diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzofel" "b/data_all_eng_slimpj/shuffled/split2/finalzzofel" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzofel" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction and Basic Idea}\\label{sec:intro}\nIt's well known in plasma physics community that for linear (waves\nand instabilities) problems we have two usual computational methods:\neigenvalue (dispersion relation) solver and initial value solver.\nFor the former, we transform the linear time derivative $\\partial\n\/\\partial t$ and spatial derivatives $\\nabla$ to spectral space\nusing $-i\\omega$ and $i{\\bf k}$; for the latter, we solve the\noriginal equations directly. In some simple case, the eigenvalue\nmethod can be reduced to analytic tractable form, e.g., many well\nknown dispersion relations are this type. However, numerical\nsolutions are always OK (except some singularity cases).\n\nUsually, the eigenvalue method is not intuitive and one needs be\ngood at theoretical derivations; the conventional initial value\nmethod is complicated in computation and cumbersome in data\nanalysis. Typically, the eigenvalue method can give all solutions of\nthe system, while the initial value method can only give the\n$\\gamma_{max}$ (most unstable) solution.\n\nCan we combine these two methods? The answer is yes. We can keep\ntime derivative $\\partial \/\\partial t$ term (and other term if have,\ne.g., kinetic $\\partial \/\\partial {\\bf v}$ term), but transform the\nlinear spatial derivatives $\\partial \/\\partial {\\bf x}$ term to\n$i{\\bf k}$, which is still an initial value method, but is solved in\nhalf (only spatial not temporal) spectral space, then also has\ncharacteristics of eigenvalue method. For example, we can highlight\nthe non-$\\gamma_{max}$ solutions.\n\nWe will show how to do it with examples for normal mode problems in\nSec.\\ref{sec:normal}. Since this method has been used for linear\ninhomogeneous eigenvalue and nonlinear problem by previous\nresearchers, we will just give some necessary descriptions with\ncitations in Sec.\\ref{sec:eigen}.\n\n\\section{Normal Mode Paradigms}\\label{sec:normal}\nIn most literatures, normal mode and eigenmode are treated as a same\nconcept since they are very similar. But, here, we distinguish them\n\\begin{description}\n \\item[Normal mode:] homogeneous\\cite{dw}, without boundary conditions;\n \\item[Eigen mode:] inhomogeneous or with boundary conditions,\n all possible solutions of the system can be expressed by\n proper sum of eigenmodes.\n\\end{description}\n\nUsing Half Spectral (HS) method, many normal mode problems can be\nreduced from PDEs to ODEs. Then can be solved extreme easily. While,\nunfortunately, for eigen mode problems, the equations are in\nlower-dimensions but still PDEs.\n\n\\subsection{Simple example}\nWe construct a simple example to show how to use this method. The\noriginal equations\n\\begin{equation} \\label{eq:simple_hs_1}\n \\left\\{ \\begin{aligned}\n \\frac{\\partial {f_1}}{\\partial t} + {u_a}\\frac{\\partial {f_1}}{\\partial x} + {u_b}\\frac{\\partial {f_2}}{\\partial x} &=\n 0, \\\\\n \\frac{\\partial {f_2}}{\\partial t} + {u_b}\\frac{\\partial {f_1}}{\\partial x} + {u_a}\\frac{\\partial {f_2}}{\\partial x} &=\n 0.\n \\end{aligned} \\right.