in what sense the discrete Cartan form for this example converges to the continuum Cartan form. Consider a node $ij \\in \\mathcal{I}[\\partial U]$ along, say, the $\\{t = t_0\\}$ edge of $\\partial U$, so that $i = 0$. We compute part of the discrete Cartan form for a boundary variation $w^{0j}$ associated to this node; namely, we compute the part associated to the derivative in the $t$ direction, since this is the normal direction along this edge. This is given by \n\\begin{align*}\n \\int_U \\sum_{k,l} \\phi_h^{kl} \\chi'_k(t) \\xi_l(x) w^{0j} \\chi'_0(t) \\xi_j(x) dt\\wedge dx &=  \\int_U \\sum_{k=0}^1 \\sum_{l=j-1}^{j+1} \\phi_h^{kl} \\chi'_k(t) \\xi_l(x) w^{0j} \\chi'_0(t) \\xi_j(x) dt\\wedge dx  \\\\\n &= \\sum_{l=j-1}^{j+1}\\frac{\\phi_h^{0l} - \\phi_h^{1l}}{\\Delta t} (\\xi_l,\\xi_j)_{L^2} w^{0j}.\n\\end{align*}\nSince $(\\xi_l,\\xi_j)_{L^2}$ for $l=j-1,j,j+1$ has total mass $\\Delta x$, this formally converges to $\\int \\frac{\\partial \\phi}{\\partial n} w dx$ (noting that the normal vector on this edge is $-\\hat{t}$). Repeating this over all nodes on $\\partial U$, this part of the discrete Cartan form formally converges to\n$$ \\int_{\\partial U} \\frac{\\partial \\phi}{\\partial n} w dS $$\n(where $dS$ is the codimension one measure on $\\partial U$), which is the continuum Cartan form. Furthermore, the other terms in the discrete Cartan form which we did not explicitly write formally converge to zero; this follows since the remaining terms formally vanish by the weak Euler--Lagrange equations in the continuum limit. To be more rigorous, for the Poisson equation, we should express the above as \n$$ \\int_{\\partial U} \\partial_n(\\phi) \\text{Tr} (w) dS, $$\nwhere $\\text{Tr}: H\\Lambda^0(X) \\rightarrow H^{1\/2}\\Lambda^0(\\partial X)$ (recall that we define $H\\Lambda^0(X)$ as the Hilbert space of square integrable functions with square integrable derivative) and $\\partial_n: \\{u \\in H\\Lambda^0(X): \\Delta u \\in L^2(X) \\} \\rightarrow H^{-1\/2}(\\partial X)$ are bounded operators. In particular, if $N'(\\phi)$ is square integrable, then $\\partial_n$ is appropriately bounded for this problem; for example, this holds if $N(\\phi)$ is polynomial in $\\phi$ with degree $p \\geq 2$ (since $X$ is compact) and hence the nonlinearity $N'(\\phi)$ can be polynomial with degree $p \\geq 1$ (with $p=1$ corresponding to the linear Helmholtz equation). In this case, convergence in $H\\Lambda^0$ of the discrete solution $\\phi_h$ to a solution $\\phi$ of the weak Euler--Lagrange equation gives weak convergence of the discrete Cartan form to the continuum Cartan form. For the wave equation, due to the metric signature, one has to be more careful regarding the definition of the relevant Sobolev spaces although the discrete Cartan form formally converges in the sense above; we aim to pursue rigorous convergence of the discrete Cartan form for evolution equations in future work.\n\\end{example}\n\nIn the next two sections, we will utilize the discrete Cartan form to state discrete analogues of multisymplecticity and Noether's theorem. We will see that these statements, involving $\\Theta^h_U$, will be in direct analogy to the continuum theorems, involving $\\Theta_U$.\n\n\\subsection{Discrete Multisymplectic Form Formula}\\label{Multisymplectic Section}\n\nWe now state a discrete analogue of the multisymplectic form formula, which generalizes the preservation of the symplectic form under the flow of a symplectic vector field. Namely, if $\\phi$ is a solution to the Euler--Lagrange equations and $V, W$ are first variations at $\\phi$ (their flow on $\\phi$ is still a solution), then \n\\begin{equation}\\label{Multisymplectic Form Formula}\n\\int_{\\partial U}(j^1\\phi)^*\\Big( j^1V \\lrcorner\\, j^1W\\lrcorner\\, \\Omega_{\\mathcal{L}} \\Big) = 0, \n\\end{equation}\nwhere $U \\subset X$ is a submanifold with smooth closed boundary (\\citet{MaPaSh1998}). The multisymplectic form formula encompasses many physical conservation laws appearing in Lagrangian field theories. For example, viewing a Lagrangian field theory in the instantaneous canonical formulation, multisymplecticity gives rise to the usual field-theoretic notion of symplecticity (\\citet{MaPaSh1998}). Furthermore, multisymplecticity encompasses the notion of reciprocity in many physical systems, relating the infinitesimal perturbation of a system by a source and the associated infinitesimal perturbation of the response by the system (see, for example, \\citet{VaLiLe2011} for Lorenz reciprocity in electromagnetism and \\citet{McAr2020} for reciprocity in semilinear elliptic PDEs, within the context of multisymplecticity). Additionally, for wave propagation problems, multisymplecticity provides a geometric formulation for the conservation of wave action (\\citet{Br1997, Br1997no2}). Since multisymplecticity is an important property of Lagrangian field theories encompassing many natural physical conservation laws, we will investigate multisymplecticity within our discretization framework. \n\nIn the literature, integrators which admit a discrete analogue of this formula are referred to as ``multisymplectic integrators''. We show that our discrete system (\\ref{DEL 1b}) admits a discrete multisymplectic form formula. The main idea of the derivation for the multisymplectic form formula is to look at second variations of the action at $\\phi$ with respect to first variations $V$ and $W$, $d^2S[\\phi]\\cdot (V,W) = 0$. More specifically, one decomposes the variation of the action into two functionals, corresponding to interior and boundary variations:\n\\begin{align*}\ndS[\\phi]\\cdot V = \\underbrace{-\\int_U (j^1\\phi)^* (j^1V \\lrcorner\\, \\Omega_{\\mathcal{L}})}_{\\equiv \\alpha_U(\\phi)\\cdot V} + \\underbrace{\\int_{\\partial U}(j^1\\phi)^*(j^1V \\lrcorner\\, \\Theta_{\\mathcal{L}})}_{=\\Theta_U(\\phi)\\cdot V}.\n\\end{align*}\nThen, $0 = d^2S[\\phi]\\cdot (V,W) = d\\alpha_U(\\phi)\\cdot (V,W) + d\\Theta_U(\\phi)\\cdot (V,W)$. The term $d\\alpha_U(\\phi)\\cdot (V,W)$ vanishes from the first variation property, so the multisymplectic form formula can be expressed\n$$ d\\Theta_U(\\phi)\\cdot (V,W) = 0, $$\nwhich is equivalent to equation (\\ref{Multisymplectic Form Formula}).\n\nIn our construction, the main impediment for a discrete analogue of the multisymplectic form formula is that a solution of the discrete equation (\\ref{DEL 1b}) does not in general satisfy an Euler--Lagrange equation locally (i.e., for arbitrary $U$) but rather integrated over a regular region $U$. Additionally, there is an additional contribution from the boundary components of the variation in the elements neighboring the boundary $T \\in \\mathcal{T}[\\partial U]$. It is in this restricted setting that we have a discrete multisymplectic form formula.\n\n\\begin{theorem}{\\textbf{(Discrete Multisymplectic Form Formula)}}\\label{Discrete Multisymplectic Form Theorem}\nLet $U$ be a regular region and let $\\phi_h$ be a solution of the local DEL (\\ref{local DEL}) and $V, W \\in \\mathfrak{X}(\\Lambda^k_h)$ be first variations for $\\phi_h$ (i.e., their flow on $\\phi_h$ still satisfies the DEL, but for arbitrary boundary variations), then\n\\begin{equation}\\label{Discrete Multisymplectic Form Formula 1}\nd\\Theta^h_U(\\phi_h)\\cdot (V,W) = 0.\n\\end{equation}\n\\begin{proof}\nDecompose the variation of the action into interior and boundary variations,\n$$ dS[\\phi_h]\\cdot V = \\underbrace{\\vphantom{\\sum_{T \\in \\mathcal{T}[\\partial U]}}-\\int_U (j^1\\phi_h)^*(j^1V_{in} \\, \\lrcorner\\, \\Omega_{\\mathcal{L}})}_{\\equiv \\alpha^h_U(\\phi_h)\\cdot V_{in}}\\ \\underbrace{- \\sum_{T \\in \\mathcal{T}[\\partial U]} \\int_T (j^1\\phi_h)^*(j^1V_{\\partial} \\, \\lrcorner\\, \\Omega_{\\mathcal{L}}) + \\int_{\\partial U}(j^1\\phi_h)^*(j^1V\\, \\lrcorner\\, \\Theta_{\\mathcal{L}})}_{=\\Theta^h_U(\\phi_h)\\cdot V}, $$\nso that $dS[\\phi_h]\\cdot V = \\alpha^h_U(\\phi_h)\\cdot V + \\Theta^h_U(\\phi_h)\\cdot V$ and hence,\n$$0 = d^2S[\\phi_h]\\cdot (V,W) = d\\alpha^h_U(\\phi_h)\\cdot (V,W) + d\\Theta^h_U(\\phi_h)\\cdot (V,W).$$\nDefine a discrete first variation as a vector field $B$ whose flow preserves the discrete Euler--Lagrange equation (\\ref{DEL 1a}), i.e., $d(\\delta S[\\phi_h]\\cdot a)\\cdot B = 0$ for any $a \\in \\mathring{\\Lambda}^k_h$; equivalently, this can be expressed as $d( \\alpha^h_U(\\phi_h)\\cdot A )\\cdot B$ for any $A \\in \\mathfrak{X}(\\mathring{\\Lambda}^k_h)$. Then, express\n$$ d\\alpha_U(\\phi_h)\\cdot (V,W) = d( \\alpha_U(\\phi_h)\\cdot V_{in})\\cdot W - d(\\alpha_U(\\phi_h)\\cdot W_{in})\\cdot V - \\alpha_U(\\phi_h)\\cdot [V_{in},W_{in}]. $$\nThe first two terms on the right hand side of the above equation vanish by the definition of first variation; furthermore, the third term vanishes by the fact that $V_{in},W_{in} \\in \\mathfrak{X}(\\mathring{\\Lambda}^k_h)$ implies $[V_{in},W_{in}] \\in \\mathfrak{X}(\\mathring{\\Lambda}^k_h)$ and the fact that $\\alpha_U(\\phi_h)$ contains $\\mathfrak{X}(\\mathring{\\Lambda}^k_h)$ in its kernel (by the discrete Euler--Lagrange equation). Hence, $d\\alpha^h_U(\\phi_h)\\cdot (V,W) = 0$. Thus, we have\n$$ d\\Theta^h_U(\\phi_h)\\cdot (V,W) = d^2S[\\phi_h]\\cdot (V,W) = 0. $$\n\\end{proof}\n\\end{theorem}\n\n\\begin{remark}\nAlthough we immediately see that the discrete multisymplectic form formula $d\\Theta^h_U(\\phi_h)\\cdot (V,W) = 0$ is in direct analogy with the continuum multisymplectic form formula $d\\Theta_U(\\phi)\\cdot (V,W) = 0$, if we write the discrete formula using the definition of the discrete Cartan form, we see that there is an additional contribution corresponding to the integration over elements $T \\in \\mathcal{T}[\\partial U]$. Although we will not write this out explicitly, we see that this additional contribution involves the linearization of the quantity $\\sum_{T \\in \\mathcal{T}[\\partial U]} \\int_T (j^1\\phi_h)^*(j^1V_{\\partial} \\, \\lrcorner\\, \\Omega_{\\mathcal{L}})$ by $W$ (and, vice versa, the linearization of $\\sum_{T \\in \\mathcal{T}[\\partial U]} \\int_T (j^1\\phi_h)^*(j^1W_{\\partial} \\, \\lrcorner\\, \\Omega_{\\mathcal{L}})$ by $V$). Since, $\\sum_{T \\in \\mathcal{T}[\\partial U]} \\int_T$ is $O(h)$ as discussed previously, we only need control of the residual associated to the linearized equations to formally show convergence of the discrete multisymplectic form formula to the continuum multisymplectic form formula.\n\nWe note that the aforementioned convergence is formal since it must also be combined appropriately with convergence of the discrete solution to a continuum weak solution using bounds on the projection. One possible method for combining these is the following observation. Since, by assumption, the projections are cochain projections, we have\n$$ d^2S_h[\\phi] \\cdot (V,W) = d^2(\\pi_h^*S)[\\phi]  \\cdot (V,W) = d^2S[\\pi_h\\phi] \\cdot (T\\pi_h \\cdot V, T\\pi_h \\cdot W). $$\nIn particular, for first variations $V,W \\in \\mathfrak{X}(Y)$ for the degenerate action, $T\\pi_h \\cdot V, T\\pi_h\\cdot W$ correspond to first variations of the discrete Euler--Lagrange equations, and the discrete multisymplectic form formula can be reinterpreted as the multisymplectic form formula for the degenerate action. Note also that for cochain projections, a simple calculation shows that $j^1( T\\pi_h \\cdot V) = T(\\pi_h^k \\times \\pi_h^{k+1}) \\cdot j^1V$, so that the terms in the integrand of the discrete multisymplectic form formula, (\\ref{Discrete Multisymplectic Form Formula 1}), are in the image of the (tangent) projections. This allows us to formulate the discrete multisymplectic formula in terms of the projection and its tangent lift, and hence more directly determine in what sense the discrete multisymplectic form formula converges as $h \\rightarrow 0$.\n\nOf course, without specifying a particular field theory and finite element spaces, we cannot proceed further to show convergence. We aim to investigate more rigorous convergence results for particular field theories in future work. See also the discussion below regarding convergence of the discrete Noether theorem to its continuum analogue. \n\n\\end{remark}\n\n\\begin{remark}\nAs noted before, the discrete Cartan form, in the case of nodal interpolating shape functions, gives precisely the discrete notion of Cartan form introduced in \\citet{MaPaSh1998}. In this case, our discrete multisymplectic form formula $d\\Theta^h_U(\\phi_h)\\cdot (V,W) = 0$ (for first variations $V,W$) gives precisely the discrete multisymplectic form formula derived in \\citet{MaPaSh1998}.\n\\end{remark}\n\n\n\n\n\\subsection{Discrete Noether's Theorem}\\label{Noether's Theorem Section}\nIn this section, we will discuss the covariant momentum map structure associated to the discrete Lagrangian structure and establish discrete analogues of Noether's theorem. \n\nLet $G$ be a Lie group with $\\mathfrak{g} := T_e G$ its Lie algebra. Suppose $G$ has a smooth group action on $Y$ via vertical bundle automorphisms; this induces a lifted action of $G$ on the associated jet bundle space. Associated to $\\eta \\in G$ are its actions $\\eta_Y, \\eta_{J^1Y}$ on the respective spaces (see \\citet{GoIsMaMo1998}). We say that a Lagrangian density $\\mathcal{L}$ is $G$-invariant if \n\\begin{equation}\\label{Equivariance}\n\\mathcal{L}(\\eta_{J^1Y} \\gamma) = \\mathcal{L}(\\gamma),\n\\end{equation}\nfor each $\\eta \\in G$, $\\gamma \\in J^1_xY$ (for every $x \\in X$). Infinitesimally, $\\delta \\mathcal{L}(\\gamma)\\cdot \\xi = 0$, for all $\\xi \\in \\mathfrak{g}$.\n\nGiven such a $G$-invariant Lagrangian, there exists a covariant momentum map  $J^\\mathcal{L}: J^1Y \\rightarrow \\mathfrak{g}^* \\otimes \\Lambda^{n}(J^1Y)$ given by\n$$ \\langle J^\\mathcal{L}, \\xi \\rangle = \\xi_{J^1Y} \\lrcorner\\, \\Theta_{\\mathcal{L}},$$\nwhere the duality pairing is between $\\mathfrak{g}$ and its dual, and for $\\xi \\in \\mathfrak{g}$, $\\xi_{J^1Y}$ is the associated infinitesimal generator. This covariant momentum map satisfies Noether's theorem, which implies a divergence form conservation law\n\\begin{equation}\\label{Noether's Theorem}\nd[(j^1\\phi)^*\\langle J^\\mathcal{L},\\xi\\rangle] = 0,\n\\end{equation}\nfor sections $\\phi$ of the configuration bundle satisfying the associated Euler--Lagrange equations. We can express equation (\\ref{Noether's Theorem}) in integral form as\n\\begin{equation}\\label{Integral Form of Noether's Theorem}\n\\Theta_U(\\phi)\\cdot \\xi_Y = \\int_{\\partial U}(j^1\\phi)^*( \\xi_{J^1Y} \\lrcorner\\, \\Theta_{\\mathcal{L}} ) = 0.\n\\end{equation}\n\n\\begin{remark}\nMore generally, one can look at symmetries of the action up to a boundary term. In this context, we say that a group $G$ is a symmetry for the theory if there exists $K: J^1(Y) \\rightarrow \\mathfrak{g}^* \\otimes \\Lambda^n J^1(Y)$ such that $ \\delta \\mathcal{L}(\\gamma)\\cdot \\xi = \\langle K,\\xi\\rangle$ for all $\\xi \\in \\mathfrak{g}$; i.e., the action transforms infinitesimally up to a total derivative. In this case, there exists an inhomogeneous momentum map $\\langle J^\\mathcal{L},\\xi\\rangle = \\xi_{J^1Y}\\lrcorner\\ \\Theta_{\\mathcal{L}} - \\langle K,\\xi\\rangle$. Analogous statements of the following discussion follow in this case.\n\\end{remark}\n\nIn the discrete setting, we would like to consider to what extent equations (\\ref{Noether's Theorem}) and (\\ref{Integral Form of Noether's Theorem}) hold. To find a discrete analogue, note that\n$$ d[(j^1\\phi)^*\\langle J^{\\mathcal{L}},\\xi\\rangle] = d[(j^1\\phi)^* (\\xi_{J^1Y} \\lrcorner\\, \\Theta_{\\mathcal{L}}) ] = \\Big(\\frac{\\delta L}{\\delta \\phi^A}(\\pounds_\\xi \\phi)^A + \\delta_\\xi L\\Big)(j^1\\phi) d^{n+1}x, $$\nwhere $\\pounds_\\xi\\phi = T\\phi \\circ \\xi_X - \\xi_Y \\circ \\phi \\in \\mathfrak{X}(Y)$ (note $\\xi_X = 0$ since we assume $G$ acts vertically, but we write the general definition since Proposition~\\ref{equivariant vertical projection} still holds in this setting), $\\delta L\/\\delta \\phi^A$ is the total variation of the Lagrangian with respect to $\\phi$, which vanishes when $\\phi$ is a solution of the Euler--Lagrange equation, and $\\delta_\\xi L$ is the variation induced by the infinitesimal action which vanishes when $\\mathcal{L}$ is section-equivariant. We see that the main obstacle is that for a solution $\\phi \\in \\Lambda^k_h$ of the discrete equations $(\\ref{DEL 1b})$, the right hand side does not vanish unless integrated over $U$ and unless the variation field $\\pounds_\\xi\\phi \\in \\Lambda^k_h$. Keeping this in mind, we state the following discrete analogue of Noether's theorem.\n\n\\begin{theorem}[Discrete Noether's Theorem]\\label{Discrete Noether Theorem}\nLet $U$ be a regular region. Let $\\mathcal{L}$ be $G$-invariant. Then, for a discrete solution $\\phi_h \\in \\Lambda^k_h$ of $(\\ref{DEL 1b})$ and $\\xi \\in \\mathfrak{g}$, \n\\begin{subequations}\n\\begin{align}\\label{Discrete Noether 1}\n\\int_U \\Big( EL(j^1\\phi_h&) \\wedge\\star (T\\pi_h \\cdot \\pounds_\\xi \\phi_h)(\\phi_h) + \\delta_\\xi L(j^1\\phi_h) \\Big) d^{n+1}x \\\\ \n&- \\sum_{T \\in \\mathcal{T}[\\partial U]} \\int_T EL(j^1\\phi_h) \\wedge\\star (T\\pi_h \\cdot \\pounds_\\xi \\phi_h)(\\phi_h)_{\\partial} = 0, \\nonumber\n\\end{align}\nwhere $EL := \\partial_2\\mathcal{L} + d^*\\partial_3\\mathcal{L}$ and $T\\pi_h$ is the lift of the projection. Furthermore, if the vertical component of $\\pounds_\\xi\\phi_h$ is in $\\Lambda^k_h$, then\n\\begin{equation}\\label{Discrete Noether 1 Cor}\n\\int_U d[(j^1\\phi_h)^*\\langle J^{\\mathcal{L}},\\xi \\rangle] - \\sum_{T \\in \\mathcal{T}[\\partial U]} \\int_T EL(j^1\\phi) \\wedge\\star (\\pounds_\\xi \\phi_h)(\\phi_h)_{\\partial} = 0.\n\\end{equation}\n\\end{subequations}\n\\begin{proof}\nThe first equation is simply re-expressing the DEL $dS[\\phi_h]\\cdot V_{in}=0$ (where $V = T\\pi_h\\cdot \\pounds_\\xi\\phi_h$) in terms of $V$ and $V_\\partial$, via $V_{in} = V - V_{\\partial}$. \n\nThe second equation $(\\ref{Discrete Noether 1 Cor})$ follows by the above discussion (expressing $d[(j^1\\phi)^*\\langle J^\\mathcal{L},\\xi \\rangle]$ in terms of the variations of $L$) and $(\\ref{Discrete Noether 1})$ since the projection leaves the subspace invariant. \n\\end{proof}\n\\end{theorem}\n\nOf course, the stronger statement (\\ref{Discrete Noether 1 Cor}) resembles the divergence form of Noether's theorem more closely than (\\ref{Discrete Noether 1}), which requires the vertical component of $\\pounds_\\xi\\phi_h$ to be in $\\Lambda^k_h$. The next proposition gives a sufficient condition for when this requirement holds.\n\n\\begin{prop}\\label{equivariant vertical projection}\nLet $\\pi_h$ be an equivariant map with respect to the action of $G$ on the configuration bundle. Then, for any section $\\phi$ of the configuration bundle and $\\xi \\in \\mathfrak{g}$,\n\\begin{equation}\n\\pounds_\\xi(\\pi_h\\phi) = T\\pi_h \\cdot \\pounds_\\xi \\phi.\n\\end{equation}\nIn particular, for $\\phi_h \\in \\Lambda^k_h$, $\\pounds_\\xi\\phi_h = T\\pi_h\\cdot \\pounds_\\xi\\phi_h$ and so in this case, (\\ref{Discrete Noether 1}) is equivalent to (\\ref{Discrete Noether 1 Cor}).\n\\begin{proof}\nFor the first statement, compute\n$$ T\\pi_h \\cdot \\pounds_\\xi \\phi = T\\pi_h \\cdot (T\\phi \\circ \\xi_X) - T\\pi_h\\cdot (\\xi_Y \\circ \\phi). $$\nNote $T\\pi_h \\cdot ( T\\phi \\circ \\xi_X)= T(\\pi_h\\phi) \\circ \\xi_X$. Furthermore, by the equivariance of $\\pi_h$, \n$$ T\\pi_h\\cdot (\\xi_Y \\circ \\phi) = \\frac{d}{dt}\\Big|_{t = 0} \\pi_h(e^{t\\xi}\\cdot\\phi) = \\frac{d}{dt}\\Big|_{t = 0}e^{t\\xi}\\cdot(\\pi_h\\phi) = \\xi_Y \\circ (\\pi_h\\phi).$$\nHence, $T\\pi_h\\cdot \\pounds_\\xi\\phi = T(\\pi_h\\phi)\\circ X - \\xi_Y\\circ \\pi_h\\phi = \\pounds_\\xi(\\pi_h\\phi)$. The statement for $\\phi_h \\in \\Lambda^k_h$ follows immediately since the projection $\\pi_h$ acts invariantly on $\\Lambda^k_h$.\n\\end{proof}\n\\end{prop}\n\nThe above proposition shows that the projection being $G$-equivariant fits naturally into the variational principle, since the variation induced by the Lie algebra action is associated to a discrete variation (in other words, since $g \\cdot (\\pi_h Y) = \\pi_h (g \\cdot Y)$, the group action restricts to the discrete field space). To see this naturality with the variational principle more explicitly, one can derive a discrete Noether theorem in the case of a $G$-equivariant projection as follows. $G$-invariance of the Lagrangian implies $G$-invariance of the action, $S_U[G \\cdot \\phi] = S_U[\\phi]$. Infinitesimally, $dS_U[\\phi]\\cdot \\xi_{Y} = 0$. Assuming $\\pi_h$ is $G$-equivariant, the flow of $\\xi_{Y}$ appropriately restricts to $\\Lambda^k_h$, so we can view $\\xi_{Y} \\in \\mathfrak{X}(\\Lambda^k_h)$. Hence, computing the variation (using equation (\\ref{Discrete Cartan Form 1})),\n\\begin{align*}\n0 &= dS_U[\\phi_h]\\cdot \\xi_Y = \\int_U (j^1\\phi_h)^*(j^1\\xi_Y \\lrcorner\\ \\Omega_{\\mathcal{L}}) + \\int_{\\partial U} (j^1\\phi_h)^* (j^1\\xi_Y \\lrcorner\\ \\Theta_{\\mathcal{L}} ) \\\\\n&= \\sum_{T \\in \\mathcal{T}[\\partial U]} \\int_T (j^1\\phi_h)^* ((j^1\\xi_Y)_{\\partial}\\ \\lrcorner\\ \\Omega_{\\mathcal{L}}) + \\int_{\\partial U} (j^1\\phi_h)^* (j^1\\xi_Y \\lrcorner\\ \\Theta_{\\mathcal{L}} )  = \\Theta^h_U (\\phi_h)\\cdot \\xi_{Y},\n\\end{align*}\nThus, in the case of a $G$-equivariant projection, the discrete Noether conservation law (\\ref{Discrete Noether 1 Cor}) can be expressed in terms of the discrete Cartan form as $\\Theta^h_U(\\phi_h)\\cdot \\xi_Y = 0$, in analogy with the continuum Noether conservation law (\\ref{Integral Form of Noether's Theorem}).\n\nThis form of the discrete Noether theorem reproduces the discrete Noether theorem of \\citet{MaPaSh1998}, in the case of nodal interpolating shape functions. Note, however, that \\citet{MaPaSh1998} make the assumption that the group $G$ acts on the nodal field values in such a way that the Lagrangian is $G$-invariant; from our perspective, this is just a particular case of $G$-equivariance of the projection arising from constructing the projection by using $G$-equivariant nodal interpolants. \n\n\\begin{remark}\\label{Weakening Group Equivariance}\nOne can weaken the assumption that the projection is equivariant. First, note that the projection does not need to be fully equivariant for Proposition \\ref{equivariant vertical projection} to hold. Rather, the projection only needs to be infinitesimally equivariant, $\\pi_h(e^{t\\xi} \\cdot \\phi) - e^{t\\xi}\\cdot \\pi_h\\phi = o(t)$ for all $\\xi \\in \\mathfrak{g}$. This weakened condition may be useful for constructing equivariant projections, since the equivariance only needs to hold to a sufficient order.\n\nFurthermore, the group equivariance $\\pi_h(g\\cdot \\phi) = g\\cdot \\pi_h\\phi$ can be weakened to $\\pi_h(g\\cdot \\phi) = \\psi_h(g)\\cdot \\pi_h\\phi$ where $\\psi_h: G \\rightarrow G$ is a differentiable group homomorphism. Since $\\psi_h$ is differentiable, it induces a Lie algebra homomorphism $\\tilde{\\psi}_h$, and the above discrete Noether theorem $\\Theta^h_U(j^1\\phi_h)\\cdot \\xi_{Y} = 0$ can be replaced by $\\Theta^h_U(j^1\\phi_h)\\cdot \\tilde{\\psi}_h(\\xi)_{Y} = 0$.  As with the other remark on weakening the notion of equivariance, this weakened notion also allows one to construct more general equivariant projections. \n\\end{remark}\n\nWe give two simple examples of group-equivariant cochain projections and subsequently remark on how one might construct more general group-equivariant cochain projections.\n\n\\begin{example}[Global Linear Group Action]\nFirst, note that although we took our field configuration bundle to be $\\Lambda^k(X)$, we could have more generally taken our fields to be vector-valued forms, corresponding to the bundle $\\Lambda^k(X) \\otimes V$ for some finite-dimensional vector space $V$. With a basis $\\{e_i\\}$ for $V$, the only modification to the discrete Euler--Lagrange (\\ref{DEL 1b}) equation is that there are $\\dim(V)$ equations corresponding to each component of the field $\\phi^i \\in \\Lambda^k(X)$ in the expansion $\\phi(x) = \\sum_i \\phi_i(x) \\otimes e_i$.\n\nSuppose a Lagrangian with such a configuration bundle is invariant under the global action by a group representation $D: G \\rightarrow GL(V)$. That is, $D$ acts on $\\phi \\in \\Lambda^k(X)\\otimes V$ as $1_{\\Lambda^k(X)} \\otimes D$:\n$$ D(g)\\phi(x) = \\sum_i\\phi_i(x) \\otimes ( D(g)e_i ), $$\nwhere $D(g)$ is independent of $x$.\n\nLet $\\pi^k_h: H\\Lambda^k \\rightarrow \\Lambda^k_h$ and $\\pi^{k+1}_h: H\\Lambda^{k+1} \\rightarrow \\Lambda^k_h$ be cochain projections, i.e., satisfying $\\pi^{k+1}_hd = d\\pi^k_h$. We can extend these to cochain projections on vector-valued forms by $\\tilde{\\pi}_h = \\pi_h \\otimes 1_V$. Furthermore, group-equivariance follows from linearity of the group action and the above definitions,\n\\begin{align*}\nD(g) \\tilde{\\pi}_h \\phi &= D(g) \\tilde{\\pi}_h \\left(\\sum_i \\phi_i \\otimes e_i\\right) = D(g) \\sum_i \\pi_h(\\phi_i)\\otimes e_i = \\sum_i \\pi_h(\\phi_i) \\otimes D(g)e_i \\\\\n&= \\tilde{\\pi}_h \\left( \\sum_i \\phi_i \\otimes D(g) e_i \\right) = \\tilde{\\pi}_h \\left( D(g) \\sum_i \\phi_i \\otimes e_i \\right) = \\tilde{\\pi}_h D(g)\\phi.