\n\\end{equation}\nHalf spectral solves\n\\begin{equation} \\label{eq:simple_hs_hs}\n \\left\\{ \\begin{aligned}\n \\partial {f_1}\/\\partial t &= - (ik{u_a}{f_1} + ik{u_b}{f_2}), \\\\\n \\partial {f_2}\/\\partial t &= - (ik{u_a}{f_1} + ik{u_b}{f_2}.\n \\end{aligned} \\right.\n\\end{equation}\nThe dispersion relation\n\\begin{equation} \\label{eq:simple_hs_dr}\n \\left\\{ \\begin{aligned}\n (\\omega - k{u_a}){f_1} = k{u_b}{f_2}, \\\\\n (\\omega - k{u_a}){f_2} = k{u_b}{f_1}.\n \\end{aligned} \\right.\n\\end{equation}\ngives, ${\\omega _ \\pm } = k({u_a} \\pm {u_b}),~{f_1} = \\pm {f_2}$.\n\nIf we assume the initial values ${f_1} = y{f_2}$, the ratio of\n$\\omega_\\pm$ is $x$ and $1-x$ respectively, then $x-(1-x)=y\n\\Rightarrow x = (y + 1)\/2$, then the ratio of the amplitudes for\n$\\omega_\\pm$ is\n\\begin{equation} \\label{eq:simple_hs_Amp}\n{{{{\\rm{A}}_ + }} \\over {{{\\rm{A}}_ - }}} = \\left| {{x \\over {1 -\nx}}} \\right| = \\left| {{{y + 1} \\over {y - 1}}} \\right|,\n\\end{equation}\nwhich means we can control the amplitudes of each modes of the\nsystem exactly by set the proper initial values.\n\n\\begin{figure}\n \\includegraphics[width=14cm]{simple_hs.pdf}\\\\\n \\caption{RK4 to solve eq.(\\ref{eq:simple_hs_hs}), as example for half spectral method, red line is dispersion relation solutions}\\label{fig:simple_hs}\n\\end{figure}\n\nA 4th order Runge-Kutta simulation of eq.(\\ref{eq:simple_hs_hs}) is\nshown in Fig.\\ref{fig:simple_hs}. We can see the frequencies and\namplitudes of each modes ($\\omega_\\pm=0.24,0.16$, $A_\\pm=0.2,0.1$)\nare exact as predict. A small mismatch should be caused by the\nnumerical discrete.\n\n\\subsection{ES1D kinetic problem}\nUsually, we have two initial value method to simulate kinetic\nproblem (e.g., Landau damping), i.e., Vlasov continuity solver and\nPIC method.\n\nTo show that half spectral method is not only for fluid problem, we\ngive a kinetic example. At this subsection, the electrostatic 1D\nLandau damping and bump-on-tail simulations are given.\n\nThe linearized equations (ion immobile) are\n\\begin{equation} \\label{eq:es1d}\n \\left\\{ \\begin{aligned}\n \\frac{\\partial \\delta f}{\\partial t} &= - ikv\\delta f + \\frac{e}{m}E\\frac{\\partial {f_0}}{\\partial v}, \\\\\n ik\\delta E &= - e\\int {\\delta fdv}.\n \\end{aligned} \\right.\n\\end{equation}\nwhich gives\n\\begin{equation} \\label{eq:es1d_df}\n\\delta f = \\frac{ieE}{m}\\frac{{\\partial _v}{f_0}}{\\omega - kv},\n\\end{equation}\ncontains both normal mode (which is independent with initial value)\nand ballistic mode (which is brought by initial value) or phase\nmixing (see e.g., \\cite{Krall1973}).\n\n\\begin{figure}\n \\includegraphics[width=12cm]{landaudamping.pdf}\\\\\n \\caption{Solve eq.(\\ref{eq:es1d}) for Landau damping}\\label{fig:landaudamping}\n\\end{figure}\n\n\\begin{figure}\n \\includegraphics[width=10cm]{bumpontail.pdf}\\\\\n \\caption{Solve eq.(\\ref{eq:es1d}) for beam-plasma instability}\\label{fig:bumpontail}\n\\end{figure}\n\nThe initial distribution function $f_0$ can be any form, e.g.,\nMaxwellian gives Landau damping, bump-on-tail gives beam-plasma\ninstabilities.\n\nFig.\\ref{fig:landaudamping} shows the simulation of Landau damping.\nOne can find very similar results (especially the fourth panel) from\nVlasov continuity simulation. However, we should notice an\nunfavorite recurrence effect\\cite{Cheng1976} caused by discrete\n$\\Delta v$, which is also found in half spectral simulation. The\nrecurrence time $T_R=2\\pi\/k\\Delta v$.\n\nFig.\\ref{fig:bumpontail} is the simulation of bump-on-tail problem.\nSince we treat linear problem, the unit of the amplitude can be\narbitrary large.\n\nOne can find, comparing with continuity solver and PIC method, using\nhalf spectral method for Landau damping simulation is extreme simple\nand can be more accurate. Comparing with numerical or analytical\ndispersion relation solver, half spectral method does not need treat\ntroublesome or confusing integral contours. Especially, when the\ninitial distribution function is not standard and the dispersion\nrelation is hard to solve, the half spectral simulation can gives an\nreasonable solution for benchmark more complicated codes.\n\n\\subsection{MHD waves}\nFor MHD waves, we solve\n\\begin{equation} \\label{eq:mhd}\n \\left\\{ \\begin{aligned}\n {{\\partial \\delta \\rho } \\over {\\partial t}} &= - i{\\rho _0}{\\bf{k}} \\cdot \\delta {\\bf{u}}, \\\\\n {\\rho _0}{{\\partial \\delta {\\bf{u}}} \\over {\\partial t}} &= {i \\over {{\\mu _0}}}({\\bf{k}} \\times \\delta {\\bf{B}}) \\times {{\\bf{B}}_0} - i{\\bf{k}}v_s^2\\delta \\rho, \\\\\n {{\\partial \\delta {\\bf{B}}} \\over {\\partial t}} &= i{\\bf{k}} \\times (\\delta {\\bf{u}} \\times {{\\bf{B}}_0}).\n \\end{aligned} \\right.\n\\end{equation}\nwhere ${{\\bf{B}}_0} = (0,0,{B_0})$, ${\\bf{k}} = (k\\sin \\theta\n,0,k\\cos \\theta )$, $v_s^2 = \\gamma {p_0}\/{\\rho _0} = \\gamma\nk{T_0}\/m$, $v_A^2 = B_0^2\/{\\mu _0}{\\rho _0}$, ${v_p} = \\omega \/k$.\n\nThree solutions are fast mode, slow mode and shear Alfv\\'en wave\n(one can find introductions of them in textbooks, e.g.,\n\\cite{Gurnett2005})\n\\begin{equation} \\label{eq:mhd_waves}\n \\left\\{ \\begin{aligned}\n v_p^2 &= {1 \\over 2}(v_A^2 + v_s^2) + {1 \\over 2}{\\left[ {{{(v_A^2 - v_s^2)}^2} + 4v_A^2v_s^2{{\\sin }^2}\\theta } \\right]^{1\/2}}, \\\\\n v_p^2 &= {1 \\over 2}(v_A^2 + v_s^2) - {1 \\over 2}{\\left[ {{{(v_A^2 - v_s^2)}^2} + 4v_A^2v_s^2{{\\sin }^2}\\theta } \\right]^{1\/2}}, \\\\\n v_p^2 &= v_A^2{\\cos ^2}\\theta.\n \\end{aligned} \\right.\n\\end{equation}\n\n\\begin{figure}\n \\includegraphics[width=15cm]{mhd_waves.pdf}\\\\\n \\caption{Solve eq.(\\ref{eq:mhd}) for ideal MHD waves, red line is dispersion relation solutions}\\label{fig:mhd_waves}\n\\end{figure}\n\nA simulation result is shown in Fig.\\ref{fig:mhd_waves}. Again, we\ncan find the simulation result exactly matches the theoretical\nsolutions. The simulation is intuitive. The frequency signal in the\nsecond panel is taken from $\\delta\\rho$ and $\\delta u_y$. If we only\nuse $\\delta\\rho$ signal, the shear Alfv\\'en wave solution will\nvanish.\n\nIf one want to go non-ideal MHD, e.g., including resistivity or\nusing anisotropic pressure $\\delta p_{\\parallel} \\neq \\delta\np_{\\bot}$, but wouldn't like to do analytical derivations, then half\nspectral method is an useful choice: it is very simple, intuitive\nand can give solutions exact enough.\n\nSince all information for linear perturbation variables is kept in\nthe simulation, we can use them for many more deeply analysis, e.g.,\nthe polarization and so on.\n\n\\subsection{EM cold plasma waves}\nEquations are\n\\begin{equation} \\label{eq:cold_waves}\n \\left\\{ \\begin{aligned}\n \\frac{\\partial \\delta {\\bf{v}_s}}{\\partial t} &= \\frac{e_s}{m_s}[\\delta {\\bf{E}}+\\delta {\\bf{v}_s}\\times\\delta {\\bf{B}}], \\\\\n \\frac{\\partial \\delta {\\bf{E}}}{\\partial t} &= ic^2{\\bf{k}} \\times \\delta {\\bf{B}} - \\delta{\\bf{J}}\/\\epsilon_0, \\\\\n \\frac{\\partial \\delta {\\bf{B}}}{\\partial t} &= -i{\\bf{k}} \\times \\delta {\\bf{E}}.