\n\\end{align*}\nA simple example of such a theory is the Schr\\\"{o}dinger equation with $V = \\mathbb{C}$, $G = U(1)$, and the group representation given by the fundamental representation of $U(1)$ in $GL(\\mathbb{C})$. The corresponding conservation law is conservation of mass in the $L^2$ norm.\n\\end{example}\n\n\\begin{example}[Yang--Mills Theory]\nAs an example of a non-global (but still linear) group action, consider Yang--Mills theories with a structure group $G$. In this setting, the field $A \\in \\Lambda^1(X) \\otimes \\mathfrak{g}$; i.e., $A$ is valued in the Lie algebra $\\mathfrak{g}$ associated to $G$ (more precisely, the field is valued in the adjoint representation of the Lie algebra). This class of theories is invariant under the linear action of $\\Lambda^0(X) \\otimes \\mathfrak{g}$ (viewed as a group under addition) on $\\Lambda^1(X) \\otimes \\mathfrak{g}$ given by\n$$ D(\\alpha) A = A + d\\alpha, $$\nfor any $\\alpha \\in \\Lambda^0(X)\\otimes \\mathfrak{g}$. Unlike the previous example, this action is local in the sense that $D(\\alpha)$ depends on the position in spacetime. \n\nNow, suppose that we have cochain projections for the sequence $H\\Lambda^0 \\overset{d}{\\rightarrow} H\\Lambda^1 \\overset{d}{\\rightarrow} H\\Lambda^2$; that is, $\\pi^2_hd = d\\pi^1_h, \\pi^1_hd = d\\pi^0_h$. Extend these to projections $\\tilde{\\pi}_h$ on $H\\Lambda \\otimes \\mathfrak{g}$ as in the previous example. The relation $\\tilde{\\pi}^2_hd = d\\tilde{\\pi}^1_h$ is required for naturality of the variational structure. On the other hand, the relation $\\tilde{\\pi}^1_hd = d\\tilde{\\pi}^0_h$ gives (weakened) group equivariance, in the following sense:\n$$ \\tilde{\\pi}^1_h( D(\\alpha)A ) = \\tilde{\\pi}^1_h(A + d\\alpha) = \\tilde{\\pi}^1_hA + \\tilde{\\pi}^1_h d\\alpha = \\tilde{\\pi}^1_hA + d\\tilde{\\pi}^0_h \\alpha = D(\\tilde{\\pi}^0_h \\alpha) \\tilde{\\pi}^1_h A. $$\nThus, as discussed in Remark \\ref{Weakening Group Equivariance}, this is an example of weakened group equivariance where, rather than full group equivariance, there is an intertwining homomorphism $\\psi_h$ from the acting group to itself. In this case, the intertwining homomorphism is $\\psi_h = \\tilde{\\pi}^0_h$.\n\nIn the continuum Hilbert space setting, the associated conservation law is the weak Gauss' law, where Gauss' law holds tested against any element of the Hilbert space. In the discrete setting, the discrete Noether's theorem gives a discrete Gauss' law, where Gauss' law holds tested against any element of the finite-dimensional subspace.  \n\n\\end{example}\n\nThe previous two examples were simple in the sense that they had a linear or global group action. Although the second example was local, the acting group is contained in the Hilbert complex of forms and group-equivariance arose from having cochain projections. \n\nTo construct group-equivariant cochain projections for more general actions, one possible method would be to utilize group-equivariant interpolation \\citep{GaLe2018,Le2019} in constructing the projection. One method to construct cochain projections from interpolants is to place an intermediate sequence between the sequence of Hilbert spaces and the sequence of finite-dimensional subspaces,\n\\[\\begin{tikzcd}\n\t{H\\Lambda^k} && {H\\Lambda^{k+1}} \\\\\n\t{} \\\\\n\t{C^k} && {C^{k+1}} \\\\\n\t\\\\\n\t{\\Lambda^k_h} && {\\Lambda^{k+1}_h}\n\t\\arrow[\"{\\sigma^k}\"', from=1-1, to=3-1]\n\t\\arrow[\"d\", from=1-1, to=1-3]\n\t\\arrow[\"{\\sigma^{k+1}}\"', from=1-3, to=3-3]\n\t\\arrow[\"{D}\", from=3-1, to=3-3]\n\t\\arrow[\"{\\mathcal{I}^k}\"', from=3-1, to=5-1]\n\t\\arrow[\"d\", from=5-1, to=5-3]\n\t\\arrow[\"{\\mathcal{I}^{k+1}}\"', from=3-3, to=5-3],\n\\end{tikzcd}\\]\nwhere $\\{\\sigma^m\\}$ are the degrees of freedom mapping into the coefficient spaces $\\{C^m\\}$, $\\{\\mathcal{I}^m\\}$ are interpolants from the coefficient spaces into the finite-dimensional subspaces, $D$ realizes $d$ in the coefficient space, and the projections are defined by $\\pi_h = \\mathcal{I} \\circ \\sigma$. The degrees of freedom must be unisolvent when restricted to the image of the interpolants. Constructing cochain projections amounts to ensuring that the top diagram commutes. Then, fixing group-equivariant interpolants $\\mathcal{I}^k, \\mathcal{I}^{k+1}$, group-equivariant cochain projections could be achieved by choosing the degrees of freedom such that they are unisolvent for this choice of interpolants and ensuring that the top diagram commutes. We will pursue such a construction in future work.\n\n\n\nTo conclude this section, we expand on the discussion in Section \\ref{Variational Structure Section} regarding how the discrete Cartan form converges to the continuum Cartan form. Of course, without specifying a specific theory (i.e., Lagrangian), we cannot completely derive rigorous bounds, but we will sketch the process. Suppose we have a solution  $\\phi_h$ of the DEL converging to a solution $\\phi$ of the Euler--Lagrange equations. As before, denote $EL = \\partial_2\\mathcal{L}+d^*\\partial_3\\mathcal{L}$. Then, the terms involving integration over $T \\in \\mathcal{T}[\\partial U]$ can be bounded as\n\\begin{align*}\n\\sum_{T \\in \\mathcal{T}[\\partial U]} & \\Big|\\int_T (\\partial_2\\mathcal{L}(j^1\\phi_h) + d^*\\partial_3\\mathcal{L}(j^1\\phi_h))\\wedge\\star V(\\phi_h)_{\\partial}\\Big| \\leq \\sum_{T\\in\\mathcal{T}[\\partial U]} \\| EL(j^1\\phi_h) \\|_{L^2(T)} \\|V(\\phi_h)_{\\partial}\\|_{L^2(T)} \\\\\n&=\\sum_{T\\in\\mathcal{T}[\\partial U]} \\| EL(j^1\\phi_h) - EL(j^1\\phi) \\|_{L^2(T)} \\|V(\\phi_h)_{\\partial}\\|_{L^2(T)} \\\\\n& \\leq \\underbrace{\\sum_{T\\in\\mathcal{T}[\\partial U]}}_{\\lesssim h^{-n}} \\underbrace{\\vphantom{\\sum_{T\\in\\mathcal{T}[\\partial U]}} \\| EL(j^1\\phi_h) - EL(j^1\\phi) \\|_{L^2(T)} }_{\\lesssim h^{n+1} \\| EL(j^1\\phi_h) - EL(j^1\\phi)\\|_{L^2(X)} } \\underbrace{\\vphantom{\\sum_{T\\in\\mathcal{T}[\\partial U]}}\\Big( \\|V(\\phi_h)_{\\partial} - V(\\phi)_{\\partial}\\|_{L^2(T)} + \\|V(\\phi)_{\\partial}\\|_{L^2(T)} \\Big)}_{\\lesssim \\|V(\\phi_h) - V(\\phi)\\|_{L^2(X)} + \\|V(\\phi)\\|_{L^2(X)} \\sim O(1) }, \\\\\n\\end{align*}\nso we see this contribution converges to zero at least linearly in $h$, assuming that the residuals $\\| EL(j^1\\phi_h) - EL(j^1\\phi) \\|_{L^2(X)}$ and $\\| V(\\phi_h) - V(\\phi)\\|_{L^2(X)}$ are at least $O(1)$, although it is often the case that the residuals will be $O(h^s)$ for some $s>0$, in which case we get $O(h^{1+s})$ convergence. In particular, given a specific theory and symmetry group, one can apply this to the discrete Noether's theorem with $V = \\xi_Y$ to show that the discrete conservation law converges to the continuum conservation law (involving only an integral over $\\partial U$).\n\n\\subsubsection{Bounds for Noether's Theorem}\nIn the previous part, we showed that solutions to the discrete Euler--Lagrange equations admit a discrete analogue of Noether's theorem, which holds over regular regions $U$. Of course, this discrete law cannot be localized smaller than a single element. So, in this part, we determine a bound on the conservation law, which will allow us to establish the sense in which $\\| d(j^1\\phi_h)^* \\langle J^{\\mathcal{L}},\\xi\\rangle \\| \\approx 0$ throughout the whole domain. We defer the proofs of the following statements to Appendix~\\ref{Appendix A}. \n\nSo far, we have only seen that the Noether current evaluated on a discrete solution satisfies a global divergence-like equality (which, as previously remarked, can be localized down to the size of a single element). However, there should be some sense in which \n$$\\|d(j^1\\phi)^*\\langle J^\\mathcal{L},\\xi\\rangle \\|\\approx 0.$$\nTo arrive at this conclusion, we determine a bound on the divergence conservation law of the Noether current of the action $S$ for a discrete solution. To answer this, we could directly measure $d[(j^1\\phi_h)^*\\langle J^\\mathcal{L},\\xi\\rangle]$ in $L^2\\Lambda^{n+1}(X)$ (where $\\phi_h$ satisfies the discrete Euler--Lagrange equations), but due to the derivative, this requires assuming $\\phi^\\xi,(\\phi_h)^\\xi \\in H\\Lambda^k$, and $(j^1\\phi)^*\\partial_3\\mathcal{L},(j^1\\phi_h)^*\\partial_3\\mathcal{L} \\in H^*\\Lambda^{k+1}$ ($H^*$ denotes square integrable with square integrable coderivative). Instead, by duality, we could weakly measure the conservation law by its action on forms with square integrable coderivatives via the following operator norm, which only requires the aforementioned terms to be square integrable. Let $\\omega \\in H\\Lambda^n$; we can view $d\\omega \\in L^2\\Lambda^{n+1}$ as a linear functional on $H^*\\Lambda^{n+1}$ via the pairing \n$$ (d\\omega, \\alpha)_{L^2\\Lambda^{n+1}(X)} = (\\omega, d^*\\alpha)_{L^2\\Lambda^{n}(X)} + \\int_{\\partial X}\\omega \\wedge \\star \\alpha , $$\nfor $\\alpha \\in H^*\\Lambda^{n+1}X$. Then, we define the following operator norms\n$$  \\|d\\omega\\|^* = \\sup_{\\alpha \\in H^*\\Lambda^{n+1}\\setminus\\{0\\}} \\frac{|(d\\omega,\\alpha)_{L^2\\Lambda^{n+1}}|}{\\|\\alpha\\|_{H^*\\Lambda^{n+1}}},$$\n$$  \\|d\\omega\\|^*_0 = \\sup_{\\alpha \\in H_0^*\\Lambda^{n+1}\\setminus\\{0\\}} \\frac{|(d\\omega,\\alpha)_{L^2\\Lambda^{n+1}}|}{\\|\\alpha\\|_{H^*\\Lambda^{n+1}}},$$\nwhere the supremum in the second norm is restricted to forms which vanish on the boundary. These gives the following set of bounds, where we view the solution $\\phi$ of the full Euler--Lagrange equation as a constant and the arbitrary field $\\psi$ as varying. For simplicity, we denote $J(\\psi,\\xi) := (j^1\\psi)^*\\langle J^\\mathcal{L},\\xi\\rangle$ for a section $\\psi$ of the configuration bundle.\n\n\\begin{prop}\\label{Bound for Noether Current}\nLet $J$ be the covariant momentum map associated to the symmetry group $G$ for the Lagrangian structure defined by $\\mathcal{L}$, and suppose $J(\\psi,\\xi) \\in H\\Lambda^n$ for arbitrary $\\psi$,$\\xi$. Let $\\phi$ be a solution to the Euler--Lagrange equation for $\\mathcal{L}$ and for each $h > 0$, let $\\phi_h$ be a solution of the discrete Euler--Lagrange equation such that $\\phi_h \\rightarrow \\phi$ in $H\\Lambda^k$. Then, for each $h > 0, \\xi \\in \\mathfrak{g}$, \n\\begin{subequations}\n\\begin{align}\\label{bound equation 1}\n\\|J(\\phi_h,\\xi) &- J(\\phi,\\xi)\\|_{L^2\\Lambda^n} + \\|dJ(\\phi_h,\\xi)\\|^*_0 \\\\ \n&\\lesssim\\ \\|\\xi_X\\|_{L^2\\mathfrak{X}}\\|\\mathcal{L}(j^1\\phi_h) - \\mathcal{L}(j^1\\phi)\\|_{L^2\\Lambda^{n+1}} \\nonumber\n\\\\ & \\qquad + \\|(\\phi_h)^\\xi - \\phi^\\xi\\|_{L^2\\Lambda^k} + \\|\\partial_3\\mathcal{L}(j^1\\phi_h) - \\partial_3\\mathcal{L}(j^1\\phi)\\|_{L^2\\Lambda^{k+1}} \\nonumber\n\\end{align}\nand\n\\begin{equation}\\label{bound equation 2}\n \\lim_{h\\rightarrow 0} \\Big(\\|J(\\phi_h,\\xi) - J(\\phi,\\xi)\\|_{L^2\\Lambda^n} + \\|dJ(\\phi_h,\\xi)\\|^*_0\\Big) = 0.\n\\end{equation}\n\\end{subequations}\n\\end{prop}\n\n\\begin{remark}\nIn the above proposition, we assumed explicitly that $\\phi_h$ converged to $\\phi$, which allows more generally for the Lagrangian to be degenerate (e.g. due to gauge freedom). .\n\\end{remark}\n\nThe above bound uses the $\\|\\cdot\\|^*_0$ norm for estimating $dJ(\\phi_h,\\xi)$ and hence does not give information about the size of $J(\\phi_h,\\xi)$ at the boundary (an approximate solution should have $J(\\phi_h,\\xi) \\approx J(\\phi,\\xi)$ along $\\partial X$ and hence the total flux of $J(\\phi_h,\\xi)$ through the boundary is approximately zero). In the subsequent bound, we show that the norm of the boundary terms can be bounded with norm of fractionally many derivatives of $J(\\phi_h,\\xi) - J(\\phi,\\xi)$ on the interior.\n\n\\begin{prop}\\label{Bound for Noether Current 2}\nAssume as in Proposition \\ref{Bound for Noether Current}. Additionally assume $X$ is a Lipschitz domain and let $\\delta \\in (0,1\/2]$. Then,\n\\begin{equation}\\label{bound equation 3} \n\\|\\text{Tr}J(\\phi_h,\\xi) - \\text{Tr}J(\\phi,\\xi)\\|_{H^\\delta\\Lambda^n(\\partial X)} + \\|dJ(\\phi_h,\\xi) \\|^* \\lesssim \\|J(\\phi_h,\\xi) - J(\\phi,\\xi)\\|_{H^{\\frac{1}{2}+\\delta}\\Lambda^{n}(X)}\n\\end{equation}\n\\end{prop}\n\n\n\\subsection{A Discrete Variational Complex}\\label{Variational Complex Section}\nThe variational bicomplex is a double complex on the spaces of differential forms over the jet bundle of a configuration bundle used to study the variational structures of Lagrangian field theories defined on this bundle (see, for example, \\citet{An1992}). The differential forms arising in Lagrangian field theory, such as the Lagrangian density, the Cartan form, and the multisymplectic form, can be interpreted as elements of this variational bicomplex. The cochain maps in this double complex are the horizontal and vertical exterior derivatives on the jet bundle, which give a geometric interpretation to the variations encountered in Lagrangian field theories. The variational bicomplex has also been extended to problems with symmetry in \\citet{KoOl2003}, and to the discrete setting for difference equations corresponding to discretizing Lagrangian field theories on a lattice in \\citet{HyMa2004}.\n\nIn this section, we interpret and summarize the results from the previous sections in terms of a discrete variational complex which arises naturally in our discrete construction and, in a sense, resembles the vertical direction of the variational bicomplex. \n\nIn our previous discussion, we saw a complex which arises from the space of discrete forms,\n$$ \\Lambda^0_h \\overset{d}{\\longrightarrow} \\Lambda^1_h \\overset{d}{\\longrightarrow} \\cdots \\overset{d}{\\longrightarrow} \\Lambda^{n}_h \\overset{d}{\\longrightarrow} \\Lambda^{n+1}_h, $$\nwhich forms a complex due to the cochain projection property. Now, consider instead the following ``vertical\" complex; consider the spaces of smooth forms on $\\Lambda^k_h$, which we denote $\\Omega (\\Lambda^k_h)$, with the ``vertical\" exterior derivative $d_v: \\Omega^m(\\Lambda^k_h) \\rightarrow \\Omega^{m+1}(\\Lambda^k_h)$ being the usual exterior derivative over the base manifold $\\Lambda^k_h$ (which is a vector space). This gives a discrete variational complex:\n\\begin{figure}[h]\\label{Discrete Variational Complex}\n\\[\\begin{tikzcd}\n\t{\\Omega^{\\dim(\\Lambda^k_h)}(\\Lambda^k_h)} \\\\\n\t\\vdots \\\\\n\t{\\Omega^1(\\Lambda^k_h)} \\\\\n\t{\\Omega^0(\\Lambda^k_h)\\ .}\n\t\\arrow[\"{d_v}\", from=4-1, to=3-1]\n\t\\arrow[\"{d_v}\", from=3-1, to=2-1]\n\t\\arrow[\"{d_v}\", from=2-1, to=1-1]\n\\end{tikzcd}\\]\n\\end{figure}\n\nNote that in the previous sections, we used $d$ to denote both the exterior derivative corresponding to the de Rham complex and the vertical exterior derivative (e.g., the multisymplectic form formula $d\\Theta^h_U(V,W) = 0$ is more precisely $d_v\\Theta^h_U(V,W) = 0$), where it was understood which was meant by the spaces where the relevant quantities were defined; however, we will distinguish the two in this section to be more precise. We call the above a vertical complex for two reasons: first, the vertical exterior derivative corresponds to differentiation with respect to the fiber values (as we will see below); furthermore, it resembles the vertical direction of the variational bicomplex. However, in our construction, there is no horizontal direction, since in the discrete setting, we are considering transgressed forms (forms integrated over a region, e.g., in the definition of the action, discrete Cartan form, and discrete multisymplectic form), so the horizontal direction collapses.  \n\nExamples of forms in the discrete variational complex include the action (restricted to $\\Lambda^k_h$) $S \\in \\Omega^0(\\Lambda^k_h)$, the discrete Cartan form $\\Theta^h \\in \\Omega^1(\\Lambda^k_h)$, and the discrete multisymplectic form $d_v\\Theta^h \\in \\Omega^2(\\Lambda^k_h)$. Let $\\{v_i\\}$ be a basis for $\\Lambda^k_h$; we then coordinatize the vector space $\\Lambda^k_h$ by the components of the expansion of any $\\phi = \\sum_i \\phi^i v_i \\in \\Lambda^k_h$, which we denote as a vector $(\\phi^i) = (\\phi^0,\\dots,\\phi^{\\dim(\\Lambda^k_h)}) \\in \\Lambda^k_h$. For example, the vertical exterior derivative of the action is\n$$ d_v S[\\phi] = \\sum_j \\frac{\\partial S[(\\phi^i)]}{\\partial \\phi^j} d_v\\phi^j. $$\nThe naturality of the variational principle and the interpretation of the weak Euler--Lagrange equations as a Galerkin variational integrator, discussed in Section \\ref{Variational Structure Section}, relate the vertical exterior derivative of $S$ to the variation of the degenerate action $S_h$. Now, let $\\Pi_i$ be the projection onto the $i^{th}$ coordinate $\\phi^i$ and let $\\mathcal{I}[\\partial U]$ denote the set of indices $i$ such that $v_i$ has nonvanishing trace on $\\partial U$. Then, for $v = (v^i) \\in \\Lambda^k_h$, we have\n\\begin{align*}\nv_{\\partial} &= \\sum_{i \\in \\mathcal{I}[\\partial U]}\\Pi_i (v), \\\\\nv_{in} &= v - v_{\\partial} = \\sum_{i \\not\\in \\mathcal{I}[\\partial U]} \\Pi_i (v).\n\\end{align*}\nRecall that we can view vector fields $V \\in \\mathfrak{X}(\\Lambda^k_h)$ as maps $V: \\Lambda^k_h \\rightarrow \\Lambda^k_h$, and we extend this to the vector fields $V_{\\partial}(\\phi) \\equiv (V(\\phi))_{\\partial}$ and $V_{in}(\\phi) \\equiv (V(\\phi))_{in}$. In particular, the discrete Cartan form in this notation is\n$$ \\Theta^h(\\phi)\\cdot V = d_vS[\\phi]\\cdot V_{\\partial}. $$\nThe variation of the action can then be expressed as\n$$ d_vS[\\phi]\\cdot V = \\text{EL}(\\phi)\\cdot V + \\Theta^h(\\phi)\\cdot V, $$\nwhere the Euler--Lagrange one-form is defined by $\\text{EL}(\\phi)\\cdot V = d_v S[\\phi]\\cdot V_{in}$. More explicitly, these can be expressed as\n\\begin{align*}\n\\Theta^h(\\phi) &=  \\sum_{j \\in \\mathcal{I}[\\partial U]} \\frac{\\partial S[(\\phi^i)]}{\\partial \\phi^j} d_v\\phi^j, \\\\\n\\text{EL}(\\phi) &=  \\sum_{j \\not\\in \\mathcal{I}[\\partial U]} \\frac{\\partial S[(\\phi^i)]}{\\partial \\phi^j} d_v\\phi^j.\n\\end{align*}\nIn particular, the discrete Euler--Lagrange equations are given by null Euler--Lagrange condition, $\\text{EL}(\\phi) = 0$ (that is, $\\text{EL}(\\phi)\\cdot V = 0$ for all $V$). Assuming a solution $\\phi$ of the null Euler--Lagrange condition, we immediately see that\n$$ d_vS[\\phi]\\cdot V = \\Theta^h(\\phi)\\cdot V, $$\nand in particular, for a symmetry of the action $d_vS[\\phi]\\cdot \\tilde{\\xi} = 0$, we have the discrete Noether's theorem $\\Theta^h(\\phi)\\cdot \\tilde{\\xi} = 0$. By taking the second exterior derivative of the action, we have\n$$ 0 = d_v^2 S[\\phi] = d_v \\text{EL}(\\phi) + d_v \\Theta^h(\\phi). $$\nThe space of first variations (at $\\phi$) is precisely the kernel of the quadratic form $d_v \\text{EL}(\\phi)$, so this gives the discrete multisymplectic form formula $d_v \\Theta^h(\\phi)(\\cdot,\\cdot) = 0$ when evaluated on first variations. Thus, the results of the previous sections can be concisely summarized in terms of the structure given by the discrete variational complex. \n\nFurthermore, this framework also encompasses the discrete variational principle with quadrature, as discussed in Remark \\ref{Remark on Quadrature}. Namely, from the discrete viewpoint, a discrete action is an element of $\\Omega^0(\\Lambda^k_h)$ and in particular, the discrete action with quadrature $\\mathbb{S}$ from (\\ref{Discrete Action with Quadrature}) is an element of $\\Omega^0(\\Lambda^k_h)$. Then, the variation of $\\mathbb{S}$ can be decomposed into interior and boundary one-forms as before,\n\\begin{align*}\nd_v \\mathbb{S}[(\\phi)] &= \\mathbb{EL}(\\phi) + \\Uptheta^h(\\phi), \\\\\n\\Uptheta^h(\\phi) &=  \\sum_{j \\in \\mathcal{I}[\\partial U]} \\frac{\\partial \\mathbb{S}[(\\phi^i)]}{\\partial \\phi^j} d_v\\phi^j, \\\\\n\\mathbb{EL}(\\phi) &=  \\sum_{j \\not\\in \\mathcal{I}[\\partial U]} \\frac{\\partial \\mathbb{S}[(\\phi^i)]}{\\partial \\phi^j} d_v\\phi^j.\n\\end{align*}\nThe discrete Euler--Lagrange equations with quadrature are given by the null Euler--Lagrange condition $\\mathbb{EL}(\\phi) = 0$, and subsequently, the discrete Noether's theorem and discrete multisymplectic form formula (in the case of quadrature) then follow analogously to before (where symmetries are with respect to $\\mathbb{S}$ and the space of first variations at $\\phi$ is the kernel of the quadratic form $d_v\\mathbb{EL}(\\phi)$). \n\n\n\\section{Canonical Semi-discretization of Lagrangian Field Theories}\\label{Semi-Discretization Section}\nTurning now to the canonical formalism of field theories, we assume that our $(n+1)$-dimensional spacetime $X$ is globally hyperbolic; i.e., $X$ contains a smooth Cauchy hypersurface $\\Sigma$ such that every infinite causal curve intersects $\\Sigma$ exactly once. It was shown in \\citet{BeSa2003} that a globally hyperbolic spacetime is diffeomorphic to the product, $X \\cong \\mathbb{R} \\times \\Sigma$. Identifying $X$ with the product, we have a slicing of the spacetime. Taking an interval $I \\subseteq \\mathbb{R}$, we have the spacelike embeddings\n$$i_t: \\Sigma \\rightarrow X$$\nfor each $t \\in I$, such that the images $\\{ \\Sigma_t := i_t(\\Sigma) \\}_{t\\in I}$ form a foliation of $X$. \n\nWe will assume our Lagrangian depends on time-dependent fields as $\\mathcal{L}(x^\\mu,\\varphi,\\dot{\\varphi},d\\varphi)$, where the field $\\varphi(t) \\in H\\Lambda^k(\\Sigma_t)$ (denoted as $\\varphi$ as opposed to the full field $\\phi$) and the exterior derivative acts on $\\Lambda^k(\\Sigma_t)$ for each $t$. \n\nWe will discuss how a semi-discretization of the variational principle gives rise to finite-dimensional Lagrangian and Hamiltonian dynamical systems (see, for example, \\citet{AbMa1978}) and subsequently discuss how the energy-momentum map structure of a canonical field theory (see \\citet{GoIsMaMo2004}) is affected by semi-discretization. \n\n\n\\subsection{Semi-discrete Euler--Lagrange Equations}\\label{Semi-Discrete EL Section}\n\nIn this section, we formally derive the semi-discrete Euler--Lagrange equations. Given our $\\Lambda^{n+1}(X)$-valued Lagrangian density, we can produce an instantaneous density by contracting with the generator of the slicing (the vector field whose flow advances time) and pulling back by the inclusion of $\\Sigma_t$ into $X$, which gives a $\\Lambda^n(\\Sigma_t)$-valued density, which we will still call $\\mathcal{L}$. Alternatively, in coordinates where the density is $L\\ dt\\wedge V(t)$ and $V(t)$ restricts to a volume form on $\\Sigma_t$, then $\\mathcal{L} = i_t^* L V(t)$.  The action in the canonical framework is given by \n\\begin{equation}\\label{Canonical Action}\nS[\\varphi] = \\int_I dt \\int_{\\Sigma_t} \\mathcal{L}(x^\\mu,\\varphi,\\dot{\\varphi},d\\varphi), \n\\end{equation}\nwhere $(x^\\mu) = (t,x^1,\\dots,x^n) = (t,x)$, and $x = (x^i)$ denotes spatial coordinates.\n\nTo derive a semi-discrete formulation of the Euler--Lagrange equations, instead of looking at arbitrary variations of the form $v(t,x)$, we instead consider variations of the form $u(t)v(x)$ where $v \\in H\\Lambda^k(\\Sigma)$ and $u \\in C_0^2(I,\\mathbb{R})$. The basic idea of the semi-discrete formulation is to allow $u$ to be arbitrary but restrict $v$ to a finite-dimensional subspace $\\Lambda^k_h$. As in the covariant case, in order to compute the variations formally without going through the Hamilton--Pontryagin principle, we will assume that the projections are cochain projections (with respect to the spatial exterior derivative $d$ on $\\Sigma$).\n\\begin{assumption}\nThe projections $\\pi^m_h: H\\Lambda^m(\\Sigma) \\rightarrow \\Lambda^m_h(\\Sigma)$ are cochain projections; i.e., that $\\pi^{k+1}_hd = d\\pi^k_h$, with respect to $d: \\Lambda^m(\\Sigma) \\rightarrow \\Lambda^{m+1}(\\Sigma)$.\n\\end{assumption}\n\n\\begin{remark}Note that we assume a finite element discretization $\\Lambda^k_h$ of the fields on the reference space $H\\Lambda^k(\\Sigma)$ (with associated projection $\\pi_h$). There are two ways to view the variations with respect to our slicing $\\{\\Sigma_t\\}$. In one way, the field variation on the reference space $v \\in \\Lambda^k_h \\subset H\\Lambda^k(\\Sigma)$ is pulled back to a field variation on a time slice $(i_t^{-1})^*v \\in H\\Lambda^k(\\Sigma_t)$, where we restrict the embedding to its image $i_t: \\Sigma \\rightarrow \\Sigma_t$. Alternatively, we can pull back forms on $\\Sigma_t$ to forms on $\\Sigma$ via $i^*$ (e.g. the Lagrangian density and its derivatives) and perform any relevant integration over the reference space $\\Sigma$. We will utilize the latter since in computation we prefer to work on one reference space. For simplicity, we will not explicitly write the pullbacks $i_t^*$ but rather implicitly incorporate it into the spacetime dependence of the Lagrangian.