\n \\end{aligned} \\right.\n\\end{equation}\nwhere $\\delta{\\bf{J}}=\\sum_{s} n_{s0}e_s\\delta {\\bf{v}_s}$. And,\n${{\\bf{B}}_0} = (0,0,{B_0})$, ${\\bf{k}} = (k\\sin \\theta ,0,k\\cos\n\\theta )$, $\\omega_{cs}=e_sB_0\/m_s$ and\n$\\omega_{ps}=n_sq_s^2\/\\epsilon_0m_s$.\n\nFor one ion species, the final dispersion relation can be reduced\n(with heavy calculations) to a fifth order equation for $\\omega^2$\n(see \\cite{Swanson2003} for details). While, using half spectral\nsimulation, this is very easy. A result is shown in\nFig.\\ref{fig:em_cold_waves}\n\n\\begin{figure}\n \\includegraphics[width=15cm]{em_cold_waves.pdf}\\\\\n \\caption{Solve eq.(\\ref{eq:cold_waves}) for EM cold plasma waves, red line is dispersion relation solutions}\\label{fig:em_cold_waves}\n\\end{figure}\n\nIf we have more than one ion species or with beams, the dispersion\nrelation can be very headache even though the problem seems not that\ncomplicated. While, using half spectral method, the problem is\nindeed still very easy.\n\n\\subsection{Summary}\nFor normal mode problem, half spectral method may be the simplest\nmethod for simulating them, which is intuitive, simple and also\nexact enough. For analytical difficult problems, this method can not\nonly be an intuitive tool but also has practical usages.\n\n\\section{Eigenmode and Nonlinear Problems}\\label{sec:eigen}\nAs claimed at the title, half spectral method can be a general\n(linear) plasma simulation method. So, we also need discuss the\neigenmode problem and some nonlinear treatments. While, it is found\nin the literatures that previous researchers have given many\nexamples of this. So, here we just give a short description and\nmention some citations.\n\n\\subsection{Tearing mode}\nEigenmode problems are similar. We take collisional tearing mode as\nexample here.\n\nA simulation matches the half spectral idea is given by Lee and\nFu\\cite{Lee1986} (see also citations of that paper). For simulation,\nwe solve\n\\begin{equation} \\label{eq:tearing}\n \\left\\{ \\begin{aligned}\n {{\\partial \\delta \\rho } \\over {\\partial t}} &= - {{\\partial \\rho } \\over {\\partial x}}\\delta {u_x} - \\rho {{\\partial \\delta {u_x}} \\over {\\partial x}} - i\\alpha \\rho \\delta\n {u_z}, \\\\\n {{\\partial \\delta {u_x}} \\over {\\partial t}} &= - {\\beta \\over {2\\rho }}{{\\partial \\delta p} \\over {\\partial x}} - {{{B_z}} \\over \\rho }{{\\partial \\delta {B_z}} \\over {\\partial x}} - {1 \\over \\rho }{{\\partial {B_z}} \\over {\\partial x}}\\delta {B_z} + i\\alpha {{{B_z}} \\over \\rho }\\delta\n {B_x}, \\\\\n {{\\partial \\delta {u_z}} \\over {\\partial t}} &= - i\\alpha {{\\beta \\delta p} \\over {2\\rho }} + {1 \\over \\rho }{{\\partial {B_z}} \\over {\\partial x}}\\delta\n {B_x}, \\\\\n {{\\partial \\delta {B_x}} \\over {\\partial t}} &= i\\alpha {B_z}\\delta {u_x} + {1 \\over {{R_m}}}{{{\\partial ^2}\\delta {B_x}} \\over {\\partial {x^2}}} - {{{\\alpha ^2}} \\over {{R_m}}}\\delta\n {B_x}, \\\\\n {{\\partial \\delta {B_z}} \\over {\\partial t}} &= - {{\\partial {B_z}} \\over {\\partial x}}\\delta {u_x} - {B_z}{{\\partial \\delta {u_x}} \\over {\\partial x}} + {1 \\over {{R_m}}}{{{\\partial ^2}\\delta {B_z}} \\over {\\partial {x^2}}} - {{{\\alpha ^2}} \\over {{R_m}}}\\delta\n {B_z}, \\\\\n {{\\partial \\delta p} \\over {\\partial t}} &= - {{\\partial p} \\over {\\partial x}}\\delta {u_x} - \\gamma p{{\\partial \\delta {u_x}} \\over {\\partial x}} - i\\alpha \\gamma p\\delta {u_z}.