\n\\end{remark}\n\n\\begin{theorem}\\label{Semi-Discrete Euler--Lagrange Theorem}\nThe (semi-discrete) Euler--Lagrange equations corresponding to the variational principle $\\delta S[\\varphi]\\cdot (uv) = 0$ for all $v \\in \\Lambda^k_h$ and $u \\in C_0^2(I,\\mathbb{R})$ are given by\n\\begin{equation}\\label{Semi-Discrete EL}\n\\frac{d}{dt}(\\partial_3\\mathcal{L},v)_{L^2\\Lambda^k(\\Sigma)} - (\\partial_2\\mathcal{L},v)_{L^2\\Lambda^k(\\Sigma)} - (\\partial_4\\mathcal{L},dv)_{L^2\\Lambda^{k+1}(\\Sigma)} = 0, \\text{ for all } v \\in \\Lambda^k_h \\text{ and } t \\in I,\n\\end{equation}\nwhere $\\mathcal{L}$ is evaluated at $(x^\\mu,\\varphi,\\dot{\\varphi},d\\varphi)$. \n\\begin{proof}\nWith $\\mathcal{L}$ evaluated at $(x^\\mu,\\varphi,\\dot{\\varphi},d\\varphi)$ (and the integration pulled back to $\\Sigma$), compute\n\\begin{align*}\n0 &= \\delta S[\\varphi]\\cdot (uv) = \\frac{d}{d\\epsilon}\\Big|_{\\epsilon = 0} S[\\phi + \\epsilon u v] \n\\\\ &= \\int_Idt\\int_{\\Sigma} \\Big[\\partial_2\\mathcal{L}\\wedge\\star u(t) v + \\partial_3\\mathcal{L}\\wedge\\star \\dot{u}(t) v + \\partial_4\\mathcal{L}\\wedge\\star u(t) dv\\Big]\n\\\\ &= \\int_Idt \\Big[ \\int_{\\Sigma} \\Big(\\partial_3\\mathcal{L}\\wedge\\star v \\Big)\\dot{u}(t) + \\int_{\\Sigma} \\Big(\\partial_2\\mathcal{L}\\wedge\\star  v + \\partial_4\\mathcal{L}\\wedge\\star dv\\Big)u(t) \\Big]\n\\\\ &= \\int_Idt \\Big[(\\partial_3\\mathcal{L},v)_{L^2}\\dot{u}(t) + (\\partial_2\\mathcal{L},v)_{L^2}u(t) + (\\partial_4\\mathcal{L},dv)_{L^2}u(t) \\Big]\n\\\\ &= -\\int_Idt \\Big[\\frac{d}{dt}(\\partial_3\\mathcal{L},v)_{L^2} - (\\partial_2\\mathcal{L},v)_{L^2} - (\\partial_4\\mathcal{L},dv)_{L^2} \\Big] u(t).\n\\end{align*}\nSince $u \\in C_0^2(I,\\mathbb{R})$ is arbitrary, the terms in the brackets vanish, which gives (\\ref{Semi-Discrete EL}).\n\\end{proof}\n\\end{theorem}\n\n\\begin{remark}\nSimilar to our discussion of the covariant case, there is a naturality relation in the variational principle when using spatial cochain projections for the semi-discrete theory. In particular,\n$$ S[\\pi_h\\varphi] = \\int_Idt \\int_\\Sigma \\mathcal{L}(x^\\mu,\\pi_h\\varphi,\\pi_h \\dot{\\varphi},d\\pi_h\\varphi) = \\int_Idt \\int_\\Sigma \\mathcal{L}(x^\\mu,\\pi_h\\varphi,\\pi_h \\dot{\\varphi},\\pi_h d\\varphi) =: S_h[\\varphi], $$\nso that the restricted variational principle can be realized as a full variational principle on a degenerate action, $\\delta S[\\pi_h\\phi]\\cdot (u\\  \\pi_hv) = \\delta S_h[\\phi]\\cdot (uv)$. Analogous to the discussion in the covariant case, the cochain property additionally removes the ambiguity of how one should discretize the spatial derivative of the field (i.e., whether one should project before or after taking the spatial derivative).\n\\end{remark}\n\nWe now show that the semi-discrete Euler--Lagrange equation (\\ref{Semi-Discrete EL}) arises from an instantaneous Lagrangian. To do this, let $\\{v_i\\}$ be a basis for $\\Lambda^k_h$. We define the instantaneous (semi-discrete) Lagrangian to be\n\\begin{equation}\\label{InstantaneousLagrangian1}\nL_h(t,\\varphi^i,\\dot{\\varphi}^i) = \\int_\\Sigma \\mathcal{L}(x^\\mu, \\varphi^iv_i, \\dot{\\varphi}^iv_i, \\varphi^i dv_i), \n\\end{equation}\nwhere $\\varphi = \\varphi^i(t) v_i \\in C^2(I, \\Lambda^k_h)$ and the associated action $S_h[\\{\\varphi^i\\}] = \\int_I dt L_h(t,\\varphi^i,\\dot{\\varphi}^i)$. We enforce the variational principle over curves $u = u^i(t)v_i \\in C^2_0(I,\\Lambda^k_h)$. The variational principle yields\n\\begin{align*}\n0 &= dS_h[\\{\\varphi^i\\}]\\cdot \\{u^j\\} = \\frac{d}{d\\epsilon}\\Big|_0 S_h[\\{\\varphi^i + \\epsilon u^j\\}] = \\sum_j \\int_I dt \\Big( \\frac{\\partial L_h}{\\partial \\varphi^j} (t,\\varphi^i,\\dot{\\varphi}^i) u^j + \\frac{\\partial L_h}{\\partial \\dot{\\varphi}^j}(t,\\varphi^i,\\dot{\\varphi}^i) \\dot{u}^j \\Big) \\\\\n&= \\sum_j \\int_I dt \\Big[ \\frac{\\partial L_h}{\\partial \\varphi^j} (t,\\varphi^i,\\dot{\\varphi}^i)  - \\frac{d}{dt} \\frac{\\partial L_h}{\\partial \\dot{\\varphi}^j}(t,\\varphi^i,\\dot{\\varphi}^i) \\Big] u^j.\n\\end{align*}\nThis holds for arbitrary $u^j \\in C^2_0(I,\\mathbb{R})$, so the term in the brackets vanishes for each $j$,\n\\begin{equation}\\label{Euler--Lagrange Equation}\n\\frac{\\partial L_h}{\\partial \\varphi^j} (t,\\varphi^i,\\dot{\\varphi}^i)  - \\frac{d}{dt} \\frac{\\partial L_h}{\\partial \\dot{\\varphi}^j}(t,\\varphi^i,\\dot{\\varphi}^i) = 0,\n\\end{equation}\nby the fundamental lemma of the calculus of variations. Expressing the derivatives of $L_h$ in terms of $\\mathcal{L}$,\n\\begin{subequations}\n\\begin{align}\n\\frac{\\partial L_h}{\\partial \\varphi^j} &= (\\partial_2\\mathcal{L},v_j)_{L^2\\Lambda^k(\\Sigma)} + (\\partial_2\\mathcal{L},dv_j)_{L^2\\Lambda^{k+1}(\\Sigma)}, \\label{Semi-Discrete Lagrangian Derivative Equation 1} \\\\\n\\frac{\\partial L_h}{\\partial \\dot{\\varphi}^j} &= (\\partial_3\\mathcal{L}, v_j)_{L^2\\Lambda^k(\\Sigma)}. \\label{Semi-Discrete Lagrangian Derivative Equation 2}\n\\end{align}\n\\end{subequations}\nSubstituting these expressions into equation (\\ref{Euler--Lagrange Equation}), we see that this is equation (\\ref{Semi-Discrete EL}) with the choice $v=v_j$. This holds for each basis form $v_j$ and hence for arbitrary $v \\in \\Lambda^k_h$. \n\nWe will now introduce a Hamiltonian structure associated with the semi-discretization and show that, in the hyperregular case, this instantaneous Lagrangian system is equivalent to an instantaneous Hamiltonian system.\n\n\\subsection{Symplectic Structure of Semi-discrete Dynamics and Hamiltonian Formulation}\\label{Semi-discrete Symplectic Structure Section}\n\nHaving derived the semi-discrete Euler--Lagrange equation (\\ref{Semi-Discrete EL}), we now relate the symplectic structure on the cotangent space of the full field space $T^*H\\Lambda^k(\\Sigma)$ to a symplectic structure on the discretized space $T^*\\Lambda^k_h$, and show that the semi-discrete Euler--Lagrange equations are equivalent to a Hamiltonian flow on $T^* \\Lambda^k_h$ if the Lagrangian is hyperregular. \n\nWe work with the reference space $\\Sigma$, since via the diffeomorphism $i_t: \\Sigma \\rightarrow \\Sigma_t$, we can pullback forms on $\\Sigma$ to $\\Sigma_t$ or vice versa (or forms on iterated exterior bundles, such as the symplectic form which is an element of $\\Lambda^2(T^*H\\Lambda^k(\\Sigma))$). On the full phase space $T^*H\\Lambda^k(\\Sigma)$, the canonical one-form $\\theta \\in \\Lambda^1(T^*H\\Lambda^k(\\Sigma))$ is given in coordinates by \n\\begin{equation}\\label{canonical form in canonical formalism}\n\\theta\\big|_{(\\varphi,\\pi)} = \\int_{\\Sigma}\\pi_A d\\varphi^A \\otimes d^nx_0\n\\end{equation}\nand the corresponding symplectic form $\\omega = -d\\theta$ is given by \n$$ \\omega\\big|_{(\\varphi,\\pi)} = \\int_{\\Sigma} (d\\varphi^A \\wedge d\\pi_A) \\otimes d^nx_0. $$\nUsing the projection map $\\pi_h: H\\Lambda^k(\\Sigma) \\rightarrow \\Lambda^k_h$, we have the pullback $\\pi_h^*: T^*\\Lambda^k_h \\rightarrow T^*H\\Lambda^k(\\Sigma)$ and the twice iterated pullback $\\pi_h^{**}: \\Lambda^p(T^*H\\Lambda^k(\\Sigma)) \\rightarrow \\Lambda^p(T^*\\Lambda^k_h)$ for any $p$. We define $\\theta_h \\equiv \\pi_h^{**}\\theta$ and $\\omega_h \\equiv \\pi_h^{**}\\omega = -d\\theta_h \\in \\Lambda^2(T^*\\Lambda^k_h)$. To find an expression for $\\theta_h$ and $\\omega_h$, we will introduce global coordinates on $T^*\\Lambda^k_h$. Let $\\{v_i\\}$ be a finite element basis for $\\Lambda^k_h$; we will use the components $\\varphi^i$ of the basis expansion $\\varphi = \\varphi^i v_i$ as the coordinates on $\\Lambda^k_h$; similarly, if we identify $T\\Lambda^k_h \\cong \\Lambda^k_h \\times \\Lambda^k_h$, then we have a basis for $T^*_\\varphi\\Lambda^k_h$ being $v^i := (\\cdot,v_i)_{L^2}$. This gives the trivialization $T^*\\Lambda^k_h \\cong \\Lambda^k_h \\times (\\Lambda^k_h)^*$ with global coordinates $(\\varphi,\\pi) \\sim (\\varphi^i,\\pi_i)$ where $\\varphi = \\varphi^i v_i$ and $\\pi = \\pi_i v^i$. We will denote these coordinates using vector notation $\\vec{\\varphi} = (\\varphi^i)$, $\\vec{\\pi} = (\\pi_i)$. \n\n\\begin{prop}\nThe $1$-form $\\theta_h$ is given in the above coordinates by\n\\begin{equation}\\label{Semi-Discrete One Form}\n\\theta_h = v^j(v_i) \\pi_j d\\varphi^i = d\\vec{\\varphi}^TM\\vec{\\pi},\n\\end{equation}\nwhere the mass matrix $M$ has components $M_i^{\\ j} := v^j(v_i) = \\int_\\Sigma v_i v_j d^nx_0.$\nFurthermore, the $2$-form $\\omega_h = -d\\theta_h$ is a symplectic form on $T^*\\Lambda^k_h$ with coordinate expression\n\\begin{equation} \\label{Semi-Discrete Symplectic Form}\n\\omega_h = d\\varphi^i \\wedge v^j(v_i) d\\pi_j = d\\vec{\\varphi}^T\\wedge M d\\vec{\\pi}.\n\\end{equation}\n\\begin{proof}\nLet $(\\varphi,\\pi) \\in T^*\\Lambda^k_h$ and $U \\in T_{(\\varphi,\\pi)}(T^*\\Lambda^k_h)$, with coordinate expression\n$$ U(\\varphi,\\pi) = \\Phi^i \\frac{\\partial}{\\partial\\varphi^i} + \\Pi_i\\frac{\\partial}{\\partial \\pi_i}. $$\nNote that $\\theta|_{(\\varphi',\\pi')}(V)$ gives the canonical pairing between the $\\partial\/\\partial \\varphi'$ component of $V$ and $\\pi'$ by equation (\\ref{canonical form in canonical formalism}). Then, since $\\pi_h^*$ is an inclusion $T^*\\Lambda^k_h \\hookrightarrow T^*H\\Lambda^k(\\Sigma)$ and hence $T\\pi_h^*$ is an inclusion on the corresponding tangent space, this gives\n\\begin{align*}\n\\theta_h|_{(\\varphi,\\pi)}(U) = \\theta|_{\\pi_h^*(\\varphi,\\pi)} (T\\pi_h^*U) = \\langle \\Phi,\\pi\\rangle = \\Phi^i \\pi_j \\int_\\Sigma v_iv_j d^nx_0 = v^j(v_i)\\pi_j \\Phi^i = v^j(v_i)\\pi_j d\\varphi^i(U).\n\\end{align*}\nEquation (\\ref{Semi-Discrete Symplectic Form}) then follows from taking (minus) the exterior derivative of equation (\\ref{Semi-Discrete One Form}).\n\nThe nondegeneracy and closedness of $\\omega_h$ clearly follow from the (global) coordinate expression (\\ref{Semi-Discrete Symplectic Form}) above. In particular, since the mass matrix $M$ is invertible (hence nondegenerate), $\\omega_h$ is nondegenerate. Closedness follows from \n$$d\\omega_h = d^2\\vec{\\varphi}^T \\wedge M d\\vec{\\pi} - d\\vec{\\varphi}^T \\wedge dM \\wedge d\\vec{\\pi} - d\\vec{\\varphi}^T\\wedge M d^2\\vec{\\pi} = 0.$$ Alternatively, $\\omega_h$ is closed as the pullback of a closed form $\\omega$. \n\\end{proof}\n\\end{prop}\n\n\\begin{remark}\nThe above defines a standard symplectic form on $T^*\\Lambda^k_h$ corresponding to the polarization $T^*\\Lambda^k_h = \\Lambda^k_h \\times (\\Lambda^k_h)^*$. To see $\\omega_h$ in standard form, we can change basis. Let $Q$ be an orthogonal matrix which diagonalizes $M$ (recall that the mass matrix is symmetric), i.e., $QMQ^T = D$. Define coordinates $\\vec{q} = Q\\vec{\\varphi}$ and $\\vec{p} = DQ\\vec{\\pi}$; then\n$$ \\omega_h = d\\vec{\\varphi}^T \\wedge M d\\vec{\\pi} = d\\vec{\\varphi}^T \\wedge Q^T DQ d\\vec{\\pi} = d(Q\\vec{\\varphi})^T \\wedge d(DQ\\vec{\\pi}) = d\\vec{q}^T \\wedge d\\vec{p}. $$\nHowever, we will work with the form of $\\omega_h$ corresponding to the finite element basis (\\ref{Semi-Discrete Symplectic Form}) since it is more directly applicable to our discretization. Also, if we chose the dual basis $l^j$ to be different from the basis $v^j = (\\cdot,v_j)$, $M$ would not necessarily be symmetric but would still define a symplectic form. This is because for our finite element method to be consistent, we require that the matrix with components $l^j(v_i)$ is invertible. Hence, it is more natural to work with the coordinates $(\\vec{\\varphi},\\vec{\\pi})$.\n\\end{remark}\n\nLet $H: T^*\\Lambda^k_h \\rightarrow \\mathbb{R}$ be the Hamiltonian of our theory, expressed in our global coordinates as $H = H(\\vec{\\varphi},\\vec{\\pi})$. The dynamics of the Hamiltonian system $(\\omega_h,H)$ is the flow generated by the Hamiltonian vector field $X_H$ satisfying $X_H \\lrcorner\\ \\omega_h = dH$, or with vector field components $X_H = (\\dot{\\varphi}^i,\\dot{\\pi}_i)$, \n\\begin{equation}\\label{Hamiltonian Dynamics for Semi-Discrete Symplectic Form}\n\\begin{cases} \\ M_{\\ k}^j\\dot{\\varphi}^k = \\frac{\\partial H}{\\partial \\pi_j}, \\\\\n\\ M_j^{\\ k}\\dot{\\pi}_k = - \\frac{\\partial H}{\\partial \\varphi^j}. \\end{cases}\n\\end{equation}\n\\begin{remark}\nIn the above, we denote row $j$ and column $k$ of $M$ as $M_j^{\\ k}$ and for $M^T$ as $M^j_{\\ k}$, which allows more generally for $M$ to not be symmetric as discussed previously. If we define $\\vec{z}$ as the concatenation of $\\vec{\\varphi}$ and $\\vec{\\pi}$, the equations (\\ref{Hamiltonian Dynamics for Semi-Discrete Symplectic Form}) can be written in skew-symmetric form\n$$ \\frac{d}{dt}\\vec{z} = J_M \\nabla_{\\vec{z}} H, $$\nwhere $J_M = \\begin{pmatrix} 0 & (M^{-1})^T \\\\ -M^{-1} & 0 \\end{pmatrix} $.\n\\end{remark}\n\n\\begin{remark}\nIn our discussion of the covariant discretization of Lagrangian field theories, we saw that the variation of the discretized action on the discrete space can be naturally related to the variation of a degenerate action on the full space.  In the semi-discrete setting, an analogous statement can be made in terms of the semi-discrete symplectic structure and a presymplectic structure on the full space. Namely, we have the symplectic form $\\omega_h \\in \\Lambda^2(T^*\\Lambda^k_h)$. Now, consider the presymplectic form $\\tilde{\\omega}_h \\in \\Lambda^2(T^*H\\Lambda^k)$ defined by $\\tilde{\\omega}_h = i_h^{**}\\omega_h$ where $i_h = (\\pi_h)^{\\dagger}: \\Lambda^k_h \\hookrightarrow H\\Lambda^k$ is the inclusion.  Clearly, $\\tilde{\\omega}_h$ is closed as the pullback of a closed form. To see that it is degenerate, for any $V,W \\in \\mathfrak{X}(T^*H\\Lambda^k)$, \n$$ \\tilde{\\omega}_h(V,W) = (i_h^{**} \\pi_h^{**}\\omega)(V,W) = \\omega(T(\\pi_h^* i_h^*)V, T(\\pi_h^* i_h^*)W). $$\nSince $i_h\\pi_h$ has a nontrivial kernel, so does $T(\\pi_h^*i_h^*) = T(i_h\\pi_h)^*$ and hence $\\tilde{\\omega}_h$ is degenerate. The flow of a vector field in the kernel of $\\tilde{\\omega}_h$, projected back to the semi-discrete space, corresponds to equivalent states in the semi-discrete setting. Quotienting the presymplectic manifold $(T^*H\\Lambda^k, \\tilde{\\omega}_h)$ by the orbits of the flow of vector fields in the kernel of $\\tilde{\\omega}_h$ gives the symplectic manifold $(T^*\\Lambda^k_h,\\omega_h)$. This relates a symplectic flow on $(T^*\\Lambda^k_h,\\omega_h)$ to an equivalence class of presymplectic flows on $(T^*H\\Lambda^k,\\tilde{\\omega}_h)$, where the equivalence class is formed by orbits of the flow of vector fields in the kernel of $\\tilde{\\omega}_h$.\n\\end{remark}\n\nWe also allow our Hamiltonian to explicitly depend on time, $H: I \\times T^*\\Lambda^k_h \\rightarrow \\mathbb{R}$, i.e., the domain of $H$ is the extended phase space $I \\times T^*\\Lambda^k_h$. The dynamics are now given by any vector field $X_H$ on the extended phase space such that $X_H\\lrcorner\\ (\\omega_h + dH\\wedge dt) = 0$ (here, $\\omega_h$ is extended to the full phase space by acting trivially in the temporal direction). If we consider the component $X^V_H$ of $X_H$ over the field space $T^*\\Lambda^k_h$, then the above is equivalent to $X^V_H \\lrcorner\\ \\omega_h = d_vH$ holding for all times; this is given again by equation (\\ref{Hamiltonian Dynamics for Semi-Discrete Symplectic Form}) but with explicit time dependence in $H$ (note that $d_vH$ is the vertical exterior derivative; in coordinates, $d_vH(t,\\varphi,\\pi) = \\frac{\\partial H}{\\partial \\varphi^i}d\\varphi^i + \\frac{\\partial H}{\\partial \\pi_j} d\\pi_j$). We could also allow explicit time dependence in $M$, but since we pullback our integration to $\\Sigma$, we view $M$ as constant and absorb the time dependence into $H$. For our setup, explicit time dependence is generally necessary since our foliation is not necessarily trivial and the Hamiltonian may be time-dependent. \n\nNow, we would like to relate the semi-discrete Euler--Lagrange equations (\\ref{Semi-Discrete EL}) to the Hamiltonian dynamics of $\\omega_h$. The first step is to produce a Hamiltonian associated to the instantaneous Lagrangian \n$$ L(t,\\varphi,\\dot{\\varphi})= \\int_{\\Sigma}\\mathcal{L}(x^\\mu,\\varphi,\\dot{\\varphi},d\\varphi). $$\nTo do this, we use the Legendre transform, which takes the form $\\pi = \\partial L\/\\partial \\dot{\\varphi}$. The pairing of $\\pi$ with a tangent vector field with components $(\\varphi,v)$ is given by computing the variation \n$$ \\langle \\pi, v \\rangle = \\left\\langle \\frac{\\partial L}{\\partial \\dot{\\varphi}}, v \\right\\rangle = \\frac{d}{d\\epsilon}\\Big|_{\\epsilon = 0}L(t,\\varphi,\\dot{\\varphi}+\\epsilon v) = (\\partial_3\\mathcal{L},v)_{L^2\\Lambda^k}. $$\nThe instantaneous Hamiltonian is given by\n$$ H(t,\\varphi,\\pi) = \\langle \\pi,\\dot{\\varphi} \\rangle - L(t,\\varphi,\\dot{\\varphi}) $$ \n(where the $\\dot{\\varphi}$ dependence is removed either by extremizing over $\\dot{\\varphi}$ or, assuming $L$ is hyperregular, by inverting the Legendre transform to get $\\dot{\\varphi}$ as a function of $(\\varphi,\\pi)$). Restricting to our finite element space $T^*\\Lambda^k_h$ gives our discrete Hamiltonian $H_h$ defined by\n$$ H_h(t,\\varphi^i,\\pi_i) = H(t,\\varphi^iv_i,\\pi_iv^j) = \\langle \\pi_jv^j,\\dot{\\varphi}^iv_i\\rangle - L(t,\\varphi^iv_i,\\dot{\\varphi}^iv_i) = M_i^{\\ j}\\pi_j\\dot{\\varphi}^i - L(t,\\varphi^iv_i,\\dot{\\varphi}^iv_i). $$\nNote that $H_h$ corresponds to the Legendre transform of the semi-discrete Lagrangian (\\ref{InstantaneousLagrangian1}), where we recall the duality pairing between $(\\varphi^j,\\pi_j)\\in T^*\\Lambda^k_h$ and $(\\varphi^i,\\dot{\\varphi}^i)\\in T\\Lambda^k_h$ is given by $M_i^{\\ j}\\pi_j\\dot{\\varphi}^i$.\n\\begin{prop}\\label{Equivalence of Semi-Discrete Formulations Prop}\nAssume that $L_h$ is hyperregular, then the dynamics associated with the Hamiltonian system $(\\omega_h,H_h)$ is equivalent to the semi-discrete Euler--Lagrange equations (\\ref{Semi-Discrete EL}). \n\\begin{proof}\nSince we assumed that $L_h$ is hyperregular, i.e., that the associated Legendre transform is a diffeomorphism $T\\Lambda^k_h \\rightarrow T^*\\Lambda^k_h$, we have $\\dot{\\varphi}^i$ as a function of $(\\varphi^j,\\pi_j)$. To verify the equivalence, we compute the equations (\\ref{Hamiltonian Dynamics for Semi-Discrete Symplectic Form}) for our given system. Compute for $L$ evaluated at $(t,\\varphi^iv_i,\\dot{\\varphi}^iv_i)$, \n\\begin{align*}\nM_j^{\\ k}\\dot{\\pi}_k &= -\\frac{\\partial H_h}{\\partial \\varphi^j} = -\\frac{\\partial}{\\partial\\varphi^j}\\Big(M_i^{\\ k}\\pi_k\\dot{\\varphi}^i - L \\Big)\n\\\\ &= -M_i^{\\ k}\\pi_k\\frac{\\partial\\dot{\\varphi}^i}{\\partial\\varphi^j} +  \\frac{\\partial}{\\partial\\varphi^j}\\int_{\\Sigma_t}\\mathcal{L}(x^\\mu,\\varphi^iv_i,\\dot{\\varphi}^iv_i,\\varphi^idv_i) \n\\\\ &= -M_i^{\\ k}\\pi_k\\frac{\\partial\\dot{\\varphi}^i}{\\partial\\varphi^j} + \\int_{\\Sigma_t}\\Big[\\partial_2\\mathcal{L}\\wedge\\star\\frac{\\partial(\\varphi^iv_i)}{\\partial\\varphi^j} + \\partial_3\\mathcal{L}\\wedge\\star\\frac{\\partial(\\dot{\\varphi}^iv_i)}{\\partial\\varphi^j} + \\partial_4\\mathcal{L}\\wedge\\star\\frac{\\partial(\\varphi^idv_i)}{\\partial\\varphi^j}\\Big]\n\\\\ &= -M_i^{\\ k}\\pi_k\\frac{\\partial\\dot{\\varphi}^i}{\\partial\\varphi^j} + \\int_{\\Sigma_t}\\Big[\\partial_2\\mathcal{L}\\wedge\\star v_j + \\partial_3\\mathcal{L}\\wedge\\star v_i \\frac{\\partial\\dot{\\varphi}^i}{\\partial\\varphi^j} + \\partial_4\\mathcal{L}\\wedge\\star dv_j\\Big]\n\\\\ &= -M_i^{\\ k}\\pi_k\\frac{\\partial\\dot{\\varphi}^i}{\\partial\\varphi^j} + (\\partial_3\\mathcal{L},v_i)\\frac{\\partial\\dot{\\varphi}^i}{\\partial\\varphi^j} + (\\partial_2\\mathcal{L},v_j)_{L^2} + (\\partial_4\\mathcal{L},dv_j)_{L^2}\n\\\\ &= (\\partial_2\\mathcal{L},v_j)_{L^2} + (\\partial_4\\mathcal{L},dv_j)_{L^2},\n\\end{align*}\nwhere in the second to last line, the first two terms cancel since $(\\partial_3\\mathcal{L},v_i) = \\langle\\pi,v_i\\rangle = \\langle\\pi_kv^k,v_i\\rangle = M_i^{\\ k}\\pi_k$. Then, note the left hand side is equivalently given by\n$$ M_j^{\\ k}\\dot{\\pi}_k = M_j^{\\ k}\\frac{d}{dt}\\pi_k = \\frac{d}{dt}\\big(M_j^{\\ k}\\pi_k\\big) = \\frac{d}{dt}\\big( \\langle v^k,v_j\\rangle\\pi_k \\big) =  \\frac{d}{dt} \\langle \\pi,v_j\\rangle = \\frac{d}{dt}(\\partial_3\\mathcal{L},v_j)_{L^2}. $$\nThus, \n$$ \\frac{d}{dt}(\\partial_3\\mathcal{L},v_j)_{L^2} = (\\partial_2\\mathcal{L},v_j)_{L^2} + (\\partial_4\\mathcal{L},dv_j)_{L^2} $$\nwhich holds for each $j$ and hence is equivalent to (\\ref{Semi-Discrete EL}). \n\\end{proof}\n\\end{prop}\n\\begin{remark}\nIn the above proposition, we assumed that $L_h$ was hyperregular for the equivalence. If $L_h$ is not hyperregular (e.g. corresponding to a degenerate field theory), the dynamics associated to $H_h$ evolve over a primary constraint surface. In this case, the dynamics of $H_h$ on the constraint surface corresponds to a (not necessarily unique) solution of the semi-discrete Euler--Lagrange equation. In this setting, the dynamics are associated to the modified Hamiltonian $\\bar{H}(\\vec{\\varphi},\\vec{\\pi},\\lambda) = H(\\vec{\\varphi},\\vec{\\pi})+ \\lambda^A\\Phi_A(\\vec{\\varphi},\\vec{\\pi})$. \n\nThe above also shows that, in the hyperregular case, the semi-discrete Euler--Lagrange equations correspond to a symplectic flow. The associated symplectic form is the pullback of $\\omega_h$ by the Legendre transform $\\mathbb{F}L_h: T\\Lambda^k_h \\rightarrow T^*\\Lambda^k_h$. In the non-regular case, the semi-discrete Euler--Lagrange equations correspond to a presymplectic flow.\n\\end{remark}\n\nTo summarize, in this section, we have pulled back the symplectic structure on $T^*H\\Lambda^k$ to $T^*\\Lambda^k_h$ and showed that the dynamics of the Hamiltonian system $(\\omega_h,H_h)$ is equivalent (in the hyperregular case) to the semi-discrete Euler--Lagrange equations of the corresponding Lagrangian system. By applying a numerical integrator for the finite-dimensional Hamiltonian system associated to $H_h$, we obtain a full discretization of the evolution problem of a field theory. \n\n\\subsection{Energy-Momentum Map}\\label{Energy-Momentum Section}\nIn this section, we examine how symmetries in the canonical formulation are affected by the semi-discretization of the field theory. In the canonical setting, the manifestation of the covariant momentum map is the energy-momentum map. If a vector in the Lie algebra of the symmetry group gives rise to an infinitesimal generator on $X$ which is transverse to the foliation, its pairing with the energy-momentum map equals the instantaneous Hamiltonian defined by that generator (the ``energy'' component). On the other hand, if the corresponding generator is tangent to the foliation, the pairing is given by the usual momentum map of the instantaneous Hamiltonian theory, corresponding to the canonical form (\\ref{canonical form in canonical formalism}) (the ``momentum'' component). We will see that, in the case of an equivariant discretization, the iterated pullback of the energy-momentum map provides the natural energy-momentum structure of the semi-discrete theory.\n\nWe start by investigating the momentum map structure of the semi-discrete theory. Let $K$ be a Lie group (with $\\mathfrak{k} := T_e K$) acting on $H\\Lambda^k$; for $\\eta \\in K$, we denote the group action $\\overline{\\eta}\\varphi := \\eta\\cdot\\varphi$ and the associated cotangent action is given by $\\widetilde{\\eta}:=(\\overline{\\eta^{-1}})^*$ (we use the same notation for these actions restricted to $\\Lambda^k_h$ and $T^*\\Lambda^k_h$, where the restriction is well-defined if the projection is group-equivariant).\n\n\\begin{prop}\\label{Momentum Map Prop}\nAssume that $K$ acts by symplectomorphisms on $(T^*H\\Lambda^k,\\omega)$; since $K$ acts by cotangent lifts on $T^*H\\Lambda^k$, it admits a canonical momentum map $J: T^*H\\Lambda^k \\rightarrow \\mathfrak{k}^*$. Furthermore, assume that the projection map $\\pi_h$ is equivariant with respect to the $K$-action on $H\\Lambda^k$ and $\\Lambda^k_h$; i.e., $\\pi_h\\bar{\\eta} \\varphi = \\bar{\\eta}\\pi_h\\varphi$. Then, $K$ acts by cotangent-lifted symplectomorphisms on $(T^*\\Lambda^k_h,\\omega_h)$ and the canonical momentum map for this action $J_h$ is given by $J_h = \\pi_h^{**}J = J \\circ \\pi_h^*$.\n\\begin{proof}\nTo see that $K$ preserves $\\omega_h$, for any $\\eta \\in K$, by equivariance, we have that\n$$ \\widetilde\\eta^*\\omega_h = (\\overline{\\eta^{-1}})^{**}\\pi_h^{**}\\omega = (\\overline{\\eta^{-1}}\\pi_h)^{**}\\omega = (\\pi_h\\overline{\\eta^{-1}})^{**}\\omega = \\pi_h^{**}(\\overline{\\eta^{-1}})^{**}\\omega = \\pi_h^{**}\\omega = \\omega_h.$$\nA similar result holds for $\\theta_h$, since $K$ preserves $\\theta$ by virtue of the fact that it acts by cotangent lifted actions.\n\nThe canonical momentum map $J$ is given by $\\langle J(\\varphi,\\pi),\\xi\\rangle = \\xi_{T^*H\\Lambda^k}(\\varphi,\\pi)\\lrcorner\\, \\theta|_{(\\varphi,\\pi)}$ for $(\\varphi,\\pi) \\in T^*H\\Lambda^k$ whereas $\\langle J_h(\\varphi,\\pi),\\xi \\rangle = \\xi_{T^*\\Lambda^k_h}(\\varphi,\\pi)\\lrcorner\\, \\theta_h|_{(\\varphi,\\pi)}$ for $(\\varphi,\\pi) \\in T^*\\Lambda^k_h$. These are both momentum maps for their respective actions since $K$ acts by cotangent lifts. Then,\n\\begin{align*}\n\\langle J_h(\\varphi,\\pi),\\xi \\rangle &= \\xi_{T^*\\Lambda^k_h}(\\varphi,\\pi)\\lrcorner\\, \\theta_h = \\xi_{T^*\\Lambda^k_h}(\\varphi,\\pi)\\lrcorner\\, \\pi_h^{**}\\theta \n\\\\ &= [T\\pi_h^*\\xi_{T^*\\Lambda^k_h}(\\varphi,\\pi)]\\lrcorner\\, \\theta\n= \\Big[T\\pi_h^*\\frac{d}{dt}\\Big|_{t = 0}\\widetilde{e^{t\\xi}}(\\varphi,\\pi)\\Big] \\lrcorner\\, \\theta  \n\\\\ &= \\Big[\\frac{d}{dt}\\Big|_{t = 0}\\pi_h^*(\\overline{e^{-t\\xi}})^*(\\varphi,\\pi)\\Big] \\lrcorner\\, \\theta  \n\\\\ &= \\Big[\\frac{d}{dt}\\Big|_{t = 0}(\\overline{e^{-t\\xi}}\\pi_h)^*(\\varphi,\\pi)\\Big] \\lrcorner\\, \\theta  \n\\\\ &= \\Big[\\frac{d}{dt}\\Big|_{t = 0}(\\pi_h\\overline{e^{-t\\xi}})^*(\\varphi,\\pi)\\Big] \\lrcorner\\, \\theta  \n\\\\ &= \\Big[\\frac{d}{dt}\\Big|_{t = 0}(\\overline{e^{-t\\xi}})^*\\pi_h^*(\\varphi,\\pi)\\Big] \\lrcorner\\, \\theta  \n\\\\ &= \\xi_{T^*H\\Lambda^k}(\\pi_h^*(\\varphi,\\pi))\\lrcorner\\, \\theta = \\langle (J\\circ\\pi_h^*)(\\varphi,\\pi),\\xi \\rangle.\n\\end{align*}\nwhere we have implicitly evaluated $\\theta_h$ at $(\\varphi,\\pi)$ and $\\theta$ at $\\pi_h^*(\\varphi,\\pi)$. Hence, $J_h = J \\circ \\pi_h^*$ or, equivalently, $J_h = \\pi_h^{**}J$.\n\\end{proof}\n\\end{prop}\n\n\\begin{remark}\nAs can be seen in the proof, one does not need full $K$-equivariance of the projection, but only infinitesimal equivariance, i.e., $\\pi_h(e^{t\\xi} \\varphi) - e^{t\\xi}\\pi_h\\varphi = o(t)$. \n\nFurthermore, one can weaken the notion of equivariance to $\\pi_h \\bar{\\eta} = \\overline{ \\psi_h(\\eta) } \\pi_h$, where $\\psi_h: K \\rightarrow K$ is a differentiable group homomorphism. In this case, if $\\tilde{\\psi}_h$ denotes the induced Lie algebra homomorphism, we can see from the above proof that the semi-discrete momentum map is related to the original momentum map via $\\langle J_h ,\\xi\\rangle = \\langle J \\circ \\pi_h^*, \\tilde{\\psi}_h(\\xi)\\rangle.$\n\nAs discussed in the covariant case, the weakening of this condition can allow us to construct more general projections.\n\\end{remark}\n\n\\begin{corollary}\nAssuming as in the proposition, if $J$ is $\\text{Ad}^*$-equivariant, then so is $J_h$.\n\\begin{proof}\nThis follows immediately from $J_h = J \\circ \\pi_h^*$, $K$-equivariance of $\\pi_h$, and the $\\text{Ad}^*$-equivariance $J \\circ \\widetilde{\\eta} = \\text{Ad}^*_\\eta J$ (where $\\text{Ad}^*_\\eta := (\\text{Ad}(\\eta^{-1}))^*)$:\n\\begin{align*} \nJ_h \\circ \\widetilde{\\eta} &= J \\circ \\pi_h^* \\circ (\\overline{\\eta^{-1}})^* =  J \\circ (\\overline{\\eta^{-1}})^* \\circ \\pi_h^*\n\\\\ &= J \\circ \\widetilde{\\eta} \\circ \\pi_h^* = (\\text{Ad}^*_\\eta J) \\circ \\pi_h^* = \\text{Ad}^*_\\eta (J \\circ \\pi_h^*) = \\text{Ad}^*_\\eta J_h,\n\\end{align*}\nwhere the equality $(\\text{Ad}^*_\\eta J) \\circ \\pi_h^* = \\text{Ad}^*_\\eta (J \\circ \\pi_h^*)$ holds since the coadjoint action acts on $J$ after it is evaluated on its input (since then it is an element of $\\mathfrak{k}^*$). In particular, $(\\text{Ad}_\\eta^*J)(\\varphi,\\pi) := \\text{Ad}_\\eta^* ( J(\\varphi,\\pi) )$, so that \n$$ ((\\text{Ad}^*_\\eta J) \\circ \\pi_h^* )(\\varphi,\\pi) = (\\text{Ad}^*_\\eta J) (\\pi_h^*(\\varphi,\\pi)) = \\text{Ad}^*_\\eta (J (\\pi_h^*(\\varphi,\\pi))) =   \\text{Ad}^*_\\eta ((J \\circ \\pi_h^*)(\\varphi,\\pi))). $$\nStated another way, this follows from associativity of the composition of functions, viewing $\\text{Ad}_\\eta^*$ as a function $\\mathfrak{k}^* \\rightarrow \\mathfrak{k}^*$.\n\\end{proof}\n\\end{corollary}\n\n\\begin{remark}\nOf course, since $K$ acts by cotangent lifts and hence by canonical symplectomorphisms, $J$ is an $\\text{Ad}^*$-equivariant momentum map, and the corollary tells us that $J_h$ is as well. However, as we remark below, one may consider more general actions which admit momentum maps, and it is not necessarily the case that those momentum maps are $Ad^*$-equivariant. The result of the previous corollary still holds in this more general setting.\n\\end{remark}\n\nThe naturality of the momentum map structures from the previous proposition and corollary can be summarized via the following commuting diagram; for any $\\eta \\in K$,\n\n\\centerline{\\xymatrix{\nT^*H\\Lambda^k \\ar[dr]^{\\widetilde{\\eta}} \\ar[dddrr]_{J} & & & & \\ar[llll]^(0.483){\\pi_h^*} T^*\\Lambda^k_h \\ar[ld]_{\\widetilde{\\eta}} \\ar[llddd]^{J_h}\n\\\\& T^*H\\Lambda^k \\ar[dr]_{J} & & \\ar[ll]^{\\pi_h^*} T^*\\Lambda^k_h \\ar[dl]^{J_h} &\n\\\\ &  & \\mathfrak{k}^* & &\n\\\\ &  & \\mathfrak{k}^* \\ar[u]^(0.6){\\text{Ad}^*_\\eta} & &.\n}}\n\n\\begin{remark} In the above proposition, we only assumed that $\\pi_h$ was equivariant with respect to the $K$-action on the configuration space, and it follows that $\\pi_h^*$ is equivariant with respect to the lifted action on the cotangent space. However, for more general actions on the cotangent space (not arising from a cotangent lift), one must instead assume $\\pi_h^*$ is equivariant with respect to this action. In this case, if the $K$-action on $T^*H\\Lambda^k$ admits a momentum map $J$, then $J_h = \\pi_h^{**}J$ is a momentum map for the action on $T^*\\Lambda^k_h$. To verify this, let $(\\varphi,\\pi) \\in T^*\\Lambda^k_h$. We know that $d\\langle J,\\xi\\rangle = i_{\\xi_{T^*H\\Lambda^k}}\\omega$. Thus, \n$$d\\langle J_h,\\xi\\rangle = \\pi_h^{**}d\\langle J,\\xi\\rangle = \\pi_h^{**}(i_{\\xi_{T^*H\\Lambda^k}}\\omega). $$\nThen, observe that by equivariance, $\\xi_{T^*H\\Lambda^k}(\\varphi,\\pi) = T\\pi^*_{h}\\xi_{T^*\\Lambda^k_h}(\\varphi,\\pi)$. Then, for any $X \\in TT^*_{(\\varphi,\\pi)}\\Lambda^k_h,$\n$$ d\\langle J_h,\\xi\\rangle (X) = (i_{\\xi_{T^*H\\Lambda^k}}\\omega)(T\\pi^*_{h}X) = \\omega(T\\pi^*_{h}\\xi_{T^*\\Lambda^k_h},T\\pi^*_{h}X) = (\\pi_h^{**}\\omega)(\\xi_{T^*\\Lambda^k_h},X) = (i_{\\xi_{T^*\\Lambda^k_h}}\\omega_h)(X), $$\nwhere the above is evaluated at $(\\varphi,\\pi)$, which verifies that $J_h$ is a momentum map. For the subsequent discussion, we will assume that $K$ acts by cotangent lifts.\n\\end{remark}\n\nWe now define the energy-momentum map and its semi-discrete counterpart. We consider vectors on $\\Sigma_t$ with both tangent components in $T\\Sigma_t$ and components transverse to the foliation, which in our adapted coordinates are in the span of $\\partial\/\\partial t$. We extend the canonical form $\\theta$ to act on vector fields on the extended phase space (in the same way we extended $\\omega_h$ in our previous discussion of time-dependence). Letting $\\tilde{\\mathcal{L}}$ denote the Lagrangian density on the full spacetime (related to the spatial density by $\\mathcal{L} = i_t^*\\partial_t\\lrcorner\\tilde{\\mathcal{L}}$), define the map $\\mathcal{J}$ from $I \\times T^*H\\Lambda^k$ to the dual of the space of vector fields on the extended phase space, via\n\\begin{equation}\\label{Energy-Momentum Map vector fields}\n\\langle \\mathfrak{J}(t,\\varphi,\\pi), V\\rangle = (V\\lrcorner\\, \\theta)(t,\\varphi,\\pi) - \\int_{\\Sigma}i_t^*V_t\\lrcorner\\, \\widetilde{\\mathcal{L}}(x^\\mu,\\varphi,\\dot{\\varphi},d\\varphi),\n\\end{equation}\nwhere we view $\\dot{\\varphi}$ as a function of $(\\varphi,\\pi)$, and where $V_t$ is the tangent-lift of the bundle projection $I \\times T^*H\\Lambda^k(\\Sigma_t) \\rightarrow I$ applied to $V$. \n\\begin{prop}\\label{Energy-Momentum Map Prop}\n$\\mathfrak{J}$ is the energy-momentum map, in the following sense:\n\\begin{enumerate}[label=(\\roman*)]\n\\item \\textbf{(Energy)} Let $\\Phi^H_t$ denote the Hamiltonian flow of $H$ and $X_H$ be the associated generator on the extended phase space; then,\n$$\\langle \\mathfrak{J}(t,\\varphi,\\pi), X_H\\rangle =  H(t,\\varphi,\\pi).$$ \n\\item \\textbf{(Momentum)} If $V$ is tangent to the foliation, then,\n$$\\langle \\mathfrak{J}(t,\\varphi,\\pi),V\\rangle = (V\\lrcorner\\, \\theta)(t,\\varphi,\\pi),$$\nand in particular, if there is a $K$-action as in Proposition (\\ref{Momentum Map Prop}) on the phase space over $\\Sigma_t$, its momentum map $J$ is given by\n$$ \\langle J(t,\\varphi,\\pi),\\xi\\rangle = \\langle \\mathfrak{J}(t,\\varphi,\\pi),\\xi_{T^*H\\Lambda^k} \\rangle,$$\nsuch that (for each fixed $t$) $d\\langle J(t,\\varphi,\\pi),\\xi\\rangle = \\xi_{T^*H\\Lambda^k}\\lrcorner\\, \\omega(t,\\varphi,\\pi)$. \n\\end{enumerate}\n\\begin{proof}\nFor the proof of (i), in local coordinates, we have \n$$ X_H = \\frac{d}{dt}\\Big|_{t=0} \\Phi_t^H(t',\\varphi,\\pi) = \\frac{\\partial}{\\partial t} + \\dot{\\varphi}^A \\frac{\\partial}{\\partial\\varphi^A} + \\dot{\\pi}_B\\frac{\\partial}{\\partial\\pi_B}, $$\nand $(X_H)_t = \\partial\/\\partial t$. Using expressions (\\ref{canonical form in canonical formalism}) and (\\ref{Energy-Momentum Map vector fields}) and the definition of the instantaneous Lagrangian density $\\mathcal{L} = i_t^*\\partial_t\\lrcorner\\, \\widetilde{\\mathcal{L}}$, \n\\begin{align*}\n\\langle \\mathfrak{J}(t,\\varphi,\\pi),X_H \\rangle &= (X_H \\lrcorner\\, \\theta)(t,\\varphi,\\pi) - \\int_{\\Sigma} i_t^*(X_H)_t\\lrcorner\\, \\widetilde{\\mathcal{L}}(x^\\mu,\\varphi,\\dot{\\varphi},d\\varphi)\n\\\\ &= \\dot{\\varphi}^A\\frac{\\partial}{\\partial\\varphi^A} \\lrcorner\\, \\Big(\\int_{\\Sigma_t} \\pi_Ad\\varphi^A\\otimes d^nx_0\\Big) - \\int_\\Sigma i_t^*\\frac{\\partial}{\\partial t}\\lrcorner\\, \\widetilde{\\mathcal{L}}(x^\\mu,\\varphi,\\dot{\\varphi},d\\varphi) \n\\\\ &= \\int_{\\Sigma_t} \\pi_A\\dot{\\varphi}^Ad^nx_0 - \\int_\\Sigma\\mathcal{L}(t,x^i,\\varphi,\\dot{\\varphi},d\\varphi) \n\\\\ &= \\langle \\pi,\\dot{\\varphi} \\rangle - L(t,\\varphi,\\dot{\\varphi}) = H(t,\\varphi,\\pi). \n\\end{align*} \n\nFor the proof of (ii), note that for $V$ tangent to the foliation, $V_t=0$, which immediately gives the first equation of (ii). Setting the vector field to an infinitesimal generator of a $K$-action gives the momentum map\n$$ \\langle \\mathfrak{J}(t,\\varphi,\\pi),\\xi_{T^*H\\Lambda^k} \\rangle = (\\xi_{T^*H\\Lambda^k} \\lrcorner\\, \\theta)(t,\\varphi,\\pi). $$\n\\end{proof}\n\\end{prop}\n\nWe now define the semi-discrete analogue of the energy-momentum map (\\ref{Energy-Momentum Map vector fields}). Define the semi-discrete energy-momentum map $\\mathcal{J}_h$ from $I \\times T^*\\Lambda^k_h$ to the dual of vector fields on the extended discrete phase space, via\n\\begin{equation}\\label{Discrete Energy-Momentum Map vector fields}\n\\langle \\mathfrak{J}_h(t,\\varphi,\\pi), V\\rangle = (V\\lrcorner\\, \\theta_h)(t,\\varphi,\\pi) - \\int_{\\Sigma}i_t^*V_{t,h}\\lrcorner\\, \\widetilde{\\mathcal{L}}_h(x^\\mu,\\varphi,\\dot{\\varphi},d\\varphi),\n\\end{equation}\nwhere $V_{t,h} = (T\\pi_h^*V)_t$ and $\\widetilde{\\mathcal{L}}_h$ is the restriction of $\\widetilde{\\mathcal{L}}$ via precomposition with $\\pi_h^*$. Of course, the analogous statement of the previous proposition holds for the semi-discrete energy-momentum map. Furthermore, $\\mathcal{J}_h$ is the restriction of $\\mathcal{J}$ in the following sense. \n\n\\begin{prop}\\label{Discrete EM Map as Restriction}\nFor $(t,\\varphi,\\pi)$ in the extended discrete phase space and $V$ a vector field over this space,\n$$ \\langle \\mathfrak{J}_h(t,\\varphi,\\pi), V \\rangle = \\langle \\mathfrak{J}(t,\\pi_h^*(\\varphi,\\pi)), T\\pi_h^* V \\rangle. $$\n\\begin{proof}\nThis follows directly from the definitions;\n\\begin{align*}\n\\langle \\mathfrak{J}(t,\\pi_h^*(\\varphi,\\pi)), T\\pi_h^* V \\rangle &= (T\\pi_h^*V \\lrcorner\\, \\theta)(t,\\pi_h^*(\\varphi,\\pi)) - \\int_{\\Sigma}i_t^{*}(T\\pi_h^*V)_t \\lrcorner\\, \\widetilde{\\mathcal{L}}(t,\\pi_h^*[(\\varphi,\\dot{\\varphi},d\\varphi)|_{(\\varphi,\\pi)}])\n\\\\ &= (V \\lrcorner\\, \\pi_h^{**}\\theta)(t,\\varphi,\\pi) - \\int_{\\Sigma}i_t^*V_{t,h}\\lrcorner\\, \\widetilde{\\mathcal{L}}_h(t,\\varphi,\\dot{\\varphi},d\\varphi)\n\\\\ &= (V\\lrcorner\\, \\theta_h)(t,\\varphi,\\pi) - \\int_{\\Sigma}i_t^*V_{t,h}\\lrcorner\\, \\widetilde{\\mathcal{L}}_h(t,\\varphi,\\dot{\\varphi},d\\varphi)\n = \\langle \\mathfrak{J}_h(t,\\varphi,\\pi),V \\rangle.\n\\end{align*}\n\\end{proof}\n\\end{prop}\nThe significance of this definition of the semi-discrete energy-momentum map is that it recovers the properties of Proposition \\ref{Energy-Momentum Map Prop} in the semi-discrete setting. \n\\begin{prop} \n\\\n\\begin{enumerate}[label=(\\roman*)]\n\\item \\textbf{(Semi-discrete Energy)}\nFor $(t,\\varphi,\\pi)$ in the extended discrete phase space,\n$$ \\langle \\mathfrak{J}_h(t,\\varphi,\\pi),X_{H_h} \\rangle = H_h(t,\\varphi,\\pi). $$\n\\item \\textbf{(Semi-discrete Momentum)}\nIf there is a $K$-action on the discrete phase space, then the momentum map $J_h$ is given by\n$$ \\langle J_h(t,\\varphi,\\pi),\\xi\\rangle = \\langle \\mathfrak{J}_h(t,\\varphi,\\pi),\\xi_{T^*\\Lambda^k_h}\\rangle. $$\nFurthermore, if the $K$-action on the discrete space arises from an action on the full space such that $\\pi_h$ is $K$-equivariant, then for any $\\xi \\in \\mathfrak{k}$, \n$$ \\langle \\mathfrak{J}_h(t,\\varphi,\\pi),\\xi_{T^*\\Lambda^k_h} \\rangle = \\langle \\mathfrak{J}(t,\\pi_h^*(\\varphi,\\pi)),\\xi_{T^*H\\Lambda^k}\\rangle . $$\n\\end{enumerate}\n\\begin{proof}\nThe first two equations follow from analogous computations to Proposition \\ref{Energy-Momentum Map Prop}. The last equation follows from the equivariance of $\\pi_h$,\n$$ T\\pi_h^* \\xi_{T^*\\Lambda^k_h}(\\varphi,\\pi) = \\xi_{T^*H\\Lambda^k}(\\pi_h^*(\\varphi,\\pi)),$$\nand Proposition \\ref{Discrete EM Map as Restriction}.\n\\end{proof}\n\\end{prop}\n\nThe significance of a semi-discrete analogue of the energy-momentum map, aside from extending the semi-discrete momentum map structure (discussed in Proposition \\ref{Momentum Map Prop}), is in determining semi-discrete analogues of Noether's second theorem, which we will pursue in subsequent work. \n\n\\subsection{Temporal Discretization of the Semi-Discrete Theory}\\label{Tensor Product Discretization Section}\nTo complete the discussion of the semi-discrete theory, we must of course discretize in time. We obtain a full discretization of the semi-discrete theory by discretizing the semi-discrete Euler--Lagrange equation (\\ref{Semi-Discrete EL}) in time via a Galerkin Lagrangian variational integrator applied to the instantaneous semi-discrete Lagrangian (\\ref{InstantaneousLagrangian1}), and show that this is equivalent to the full spacetime DEL (\\ref{DEL 1a}) with tensor product elements. The associated finite element on the full spacetime is a tensor product mesh, obtained by discretizing the space $\\Sigma$ and extendeding these elements in time by a partition of $I$. Of course, this is not the most general setup for a spacetime discretization, but often one wishes to discretize in time separately; for example, by choosing the appropriate temporal basis functions, the computation becomes local in time so that one can time march the solution from the initial data, instead of solving the entire DEL on the spacetime grid. Furthermore, there are constructions of cochain projections for tensor product elements (\\citet{Ar2018}) so that with these finite element spaces, the naturality of the variational principle discussed in Section 2 carries over in the tensor product setting. \n\n\\begin{remark}\nThere is a slight subtlety here when comparing to the covariant theory on the full spacetime $X$. In the covariant theory, we consider $k$-forms on $X$, $\\Lambda^kX$, whereas here we are considering $k$-forms on $\\Sigma$, $\\Lambda^k(\\Sigma)$. Letting $\\pi_1: \\mathbb{R} \\times \\Sigma \\rightarrow \\mathbb{R}, \\pi_2: \\mathbb{R} \\times \\Sigma \\rightarrow \\Sigma$ be the projections, we have pointwise,\n$$ \\wedge^k(T^*X) = \\wedge^kT^*(\\mathbb{R}\\times\\Sigma) \\cong \\Big(\\pi_1^*(\\wedge^0T^*\\mathbb{R}) \\wedge \\pi_2^*(\\wedge^kT^*\\Sigma)\\Big) \\oplus \\Big( \\pi_1^*(\\wedge^1T^*\\mathbb{R}) \\wedge \\pi_2^*(\\wedge^{k-1}T^*\\Sigma) \\Big). $$\nThis congruence does not hold at the level of sections: to see this in coordinates $(t,x)$ on $\\mathbb{R} \\times \\Sigma$, we have forms which look like $f(t) g(x) dx^{j_1}\\wedge\\dots\\wedge dx^{j_{k}}, f(t) dt \\wedge g(x) dx^{j_1}\\wedge\\dots\\wedge dx^{j_{k-1}}$ which cannot give a form which looks like e.g. $h(t,x) dt \\wedge dx^{j_1}\\wedge\\dots\\wedge dx^{j_{k-1}}$ where $h$ is some function that cannot be expressed as a product $f(t)g(x)$. However, we are assuming time-dependent fields $\\varphi: t \\mapsto  H\\Lambda^k(\\Sigma)$ so we do have the forms which look like $\\varphi(t) = g(t,x)dx^{j_1}\\wedge\\dots\\wedge dx^{j_k}$. Thus, we only need to consider multiple fields to obtain full generality $\\varphi_1: t \\mapsto H\\Lambda^{k}(\\Sigma), \\varphi_2: t \\mapsto H\\Lambda^{k-1}(\\Sigma)$ (here we are identifying $C^\\infty\\mathbb{R} = \\Lambda^0\\mathbb{R} \\cong \\Lambda^1\\mathbb{R}$, so by $\\varphi_2(t)$ we really mean $\\varphi_2(t)dt$). Of course, this issue does not arise for scalar functions, so for simplicity, in this section, we will consider scalar functions (i.e., $k=0$), although the discussion generalizes to the case of arbitrary $k$; one just needs to consider multiple fields. \n\\end{remark}\n\nAssume the setup as in the discussion of the semi-discrete theory. Furthermore, assume that we have a finite element discretization of $H_0(I)$ (the space of square integrable functions in time with square integrable derivative, which vanish on $\\partial I$) with basis functions $\\{w_\\alpha \\}$. Recall the instantaneous semi-discrete Lagrangian (\\ref{InstantaneousLagrangian1}) is a function of the curves $\\varphi^i(t),\\dot{\\varphi}^i(t)$ which are the coefficients of the expansions of $\\varphi(t),\\dot{\\varphi}(t) \\in \\Lambda^0_h$ relative to the basis $\\{v_i\\}$ of $\\Lambda^0_h$. Using the basis $\\{w_\\alpha\\}$, we discretize these curves as \n\\begin{align*}\n\\varphi^i(t) = (\\varphi^i)^{\\alpha} w_\\alpha(t),\n\\end{align*}\nwhere $\\varphi(t,x) = (\\varphi^i)^\\alpha w_\\alpha(t)v_i(x)$ in this notation. We consider the associated fully discrete action as a function of the coefficients,\n$$ S[\\{(\\varphi^i)^{\\alpha}\\}] = \\int_I dt L_h(t,\\varphi^i(t), \\dot{\\varphi}^i(t)) = \\int_I dt L_h(t,(\\varphi^i)^{\\alpha} w_\\alpha, (\\varphi^i)^{\\alpha} \\dot{w}_\\alpha). $$\nEnforcing the discrete variational principle in time gives the weak form of the Euler--Lagrange equations,\n$$ 0 = \\frac{\\delta S}{\\delta (\\varphi^i)^\\alpha} = \\left(\\frac{\\partial L_h}{\\partial \\varphi^i}, w_\\alpha\\right)_{L^2(I)} + \\left(\\frac{\\partial L_h}{\\partial \\dot{\\varphi}^i},\\dot{w}_\\alpha\\right)_{L^2(I)}. $$\nSubstituting equations (\\ref{Semi-Discrete Lagrangian Derivative Equation 1}) and (\\ref{Semi-Discrete Lagrangian Derivative Equation 2}) gives\n\\begin{align*}\n0 &=  -((\\partial_3\\mathcal{L},v_i)_{L^2(\\Sigma)}, \\dot{w}_\\alpha)_{L^2(I)} - ((\\partial_2\\mathcal{L},v_i)_{L^2(\\Sigma)},w_\\alpha)_{L^2(I)} - ((\\partial_4\\mathcal{L},dv_i)_{L^2\\Lambda^1(\\Sigma)}, w_\\alpha)_{L^2(I)} \\\\ \n&= -(\\partial_3\\mathcal{L}, v_i \\dot{w}_\\alpha)_{L^2(\\Sigma \\times I)} - (\\partial_2\\mathcal{L},v_iw_\\alpha)_{L^2(\\Sigma \\times I)} - (\\partial_4\\mathcal{L},(dv_i) w_\\alpha)_{L^2\\Lambda^1(\\Sigma) \\times L^2(I)} \\\\\n&= -\\left(\\frac{\\partial \\mathcal{L}}{\\partial\\dot{\\varphi}}, v_i \\dot{w}_\\alpha\\right)_{L^2(\\Sigma \\times I)} - \\left(\\frac{\\partial \\mathcal{L}}{\\partial \\varphi},v_iw_\\alpha\\right)_{L^2(\\Sigma \\times I)} - \\left(\\frac{\\partial \\mathcal{L}}{\\partial (d\\varphi)},(dv_i) w_\\alpha\\right)_{L^2\\Lambda^1(\\Sigma) \\times L^2(I)}.\n\\end{align*}\nNote that these equations can also be obtained directly from the semi-discrete Euler--Lagrange equations (\\ref{Semi-Discrete EL}) by applying the Galerkin method in time with respect to the basis $\\{w_\\alpha\\}$. Here, $d$ denotes the spatial exterior derivative on $\\Sigma$. If $d_t$ denotes the temporal exterior derivative and we identity functions on $I$ with one-forms on $I$, we have $\\dot{w}_\\alpha \\cong d_t w_\\alpha$. If $d_T = d + d_t$ denotes the total exterior derivative on $\\Sigma \\times I$, then $d_T(v_i w_\\alpha) = (dv_i)w_\\alpha + v_i d_t w_\\alpha.$ We now view the time-dependent function $\\varphi: t \\mapsto \\varphi(t)$ as a function $\\phi$ on spacetime, so the above can be written\n\\begin{align*}\n0 &= - \\left(\\frac{\\partial \\mathcal{L}}{\\partial \\phi},v_iw_\\alpha\\right)_{L^2(\\Sigma \\times I)} - \\left(\\frac{\\partial \\mathcal{L}}{\\partial (d\\phi)},(dv_i) w_\\alpha\\right)_{L^2\\Lambda^1(\\Sigma) \\times L^2(I)}  -\\left(\\frac{\\partial \\mathcal{L}}{\\partial(d_t\\phi)}, v_i d_tw_\\alpha\\right)_{L^2(\\Sigma) \\times L^2\\Lambda^1(I)} \\\\\n&= - \\left(\\frac{\\partial \\mathcal{L}}{\\partial \\phi},v_iw_\\alpha\\right)_{L^2(\\Sigma \\times I)} - \\left(\\frac{\\partial \\mathcal{L}}{\\partial (d_T\\phi)},d_T(v_iw_\\alpha)\\right)_{L^2\\Lambda^1(\\Sigma \\times I) },\n\\end{align*}\nwhich is the DEL (\\ref{DEL 1a}) with tensor product basis $\\{v_iw_\\alpha\\}$. \n\nNote that this result can also be obtained from the semi-discrete Hamiltonian setting, using the fact that the semi-discrete Hamiltonian and semi-discrete Lagrangian formulations are equivalent in the hyperregular case by Proposition \\ref{Equivalence of Semi-Discrete Formulations Prop}, and the fact that generalized Lagrangian variational integrators and generalized Hamiltonian variational integrators are equivalent in the hyperregular case, as established in \\citet{LeZh2009}.\n\n\\section{Conclusion and Future Directions}\nIn this paper, we showed how discretizing the variational principle for Lagrangian field theories using finite element cochain projections naturally gives rise to a discrete variational structure which is analogous to the continuum variational structure; namely, the discrete variational structure is encoded by the discrete Cartan form. Our discrete Cartan form generalizes the discrete Cartan form introduced by \\citet{MaPaSh1998} to more general finite element spaces within the finite element exterior calculus framework. Using the discrete Cartan form, we expressed a discrete multisymplectic form formula and a discrete Noether theorem in direct analogy to their continuum counterparts. Furthermore, we studied semi-discretization of Lagrangian PDEs by spatial cochain projections, showing that such semi-discretization gives rise to semi-discrete symplectic, Hamiltonian, and energy-momentum map structures. Finally, we related the methods obtained by covariant discretization and canonical semi-discretization in the case of tensor product finite elements. \n\nIn the paper, we outlined several possible research directions, including studying particular field theories and showing rigorous convergence of the discrete Cartan form, constructing group-equivariant cochain projections, and establishing a discrete Noether's second theorem utilizing the semi-discrete energy-momentum map. Another natural research direction would be to extend the discrete variational structures presented here to the discontinuous Galerkin setting and compare them with the results obtained in the multisymplectic Hamiltonian setting by  \\citet{McAr2020}. In particular, we expect that in this setting, the discrete Cartan form would only involve integration over $\\partial U$, since boundary variations can be localized to codimension-one simplices, unlike for conforming finite element spaces. Furthermore, we aim to investigate how the discrete variational structures presented in this paper (in the conforming setting and extended to the discontinuous Galerkin setting) can be used to provide a geometric variational framework for studying lattice field theories, building on the discrete variational framework for lattice field theories initiated in \\citet{ArZa2014}. \n\n\\section*{Acknowledgements}\nBT was supported by the NSF Graduate Research Fellowship DGE-2038238, and by NSF under grants DMS-1411792, DMS-1813635. ML was supported by NSF under grants DMS-1411792, DMS-1345013, DMS-1813635, by AFOSR under grant FA9550-18-1-0288, and by the DoD under grant 13106725 (Newton Award for Transformative Ideas during the COVID-19 Pandemic). \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Auxiliary tools}\n\\subsection{Proof of Proposition~\\ref{fct:expres}}\n\\begin{proof}\nThere are at most $2^{\\bar{\\alpha}}$ possible inputs to function $f_1$. Hence, since it is deterministic, there are at most $2^{\\bar{\\alpha}}$ possible outputs. Take the family of all subsets of $N$ of size at most $N$ and define the partition of this family into subfamilies -- \\dk{each consisting of sets}\nwith \nthe same value of $f_1$. This partition has at most $2^{\\bar{\\alpha}}$ elements, \\dk{because this is the size of the domain of $f_1$}. Fix an arbitrary ordering of this partition and enumerate its elements. Each subfamily receives a unique label with at most $\\bar{\\alpha}$ bits. Let feedback function $f_2$ for each set $K$ return the label of the subfamily to which $K$ belongs. Such feedback function has expressiveness at most $\\bar{\\alpha}$ and clearly it \nsatisfies the property required from $f_2$ in the statement of the fact.\n\\end{proof}\n\n\\subsection{Proof of Proposition~\\ref{fct:technique}}\n\\begin{proof}\nLet $\\mathcal{S}_{adv}$ denotes the set of all strategies of $\\alpha$-Malicious Adversary\\xspace. From the definition of the adversary, we have $\\mathsf{Adv}\\xspace(Q_{\\tau} \\cap K_1, \\tau) = Q_{\\tau} \\cap K_1$ and $\\mathsf{Adv}\\xspace(Q_{\\tau} \\cap K_2, \\tau) = Q_{\\tau} \\cap K_2$. Hence position $\\tau$ of feedback vector $\\mathcal{F}(K_1,\\mathsf{Adv}\\xspace)$ equals to $\\mathsf{Feed}\\xspace(Q_{\\tau} \\cap K_1)$ for any strategy of the adversary. \nSimilarly position $\\tau$ of feedback vector $\\mathcal{F}(K_2,\\mathsf{Adv}\\xspace)$ equals to $\\mathsf{Feed}\\xspace(Q_{\\tau} \\cap K_2)$ for any strategy of the adversary. Since $\\mathsf{Feed}\\xspace(Q_{\\tau} \\cap K_1) \\neq \\mathsf{Feed}\\xspace(Q_{\\tau} \\cap K_2)$, then \n\\dk{$\\{\\mathcal{F}(K_1,\\mathsf{Adv}\\xspace) : \\mathsf{Adv}\\xspace \\in \\mathcal{S}_{adv}(\\mathcal{Q},K_1)\\} \\cap  \\{\\mathcal{F}(K_2,\\mathsf{Adv}\\xspace) : \\mathsf{Adv}\\xspace \\in \\mathcal{S}_{adv}(\\mathcal{Q},K_2)\\} = \\emptyset$.