\n \\end{aligned} \\right.\n\\end{equation}\nwhere parameter are $\\alpha = kl$, $\\beta = 2{\\mu _0}{p_\\infty\n}\/B_\\infty ^2$, ${R_m} = {v_A}l\/\\eta$, $\\gamma$. The normalization\nunit are ${B_\\infty }$, ${\\rho _\\infty }$, ${v_A} = B_\\infty ^2\/{\\mu\n_0}{\\rho _\\infty }$, ${p_\\infty }$, $l$, ${t_0} = l\/{v_A}$.\n\nThe results are very well in that paper\\cite{Fu1995}. So, we can\ntrust that half spectral method is also good for eigenmode problem.\n\nA bad thing is that, for eigenmode problem, we still need solve\nPDEs.\n\n\\subsection{Hasegawa-Mima equation}\nIn fact, an usual way for nonlinear drift wave turbulence by\nHasegawa-Mima equation\\cite{Hasegawa1978} simulation is doing in\nspectral space, which is exact the half spectral idea of this\nmanuscript. A very detailed introduction can be found in Waltz's\nlecture notes\\cite{Waltz1986}. To give a rough impression, one of\nthe equations is shown below\n\\begin{equation}\\label{eq:hm}\n \\frac{d}{dt}\\phi_k(t)=(-i\\omega_{0k}+\\gamma_k)\\phi_k(t)+\\frac{1}{2}\\sum_{k_1k_2}\\delta(k-k_1-k_2)V_{kk_1k_2}\\phi_{k_1}(t)\\phi_{k_2}(t).\n\\end{equation}\n\nWe should comment here, for nonlinear problem, we need sum all $k$\nmodes and the complex conjugate should also be kept.\n\n\\section{Summary and Comments}\nIn this manuscript, we discussed the idea of half spectral method,\nand showed that it can be a general simulation method. However, this\nidea maybe not new, especially for eigenmode and nonlinear problem,\nmany previous researchers used it. I don't know whether this method\nis new for normal mode problems yet. And the name {\\it half\nspectral} using here is just for convenient.\n\nFor tokamak (or other problems with strong guide field) simulation,\na similar method is called flux tube (e.g., \\cite{Peeters2009}),\nwhich using the idea of symmetry to reduce dimensions because the\nphysics is mainly along magnetic field line then we can take it as\none coordinate. Poloidal and toroidal mode numbers $m$ and $n$ are\noften used in flux tube simulation. I haven't checked whether the\nequations for flux tube simulation are the same for half spectral\nsimulation yet. They may be exact the same or at least very similar.\n\nBut, we can tell that spectral or pseudo spectral method are\ndifferent from this half spectral method, because that they need\ntransform back to real space. For a same simulation, spectral or\npseudo spectral method can be alternative by other discrete methods\n(e.g., finite difference) and won't influent the simulation results.\nWhile, half spectral method is independent on discrete methods.\n\nThis idea is partly inspired by ``{\\it A 2D Hasegawa-Mima Model of\nElectrostatic Drift Wave Turbulence}\" simulation by Deng ZHAO (PKU,\n2011) and P239C course project (UCI, 2010) ``{\\it Cold plasma-warm\nbeam interaction}\" given by Prof. Liu CHEN. The basic idea is the\nsame. The new here is that we find this method can be generalized.\n\nIf we do not care that whether this idea is new or not, we can find\nhalf spectral method is simple, interesting and useful. MATLAB codes\nfor this manuscript are given in attached files.