}\n\\end{proof}\n\n\\subsection{Proof of Fact~\\ref{clm:oddeven}}\n\\begin{proof}\nWe have:\n\\begin{align*}\n(1-2p)^n = ((1-p) - p)^n &= \\sum_{k=0}^n {n \\choose k} (-p)^k (1-p)^{n-k} \\\\\n& = \\sum_{k = 0}^{\\lceil n\/2 \\rceil} {n \\choose 2k}p^{2k}(1-p)^{n-2k} - \\sum_{k = 0}^{\\lceil n\/2 \\rceil} {n \\choose 2k+1}p^{2k+1}(1-p)^{n-2k-1} \\\\\n& = \\Pr{X \\text{ is even}} - \\Pr{X \\text{ is odd}}. \n\\end{align*}\nAn since $1= \\Pr{X \\text{ is even}} + \\Pr{X \\text{ is odd}} $, we get $\\Pr{X \\text{ is odd}} = (1 - (1-2p)^n)\/2$.\n\\end{proof}\n\n\n\n\n\n\\subsection{Minimal expressiveness -- Binary feedback}\n\\label{sec:binary}\n\n\n\n\n\n\n\n\n\n\n\nWe first show (Lemma~\\ref{lem:telescope}) an upper bound on length of a sequence that distinguishes any pair of sets satisfying a certain size restriction. \nThis length is inversely proportional to the product of capacity $\\alpha$ and the lower bound on the size of the symmetric difference between the sets, denoted by $\\delta$.\nThis proof is based on analyzing a certain Separation Property of a sequence of random queries drawn from specific\nprobabilistic distribution, and showing that it yields distinguishing between two sets $K_1,K_2$ with a large probability, sufficient\nto derandomize it.\nIn the second step (Lemma~\\ref{lem:binarydelta}), we show how to remove the size restrictions from the result.\nFinally, Theorem~\\ref{thm:binary} will follow directly from Lemma~\\ref{lem:binarydelta} applied for $\\delta=1$.\n\nIn Lemma~\\ref{lem:binarydelta} we will need the following notation and basic facts.\n\\paragraph{Basic notation and tools}\nWe will use the following notation for the symmetric difference \nof \ntwo sets $A \\;\\triangle \\; B = (A \\setminus B) \\cup (B \\setminus A)$. In our proofs, we also use the following two elementary~facts:\n\\begin{fact}\n\\label{clm:oddeven}\nLet $X \\sim \\mathsf{Binomial}(n,p)$, then $\\Pr{X \\text{ is odd}} = \\frac{1}{2} - \\frac{1}{2} (1-2p)^n$.\n\\end{fact}\n\\dk{The proof of Fact~\\ref{clm:oddeven} can be found in the appendix.} The following fact can be found {\\it e.g.}\\xspace, in \\cite[(p. 34 eq. 6)]{mitrinovic1970analytic}.\n\\begin{fact}\n\\label{clm:bern_rev}\nFor any $0 \\leq x \\leq 1$ and $n \\in \\mathbf{N}_+$:\n$\n(1-x)^n \\leq 1- nx + \\frac12 n(n-1)x^2.\n$\n\\end{fact}\n\n\\paragraph{\\dk{Main technical tools}}\n\\dk{We first show how to construct sequences distinguishing pairs of sets $K_1,K_2$ satisfying specific conditions.}\n\n\\begin{lemma}\n\\label{lem:telescope}\nFor any $1 \\leq \\delta \\leq k\/\\alpha$ and if $k \\geq \\alpha$, there exists a sequence of  $O\\left(\\frac{k^2}{\\alpha \\delta} \\cdot \\log (n\/k)\\right)$ sets $\\mathcal{Q}$ such that for any two sets $K_1,K_2\\subseteq N$ satisfying $k \\geq |K_1| \\geq k\/2$ and $|K_1| \\geq |K_2|$ and $|K_1 \\;\\triangle \\; K_2| \\geq \\delta$ there exists $Q \\in \\mathcal{Q}$ that satisfies $|Q\\cap K_1| \\leq \\alpha$, $|Q\\cap K_2|\\leq \\alpha$ and $\\feed{\\alpha}(Q \\cap K_1) \\neq \\feed{\\alpha}(Q \\cap K_1)$.\n\n\n\n\\end{lemma}\n\\begin{proof}\nWe will show this result using the probabilistic method. \nMore precisely, we first define a sequence of random queries $\\mathcal{Q}$ of length $O(\\frac{k^2}{\\alpha\\delta} \\cdot \\log (n\/k))$.\nNext, we fix any two different sets $K_1,K_2 \\subseteq N$ whose cardinalities satisfy the conditions of the lemma. Recall that, \\dk{by Proposition~\\ref{fct:technique},} a query $Q \\in \\mathcal{Q}$ \\emph{distinguishes} $K_1$ from $K_2$ if it satisfies the three conditions from the statement of the Lemma: $|Q\\cap K_1| \\leq \\alpha$, $|Q\\cap K_2|\\leq \\alpha$ and $\\feed{\\alpha}(Q \\cap K_1) \\neq \\feed{\\alpha}(Q \\cap K_1)$. We will compute the probability that no query from sequence $\\mathcal{Q}$ distinguishes the considered sets $K_1$ and $K_2$.\nThen, we apply the union bound over all pairs of $K_1,K_2$ and take the complementary event, which, as we show, holds with a positive probability. This implies existence of the sought query sequence. The details follow.\n\n\\noindent\n{\\em Definition of random sequence $\\mathcal{Q}$.}\nWe define a sequence of probabilities $\\mathcal{P}$ of length $O(\\frac{k^2}{\\alpha\\delta} \\cdot \\log (n\/k))$ as probability $ \\frac{\\alpha}{16k}$ repeated $\\left\\lceil \\frac{150 k^2}{\\alpha\\delta} \\cdot \\log \\frac{4n}{k} \\right\\rceil$ times.\nWe define $Q_i$, an $i$-th element of the sequence $\\mathcal{Q}$, as a set generated by including each element of $N$ independently with the $i$-th probability from sequence $\\mathcal{P}$.\n\n\\noindent\n{\\em Proving Separation Property.} \nConsider two different sets $K_1,K_2 \\subseteq N$ whose cardinalities satisfy the conditions of the lemma.\nLet $S = K_1 \\;\\triangle \\; K_2$ be the symmetric difference of $K_1$ and $K_2$. Note that $2k \\geq |K_1 \\cup K_2| \\geq |S| \\geq \\delta$. We also know, by the assumed restriction on the size of $K_1$, that $|K_1 \\cup K_2| \\geq k\/2$. \nWe want to show the following:\n\\begin{quote}\n{\\bf\\em  Separation Property:} for any positive integer $i\\leq |\\mathcal{Q}|\/2$ and for some constant $c > 0$, the probability that query $Q_i \\in \\mathcal{Q}$,\ndistinguishes $K_1$ and $K_2$ is at least $c\\alpha\\delta\/k$.\n\\end{quote}\nBefore proving the Separation Property we need the following technical claim.\n\n\\noindent\n{\\bf Claim.} For any $Q_j\\in \\mathcal{Q}$, where $j\\leq |\\mathcal{Q}|$:\n\\begin{align*}\n\\Pr{ Q_j \\text{ distinguishes } K_1 \\text{ from } K_2} & \\geq \\Pr{|(K_1 \\cup K_2) \\cap Q_j| \\leq \\alpha \\text{ and } |S \\cap Q_j| \\text{ is odd}} \\ .\n\\end{align*}\n\n\\noindent\n{\\em Proof of the Claim.}\nConsider a query $Q_j$, for some $j\\leq |\\mathcal{Q}|$, from the random sequence $\\mathcal{Q}$, and assume that the event ``$|(K_1 \\cup K_2) \\cap Q_j| \\leq \\alpha \\text{ and } |S \\cap Q_j| \\text{ is odd}$'' holds. It follows from $|(K_1\\cup K_2) \\cap Q_j| \\leq \\alpha$ that $|K_1 \\cap Q_j| \\leq \\alpha$ and $|K_2 \\cap Q_j| \\leq \\alpha$. Moreover, since the event also implies that $|S \\cap Q_j|$ is odd, then $|K_1 \\cap Q_j| \\neq |K_2 \\cap Q_j| \\mod 2$. Hence, $\\feed{\\alpha}(K_1 \\cap Q_j) \\neq \\feed{\\alpha}(K_2 \\cap Q_j)$.\nThis completes the proof of the Claim. \\ $\\blacksquare$\n\n\n\n\n\n\nWe continue the proof of the Separation Property. \nIn the case, where $\\alpha \\geq \\log\\frac{256k}{7\\alpha \\delta}$ we have, that:\n\\[ \n\\Pr{|(K_1 \\cup K_2) \\cap Q| \\leq \\alpha \\text{ and } |S \\cap Q| \\text{ is odd}}  \\geq \\Pr{|S \\cap Q| \\text{ is odd}}  - \\Pr{|(K_1 \\cup K_2) \\cap Q| > \\alpha}\n\\ .\n\\]\nRandom variable $H_2 = |(K_1 \\cup K_2) \\cap Q|$ is distributed according to the Binomial distribution with parameters $|K_1 \\cup K_2|$ and $p$, and $\\E{H_2} = p |K_1 \\cup K_2| \\leq 2pk = \\alpha \/8$. Then, by Chernoff bound (c.f.,~\\cite{SURV}):\n\\[\n\\Pr{H_2 \\geq  \\alpha} \\leq 2^{-\\alpha} \\ .\n\\]\nRandom variable $|S \\cap Q|$ is also distributed according to the Binomial distribution with parameters $|S|$ and~$p$. By Fact~\\ref{clm:oddeven} we have\n\\[\n\\Pr{|S \\cap Q| \\text{ is odd}} = \\frac{1}{2} - \\frac{1}{2}(1-2p)^{|S|} \\ .\n\\]\nTerm $\\frac{1}{2}(1-2p)^{|S|}$ is maximized, when $|S|$ is minimized, which gives us:\n\\[\n\\Pr{|S \\cap Q| \\text{ is odd}} \\geq \\frac{1}{2} - \\frac{1}{2}(1-2p)^{\\delta} \\ .\n\\]\nUsing Fact~\\ref{clm:bern_rev}, we get:\n\\begin{align*}\n\\frac{1}{2} - \\frac{1}{2}(1-2p)^{\\delta} \\geq \\frac12 - \\frac12\\left(1- 2p\\delta + 4\\delta(\\delta-1)p^2\\right) = \\frac{\\alpha \\delta}{16k} - \\frac{\\delta(\\delta-1) \\alpha^2}{128 k} = \\frac{\\alpha \\delta}{16k}\\left(1 - \\frac{(\\delta - 1)\\alpha}{8k}\\right) \\ ,\n\\end{align*}\nand knowing that $\\delta < k\/\\alpha$ we get $\\left(1 - \\frac{(\\delta - 1)\\alpha}{8k}\\right) \\geq 7\/8$. Finally, combining the above and knowing that $\\alpha \\geq \\log\\frac{256k}{7\\alpha \\delta}$,\nwe get:\n\\[\n \\Pr{|S \\cap Q| \\text{ is odd}}  - \\Pr{|(K_1 \\cup K_2) \\cap Q| > \\alpha}\n\\geq \\frac{7\\alpha\\delta}{128k}  -  2^{-\\alpha} \\geq \n\\frac{7\\alpha \\delta}{256k} \\geq \\frac{\\alpha \\delta}{50k} \\ .\n\\]\n\nIn the second case assume, that $\\alpha \\leq \\log\\frac{256k}{7\\alpha \\delta}$. In this case we have:\n\\begin{align*}\n\\Pr{ Q_j \\text{ distinguishes } K_1 \\text{ from } K_2} & \\geq \\Pr{|(K_1 \\cup K_2) \\cap Q_j| \\leq \\alpha \\text{ and } |S \\cap Q_j| \\text{ is odd}} \\\\\n&  \\geq \\Pr{|(K_1 \\cap K_2) \\cap Q_j| \\leq \\alpha - 1 \\text{ and } |S \\cap Q_j| = 1} \\\\\n& \\geq \\Pr{|(K_1 \\cap K_2) \\cap Q_j| \\leq \\alpha - 1 } \\cdot \\Pr{|S \\cap Q_j| = 1},\n\\end{align*}\nwhere the last equality is true, because sets $S$ and $K_1\\cap K_2$ are disjoint. \n\nRandom variable $H_1 = |(K_1 \\cap K_2) \\cap Q|$ is distributed according to the Binomial distribution with parameters $|K_1 \\cap K_2|$ and $p$, and $\\E{H_1} = p |K_1 \\cap K_2| \\leq pk = \\alpha \/16$. Then, by Markov inequality $\\Pr{H_2 \\geq  \\alpha} \\leq 1\/16$.\nWe want to lowerbound term \n\\[\n\\Pr{|S \\cap Q_j| = 1} = p  |S| (1-p)^{|S|-1} = \\frac{\\alpha|S|}{16k}\\cdot \\left(1-\\frac{\\alpha}{16k}\\right)^{|S|-1}\n\\]\nIf $|S| \\leq 16k \/ \\alpha$, then $(1-\\alpha\/(16k))^{|S|-1} \\geq e^{-1}$ and:\n\\[\n\\Pr{|S \\cap Q_j| = 1} \\geq \\frac{\\alpha\\delta}{16 e k}.\n\\] \nHence $\\Pr{ Q_j \\text{ distinguishes } K_1 \\text{ from } K_2} \\geq \\frac{15}{16} \\cdot \\frac{\\alpha\\delta}{16 e k} \\geq \\frac{\\alpha\\delta}{50 k}$.\n\nIf $|S| > 16 k \/ \\alpha$, then $p|S| \\geq 1$ and  (knowing that $|S| \\leq 2k$), we get:\n \\[\n\\left(1-\\frac{\\alpha}{16k}\\right)^{|S|-1} \\geq \\left(1-\\frac{\\alpha}{16k}\\right)^{\\left(\\frac{16k}{\\alpha} -1\\right) \\frac{\\alpha}{8} + \\frac{\\alpha}{8}} \\geq e^{-\\alpha \/ 8} \\cdot e^{-\\alpha \/ 8} \\geq e^{-\\alpha} \\geq  \\frac{7\\alpha \\delta}{256k}.\n\\]\nHence, also in this case, we get $\\Pr{ Q_j \\text{ distinguishes } K_1 \\text{ from } K_2} \\geq \\frac{15}{16} \\cdot \\frac{7\\alpha \\delta}{256k} \\geq \\frac{\\alpha\\delta}{50 k}$.\n\n\nThis completes the proof of the Separation Property for $c=1\/50$.\n\n\n\n\\noindent\n{\\em Computing the probability of $\\mathcal{Q}$ distinguishing $K_1$ from $K_2$.}\nBy the proven Separation Property for $c=1\/50$ and by independence of selection of each query in the sequence $\\mathcal{Q}$ of length $150k^2\/(\\alpha\\delta)\\cdot \\log(4n\/k)$, the probability that \n$\\mathcal{Q}$ fails to distinguish $K_1$ from $K_2$ is\nat most:\n\\[\n\\left(1-\\frac{\\alpha\\delta}{50 k}\\right)^{150k^2\/(\\alpha\\delta)\\cdot \\log(4n\/k)} \n=\n\\left(1-\\frac{\\alpha\\delta}{50 k}\\right)^{50k\/(\\alpha\\delta)\\cdot 3k\\log(4n\/k)} \n\\leq e^{-3k\\log(4n\/k)} = \\left(\\frac{4n}{k}\\right)^{-3k} \\ .\n\\]\n{\\em Applying the union bound and probabilistic argument.}\nThe number of possible pairs of sets $K_1,K_2$  for the case  $k \\leq n\/2$ can be upper bounded as follows: \n\\[\n\\left(\\sum_{i=1}^k {n\\choose i}\\right)^2 \\leq k^2 {n\\choose k}^2 \\leq k^2\\left(\\frac{en}{k}\\right)^{2k} \\ .\n\\]\nIf $n \\geq k \\geq n\/2$ the number of possible pairs $K_1,K_2$ can be simply upper bounded by:\n\\[\n2^n \\cdot 2^n \\leq \\left(\\frac{4n}{k}\\right)^{2k} \\ .\n\\]\nIn both cases the number of possible pairs of $K_1,K_2$ is upper bounded by \n $$ k^2 \\cdot \\left(\\frac{4n}{k}\\right)^{2k} \\ .$$\nThus, using the Union Bound, the probability that some pair of sets is not distinguished by $\\mathcal{Q}$ is at most:\n\\[\n\\left(\\frac{4n}{k}\\right)^{-3k} \\cdot k^2 \\cdot \\left(\\frac{4n}{k}\\right)^{2k} \\leq 4^{-k} \\cdot k^2  < 1 \\ .\n\\]\nHence, there is a positive probability of the complementary event that there exists a sequence of length $O(\\frac{k^2}{\\alpha\\delta} \\cdot \\log (n\/k))$ that distinguishes any pair $K_1$ and $K_2$ (satisfying the conditions of the lemma) under the $\\feed{\\alpha}$ feedback function, and by the probabilistic argument -- such a sequence exists.\n\\end{proof}\n\n\n\\dpa{\nIn the next lemma we show that it is possible to also distinguish sets of size at most $\\alpha$.\n\\begin{lemma}\n\\label{lem:telescope2}\nIf $k \\leq \\alpha$, there exists a sequence of  $O\\left(k \\cdot \\log (n\/k)\\right)$ sets $\\mathcal{Q}$ such that for any two sets $K_1,K_2\\subseteq N$ satisfying $k \\geq |K_1|$, $k \\geq |K_2|$  and $K_1 \\neq K_2$ there exists $Q \\in \\mathcal{Q}$ that satisfies $\\feed{\\alpha}(Q \\cap K_1) \\neq \\feed{\\alpha}(Q \\cap K_1)$.\n\\end{lemma}\n\\begin{proof}\nThe proof follows in a similar vein as proof of Lemma~\\ref{lem:telescope}.\nWe construct a sequence of  $\\lceil 3k \\log (4n\/k) \\rceil$ queries $\\mathcal{Q}$ as follows: $i$-th element of the sequence is generated by including each element of $N$ independently with probability $\\frac12$. Since $|K_1| \\leq \\alpha$ and $|K_2| \\leq \\alpha$, then \n$\\Pr{ Q_j \\text{ distinguishes } K_1 \\text{ from } K_2} \\geq \\Pr{|(K_1 \\;\\triangle \\; K_2) \\cap Q_j| \\text{ is odd}}$. By Fact~\\ref{clm:oddeven} we have $\\Pr{|(K_1 \\;\\triangle \\; K_2) \\cap Q_j| \\text{ is odd}} = \\frac12$.\n\n\nSimilarly as in Lemma~\\ref{lem:telescope}, the number of possible pairs of $K_1,K_2$ can be upper bounded by $ k^2 \\cdot \\left(\\frac{4n}{k}\\right)^{2k}$.\nThus, using the Union Bound, the probability that some pair of sets is not distinguished by $\\mathcal{Q}$ is at most:\n\\[\n\\left(\\frac{4n}{k}\\right)^{-3k} \\cdot k^2 \\cdot \\left(\\frac{4n}{k}\\right)^{2k} \\leq 4^{-k} \\cdot k^2  < 1 \\ .\n\\]\nHence, there is a positive probability of the complementary event that there exists a sequence of length $O(k \\cdot \\log (n\/k))$ that distinguishes any pair $K_1$ and $K_2$ (satisfying the conditions of the lemma) under the $\\feed{\\alpha}$ feedback function, and by the probabilistic argument -- such a sequence exists.\n\\end{proof}\n}\n\n\nIn the next lemma we show that the sequences constructed in Lemma~\\ref{lem:telescope} and Lemma~\\ref{lem:telescope2} could be concatenated in order to obtain a sequence that distinguishes sets without the lower restriction on their sizes.\n\n\\begin{lemma}\n\\label{lem:binarydelta}\nThere exists a sequence $\\mathcal{Q}$  of length  $O\\left(\\left(k+\\frac{k^2}{\\alpha \\delta}\\right) \\cdot \\log (n\/k)\\right)$ for any $1 \\leq \\delta \\leq \\max\\{k\/\\alpha,1\\}$, such that for any sets $K_1,K_2\\subseteq N$ satisfying $|K_1|,|K_2| \\leq k$ and $|K_1 \\;\\triangle \\; K_2| \\geq \\delta$ there exists $Q \\in \\mathcal{Q}$ that satisfies $|Q\\cap K_1| \\leq \\alpha$, $|Q\\cap K_2|\\leq \\alpha$ and $\\feed{\\alpha}(Q \\cap K_1) \\neq \\feed{\\alpha}(Q \\cap K_1)$.\n\\end{lemma}\n\\begin{proof}\nAssume that $k$ is a power of $2$ (if it is not, we can increase $k$ to the closest power of $2$ without increasing the asymptotic complexity of our sequence). From Lemma~\\ref{lem:telescope}, we have that there exists a sequence of length $c k^2 \\log(n\/k) \/ (\\alpha\\delta)$, for some constant~$c$, distinguishing any two sets $K_1,K_2$ satisfying $|K_1| \\geq |K_2|$ and $k\\geq |K_1| \\geq k\/2$ and $|K_1 \\;\\triangle \\; K_2| \\geq \\delta$. We want to show that such a sequence exists for any pair of sets of size at most $k$. We call the sequences from Lemma~\\ref{lem:telescope} applied to parameter $k\/2^i$ instead of $k$ as $\\mathcal{Q}_i$. By concatenating such sequences for $i = 0,1,\\dots,\\lfloor \\log_2 (k\/\\alpha) \\rfloor$ and with sequence $\\hat{\\mathcal{Q}}$ from Lemma~\\ref{lem:telescope2}, we obtain sequence $\\mathcal{Q}$ of length:\n$\n\\lceil 3k \\log (4n\/k) \\rceil + \\sum_{i = 0}^{\\lfloor\\log_2 (k\/\\alpha) \\rfloor} c\\frac{k^2\\log(2^in\/k)}{4^i\\alpha\\delta } \\in O((k+\\frac{k^2}{\\alpha\\delta}) \\cdot \\log(n\/k) )$.\nTake any two sets $K_1$ and $K_2$ such that $|K_1|, |K_2| \\leq k$ and $|K_1 \\;\\triangle \\; K_2| \\geq \\delta$. Without loss of generality assume that $|K_1| \\geq |K_2|$.\nIf $|K_1| \\leq \\alpha$, then the pair is distinguished by $\\hat{\\mathcal{Q}}$ due to Lemma~\\ref{lem:telescope2}. Otherwise,\nwe find such $i$, that $k2^{-i}\\geq |K_1| \\geq k 2^{-i-1}$. By Lemma~\\ref{lem:telescope}, sequence $\\mathcal{Q}_i$ distinguishes $K_1$ from $K_2$. Since $\\mathcal{Q}$ contains $\\mathcal{Q}_i$ as subsequence, then $\\mathcal{Q}$ also distinguishes $K_1$ from $K_2$.\n\\end{proof}\n\nAs mentioned earlier, Theorem~\\ref{thm:binary} follows directly from Lemma~\\ref{lem:binarydelta} applied for $\\delta=1$.\nIt is worth mentioning that Lemma~\\ref{lem:binarydelta}, based on technical development in Lemma~\\ref{lem:telescope},  could be seen as more universal tool that could be applied\nto the analysis of other feedbacks related to or using parity as its part, c.f., Section~\\ref{sec:geberal-feedback}.\n\n\\dpa{\n\\paragraph{Randomized counterpart construction}\n\\dk{First} \nnote that the explicit randomized construction used in the proof of Lemma~\\ref{lem:telescope} leads directly to the following corollary:\n\n\\begin{corollary}\n\\label{cor:random1}\nThere exists an \\dk{explicit} randomized algorithm that generates a sequence $\\mathcal{Q}$ of $O(\\frac{k^2}{\\alpha\\delta} \\cdot \\log (n\/k))$ sets such that with probability at least $1 - k^2 \/ 4^{-k}$ \\dk{the following holds:} for any sets  $K_1,K_2\\subseteq N$ \n\\dk{such that}\n$k \\geq |K_1| \\geq k\/2$ and $|K_1| \\geq |K_2|$ and $|K_1 \\;\\triangle \\; K_2| \\geq \\delta$, there exists $Q \\in \\mathcal{Q}$ that satisfies $|Q\\cap K_1| \\leq \\alpha$, $|Q\\cap K_2|\\leq \\alpha$ and \n\\dk{$\\feed{\\alpha}(Q \\cap K_1) \\neq \\feed{\\alpha}(Q \\cap K_2)$.}\n\\end{corollary}\n\nA randomized algorithm generating a concatenation of sequences \\dk{$\\mathcal{Q}_i$, taken} from Corollary~\\ref{cor:random1} \\dk{for parameters $k\/2^i$,} in the same manner as in Lemma~\\ref{lem:binarydelta} for $\\delta = 1$, results in an explicit randomized algorithm for $(n, k)$-Group-Testing\\xspace under $\\alpha$-Malicious Adversary\\xspace. It is easy to see that if each of $\\lfloor \\log_2 k \\rfloor$ concatenated sequences $\\mathcal{Q}_i$ does not fail ({\\it i.e.,}\\xspace it does distinguish all pairs of sets of certain sizes), then the resulting sequence distinguishes all pairs of sets of sizes at most $k$. \n\n\\begin{corollary}\n\\label{cor:binary_random}\nUnder $\\feed{\\alpha}$ feedback and under adaptive $\\alpha$-Malicious Adversary\\xspace, there exists an explicit randomized solution to $(n, k)$-Group-Testing\\xspace with query complexity\\\\  \\dk{$O\\left(\\left(k+\\frac{k^2}{\\alpha}\\right) \\cdot \\log \\frac{n}{k}\\log 1\/c\\right)$} working with probability at least  $1 - c$, for any \n$c\\in (0,1)$.\n\\end{corollary}\n\\begin{proof}\nThe probability that a single of the sequences concatenated in Lemma~\\ref{lem:binarydelta} fails to distinguish all sets of certain sizes is at most:\n\\[\nk^2 \\cdot 4^{-k} + \\sum_{i=0}^{\\lfloor \\log_2 (k\/\\alpha)\\rfloor} (k\/2^i)^2 4^{-(k\/2^i)} \\leq \\frac14 + \\sum_{j = 1}^{\\infty} j^2 4^{-j} = \\frac{107}{108}\n\\ ,\n\\]\nbecause $\\sum_{j = 1}^{\\infty} j^2 4^{-j} = 20\/27$. If we concatenate $\\lceil \\log_{108\/107} 1\/c \\rceil$ independently generated such sequences, we get a sequence that distinguishes all sets with probability at least $1-c$.\n\\end{proof}\n}\n\n\n\\section{Discussion of results and open directions}\n\\label{sec:future}\nWe conclude the paper with four promising future directions. \n\\paragraph{Sparsity}\nIn addition to the query complexity, there are two additional metrics of Group Testing\\xspace solutions that are \n\\dk{studied i\n}\nliterature. These parameters are: the maximum number of queries to which \n\\dk{an\nelement belongs to (typically denoted by $w$) \nand the maximum size of a query (typically denoted by $\\rho$).} The interplay between all these three parameters, \\dk{i.e., query complexity, $w$ and $\\rho$,} was carefully studied in~\\cite{Inan2020}  in case of the Beeping feedback,\n\\dk{and in some other recent works \\cite{HKK20,Johnson2020} the sparsity of some particular selectors was established and discussed.}\nIt is possible to derive bounds on parameters $w$ and $\\rho$ also for the query sequences considered in this paper. In particular, the sequence in Theorem~\\ref{thm:binary} under the $\\feed{\\alpha}$ feedback has $w = O(k \\log(n\/k))$ and $\\rho = O(n\\alpha \/ k)$, where both bounds can be obtained by a small modification of the analysis in Section~\\ref{sec:binary}. An interesting future direction would be to study tradeoffs between query complexity and the values of $w,\\rho$ for different feedback models, \\dk{in particular, for different capacity $\\alpha$ and expressiveness $\\beta$.} \n\n\n\n\\paragraph{Randomness}\nA popular line of research in Group Testing\\xspace is to consider randomized solutions~\\cite{coja2020information, Bondorf21, Johnson2020}. While in this work we focus on deterministic solutions, some of our algorithms \n\\dk{have their simply constructed randomized counterparts, also presented in this work.}\nRandomized algorithms defined in this way \n\\dk{correctly distinguish}\n{\\em all} sets~$K$.\nThis can be contrasted with existing solutions \\dk{that, typically, have weaker guarantees: with some probability, to correctly distinguish a} \n{\\em randomly chosen} set~$K$ \\dk{from other sets of size at most~$k$, or to correctly identify each element only with some probability (resulting in some false-positives and\/or false-negatives with non-zero probability)}.\n\\dk{Moreover, they typically work against a weaker non-adaptive version of an adversary, who has to choose the unknown set~$K$ before the random choices of the algorithm.}\nAn interesting future direction would be to \ninvestigate \n\\dk{how different probabilistic guarantees and types of adversaries influence the\nquery complexity of generalized Group Testing.} \n\\dk{Another intriguing question is how random perturbations of the feedback function (see e.g.,~\\cite{Scarlett2020}) affect the query complexity.\nFinally, designing efficient coding (i.e., constructing queries) and decoding (i.e., reconstructing set $K$ from the feedback) algorithms, working in polynomial time, is a challenging open direction, sometimes even for randomized algorithms (c.f., Section~\\ref{sec:geberal-feedback} with $\\abfeed{\\alpha,\\beta}(X)$ feedback).} \n\n\n\n\n\n\n\\paragraph{Other feedbacks}\n\nThe third direction, motivated by subtle examples of the considered $(\\alpha,2\\lceil\\log_2 n\\rceil)$-feedbacks of different\nquery complexity in Section~\\ref{sec:case-study}, is to study other specific well-motivated classes of $(\\alpha,\\beta)$-feedbacks and their complexities.\nAlthough \\dk{all $(\\alpha,\\beta)$-feedbacks} have to observe the universal lower bounds, \\dk{such as the one in Theorem~\\ref{thm:fulllower},}\ntheir actual query complexity might be asymptotically larger.\n\n\\paragraph{Other adversaries}\nObserve that in our proofs of the lower bounds, Theorems~\\ref{thm:fulllower} and \\ref{thm:lower-2min}, we use a weak $\\alpha$-Honest Adversary\\xspace. This makes our lower bounds stronger and suggests that in case of deterministic non-adaptive algorithms, the adversary that uses \\dk{{\\em some}} fixed function $\\mathsf{Adv}\\xspace$ \\dk{may have} similar power to the one being allowed to return arbitrary subsets. \n\\dk{What actually follows from our results is that this adversarial impact may be similar for the best feedbacks in the class of $(\\alpha,\\beta)$-feedbacks, but does not necessarily tell us about the impact for a specific feedback function.} \nThis opens an interesting direction of studying the impact of\nadversarial power, \\dk{and more generally non-adaptiveness and ``maliciousness'',} to the Group Testing\\xspace problem, \\dk{not only for general classes of $(\\alpha,\\beta)$-feedbacks (universal lower bounds, matching by upper bounds obtained for some $(\\alpha,\\beta)$-feedbacks), but also for specific well-motivated feedback functions.}\n\n\\section{Motivation, previous and related work}\n\\label{sec:related-future}\n\nThe problem of Group Testing\\xspace (and related equivalent problems such as coin weighting) has been considered in various feedback models. In this section we present details of implementation of some classical feedback models in our framework. Our framework, with two parameters of feedback $\\alpha$ and~$\\beta$, allows, among others, a comparison of results in different models, for a discussion about what is the best utilization of feedback output bits, and for comparison and generalization of existing results obtained for specific feedbacks, \\dk{c.f., Table~\\ref{tab1}.}\n\n\\paragraph{Beeping model and shared channel communication}\nBeeping feedback model is a standard model considered in most of the Group Testing\\xspace literature~\\cite{duhwang}, where the feedback tells whether the intersection between query $Q$ and set $K$ is empty or not.  Solutions to Group Testing\\xspace in this feedback model have direct applications to conflict resolution on a multiple access channel and broadcast in unknown radio networks, c.f.,~\\cite{ClementiMS01}.\n\nObserve that in\n Beeping\nfeedback model, the feedback returns $0$ if the intersection is empty and $1$ otherwise. Thus beeping feedback is a $(1,1)$-feedback.\n\nIn this feedback model, the Group Testing\\xspace problem is known to be solvable using $O(k^2\\log(n\/k))$~\\cite{BonisGV03} queries and an explicit construction of length $O(k^2 \\log^2 n)$~\\cite{kautz1964nonrandom} exists. Best known lower bound (for $k < \\sqrt{n}$) is $\\Omega(k^2 \\log n\/ \\log k )$~\\cite{ClementiMS01}. \n\nA related model, where the feedback equals NULL if the intersection is of size $0$, the identifier of the element, if the intersection is of size $1$ and a value COLLISION otherwise, can be seen as $(2,\\log n)$-feedback. This model is applicable to communication on shared channel and has been an area of extensive research. The solutions in literature include adaptive algorithms~\\cite{capetanakis1979generalized, capetanakis1979tree}, semi-oblivious algorithms where an element can deactivate after successful transmission~\\cite{KomlosG85, greenberg1985lower} (see surveys~\\cite{gallager1985perspective,chlebus2017randomized} for more details on results in this model).\n\\dk{As mentioned earlier, some of the previous works also consider non-adaptive adversarial component of the feedback, c.f.,~\\cite{Bar-YehudaGI92}.}\n\n\\paragraph{Finite-field additive radio network}\nIn this model, the feedback to a query is a parity of the size of the intersection between set $K$ and a query. \nOne can observe that using $BCC$-codes of length $O(k \\log \\frac{n}{k})$~\\cite{Censor-HillelHL15} it is possible to design a sequence of \n queries of the same length, that solves $(n, k)$-Group-Testing\\xspace in this model. \n\nThis construction solves $(n, k)$-Group-Testing\\xspace with $O(k \\log \\frac{n}{k})$ in a $\\feed{k}$ feedback model (which is an example of $(k,1)$-feedback), because by the definition of $BCC$-codes any bit-wise XOR of up to $k$ codewords is unique.  The construction of BCC codes has also been applied to solutions of standard communication problems (such as broadcast) in specific models of communication networks~\\cite{Censor-HillelHL15}. \n \n \n \n \n We note that $\\feed{\\alpha}$ feedback for borderline value of $\\alpha = k$ corresponds to the setting considered in~\\cite{Censor-HillelHL15}. In this case our algorithm matches the best known upper bound, hence our proposed feedback function and our algorithm are a valid generalization, showing the smooth transition of query complexity\nbetween settings of $\\alpha = \\log k$ and $\\alpha = k$ in a pace inversely proportional to the feedback capacity $\\alpha$.\n\n \n \\paragraph{Coin weighting}\n The problem of coin weighting is exactly the Group Testing\\xspace problem with a different feedback. In the coin weighting problem, we have a set of $n$ coins of two distinct weights $w_0$ (true coin) and $w_1$ (counterfeit coin), out of which up to $k$ are counterfeit ones. We are allowed to weight any subset of coins in a spring scale, hence we can deduce the number of counterfeit coins in each weighting. The task is to identify all the counterfeit coins. \n \n The coin weighting can be implemented in our framework as a $(k,\\log k)$-feedback, where the feedback returns the size of the intersection between the query and the set $K$. The problem is solvable with $O(k\\log (n\/k)\/ \\log k)$~\\cite{GrebinskiK00} queries. \n \nBounds for both $BCC$ codes and non-adaptive coin weighting are tight, thus increasing the number of output bits from $1$ to $\\log k$ results in decrease in query complexity by a factor of $\\log k$.\n\n\\paragraph{Threshold Group Testing\\xspace}\nIn this variant of Group Testing\\xspace introduced in~\\cite{DAMASCHKE}, a number of thresholds $0 < t_1 \\leq t_2 \\leq \\dots \\leq t_s$ are defined. Thresholds divide the set $[k]$ into set of discrete intervals $[0,t_1), [t_1,t_2),\\dots, [t_{s-1},t_{s}),[t_s,k]$. The feedback to query $Q$ is the index of the interval to which $|K \\cap Q|$ belongs. This feedback can be implemented as a $(t_s + 1, \\log s)$-feedback. An upper bound for a single threshold $t$ of approximately $O(\\frac{k^2}{\\sqrt{t}} \\log\\frac{n}{k})$~\\cite{MarcoJKRS20} suggests that single threshold feedback is probably not the optimal feedback (according to our parameters) since we know that $(t,1)$-feedbacks can lead to query complexity $O(\\frac{k^2}{t} \\log\\frac{n}{k})$. On the other hand, in~\\cite{MarcoJK19} the authors analyze a feedback with $\\sqrt{k \\log k}$ thresholds out of which maximum threshold is $\\Omega(k)$, which in our framework translates to a $(k, \\log k)$-feedback. Result in~\\cite{MarcoJK19} is an algorithm with query complexity $O(\\frac{k}{\\log k} \\cdot \\log \\frac{n}{k})$, which is \\dk{\nlogarithmically far from $O\\left(\\frac{k}{\\log k} \\log^2 n\\right)$ obtained from our generic upper bound $O\\left(\\frac{k^2}{\\alpha \\beta} \\log^2 n\\right)$\nin Theorem~\\ref{thm:generalupper} instantiated for $\\alpha = k$, $\\beta = \\log k$.} \n\n\n\\paragraph{Other related results}\nThe problem of Group Testing\\xspace has been recently discussed from different perspectives. Some papers consider  different models of generating (or constraining) the subset $K$. This may lead to critically different optimal strategies, even for non-adaptive settings.  In~\\cite{Aldridge2019} the author considers the model, wherein each element is included in $K$ with a fixed probability $p$ -- we need $\\Omega(n)$ tests to have error probability tending to zero. Somehow related randomized model has been discussed in~\\cite{BonisV17}, wherein the algorithm may fail on a small fraction of inputs. \nIn \\cite{Inan2020} the authors consider ``sparse'' Group Testing\\xspace, where the size of each query is limited. They also consider settings wherein each element can be included in a limited number of queries. \n\n\n\n\n\\subsection{Full feedback}\n\nWe define the following $(\\alpha,\\alpha\\log n)$-feedback function.\n\\[\nF(S) = S.\n\\]\nWe will call such feedback function \\fullfeed{\\alpha}. Here the feedback function returns the set $S$ (understood as the set of identifiers of the elements). Such a set can be encoded using $\\alpha\\log_2 n$ bits.\n\n\n\\begin{lemma}\n\\label{lem:fullupper}\nIf $k < \\sqrt{n}$ and $\\alpha \\geq 18 \\log k$, then there exists a family $\\mathcal{A}$ of $t=O((k^2 \/ \\alpha^2) \\cdot \\log (n\/k))$ subsets of $N$\\Marek{Jest tez niejednoznacznosc z $N$ porownujac z innymi rozdzialami} with the property that for any set $S\\subset N$ such that $|S| \\leq k$ and any element $s\\in S$, there exists  $A \\in \\mathcal{A}$ with the property that $s \\in A$ and $|S \\cap A| \\leq \\alpha$.\n\\end{lemma}\n\\begin{proof}\n\n\nIn other words we need to show that there  exists a family of $t$ subsets of $\\mathcal{A}$ such that for any  $|S| \\leq k$\n\n Such that if at most $S$ \nWe need to show that \nLet  $h =\\lceil 32 \\log(3n\/k) \\rceil$. We construct a $6h \\cdot k^2 \/ \\alpha^2\\times n$ binary matrix, where each entry is $1$ with probability $\\frac{\\alpha}{6 \\cdot k}$ and $0$ otherwise. We will show that there exists a matrix satisfying the following two claims:\n\\subparagraph{Claim 1} With probability at least $2\/3$ each column has at least $\\frac{hk}{2\\alpha} $ ones.\\\\\nFor a fixed column $v$, the number of ones $N_v$ in the column is a sum of Bernoulli random variables. We have $\\E{N_v} = \\frac{hk}{\\alpha}$ hence we can bound the probability that this event fails for a fixed $v$ as follows:\n\\[\n\\Pr{N_v < h\\frac{k}{2\\alpha}} \\leq e^{-\\frac{hk}{16\\alpha}} <\\frac{1}{3n},\n\\]\nwhere the last inequality follows from the fact that, $\\frac{hk}{16\\alpha} \\geq 32 \\log(3n\/k) \/ 16  \\geq \\log(3n) $ (because $k < \\sqrt{n}$).\n\n\n\nTaking the union bound over all $n$ columns, completes the proof of the claim.\n\\subparagraph{Claim 2} With probability at least $2\/3$ for every subset of $\\frac{hk}{2\\alpha}$ rows and any subset of $k$ columns, the resulting $\\frac{hk}{2\\alpha} \\cdot \\log(n\/k) \\times k$ matrix has at least one row with at most $\\alpha$ ones.\n\nFor any fixed subset of $k$ columns and $\\frac{hk}{\\alpha}$ rows, the number of ones $X_r$ in a row $r$ of the resulting matrix is a sum of $k$ binomial random variables. We have $\\E{X_r} = \\frac{\\alpha}{6}$, we have by Chernoff bound\n\\[\n\\Pr{X_r > \\alpha} <\\Pr{X > 6 \\E{X}} < e^{-\\alpha}.\n\\]\nThe probability that this happens in each of $\\frac{hk}{2\\alpha}$ rows is at most:\n\\[\ne^{-\\alpha \\cdot \\frac{hk}{2\\alpha}} =e^{-\\frac{hk}{2}}.\n\\]\nThe logarithm of the number of possible combinations of rows and columns in this claim is:\n\\[\n\\log \\left({n \\choose k} \\cdot {\\frac{6h k^2}{\\alpha^2}\\choose \\frac{hk}{\\alpha}}\\right) \\leq k \\log\\left(\\frac{en}{k}\\right) + \\frac{hk }{\\alpha} \\log\\left(\\frac{6 e k}{\\alpha}\\right),\n\\]\nand since $\\alpha > 18 \\log k$, we obtain:\n\\begin{align*}\n\\log \\left(e^{-\\frac{hk}{2}} \\cdot {n \\choose k} \\cdot {6h \\cdot \\frac{k^2}{\\alpha^2}\\choose h \\cdot \\frac{k}{\\alpha}}\\right) &\\leq -\\frac{hk}{2}  + k \\log\\left(\\frac{en}{k}\\right) + \\frac{hk }{\\alpha} \\log\\left(\\frac{6 e k}{\\alpha}\\right)\\\\\n& \\leq - \\frac{hk}{2}   + k \\log\\left(\\frac{en}{k}\\right) + \\frac{hk \\log k}{18 \\log k} \\leq -\\frac{4hk}{9} + k \\log\\left(\\frac{en}{k}\\right) \\\\\n& \\leq - 14 k \\log\\frac{3n}{k} + k\\log\\frac{en}{k} \\leq -13 k\\log\\frac{3n}{k} < \\log\\frac{1}{3}.\n\\end{align*}\nHence:\n\\[\ne^{-k \\cdot 32 \\log(3n\/k)} \\cdot {n \\choose k} \\cdot {\\frac{6h k^2}{\\alpha^2}\\choose \\frac{hk}{\\alpha}} <\\frac{1}{3}.\n\\]\nThus, by using Union bound this completes the proof of the claim and we observe by using the probabilistic method, that there exists a matrix $M$ satisfying both claims.\n\nTo construct a family $A_1, A_2,\\dots, A_t$, with $t \\in O((k^2 \/ \\alpha^2) \\cdot \\log (n\/k))$ out of the matrix we simply include $i$-th element in set $j$ if entry in $j$-th row and $i$-th column of matrix $M$ is $1$. Claim $1$ implies that each element of $N$ belongs to at least $\\frac{hk}{2\\alpha}$ sets and claim $2$ implies that regardless of the choice of $K$, every subfamily of $\\frac{hk}{2\\alpha}$ sets contains a set with at most $\\alpha$ elements from $K$. \n\\end{proof}\n\n\\begin{theorem}\n\\label{thm:fullupper}\nUnder the $\\fullfeed{\\alpha}$ feedback model, if $\\sqrt{n} > k > \\alpha > 18 \\log k$, there exists a  family solving the $k$-group testing problem with $t \\in O((k^2 \/ \\alpha^2) \\cdot \\log (n\/k))$.\n\\end{theorem}\n\\begin{proof}\nThe theorem is a direct consequence of Lemma~\\ref{lem:fullupper}. We simply observe that under the $\\fullfeed{\\alpha}$ feedback model, family $\\mathcal{A}$ solves the $k$-group testing problem, because the feedback returns at least once the identifier of each element of $S$.\n \\end{proof}\n\n\\section{Lower bound}\n\\label{sec:lower}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:fulllower}]\nAn $(n,k,k)$-selector is a sequence of queries $Q_1,Q_2,\\dots,Q_t$ such that for any $|K| \\leq k$ and any element $x \\in K$, for some query $Q$ we have $Q\\cap K = \\{x\\}$. An $(n,k,k)$-selector is known to have query complexity $\\Omega(\\min\\{n, (k^2\/\\log k) \\cdot \\log n\\})$~\\cite{ClementiMS01} and \nis known to exist with query complexity $O(k^2 \\log n)$~\\cite{erdos1985families}. Assume that $n$ is sufficiently large and let $c_1$ be such a constant that $(n,k,k)$-selector of query complexity $t_1 \\leq c_1 (k^2\/\\log k) \\cdot \\log n $ does not exist; $c_1$ is well-defined by~\\cite{ClementiMS01}. Let $c_2$ be a constant such that $(n,\\alpha+1,\\alpha+1)$-selector of size $t_2 \\leq c_2 \\alpha^2 \\log n$ exists;\n$c_2$ is well-defined by~\\cite{erdos1985families}. Let $\\mathcal{R}= \\langle R_1, R_2, \\dots, R_{t_2}\\rangle$ be such a selector.\n\nThe proof of the theorem is by contradiction. Assume that there exists a sequence $\\mathcal{Q} = \\langle Q_1,Q_2,\\dots, Q_t\\rangle$ solving $(n, k)$-Group-Testing\\xspace with some $(\\alpha,\\beta)$-feedback function, such that the length of the sequence is $t \\leq \\frac{c_1}{c_2}\\cdot \\frac{k^2}{\\alpha^2} \\log^{-1} k$.\n\nFirst we show the following fact: for any set $K \\subset N$ of size $|K|\\leq k$ and any element $x \\in K$ there exists a set $Q$ in sequence $\\mathcal{Q}$ such that $|K\\cap Q| \\leq \\alpha + 1$. Assume the contrary and fix $K$ and $x$ that violate the fact. Observe that sets $K$ and $K\\setminus \\{x\\}$ may produce the same feedback for any $(\\alpha,\\beta)$-feedback function (regardless of the value of $\\beta$). This is because for any $Q_i$ such that $x\\in Q_i$ we have $|Q_i \\cap K| \\geq \\alpha + 2$ and $|Q_i \\cap (K\\setminus \\{x\\})| \\geq \\alpha + 1$. Hence, in the case of $Q_i \\cap K$ the $\\alpha$-Honest Adversary\\xspace may provide to the feedback function the same $\\alpha$ elements as in $Q_i \\cap (K\\setminus \\{x\\})$. Since the feedback function is deterministic, the results will be the same, which is a contradiction with the fact that $\\mathcal{Q}$ solves the $(n, k)$-Group-Testing\\xspace problem.\n\nNext we transform $\\mathcal{Q}$ into an $(n,k,k)$-selector: we take the $(n,\\alpha+1,\\alpha+1)$-selector $\\mathcal{R}= \\langle R_1, R_2, \\dots, R_{t_2}\\rangle$ and construct a sequence \n$\\mathcal{S} = \\langle Q \\cap R \\text{, for each } R = R_1, R_2, \\dots, R_{t_2}   \\text{, for each } Q = Q_1, Q_2, \\dots, Q_t \\rangle$.\nThe obtained family $\\mathcal{C}$ has $t \\cdot t_2 \\leq t_1$ queries.\nWe prove that it is also an $(n,k,k)$-selector.\nFor any set $K$ with $|K| \\leq k$ and any $s \\in S$ there exists, by the property of the sequence $\\mathcal{Q}$ proved above, a set $Q$ in sequence $\\mathcal{Q}$ such that $|Q \\cap K| \\leq \\alpha + 1$. Now, since $\\mathcal{R}$ is an $(n,\\alpha+1,\\alpha+1)$-selector, there exists a set $R$ in $\\mathcal{R}$ such that $R\\cap (Q \\cap K) = \\{x\\}$. By the construction of $\\mathcal{S}$, set $Q\\cap R$ belongs to sequence $\\mathcal{S}$, hence element $x$ is selected by family $\\mathcal{S}$. Hence $\\mathcal{S}$ is an $(n,k,k)$-selector. We know however that $(n,k,k)$-selector of length at most $t_1$ does not exist, and thus obtain a contraction showing that such a family $\\mathcal{Q}$ cannot exist.\n\\end{proof}\n\\dpa{\n\\paragraph{Randomized counterpart \\dk{result}}\nIn Theorem~\\ref{thm:fulllower} we show that any sequence that solves $(n, k)$-Group-Testing\\xspace must have length of at least $\\Omega\\left(\\frac{k^2}{\\alpha^2} \\log^{-1} k\\right)$. Thus, a randomized algorithm generating sequences that solve $(n, k)$-Group-Testing\\xspace with at least a constant probability must have expected query complexity of $\\Omega\\left(\\frac{k^2}{\\alpha^2} \\log^{-1} k\\right)$, \\dk{since each correct sequence must have such length (by Theorem~\\ref{thm:fulllower}).}\n\\begin{corollary}\nIf $n > k^2\\log n\/\\log k$, then any randomized solution to $(n, k)$-Group-Testing\\xspace under any $(\\alpha,\\beta)$-feedback has expected query complexity \n$\\Omega\\left(\\frac{k^2}{\\alpha^2} \\log^{-1} k\\right)$ for some adaptive $\\alpha$-Honest Adversary\\xspace.\n\\end{corollary}\n}\n\\subsection{Maximum expressiveness -- Full feedback}\n\nIn this section we consider $\\fullfeed{\\alpha}$ feedback. Using it, we show that larger expressiveness of feedback allows for smaller query complexity. The following lemma is independent of any feedback function and shows that there exists a query sequence that \\emph{$\\alpha$-isolates} each element of $K$, in the following sense: for any set $K$ of size at most $k$ and each element in $K$, there exists a query such that this element and at most $\\alpha-1$ other elements from $K$ belong to this query.\n\n\\begin{lemma}\n\\label{lem:fullupper}\nIf $\\alpha \\geq 9 \\log k$ and $k \\geq \\alpha$, then there exists a sequence $\\mathcal{Q}$ of $t=O((k^2 \/ \\alpha^2) \\cdot \\log n)$ subsets of $N$ with the property that for any set $K \\subseteq N$ such that $|K| \\leq k$ and any element $x\\in K$, there exists  $Q \\in \\mathcal{Q}$ with the property that $x \\in Q$ and $|K \\cap Q| \\leq \\alpha$.\n\\end{lemma}\n\\begin{proof}\nWe prove existence of such family $\\mathcal{Q}$ by a probabilistic argument. \nLet  $h =\\lceil 16 \\log (3n) \\rceil$.  The family  $\\mathcal{Q}$ consists of \n$t=6h\\cdot x \\cdot k^2 \/ \\alpha^2$  subsets denoted as $Q_1,\\ldots, Q_t$. For each $i=1,\\ldots, t$ the set $Q_i$ is generated in the following manner: each element $x\\in N$ belongs to $Q_i$ with probability  $\\frac{\\alpha}{6 \\cdot k}$. All the random choices are independent over all elements and subsets. \n\n\n\\subparagraph{Claim 1.} With probability at least $2\/3$ each element of $N$ belongs to at least $\\frac{ k}{2\\alpha} $ queries in the sequence $\\mathcal{Q}$.\n\n\\textit{Proof of Claim 1.}\nFor a fixed $x\\in N$, let $L_x=|\\{i\\in [t]: x \\in A_i\\}|$. \nClearly, $L_x$ is a sum of  Bernoulli trials and $\\E{L_x} = \\frac{h x k}{\\alpha}$. Due to  the independence of random inclusion of consecutive elements we can use a standard Chernoff bound~\\cite{SURV} to get\n$\n\\Pr{L_x < h\\frac{k}{2\\alpha}} \\leq e^{-\\frac{hk}{16\\alpha}} <\\frac{1}{3n}\n$,\nwhere the last inequality follows from the fact that, $\\frac{hk}{16\\alpha} \\geq \\log (3n)$.\nUsing the union bound over all $n$ possible elements $v \\in N$ we get Claim 1. $\\blacksquare$ \n\n\n\n\n\nConsider a sub-sequence of queries from $\\mathcal{Q}$, and from all these queries we remove all elements that do not belong to $K$,\nnamely: $\\mathcal{Q}_{K,T}=\\{Q_i\\cap K: Q_i \\in \\mathcal{Q} \\ \\& \\ i \\in T\\}$. \n\n\\subparagraph{Claim 2.} With probability at least $2\/3$, for any choice of $K$ with $k$ elements and any  $T$ of \n$\\frac{hk}{2\\alpha}$ indices, the resulting sequence $\\mathcal{Q}_{K,T}$ contains a set with at most $\\alpha$ elements.\n\n\n\\textit{Proof of Claim 2.}\nLet us fix  any subset  $K\\subseteq N$ and a set $T$ with proper cardinalities. In any fixed set $Q^' \\in  \\mathcal{Q}_{K,T}$ we define its number of elements as $X_{Q^'}$. Clearly, $X_{Q^'}$ is a sum of Bernoulli random variables. \n\nWe have $\\E{X_{Q^'}} = \\frac{\\alpha}{6}$ and by the  Chernoff bound we get\n$\n\\Pr{X_{Q^'} > \\alpha} <\\Pr{X > 6 \\E{X}} < e^{-\\alpha} \n$.\nDue to independence of choices elements in different queries, the probability that the number of elements is greater than $\\alpha$ in all $\\frac{hxk}{2\\alpha}$ sets in $\\mathcal{Q}_{K,T}$ is at most\n$\ne^{-\\alpha \\cdot \\frac{hk}{2\\alpha}} =e^{-\\frac{hk}{2}} \n$.\nRecall that the above reasoning was performed for a fixed choice of sets $K$ and $T$. To apply a union bound argument one needs to multiply the above value by the number of all possible choices of sets $K$ and~$T$. The logarithm of the number of possible combinations of $K$ and $T$ equals~to:\n\\[\n\\log \\left({n \\choose k} \\cdot {\\frac{6h k^2}{\\alpha^2}\\choose \\frac{hk}{2\\alpha}}\\right) \\leq k \\log\\left(\\frac{en}{k}\\right) + \\frac{hxk}{2\\alpha} \\log\\left(\\frac{6 e k}{\\alpha}\\right),\n\\]\nand since $\\alpha > 9 \\log k$ we obtain the logarithm of the union-bounded probability multiplied by the number of choices we get:\n\\begin{align*}\n\\log \\left(e^{-\\frac{hk}{2}} \\cdot {n \\choose k} \\cdot {6h \\cdot \\frac{k^2}{\\alpha^2}\\choose h \\cdot \\frac{k}{2\\alpha}}\\right) &\\leq -\\frac{hk}{2}  + k \\log\\left(\\frac{en}{k}\\right) + \\frac{hk }{2\\alpha} \\log\\left(\\frac{6 e k}{\\alpha}\\right)\\\\\n& \\leq - \\frac{hk}{2}   + k \\log\\left(\\frac{en}{k}\\right) + \\frac{hk \\log k}{18 \\log k} \\leq -\\frac{4hk}{9} + k \\log\\left(\\frac{en}{k}\\right) \\\\\n& \\leq - 14 k \\log\\frac{3n}{k} + k\\log\\frac{en}{k} \\leq -13 k\\log\\frac{3n}{k} < \\log\\frac{1}{3}\n\\ .\n\\end{align*}\nHence,\n$\ne^{-k \\cdot 32 \\log(3n\/k)} \\cdot {n \\choose k} \\cdot {\\frac{6h k^2}{\\alpha^2}\\choose \\frac{hk}{\\alpha}} <\\frac{1}{3} \n$.\nThis concludes the proof of Claim 2. $\\blacksquare$\n\nObserve that with probability at least $1\/3$, a randomly chosen family $\\mathcal{Q}$ simultaneously meets conditions described in Claim 1 and Claim 2, by the union bound. Consequently, with probability at least $1\/3$, in the randomly generated family $\\mathcal{Q}$ for any set $K$ of size $k$ and every sub-sequence $T$ of $\\frac{hk}{\\alpha}$ queries from $\\mathcal{Q}$ there is at least one query $Q_i$ such that $|Q_i \\cap K| \\leq \\alpha$, for some $i\\in T$. Hence, such a family $\\mathcal{Q}$ exists, by straightforward probabilistic argument. \nFinally, observe that since $\\mathcal{Q}$ works for any set $K$ of exactly $k$ elements, then it also does \nfor any $K$ such that $|K| \\leq k$. \n\\end{proof}\n\n\nInterestingly, sequence $\\mathcal{Q}$ from Lemma~\\ref{lem:fullupper} with parameters $n,k$ does \\textbf{not} distinguish all pairs of sets $K_1,K_2$ of size at most $k$. We only know, that each element $x \\in K_1$ belongs to some query $Q_{\\tau} \\in \\mathcal{Q}$, with $|K_1 \\cap Q_{\\tau}| \\leq \\alpha$. But we may have $|K_2 \\cap Q_{\\tau}| > \\alpha$ and the $\\alpha$-Malicious Adversary\\xspace may force the feedbacks to be equal on this position for sets $K_1$ and $K_2$. To solve this problem, in the proof of Theorem~\\ref{thm:fullupper}, we take the sequence from Lemma~\\ref{lem:fullupper} with parameters $n,2k$, and use it for set $K = K_1 \\cup K_2$.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:fullupper}] \n\n\n\nThe component $\\frac{n}{\\alpha}$ follows from the fact, that a simple selector, where each element belongs to one query and each query contains $\\alpha$ elements (except the last query that contains at most $\\alpha$) has query complexity $O(\\frac{n}{\\alpha})$ and solves $(n, k)$-Group-Testing\\xspace under the $\\fullfeed{\\alpha}$ feedback and works under $\\alpha$-Malicious Adversary\\xspace. \nThe first part of theorem is a consequence of Lemma~\\ref{lem:fullupper}. Specifically, we take the family $\\mathcal{Q}$ from Lemma~\\ref{lem:fullupper} with parameters $n, 2k$. We observe that for any two sets $K_1$, $K_2$, with $|K_1|, |K_2|\\leq \\alpha$ and $K_1 \\neq K_2$, we have $|K_1 \\cup K_2| \\leq 2k$ and  $K_1 \\;\\triangle \\; K_2 \\neq \\emptyset$. Take any $x \\in K_1 \\;\\triangle \\; K_2 $ and observe that due to Lemma~\\ref{lem:fullupper} there is a query $Q\\in \\mathcal{Q}$ such that $x\\in Q$ and $|Q\\cap (K_1 \\cup K_2)| \\leq \\alpha$. Hence, $|Q\\cap K_1| \\leq \\alpha$, $|Q\\cap K_2| \\leq \\alpha$ and $\\fullfeed{\\alpha}(Q\\cap K_1) \\neq \\fullfeed{\\alpha}(Q\\cap K_2)$. Consequently, $\\mathcal{Q}$ solves $(n, k)$-Group-Testing\\xspace under $\\alpha$-Malicious Adversary\\xspace, by Proposition~\\ref{fct:technique}.\n\n\nThe second part of the theorem follows from the fact that we can use the result from Theorem~\\ref{thm:binary} and obtain a sequence of length $O(\\frac{k^2}{\\alpha} \\log(n\/k))$ (this results does not require the assumption on $\\alpha$ and also works under $\\alpha$-Malicious Adversary\\xspace). Note that we do not need the $O(k \\log(n\/k))$ component here because under $\\fullfeed{\\alpha}$ in the case where $k \\leq \\alpha$, the problem is solvable using a single query.\n \\end{proof}\n\n\n\\dpa{\n\\paragraph{Randomized counterpart construction}\nIn the proof of Lemma~\\ref{lem:fullupper} we construct a sequence at random and show that it satisfies a certain condition with probability at least $1\/3$. Clearly, from this we can obtain an explicit randomized construction that succeeds with probability~$1\/3$, \\dk{and by iterating it a certain number of times we get the following:}\n\n\\begin{corollary}\nUnder $\\fullfeed{\\alpha}$ feedback and under adaptive $\\alpha$-Malicious Adversary\\xspace, there exists an explicit randomized solution to $(n, k)$-Group-Testing\\xspace with query complexity \n\\begin{align}\\nonumber\n& \\dk{O\\left(\\frac{k^2}{\\alpha\\beta}\\left(\\frac{\\beta}{\\alpha} + \\log n\\right)\\cdot \\log n \\cdot \\log 1\/c\\right)} & \\text{if } \\alpha > 18 \\log k, \\\\\\nonumber\n& O\\left(\\frac{k^2}{\\alpha} \\cdot \\log\\frac{n}{k} \\dk{\\cdot \\log 1\/c}\\right) & \\text{otherwise}.\n\\end{align}\n working with probability at least  $1 - c$, for any \n$c\\in (0,1)$.\n\\end{corollary}\n\\begin{proof}\nWe first observe that if $k\\leq \\alpha$ then, because of the $\\fullfeed{\\alpha}$ feedback, a single query containing all elements from set $N$ solves $(n, k)$-Group-Testing\\xspace. Hence, we focus on case $k \\geq \\alpha$. If $\\alpha \\leq 18 \\log k$, then we can use the result from Corollary~\\ref{cor:binary_random} and obtain a \\dk{desired} sequence of length $O((k + \\frac{k^2}{\\alpha}) \\log(n\/k)\\dk{\\log 1\/c})$ \\dk{with probability of success at least $1-c$}, which becomes $O(\\frac{k^2}{\\alpha} \\log(n\/k)\\dk{\\log 1\/c})$, because $k \\geq \\alpha$. Finally if $18\\log k < \\alpha \\le k$, we can use the construction from the proof of Lemma~\\ref{lem:fullupper} that fails with probability at most $1\/3$. By repeating it $\\lceil \\log_3 1\/c\\rceil$ times, \\dk{independently, and concatenating the resulting sequences} we obtain a desired probability of success.\n\\end{proof}\n}\n\n\n\n\\subsection{General feedback}\n\\label{sec:geberal-feedback}\n\nIn our construction of $\\abfeed{\\alpha,\\beta}(X)$ (introduced in Definition~\\ref{def:feedbacks}) we use the following \ncode, where notation $\\bigoplus S$ denotes bit-wise XOR of all the elements of set $S$. Such a code was defined in \\cite{Censor-HillelHL15} and its explicit construction can be found in \\cite{roth2006introduction}.\n\\begin{definition}\n\\label{def:BCC}\n An $[n, \\beta, \\gamma]$-BCC-code is a set $C\\subseteq \\{0,1\\}^\\beta$ of size $|C| = n$ such that for any two subsets $S_1,S_2 \\subseteq C$ (with $S_1\\neq S_2$) of sizes $|S_1|,|S_2| \\leq \\gamma$ it holds that $\\bigoplus S_1 \\neq \\bigoplus S2$.\n\\end{definition}\n\n\n\\begin{lemma}{\\cite[Lemma 2]{Censor-HillelHL15}} \n\\label{lem:bcc}\nThere exist $[n,\\beta , \\gamma]$-BCC codes with $\\beta = O(\\gamma \\log n)$. \n\\end{lemma}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:generalupper}]\nFirst we prove the part of the theorem that works under the assumption $\\alpha > 18 \\log k$. We denote $\\beta^' =\\min\\left\\{\\left\\lfloor \\frac{\\beta - 1}{c \\log n}\\right\\rfloor,\\alpha \\right\\}$. We note that if $\\beta < \\log n$, then we get $\\beta^' = 0$ but the result follows simply by choosing $\\mathcal{Q}$ as the sequence from Theorem~\\ref{thm:binary}.\nNote that we must have $\\beta^' \\leq \\alpha$ because input set $X$ cannot contain more than $\\alpha$ elements.\n\nAssume that $\\beta > \\log n$ and let us take the family  from Lemma~\\ref{lem:binarydelta} with parameter $\\delta = \\beta^'$ and concatenate it with the family from Lemma~\\ref{lem:fullupper} with parameters $2k$ and $n\n. Observe, that this resulting family $\\mathcal{Q}$ (composed of two parts $\\mathcal{Q}_1$ and $\\mathcal{Q}_2$) has length $ O\\left(\\frac{k^2}{\\alpha\\beta^'} \\cdot \\log n\\right) = O\\left(\\max\\left\\{\\frac{k^2}{\\alpha^2}\\log n,\\frac{k^2}{\\alpha \\beta}\\log^2 n\\right\\}\\right)$. We will show that this family distinguishes under $\\abfeed{\\alpha,\\beta}$  feedback model, any two sets $K_1$, $K_2$ satisfying $|K_1|,|K_2| \\leq k$.\nWe will consider two cases.\n\nIn the case, where $|K_1 \\;\\triangle \\; K_2| < \\beta^'$, we pick an arbitrary element $s \\in K_1 \\;\\triangle \\; K_2$. Without loss of generality assume that $s \\in K_1$. By Lemma~\\ref{lem:fullupper} in some $Q \\in \\mathcal{Q}_2$ we have $|Q\\cap (K_1 \\cup K_2)| \\leq \\alpha$ and $s \\in Q$. \n\nWe want to show that:\n\\[\n\\bigoplus_{s \\in Q \\cap K_1} \\mathsf{BCC}\\xspace(s)  \\neq \\bigoplus_{s \\in Q\\cap K_2} \\mathsf{BCC}\\xspace(s).\n\\]\nWe denote $K_1^' = (K_1 \\setminus K_2) \\cap Q$ and $K_2^' = (K_2 \\setminus K_1) \\cap Q$ and $T = K_1\\cap K_2 \\cap Q$. We know that $K_1^' \\neq \\emptyset$, because $s \\in K_1^'$ and also $K_1^' \\neq K_2^'$ because $s \\notin K_2^'$. Since $|K_1 \\;\\triangle \\; K_2| < \\beta^'$, and $K_1^',K_2^' \\subset K_1 \\;\\triangle \\; K_2$ then also $|K_1^'|,|K_2^'| <\\beta^'$. By the definition of BCC codes~\\cite[Lemma 2]{Censor-HillelHL15} we have that:\n$\n\\bigoplus_{s \\in K_1^' } \\mathsf{BCC}\\xspace(s)  \\neq \\bigoplus_{s \\in K_2^' } \\mathsf{BCC}\\xspace(s).\n$\nUsing the properties of operation XOR:\n\\[\n\\bigoplus_{s \\in Q\\cap K_1} \\mathsf{BCC}\\xspace(s) = \\bigoplus_{s \\in K_1^' } \\mathsf{BCC}\\xspace(s) \\oplus \\bigoplus_{s \\in T} \\mathsf{BCC}\\xspace(s) \\neq \\bigoplus_{s \\in K_2^' } \\mathsf{BCC}\\xspace(s) \\oplus \\bigoplus_{s \\in T} \\mathsf{BCC}\\xspace(s) = \\bigoplus_{s \\in Q\\cap K_2} \\mathsf{BCC}\\xspace(s).\n\\]\n\nHence, if $|K_1\\;\\triangle \\; K_2| \\leq \\beta^'$ then there exists $Q$ such that $|Q\\cap (K_1\\cup K_2)|\\leq \\alpha $. Consequently, $|Q\\cap K_1|\\leq \\alpha$ and $|Q\\cap K_2|\\leq \\alpha$.\nWe have obtained that $\\abfeed{\\alpha,\\beta}(Q\\cap K_1) \\neq \\abfeed{\\alpha,\\beta}(Q\\cap K_2)$, thus $Q$ distinguishes $K_1$ and $K_2$.\n\nIf $|K_1 \\;\\triangle \\; K_2| \\geq \\beta^'$, then by Lemma~\\ref{lem:binarydelta} the first part of our sequence $\\mathcal{Q}_1$ distingushes $K_1$ and $K_2$ under the binary feedback. Since $\\abfeed{\\alpha,\\beta}$ includes the binary feedback, our sequence $\\mathcal{Q}$ distinguishes $K_1$ from $K_2$. \n\nAfter considering both cases, we have that sequence $\\mathcal{Q}$ distinguishes any two sets $K_1,K_2$ of size at most $\\alpha$ and thus we can use Proposition~\\ref{fct:technique} and obtain  that $\\mathcal{Q}$ solves the $(n,k)$-Group-Testing problem under $\\alpha$-Malicious Adversary\\xspace.\n\n\n\n\nThe second part of the theorem follows from the fact that we can use the result from Theorem~\\ref{thm:binary} and obtain a sequence of length $O(\\frac{k^2}{\\alpha} \\log(n\/k))$ (this results does not require the assumption on $\\alpha$). \n\\end{proof}\n\n\\dpa{\n\\paragraph{Randomized counterpart construction}\n\\dk{Unlike in previous sections, for $\\abfeed{\\alpha,\\beta}(X)$ feedback it is not simple to provide an explicit algorithm, even randomized.\nThis is because} an explicit (even randomized) construction of BCC-codes is not known. \n\\dk{Therefore, we suggest this problem as one of interesting open directions.}\n}\n\\section{Introduction}\n\nGroup Testing\\xspace, introduced by \n\\cite{dorfman1943detection}, is an inference problem, where the goal is to identify, by asking queries, all elements of an unknown set $K$. All we initially know about set $K$ is that $|K| \\leq k$ and that it is a subset of some much larger set $N$ with $|N| = n$, for given parameters $k,n$. To learn set $K$, we must have answers to the queries that provide some information about set $K$. In our model, the answer to a query $Q$ depends on the intersection between $K$ and $Q$ and equals to $\\mathsf{Feed}\\xspace(K \\cap Q)$, where $\\mathsf{Feed}\\xspace$ is some known pre-defined feedback function. The sequence of queries is a correct solution to Group Testing\\xspace if and only if for any  two different sets $K_1, K_2$ such that $|K_1|,|K_2| \\le k$,\n\\dk{the sequence of feedback answers computed for sets $K_1$ and $K_2$ are different; we say then that the sequence of queries distinguishes any pair of sets, or identifies any set of size at most $k$.\\footnote{%\n\\dk{In this work we abstract from \ncomputational efficiency\nof decoding of sets,\nwhich is a large research area by itself, c.f.,~\\cite{Aldridge2014}.}}}\nThe objective is, \nfor a given deterministic feedback function $\\mathsf{Feed}\\xspace(\\cdot)$,\nto find \na sequence of queries that identifies any set $K$ of size at most $k$ and the length of this sequence, called query complexity, will be shortest possible. In particular, we are interested in algorithms that have query complexity logarithmic in $n=|N|$ and polynomial in $k=|K|$. In \nthe classic \nvariant, \nstudied in most of the existing relevant literature,\nthe function $\\mathsf{Feed}\\xspace$ simply answers whether the intersection between $K$ and $Q$ is empty or not, \\dk{while another popular feedback returns the size of the intersection~\\cite{duhwang,gallager1985perspective}.} These variants were applied in many domains, including pattern matching~\\cite{clifford2010pattern, Indyk97}, compressed sensing~\\cite{cormode2006combinatorial}, streaming algorithms~\\cite{cormode2005s} and graph reconstruction~\\cite{choi2010optimal,GrebinskiK00}\n or even accelerating computations in neural networks~\\cite{Liang2021}. Though one of the most prominent examples of applications of Group Testing\\xspace is in conflict resolution in communication networks~\\cite{capetanakis1979generalized,capetanakis1979tree,chlebus2017randomized,gallager1985perspective,greenberg1987estimating,greenberg1985lower,KomlosG85,massey1981collision,wolf1985born}.\n\nThere is a large body of literature introducing new variants of the Group Testing\\xspace~\\cite{Censor-HillelHL15,GrebinskiK00,Bshouty09,MarcoJKRS20,MarcoJK19}, which could be \\dk{simply} viewed as different feedback functions applied to \\dk{some generic} Group Testing\\xspace framework. Therefore, in this paper we aim at \\dk{designing such a universal\nframework allowing a holistic view at many previous modifications of Group Testing\\xspace setting,\nand\nstudy the dependence of the query complexity \non two identified fundamental parameters of the feedback function:}\n\\begin{description}\n\\item[\\emph{Capacity:}] this parameter denotes the maximum set size that can be processed by the feedback function. \n\\dk{In other words, the domain of the feedback function with capacity $\\alpha$ is the family of all subsets of $N$ of size at most $\\alpha$.} \nThe capacity is denoted by $\\alpha$ throughout this paper and varies between $1$~and~$k$.\n\\item [\\emph{Expressiveness:}] this parameter denotes the number of output bits of the feedback function. \\\\\nIt is denoted by $\\beta$ throughout this paper, and varies between $1$ and $\\bar{\\alpha}$, where the latter denotes a binary logarithm of the number of all subsets of $N$ of size at most $\\alpha$, \\dk{i.e., $\\bar{\\alpha}=\\log_2 \\sum_{i=0}^{\\alpha} \\binom{n}{i}$.}\n\\end{description}\n\n\nA feedback function with capacity $\\alpha$ and expressiveness $\\beta$ is called an {\\em $(\\alpha,\\beta)$-feedback}.\n\nIf for some query $Q$, $|Q \\cap K| > \\alpha$, then the intersection set $Q\\cap K$ cannot be passed directly to the feedback function, \\dk{because $(\\alpha,\\beta)$-feedback functions are not defined for such sets.} \nTherefore, in our framework we resolve this issue by presence of an adversary \\dk{-- a non-deterministic feature which,} for such queries with large intersection, \nselects a set with at most $\\alpha$ elements\nand the answer to query $Q$ is the feedback on this set. \nWe consider different models of an adversary, \\dk{including a powerful Malicious Adversary, who could ``fool'' the feedback with {\\em arbitrary sets} of at most $\\alpha$ elements of $N$,\\footnote{%\n\\dk{One could also assume that the Malicious Adversary could give {\\em any} input to the feedback function in case of exceeded capacity --\nthis could however require re-definition of the feedback function to handle \nnon-valid input sets.}\n\n\\dk{Example of an adversary, widely but implicitly considered in the literature, is the mechanism of feedback in radio networks or multiple access channels, when the feedback provides answers ``collision'' or ``silence'' or an arbitrary element if two or more neighbors of a communication device transmit simultaneously, c.f.,~\\cite{Bar-YehudaGI92,chlebus2017randomized}.}\n}\nand more benign Honest Adversary, who always returns {\\em some subset} of the intersection.}\n\n\nClearly, increasing \nthe capacity parameter $\\alpha$ or expressiveness $\\beta$ increases the number of $(\\alpha,\\beta)$-feedback functions,\nand \\dk{thus} should decrease the query complexity of the best feedbacks in this family. \nBut what is the asymptotic \\dk{pace of this query complexity decrease?} \n\\dk{Is there a substantial difference in query complexity of Group Testing\\xspace under Honest and Malicious adversaries?}\nAre there better and worse feedback functions for given $\\alpha,\\beta$, i.e., resulting in smaller (resp., larger) query complexity?\nThis paper provides partial answers to \nthese questions.\n\n\n\n\n\n\\paragraph{Document structure}\nIn Section~\\ref{sec:model} we formally define the general framework of $(n, k)$-Group-Testing\\xspace,\n\\dk{including\n$(\\alpha,\\beta)$-feedback functions and adversaries,} and outline the contribution of the paper.\nThen we discuss a related work on specific feedbacks in Section~\\ref{sec:related-future}.\nIn Section~\\ref{sec:upper} we prove upper bounds on the query complexity for efficient feedbacks with minimal, maximal and general expressiveness $\\beta$ and any capacity $\\alpha$, \\dk{under powerful Malicious Adversary}. Section~\\ref{sec:lower} presents a lower bound for feedbacks with maximum expressiveness, i.e., $(\\alpha,\\bar{\\alpha})$-feedbacks, \\dk{which holds even for more benign Honest adversaries.} A case study of two $(\\alpha,2\\log n)$-feedback functions with (provably) substantially different query performance is given in Section~\\ref{sec:case-study}. \nDiscussion of results from perspective of future directions is given in~Section~\\ref{sec:future}. \n\\section{Generalized framework and our contribution}\n\\label{sec:model}\n\n\\dk{As we will discuss in Sections~\\ref{sec:technical-results} and~\\ref{sec:related-future},\nmany previously considered variants of $(n, k)$-Group-Testing\\xspace problem could be expressed by, and their query\ncomplexities depend on, specific parameters of the feedback to the queries. Here,}\nwe formally introduce \\dk{generalized framework, including} families of $(\\alpha,\\beta)$-feedbacks, where $\\alpha$ is the feedback capacity\nwhile $\\beta$ is its expressiveness, \\dk{and adversaries that provide input to the feedback function in case the intersection has more than $\\alpha$ elements.} \n\\dk{We consider {\\em non-adaptive deterministic} solutions, \nin which subsequent queries do not depend on the feedback from the previous ones nor on random bits. \nThis class is very popular in the literature, due to its applicability and relevance to coding~\\cite{kautz1964nonrandom} and information theory~\\cite{erdos1963two,aldridge2019group}.}\n\n\\dk{In subsequent technical sections, we will be studying} query complexity of the whole classes of $(\\alpha,\\beta)$-feedbacks,\ndepending on parameters $n,k,\\alpha,\\beta$ \\dk{and specific adversary}, as well as several interesting sub-classes.  \n\\dk{We will also discuss randomized counterparts of our deterministic solutions, c.f., Definition~\\ref{def:randomized}, as in some cases they could be computed more~efficiently.}\n\n\nWe assume that the universe of all elements $N$, with $|N| = n$, is enumerated with integers $1,2,\\dots,n$. Throughout the paper we will associate an element with its identifier. \n\n\\paragraph{Specification of generalized Group Testing\\xspace framework}\n\\begin{definition}\n\\label{def:framework}\nThe generalized $(n, k)$-Group-Testing\\xspace framework is defined as follows:\n\\begin{enumerate\n\\item An $(n, k)$-Group-Testing\\xspace Algorithm is defined as a sequence of queries $\\mathcal{Q}=\\mathcal{Q}^{n,k} = \\langle Q_1,Q_2,\\dots,Q_t\\rangle$ depending on $n$, $k$, where each query is an arbitrary subset of $N$.\n\\dk{The sequence length $t$ is called a query complexity of the sequence\/algorithm.}\\footnote{%\n\\dk{Due to the scope of this paper, our definition considers non-adaptive algorithms, i.e., in which the sequence of queries is fixed in advance.\nHowever, an analogous framework can be defined for adaptive algorithm, in which consecutive queries are defined\nbased on the partial feedback vector, i.e., feedbacks on the preceding queries.}}\n\n\\item An adversary is defined as an entity that performs two actions. Firstly, it chooses set $K$ as an arbitrary subset of $N$ with $|K|\\leq k$. Secondly, it defines a\n function $\\mathsf{Adv}\\xspace(X,i \\mid \\mathcal{Q},K)$, for every $X \\subseteq N$, and every $i \\in \\{1,\\dots, |\\mathcal{Q}|\\}$, where $i$ denotes the index\\footnote{\\dk{This means that the adversary receives not only the whole sequence $\\mathcal{Q}$ but also the step number; hence, may output different values for two identical intersection sets but obtained for different queues.}} in sequence $\\mathcal{Q}$. \n This function must satisfy:\n \\begin{itemize}\n\\item\n$\\mathsf{Adv}\\xspace(X,i \\mid \\mathcal{Q}, K))\\subseteq N$,\n\\item\n$|\\mathsf{Adv}\\xspace(X,i \\mid \\mathcal{Q}, K))| \\leq \\alpha$,\n\\item\n $\\mathsf{Adv}\\xspace(X,i  \\mid \\mathcal{Q}, K) = X$, if $|X| \\leq \\alpha$.\n \\end{itemize}\n\\item An adversary strategy, \\dk{under a given query sequence $\\mathcal{Q}$ and a set $K$ fixed by an adversary\n} is defined as \n\\dk{a}\nfunction $\\mathsf{Adv}\\xspace(X,i \\mid \\mathcal{Q}, K)$ of two arguments: set $X \\subset N$ and index $i \\in \\{1,2,\\dots, |\\mathcal{Q}|\\}$. \n\\dk{$\\mathcal{S}_{adv}(\\mathcal{Q},K)$ denotes the set of all adversarial strategies under a given query sequence $\\mathcal{Q}$ and a set $K$ fixed by an adversary, and $\\mathcal{S}_{adv}\\dk{(\\mathcal{Q},\\cdot)=\\{\\mathcal{S}_{adv}(\\mathcal{Q},K)\\}_{K\\subseteq N, |K|\\le k}}$ is the set of all possible strategies of the adversary over sets $K$ of at most $k$ elements.\n\\item An $(\\alpha,\\beta)$-feedback function $\\mathsf{Feed}\\xspace$ is a function that takes as an input any subset of $N$ with at most $\\alpha$ elements and \n\\dk{outputs}\na binary vector of $\\beta$ bits.\n\\item A feedback vector is defined as a sequence of outputs of the feedback function on the intersections between $K$ and the subsequent queries $Q_1,Q_2,\\dots,Q_t$:\n\\begin{align*}\n\\mathcal{F}(K,\\mathsf{Adv}\\xspace) = &\\langle \\mathsf{Feed}\\xspace(\\mathsf{Adv}\\xspace(Q_1 \\cap K, 1 \\mid \\mathcal{Q}, K)), \\mathsf{Feed}\\xspace(\\mathsf{Adv}\\xspace(Q_2\\cap K, 2 \\mid \\mathcal{Q}, K)),\\dots,\\\\& \\mathsf{Feed}\\xspace(\\mathsf{Adv}\\xspace(Q_t \\cap K, t \\mid \\mathcal{Q}, K))\\rangle \\ ,\n\\end{align*}\n\n\\item For any fixed $n,k$, we say that a sequence of queries $\\mathcal{Q}$ solves $(n, k)$-Group-Testing\\xspace problem under some adversary with the set of possible strategies $\\mathcal{S}_{adv}\\dk{(\\mathcal{Q},\\cdot)}$\nif we have:\n\\begin{equation}\\nonumber\n \\bigforall_{\\substack{K_1, K_2 \\subset N\\\\ |K_1|, |K_2| \\leq \\alpha \\\\ K_1 \\neq K_2}} \\{\\mathcal{F}(K_1,\\mathsf{Adv}\\xspace) : \\mathsf{Adv}\\xspace \\in \\mathcal{S}_{adv}\\dk{(\\mathcal{Q},K_1)}\\} \\cap  \\{\\mathcal{F}(K_2,\\mathsf{Adv}\\xspace) : \\mathsf{Adv}\\xspace \\in \\mathcal{S}_{adv}\\dk{(\\mathcal{Q},K_2)}\\} = \\emptyset \n \\end{equation}\n\\dk{In other words, the sets of possible (under the given adversary) feedback vectors for two different sets $K_1,K_2$ are disjoint.}\n \\end{enumerate}\n\\end{definition}\n\n\n\\remove{\n-------------------------------------[REMOVE OLD DEFINITION]------------------------\\\\\n The {\\em $(n, k)$-Group-Testing\\xspace} algorithm in the model with $(\\alpha,\\beta)$-feedback is a sequence of queries $Q_1, Q_2, \\dots, Q_t$, where each $Q_\\tau \\subset N$. An $(\\alpha,\\beta)$-feedback function $\\mathsf{Feed}\\xspace$ is a function that takes as an input any subset of $N$ with at most $\\alpha$ elements and returns a binary vector of $\\beta$ bits as its output. A \\emph{feedback vector} is defined as a sequence of outputs of the feedback function on the intersections between $K$ and the subsequent queries $Q_1,Q_2,\\dots,Q_t$:\n\n($\\mathsf{Adv}\\xspace(Q_1 \\cap K | K,\\mathcal{Q})$)\n\\[\n\\mathcal{F}(K) = \\langle \\mathsf{Feed}\\xspace(\\mathsf{Adv}\\xspace(Q_1 \\cap K)), \\mathsf{Feed}\\xspace(\\mathsf{Adv}\\xspace(Q_2 \\cap K)),\\dots,\\mathsf{Feed}\\xspace(\\mathsf{Adv}\\xspace(Q_t \\cap K))\\rangle \\ ,\n\\]\nwhere $\\mathsf{Adv}\\xspace$ is an action of the adversary, for any subset of $N$.\n, observing the following rules: $\\mathsf{Adv}\\xspace(S) \\subseteq S$ and $|\\mathsf{Adv}\\xspace(S)| = \\min\\{|S|, \\alpha\\} $. \nIn other words (HERE DEFINITION OF AN ADVERSARY), the adversary is in the ``middle'' between $K \\cap Q$ and feedback. If $|K\\cap Q| \\leq \\alpha$, then the adversary has to pass the whole set $K \\cap Q$ to the feedback function. On the other hand if $|K\\cap Q| > \\alpha$, then the adversary, depending on the model (see Definition~\\ref{def:adversaries}), can pass to the feedback function an arbitrary subset of $K\\cap Q$ of size exactly~$\\alpha$ or a completely arbitrary subset of at most elements from the whole domain $N$. We remark that the adversary may be adaptive and does not have to pass the same set even if two queries yield identical intersections larger than $\\alpha$. \n\n\n A sequence of queries $\\mathcal{Q} = \\langle Q_1, Q_2, \\dots, Q_t \\rangle$\n is said to {\\em solve $(n, k)$-Group-Testing\\xspace} if regardless of the actions of the adversary and for any two different sets $K_1 \\neq K_2$ such that $|K_1|, |K_2| \\leq k$ we have $\\mathcal{F}(K_1) \\neq \\mathcal{F}(K_2)$. In other words, the feedback vector for any two sets $K_1, K^\"$ is different UNDER A GIVEN TYPE OF ADVERSARY IF regardless of the strategy of the (WHICH ADVERSARY?) adversary ({\\it i.e.,}\\xspace which $\\alpha$ elements it chooses from the intersections of sets with queries if they are larger than $\\alpha$).\\footnote{%\nFormally, each set $K$ may produce several different feedback vectors, depending on possibly different adversarial strategy \nwhen feedback capacity is exceeded ($|K\\cap Q|>\\alpha$). Therefore, the solution to the $(n, k)$-Group-Testing\\xspace problem must assure\nthat the families of admissible feedback vectors obtained for any sets $K_1,K_2$ are disjoint. It also implies that each set $K$\ncould be correctly encoded based on any feedback vector that could be obtained for this set.}\nWhen $\\mathcal{F}(K_1)$ differs from $\\mathcal{F}(K_2)$ on some position corresponding to query $Q$ we say that query $Q$\\emph{distinguishes} sets $K_1$ and $K_2$ under feedback $\\mathsf{Feed}\\xspace$ and adversary $\\mathsf{Adv}\\xspace$. \n\\\\-------------------------------------------------------------\\\\\n}\n\n\\dk{In the above framework, the adversary could be deterministic (if the set of strategies $\\mathcal{S}_{adv}(\\mathcal{Q},K)$ for any given $\\mathcal{Q},K$ is a single function, e.g., always passing an empty set to the feedback function for intersections larger than $\\alpha$) or non-deterministic (otherwise). The feedback function is always deterministic. \nObserve also that in case of non-adaptive algorithms considered in this work the order of queries does not matter from perspective of query complexity, but \nhelps in the analysis\nto relate queries with their corresponding feedbacks in the feedback vector.}\n\n\\paragraph{Decoding of elements}\n\\dk{It follows from our Definition~\\ref{def:framework}, point 6, of solving $(n, k)$-Group-Testing\\xspace problem that elements of the hidden set $K$ could be enlisted.\nA straightforward, though not computationally efficient way, would be to consider all possible sets $K$ of size at most $k$; then, for each of them -- consider a family of all possible adversarial strategies and compute\nfeedback vectors for them; finally, one could find among them a matching copy of the actual feedback vector.\nThis copy is in some computed family corresponding to a set $K$, which is the actual hidden set to be enlisted.\nThe correctness of this solution follows directly from Definition~\\ref{def:framework}, point 6: all possible feedback vectors obtained for all possible adversarial strategies are disjoint \nfor different sets $K$ \nof size at most $k$.\nIn this work we do not study more efficient decoding algorithms than the above mentioned method -- this topic could be an interesting and challenging future direction.}\n\n\n\n\n\\paragraph{Maximum capacity}\nThe intersection between a query and set $K$ has always at most $k$ elements, hence having $\\alpha$ larger than $k$ does not increase the power of the model compared to the case of $k = \\alpha$. Therefore, in all our results we assume that $k \\geq \\alpha$ (for a setting $\\alpha > k$ one could use a sequence of queries for $k = \\alpha$).\n\n\\paragraph{Maximum expressiveness}\nSimilarly, we may restrict our considerations to $\\beta \\leq \\bar{\\alpha}$ because of the following fact.\n\\begin{proposition}\n\\label{fct:expres}\nFor any feedback function $f_1$, there exists a feedback function $f_2$ with expressiveness $\\beta \\leq \\bar{\\alpha}$ such that for any two sets $K_1, K_2 \\subseteq N$, with $|K_1|, |K_2| \\leq \\alpha$, \n\\[\nf_1(K_1) = f_1(K_2) \\Leftrightarrow f_2(K_1) = f_2(K_2) \\ .\n\\]\n\\end{proposition}\n\n\n\n\n\n\n\n\\paragraph{Adversaries and feedback functions}\n\\label{results}\n\n\nIn this paper we consider the following adversaries and feedback functions. Note that one could consider also other types of adversaries and feedback~functions.\n\\begin{definition}\n\\label{def:adversaries}\nWe define the following two adversary \n\\dk{types:}\n\\begin{enumerate}\n\\item $\\alpha$-Malicious Adversary\\xspace. This adversary,  whenever for some query $Q$ we have $|Q\\cap K| > \\alpha$, choses an arbitrary subset of at most $\\alpha$ elements from set $N$ and passes this set to the feedback function. Effectively, such adversary has the power to choose an arbitrary\n value of feedback for queries that intersect with the hidden set $K$ on more than~$\\alpha$~elements.\n\\item $\\alpha$-Honest Adversary\\xspace.  This adversary, whenever for some query $Q$ we have $|Q\\cap K| > \\alpha$, choses a subset of exactly $\\alpha$ elements from set $Q\\cap K$ and passes this set to the feedback function.  \n\n\\begin{itemize}\n\\item $\\alpha$-Honest $x$-Avoiding Adversary\\xspace, is a special case of $\\alpha$-Honest Adversary\\xspace that for some element $x\\in N$, if $x \\in K \\cup Q$ and $|K\\cup Q| > \\alpha$ then the set chosen by the adversary does \\textbf{not} contain element $x$. In other words it hides element $x$, whenever possible. \n\\end{itemize}\n\n\\end{enumerate}\n\\end{definition}\n\n\\begin{definition}\n\\label{def:feedbacks}\nWe define the following three feedback functions:\n\\begin{enumerate}\n\\item $\\feed{\\alpha}(X) = (|X|\\text{ mod } 2)$. It is an $(\\alpha,1)$-feedback, function because the returned value can be encoded on one bit.  \n\\item $\\fullfeed{\\alpha}(X) = X$. It is an $(\\alpha,\\bar{\\alpha})$-feedback, as any subset of $N$ of at most $\\alpha$ elements can be encoded by $\\bar{\\alpha}$~bits.\n\\item $\\abfeed{\\alpha,\\beta}(X) = \\left(|X| \\text{ mod } 2\\right) \\bigparallel \\left(\\bigoplus_{x \\in X} \\mathsf{BCC}(x)\\right),$ where $\\mathsf{BCC}(x)$ is an  $\\left[n,\\beta- 1, \\left\\lfloor \\frac{\\beta - 1}{c \\log \\frac{n}{k}}\\right\\rfloor\\right]$-BCC code of element $x$, \\dk{c.f., Definition~\\ref{def:BCC},} and $c$ is a constant from~\\cite[Lemma 2]{Censor-HillelHL15}\n$\\bigoplus$ denotes bitwise XOR operation and $\\bigparallel$ denotes concatenation of vectors. It is an $(\\alpha,\\beta)$-feedback, because $BCC$ code uses $\\beta-1$ bits and the remaining bit denotes the parity of $|X|$.\n\\end{enumerate}\n\\end{definition}\n\n\n\\dk{The above Definition~\\ref{def:adversaries} of adversaries and the third defined feedback function in Definition~\\ref{def:feedbacks} have not been considered in the Group Testing\\xspace literature, to the best of our knowledge. \nWe will derive upper bounds under the strongest of the defined adversaries, $\\alpha$-Malicious Adversary\\xspace, while we also prove nearly matching lower bound(s) that holds also under the weaker $\\alpha$-Honest $x$-Avoiding Adversary\\xspace; thus, the power of the adversary does not have a substantial impact on the query complexity of Group Testing\\xspace.\n\nThe next observation specifies useful criteria for the analysis of algorithms against $\\alpha$-Malicious Adversary\\xspace, which we will apply in all our proofs of upper bounds.}\n\n\\begin{proposition}\n\\label{fct:technique}\nFix any $n,k$. If for query sequence $\\mathcal{Q}^{n,k}= \\left\\langle Q_{i}\\right\\rangle_{i=1}^{t}$ we have that for any $K_1,K_2 \\subset N$, with $|K_1|, |K_2| \\leq \\alpha$ and $K_1 \\neq K_2$: \n\\[\n\\bigexists_{\\tau} |Q_{\\tau} \\cap K_1| \\leq \\alpha  \\wedge  |Q_{\\tau} \\cap K_2| \\leq \\alpha \\wedge \\mathsf{Feed}\\xspace(Q_{\\tau} \\cap K_1) \\neq \\mathsf{Feed}\\xspace(Q_{\\tau} \\cap K_2) \\ ,\n\\]\nthen $\\mathcal{Q}^{n,k}$ solves $(n, k)$-Group-Testing\\xspace under $\\alpha$-Malicious Adversary\\xspace.\n\\end{proposition}\nIn the following we will say that a query $Q_{\\tau}$ \\emph{distinguishes} sets $K_1$  and $K_2$ under some feedback function $\\mathsf{Feed}\\xspace$ if $|Q_{\\tau}\\cap K_1| \\leq \\alpha$, $|Q_{\\tau}\\cap K_2|\\leq \\alpha$ and $\\mathsf{Feed}\\xspace(Q_{\\tau} \\cap K_1) \\neq \\mathsf{Feed}\\xspace(Q \\cap K_2)$.\n\\subsection{Technical results}\n\\label{sec:technical-results}\n\\paragraph{Binary feedback} First, we consider feedbacks with minimum possible expressiveness, namely, \nreturning\nonly one bit of information. In this setting we have to answer the question of \\emph{What is the most useful bit of information about a set of elements?} It turns out that a parity bit allows us to obtain an efficient solution in the family of $(\\alpha,1)$-feedbacks. Interestingly, this result, and all our other upper bounds, hold for the strongest adversary.\n\\begin{theorem}\n\\label{thm:binary}\nUnder $\\feed{\\alpha}$ feedback and under $\\alpha$-Malicious Adversary\\xspace, there exists a \\dk{deterministic} solution to $(n, k)$-Group-Testing\\xspace with query complexity $O\\left(\\left(k +\\frac{k^2}{\\alpha}\\right) \\cdot \\log \\frac{n}{k}\\right)$.\n\\end{theorem}\nThe proof is based on derandomization of random queries drawn from different random distribution, after proving\nthat these queries satisfy a certain Separation Property (formulated and proved in Lemma~\\ref{lem:telescope}).\n\n\\paragraph{Full feedback} \nOur second result is in the setting with\nthe \nmaximum possible expressiveness $\\beta=\\bar{\\alpha}=\\Theta(\\alpha\\log(n\/\\alpha))$, i.e., sufficient to return all identifiers of any set of size at most $\\alpha$. \nWe show that maximum expressiveness allows to design algorithms with small query complexity $O\\left(\\min\\left\\{\\frac{n}{\\alpha},\\frac{k^2}{\\alpha^2} \\cdot \\log^c n\\right\\}\\right)$ for some $c\\in [1,2]$, \\dk{more precisely:} \n\\begin{theorem}\n\\label{thm:fullupper}\nUnder $\\fullfeed{\\alpha}$ feedback and under $\\alpha$-Malicious Adversary\\xspace, \nthere exists a \\dk{deterministic} solution to $(n, k)$-Group-Testing\\xspace with query complexity:\n\\begin{align}\\nonumber\n& O\\left(\\min\\left\\{\\frac{n}{\\alpha},\\frac{k^2}{\\alpha^2} \\cdot \\log n\\right\\}\\right) & \\text{if } \\alpha > 18 \\log k, \\\\\\nonumber\n& O\\left(\\min\\left\\{\\frac{n}{\\alpha},\\frac{k^2}{\\alpha} \\cdot \\log\\frac{n}{k}\\right\\}\\right) & \\text{otherwise}.\n\\end{align}\n\\end{theorem}\nThe proof is via derandomization of a random sequence of queries $\\mathcal{Q}$, from which we require \nto simultaneously satisfy two conditions: on the number of queries containing a specific element, and on the sizes of the intersections of queries from any subset of $\\mathcal{Q}$ of certain size and any possible instantiation of set $K$.\n\nInterestingly, for $\\alpha =\\omega(\\sqrt{k\\log n})$, the obtained query complexity is sublinear in $k$. This can be contrasted with an $\\Omega(k \\log(n\/k))$ lower bound for classical Group Testing\\xspace{} \\dk{(i.e., for $\\alpha=O(1)$)} that holds also for any randomized algorithm working with non-vanishing probability~\\cite{coja2020information}. \\dk{This proves the impact of feedback capacity on query complexity.}\n\n\n\\paragraph{General feedback} \nAfter considering both extreme values of $\\beta$ we study the general case, where a feedback needs to work for an arbitrary $1\\le \\beta\\le \\bar{\\alpha}$. In this case our first contribution is a design of  a more sophisticated general feedback function $\\abfeed{\\alpha,\\beta}$, \nc.f., Definition~\\ref{def:feedbacks}, which works for almost any $\\alpha, \\beta$. Our proposed feedback is a concatenation of a specific code (called BCC code) with an additional parity bit. Under this feedback we obtain the main result of the paper:\n\\begin{theorem}\n\\label{thm:generalupper}\nUnder $\\abfeed{\\alpha,\\beta}$ feedback and under \n\\dk{$\\alpha$-Malicious Adversary\\xspace,}\nthere exists a \\dk{deterministic} solution to $(n, k)$-Group-Testing\\xspace with query complexity $O\\left(\\frac{k^2}{\\alpha\\beta}\\log^{c+1} n\\right)$ for some $c\\in [1,2]$,\n\\dk{more precisely:}\n\\begin{align}\\nonumber\n& O\\left(\\frac{k^2}{\\alpha\\beta}\\log n\\left(\\frac{\\beta}{\\alpha} + \\log n\\right)\\right) & \\text{if } \\alpha > 18 \\log k, \\\\\\nonumber\n& O\\left(\\frac{k^2}{\\alpha} \\cdot \\log\\frac{n}{k}\\right) & \\text{otherwise}.\n\\end{align}\n\n\\end{theorem}\nOur main result shows that the query complexity decreases linearly with $\\alpha$ and with $\\beta$. Intuitively factor $\\frac{k}{\\alpha}$ in our complexity comes from \\emph{congestion}, since the feedback function has capacity to serve at most $\\alpha$ elements out of $k$ in a single query. The second factor $\\frac{k \\log\\frac{n}{k}}{\\beta} \\approx \\frac{\\log {n \\choose k}}{\\beta}$ comes from the information-theoretic bound that we need $\\log_2 {n \\choose k}$ bits to uniquely encode any subset of $k$ elements and the fact that the feedback function provides only $\\beta$ bits per round. \nWhat is surprising and challenging to prove is that the query complexity of efficient (but not all!) $(\\alpha,\\beta)$-feedbacks is (close to) a multiplication of these two characteristics.\n\nThe proof combines ideas from the analysis of the binary feedback and full feedback. In the binary feedback case we observe that sets that differ on many elements can be distinguished quickly using the parity feedback. On the other hand, sets that differ only on few elements are handled using a combination of full feedback algorithm with a specific coding to encapsulate the feedback into $\\beta$ bits.\n\n\\paragraph{Lower bound}\nWe show a lower bound that proves that our upper bound shown in Theorem~\\ref{thm:fullupper} is optimal up to polylogarithmic factor, for any $\\alpha$.\n\\dk{It holds} even for a weaker adversary, \\dk{$\\alpha$-Honest Adversary\\xspace, or more specifically, for its sub-type of $\\alpha$-Honest $x$-Avoiding Adversary\\xspace. Thus, it also holds for the stronger $\\alpha$-Malicious Adversary\\xspace, for which all our algorithms are~analyzed.}\n\n\n\\begin{theorem}\n\\label{thm:fulllower}\nIf $n > k^2\\log n\/\\log k$, then any \\dk{deterministic} solution to $(n, k)$-Group-Testing\\xspace under any $(\\alpha,\\beta)$-feedback has query complexity \n$\\Omega\\left(\\frac{k^2}{\\alpha^2} \\log^{-1} k\\right)$ for some $\\alpha$-Honest Adversary\\xspace.\n\\end{theorem}\n\n\\dk{\nThe proof of Theorem~\\ref{thm:fulllower} is by transformation of our generalized Group Testing\\xspace framework to selectors -- structures studied in related literature, formally defined in Section~\\ref{sec:lower}. We show that if there were shorter query sequences, there would exist selectors violating some of their lower bound. This transformation is however possible only in one way, as we will show in the next result.}\n\n\\paragraph{Minimum Elements feedbacks} Our two final results show that designing an efficient feedback function is very subtle. We show that a reasonable $(\\alpha,2\\log n)$-feedback function that returns two minimal elements from the set leads to very large query complexity of $\\Omega(\\min\\{n,k^2\\})$ if we restrict the function to return the elements \\emph{in fixed order}, c.f., Theorem~\\ref{thm:lower-2min}. Without this restriction it is possible to obtain feedback function for which there exists a \\dk{deterministic} algorithm with query complexity $O\\left(\\frac{k^2}{\\alpha} \\cdot \\log \\frac{n}{k}\\right)$, \nc.f., Corollary~\\ref{cor:upper-2min}.\n\n\\dk{Theorem~\\ref{thm:lower-2min} with Corollary~\\ref{cor:upper-2min} provide an argument that there is no universal reduction between  selectors and our general Group Testing\\xspace framework, \nas both the considered feedback functions have the same parameters $\\alpha$ and $\\beta$ and differ only (slightly) in the definition of the feedback function, yet having query complexities different nearly by factor $\\alpha$. Thus, our general framework is provably more complex than the theory of selectors.}\n\n\n\\begin{table}\n\t\\centering\n\t\\begin{tabular}{llll}\n\t\t\\toprule\n\t\t$\\alpha$ & $\\beta$ & Upper bound & Lower bound \\\\\\midrule\n\t\t$1$ & $1$ & $O\\left(k^2 \\log\\frac{n}{k}\\right)$~\\cite{BonisGV03} & $\\Omega\\left(k^2 \\frac{\\log n}{\\log k} \\right)$~\\cite{ClementiMS01} \\\\\\midrule\n\t\t$k$ & $1$ & $O(k \\log \\frac{n}{k})$~\\cite{Censor-HillelHL15} & $\\Omega(k \\log \\frac{n}{k})$~\\cite{Censor-HillelHL15}  \\\\\\midrule\n\t\t$k$ & $\\log k$ & $O\\left(k \\frac{\\log \\frac{n}{k}}{\\log k}\\right)$~\\cite{GrebinskiK00} &$\\Omega\\left(k \\frac{\\log \\frac{n}{k}}{\\log k}\\right)$ ~\\cite{djackov1975search, lindstrom1975determining}\\\\\\midrule\n\t\t$*$  & $1$ & $O\\left(\\left(k + \\frac{k^2}{\\alpha}\\right) \\log\\frac{n}{k} \\right)$ Thm~\\ref{thm:binary} &$\\Omega\\left(\\frac{k^2}{\\alpha^2} \\log^{-1}k \\right)$ Thm~\\ref{thm:fulllower} \\\\\\midrule\n\n\t$*$  & $\\bar{\\alpha}$ & $O\\left(\\min\\left\\{\\frac{n}{\\alpha},\\frac{k^2}{\\alpha^2} \\log n  \\right\\}\\right)$ Thm~\\ref{thm:fullupper} & $\\Omega\\left(\\frac{k^2}{\\alpha^2} \\log^{-1}k \\right)$ Thm~\\ref{thm:fulllower} \\\\\\midrule\n\t\t$*$  & $*$ & $O\\left(\\frac{k^2}{\\alpha\\beta} \\log^2 n \\right)$ Thm~\\ref{thm:generalupper} & $\\Omega\\left(\\frac{k^2}{\\alpha^2} \\log^{-1}k \\right)$ Thm~\\ref{thm:fulllower}\\\\\t\t\\bottomrule\n\t\\end{tabular}\n\t\\caption{\\label{tab1} Results\n\ton non-adaptive $(n, k)$-Group-Testing\\xspace with $(\\alpha,\\beta)$-feedback. The upper bound column states query complexity of the best found $(\\alpha,\\beta)$-feedback found for \tparameters $\\alpha,\\beta$ fixed in the first two columns; as we will show, not all $(\\alpha,\\beta)$-feedbacks could reach that complexity. Symbol $*$ stands for any valid \nvalue of the parameter, and $\\bar{\\alpha}$ stands for a ceiling of the binary logarithm of the number of all subsets of $N$ of size at most $\\alpha$. We display results from Theorem~\\ref{thm:fullupper} and~\\ref{thm:generalupper} in regime $\\alpha > 18 \\log k$, however our theorems cover the whole range of $\\alpha$. }\n\t\\end{table}\n\n\nTable~\\ref{tab1} presents our main \\dk{deterministic} results in comparison to the most related previous work on specific feedback~functions.\n\\dk{In this work we also analyze explicitly constructed randomized counterparts of the deterministic results.}\n\n\\dpa{\n\\begin{definition}\n\\label{def:randomized}\nA randomized algorithm solves $(n, k)$-Group-Testing\\xspace against Adaptive Adversary with probability $1-c$, for some $0\\leq  c < 1$, if with probability $1-c$ it generates a sequence of queries $\\mathcal{Q}$ that solves $(n, k)$-Group-Testing\\xspace according to Defintion~\\ref{def:framework}.\n\\end{definition}\nNote that in Definition~\\ref{def:randomized} the adversary is assumed to know sequence $\\mathcal{Q}$ (see Defintion~\\ref{def:framework}(3)). Hence, \n\\dk{our analysis' of randomized counterparts of deterministic solutions also hold}\nagainst \\emph{Adaptive Adversary}. This is to distinguish from the case, where the adversary does not know all the queries when choosing set $K$~\\cite{BonisV17, bay2020optimal}.}\n\n\n\n\n\n  \n\n\n\n\\section{Lower bound}\n\nConsider first a binary feedback, $\\beta=1$.\n\nConsider queries $Q_1,\\ldots,Q_m$, for some $m=c\\cdot \\frac{k^2}{\\alpha^{3\/2}}$, for some constant $c>0$, that solve $(n,k)$-Group-Testing with $\\alpha$-oracle and $\\beta$-feedback.\nWe may assume that all queries have more than $1$ element; otherwise, we could exclude the singletons and deal with the remaining queries.\nConsider a randomly selected pair $\\{v,w\\}$ of different elements from $[n]$. \nFor any integer $1\\le m$, let $X_i(v,w)$ be equal to the size of $Q_i$ if $|Q_i\\cap \\{v,w\\}|=1$ and $0$ otherwise,\nand let $Y(v,w)=\\{i: X_i(v,w)>0\\}$.\n\n\\begin{lemma}\n\\label{l:expectedX}\n$E[\\sum_{i=1}^m X_i(v,w)] \\le ???$ and $E[Y(v,w)] ???$.\n\\end{lemma}\n\n\\begin{proof}\n$E[\\sum_{i=1}^m X_i(v,w)] = \\sum_{i=1}^m |Q_i|\\cdot \\frac{2|Q_i|(n-|Q_i|)}{n(n-1)}$\n\\end{proof}\n\nIt follows from Lemma~\\ref{l:expectedX} and the Markov inequality, that there is a pair $\\{v,w\\}$ such that\n$\\sum_{i=1}^m X_i(v,w)] \\le ???$ and $Y(v,w) ???$.\n\n\n\\section{Upper bounds}\\label{sec:upper}\n\n\\input{binary-feedback-upper}\n\n\\input{full-feedback-upper}\n\n\\input{general-feedback-upper}\n\n\n\\input{full-feedback-lower}\n\\input{min-feedback}\n\n\n\\input{discussion}\n\\bibliographystyle{abbrv}\n\n\\section{Some $(\\alpha,\\beta)$-feedbacks are better than others}\n\\label{sec:case-study}\n\\dk{One could be tempted to develop a similar universal reduction, \\dk{as in the proof of the lower bound in Section~\\ref{sec:lower}, also} for upper bounds -- between a setting with any $(\\alpha,\\beta)$-feedback \nand some strong selectors. However, in this section we show that such a reduction does not exist: we define two seemingly very similar feedback functions with the same values of $\\alpha, \\beta$ and show that the resulting query complexities for these two feedbacks are asymptotically very different.}\n\nConsider the following two feedback functions, both being $(\\alpha,\\beta)$-feedbacks for any $\\alpha\\le k$ and $\\beta = 2 \\lceil\\log_2 n \\rceil$. In the following we associate each element with its identifier. We assume that each identifier has exactly $\\lceil \\log_2 n \\rceil$ bits and is different from the string of~all~zeros.\n\\[\nF_1(S) =\n\\begin{cases}\n(\\min S) \\bigparallel (\\min(S \\setminus \\{\\min S\\}), \\quad \\text{if }2\\leq |S| \\leq \\alpha \\\\\n(\\min S) \\bigparallel 00\\dots 0, \\quad \\text{if } |S| = 1~. \\\\\n\\end{cases}\n\\]\n\\[\nF_2(S) =\n\\begin{cases}\n(\\min S) \\bigparallel  \\min(S \\setminus \\{\\min S\\}), \\quad \\text{if }2\\leq |S| \\leq \\alpha \\text{ and $|S|$ is odd} \\\\\n(\\min(S \\setminus \\{\\min S \\})) \\bigparallel  \\min S , \\quad \\text{if }2\\leq |S| \\leq \\alpha  \\text{ and $|S|$ is even} \\\\\n(\\min S) \\bigparallel 00\\dots 0, \\quad \\text{if } |S| = 1~. \\\\\n\\end{cases}\n\\]\nWe show that the query complexity of $(n, k)$-Group-Testing\\xspace with feedback $F_2$ is substantially lower than with feedback $F_1$.\n\n\n\\begin{corollary}\n\\label{cor:upper-2min}\nFor any $\\alpha \\leq k$, query complexity of $(n, k)$-Group-Testing\\xspace under feedback $F_2$ under $\\alpha$-Malicious Adversary\\xspace is $O\\left(\\frac{k^2}{\\alpha} \\log (n\/k) \\right)$.\n\\end{corollary}\n\n\\begin{proof}\nWe can see that under feedback $F_2$ we can deduce the parity bit from the feedback from every request with intersection at most $\\alpha$, by checking the order of the two outputted elements (note that if there is only one or no outputted elements, the parity is obvious). Hence, the corollary is a direct consequence of Theorem~\\ref{thm:binary}.\n\\end{proof}\n\\paragraph{Remark}\nAn unexpected inspiration for the feedback $F_2$ is a type of move, called \\emph{count signal}, used in contract bridge. Contract bridge is a card game, where players play in pairs (but without seeing non-revealed cards of other players) and sometimes it is crucial to exchange some information between the partners about their cards. The only way to disclose is by revealing (playing) the cards, but the \\emph{order} in which the cards are played can have some meaning. A count signal is exactly the $F_2$ feedback, where the parity of one player's cards (in some particular suit) is disclosed by the order in which he\/she plays the cards. For example a player holding $\\diamondsuit Q 963$ (meaning Queen, 9, 6, 3 in diamonds) plays $6$ and then $3$ to show even number of cards in diamonds. \n\n\nOn the other hand, returning the same two minimal values as in $F_2$ but always in order, results in a dramatic increase in the query complexity, \\dk{even under Honest Adversary.}\n\n\\begin{theorem}\n\\label{thm:lower-2min}\nFor any $k, \\alpha$, query complexity of $(n, k)$-Group-Testing\\xspace under feedback $F_1$ is $\\Omega(\\min\\{n,k^2\\})$ under $\\alpha$-Honest Adversary\\xspace.\n\\end{theorem}\n\\begin{proof}\nAssume, that we have a sequence of queries solving $(n, k)$-Group-Testing\\xspace. Let us denote the queries by $Q_1,Q_2,\\dots, Q_t$, where $t$ is the number of queries. Recall that $N$ denotes the set of all the elements.\n\nIf $k \\geq n\/2$, then assuming that we had $t < k \/ 2$, take an arbitrary set of $k$ elements $K \\subset N$. And observe that feedback to each query reveals at most two identifiers. The total number of identifiers revealed is $t \\cdot 2 < k$. Hence there is an element $x \\in K$ that is never returned by the feedback. It is easy to see that by the properties of the feedback function $F_1$, the feedbacks for set $K \\setminus \\{x\\}$ would be identical as for set $K$ for each query. Hence if $k \\geq n\/2$ we must have $t \\geq k\/2 \\geq n\/4$.\n\n \nLet us now consider the more interesting range of $k < n\/2$. We define sets of indices $T_{>1} = \\{\\tau :  |Q_\\tau| > 1\\}, T_{= 1} = \\{\\tau :  |Q_\\tau|  = 1\\}$. Denote the following set of elements:\n\\[\nM = \\bigcup_{\\tau \\in T_{=1}} Q_\\tau \\cup \\bigcup_{\\tau \\in T_{>1}}\\left( \\{ \\min Q_{\\tau} \\} \\cup \\{ \\min (Q_{\\tau} \\setminus \\min Q_{\\tau})\\} \\right)\n\\ .\n\\]\nWe know that $|M| \\leq 2t$, because we take at most $2$ elements from each query. Denote set $R= N \\setminus M$. Set $R$ are the elements from $N$ that are not smallest (or second smallest) in any of the queries.  \n\nIf we have $|R| < k$, then $n - 2t < k$ and since $k \\leq n\/2$ we have $t \\geq n\/4$. \n\nAssume that $|R| \\geq k$, consider $R$ ordered in the decreasing order of identifiers. Denote this ordering as $r_1,r_2,\\dots$ and let $R_{i,j} = \\{r_i,r_2,\\dots, r_j\\}$. Denote the indices of queries that include element $r_i$ as $T^{(r_i)}_{> 1} = \\{\\tau \\in T_{>1} | r_i \\in Q_\\tau\\}$.\n\nFor any $j \\in N$ and $i=1,2,\\dots,j$, define two sets of query indices:\n\\[\nA_j(i) = \\{\\tau \\in T^{r_i}_{>1}, |Q_{\\tau} \\cap R_{i+1,j}| = 0 \\}~,\n\\]\n\\[\nB_j(i) = \\{\\tau \\in T^{r_i}_{>1}, |Q_{\\tau} \\cap R_{i+1,j}| = 1 \\}~.\n\\]\nWe will prove the following: \\\\\n\\textit{Claim 1: For any $j,i$ we have $2 \\cdot |A_j(i)| + |B_j(i)| > k - j$.}\n\nAssume on the contrary that this does not hold for some particular $j,i$ and observe that then we can find for every query $Q_\\tau$ for $\\tau \\in A_j(i)$ two elements that belong to $Q_\\tau$ and are smaller than $r_i$. Take such two elements for each $\\tau \\in A_j(i)$. We have $2|A_j(i)|$ elements, call this set $A$. For every $\\tau \\in B_j(i)$ find one element that belongs to $Q_\\tau$ and is smaller than $r_i$. We take $|B_j(i)|$ such elements (one for each of $B_j(i)$) and call this set $B$. Consider two sets \n\\[\nS = R_{1,j} \\cup A \\cup B~,\n\\]\n\\[\nS^' = R_{1,j} \\cup A \\cup B \\setminus \\{r_i\\}~.\n\\]\nObserve that $|S| \\leq |S^'| \\leq  j + k -j = k$. We will compare the feedbacks for $S$ and $S^'$ and show that the feedbacks are identical for each query.\nNote that for every $\\tau \\in T^{(r_i)}_{> 1}$, $S^' \\cap Q_\\tau$ has at least two elements that are smaller than $r_i$. Hence if $|S \\cap Q_\\tau| \\leq \\alpha$ then surely $F_1(S^' \\cap Q_\\tau) = F_1(S \\cap Q_\\tau)$. If $|S \\cap Q_\\tau| > \\alpha$, there is a simple strategy of an adversary to ensure equal feedbacks. The adversary selects an arbitrary set $X$ with $|X| = \\alpha$ satisfying, $X \\subset S^' \\cap Q_\\tau$ and  $X \\subset S \\cap Q_\\tau$ and passes $X$ to the feedback function. Hence the feedback in step $\\tau$ is identical for both $S$ and $S^'$. Note that, since $r_i \\notin M$, then $r_i$ does not belong to any other query than the queries with indices in $ T^{(r_i)}_{> 1}$, hence we cannot distinguish $S$ from $S^'$. This means that the query sequence does not solve the set learning problem. We obtained a contradiction, which proves the claim.\n\\\\\nIn the next claim we prove that sets $A_j(i)$ and $B_j(i)$ are disjoint.\n\\\\\n\\textit{Claim 2: For any $j$, we have $A_j(i) \\cap A_j(i^') = \\emptyset$ and $B_j(i) \\cap B_j(i^') = \\emptyset$, for $i,i^' \\leq j$ and $i \\neq i^'$.}\n\n\nAssume on the contrary that for some $j$ and $i,i^' \\leq j$ we have $A_j(i) \\cap A_j(i^') \\neq \\emptyset$ and take arbitrary $\\tau^* \\in A_j(i) \\cap A_j(i^')$. Assume without loss of generality that $i < i^'$. By the definition of sets $A$ we have $r_i \\in Q_{\\tau^*}$ and $r_{i^'} \\in Q_{\\tau^*}$. Since $i<i^' \\leq j$ we also have $r_{i^'} \\in R_{i+1,j}$, hence $|Q_{\\tau^*} \\cap R_{i+1,j}| \\geq 1$ and $\\tau^* \\notin A_j(i)$ a contradiction. \nNow if $B_j(i) \\cap B_j(i^') \\neq \\emptyset$ then similarly take $\\tau^* \\in B_j(i) \\cap B_j(i^')$ and assume $i < i^'$. We have $r_i, r_{i^'} \\in Q_{\\tau^*}$. We know that $|Q_{\\tau^*} \\cap R_{i^' +1,j}| = 1$ thus $|Q_{\\tau^*} \\cap R_{i^',j}| = 2$. Which implies that $|Q_{\\tau^*} \\cap R_{i +1,j}| \\geq 2$ and $\\tau^* \\notin B_j(i)$. We obtained a contradiction proving the claim.\n\n\nWe fix $j^* = \\lfloor k\/2 \\rfloor$. From Claim 1 we have $2 \\cdot |A_{j^*}(i)| + |B_{j^*}(i)| > k \/ 2$ for each $i = 1,2,\\dots,j$. Sets $A_j(i)$ contain indices of queries hence using Claim 2 we get: \n\n\\[\nt \\geq \\left | \\bigcup_{i =1}^{j^*} A_{j^*}(i) \\right | =  \\sum_{i =1}^{j^*} \\left |A_{j^*}(i) \\right |~,\n\\]\n\\[\nt \\geq \\left | \\bigcup_{i =1}^{j^*} B_{j^*}(i) \\right | =  \\sum_{i =1}^{j^*}  \\left |B_j(i) \\right |~.\n\\]\nAdding up the above inequalities gives us:\n\\[\n3t \\geq 2\\cdot \\sum_{i =1}^{j^*}  \\left |A_{j^*}(i) \\right | +   \\sum_{i =1}^{j^*} \\left |B_{j^*}(i) \\right | = \\sum_{i=1}^{j^*} (2 |A_{j^*}(i) | + \\left |B_{j^*}(i) \\right |)  \\geq j^* \\cdot k \/ 2 \\geq k^2 \/ 4 - k\/2\n\\ .\n\\]\nThus, finally we get $t \\geq k^2 \/ 12 - k\/6$.\n\\end{proof}\n\n\\dpa{\n\\paragraph{Randomized counterpart \\dk{result}}\nIn Theorem~\\ref{thm:lower-2min} we show that any sequence that solves $(n, k)$-Group-Testing\\xspace under feedback $F_1$ under $\\alpha$-Honest Adversary\\xspace must have length of at least $\\Omega\\left(\\min\\{n,k^2\\}\\right)$. Thus, a randomized algorithm generating sequences that solve $(n, k)$-Group-Testing\\xspace under this feedback with at least a constant probability must have expected query complexity of $\\Omega\\left(\\min\\{n,k^2\\}\\right)$, \\dk{since each correct sequence must have such length (by Theorem~\\ref{thm:lower-2min}).}\n\\begin{corollary}\nFor any $k, \\alpha$, query complexity of any randomized solution to $(n, k)$-Group-Testing\\xspace under feedback $F_1$ is $\\Omega(\\min\\{n,k^2\\})$ under adaptive $\\alpha$-Honest Adversary\\xspace.\n\\end{corollary}\n}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}