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nChaotic advection (or Lagrangian chaos) was introduced by Aref (1988) to qualify a motion in which it is possible to generate chaotic trajectories even if the flow is laminar. Such phenomena are observed in a wide range of physical systems (Ottino (1989)) and has fundamental applications for instance in geophysical flow (Behringer, et al. (1991); Brown and Smith (1991)). \nChaotic advection was mostly studied for two dimensional unsteady flow \n(Solomon and Gollub (1988); Solomon, et al. (2003); Camassa and Wiggins (1991)). \nThe advantage of these studies is that it uses the theory of dynamical systems and in particular the Hamiltonian theory for incompressible flows; the phase space being the physical space in this case. For these motions, experiments can be implemented in a laboratory. \\\\\nEven though chaos is\nsometimes preferable in mixing problems (Benzekri, et al.(2006)), we usually want to suppress it for instance in chaotic advection.\nRecently, a method to control chaotic diffusion in Hamiltonian dynamical systems was proposed by (Vittot, et al. (2005); Chandre, et al.(2005)). \nIt was shown that it is possible to prevent chaotic diffusion by adding a small term to the Hamiltonian. This technique was used to a model describing anomalous transport in magnetized plasmas, and applied experimentally on a beam of electrons produced by a long Travelling Wave Tube (Chandre, et al.(2005)).\\\\\nOur goal in this paper is to use this method of control of Hamiltonian chaos for the problem of chaotic advection in a two dimensional time periodic flow. We apply this method within the framework of an experiment using magneto-hydrodynamic technique which shows that particle trajectories in a time periodic flow are chaotic. More precisely, the experiment consists in an electric current passing through a thin layer of an electrolytic solution with a free surface. The dynamics of passive particles of a time periodic two dimensional and incompressible flow is Hamiltonian. The Hamiltonian modeling the flow was derived ad hoc in (Solomon and Gollub (1988); Solomon, et al. (2003)). A comparison was made with the experiment to validate the model. The equation of motion of these passive particles comes from a Hamiltonian of $1.5$ degrees of freedom that we study in the limit of weak amplitude oscillations.\\\\\n The model proposed to describe this phenomena is \nbased on the following streamfunction:\n\\begin{eqnarray}\n\\label{1}\n\\Psi_{\\epsilon}(x,y,t)=\\alpha \\sin(x + \\epsilon \\sin t) \\sin y,\t\n\\end{eqnarray}\nwhere \n$\\alpha$ is the maximal vertical velocity in the flow, \n$\\epsilon$ is the amplitude of the lateral oscillations of the velocity field. \nThe current interacts with an alternative magnetic field produced by magnets below the fluid. A chain of vortices are then observed. Time periodic dependence is imposed externally with a plunger that oscillates slowly up and down, displacing the flow laterally, i.e., in the direction perpendicular to the roll axes, giving rise to chaotic advection. \nThe same phenomenon is observed in Rayleigh-B\\'enard convection.\nFor the steady state, rolls are formed periodically. It has been shown that by imposing a sinusoidal flow, chaotic advection occurs.\\\\\nUnder the perturbation, i.e. for $\\epsilon\\neq0$, the vertical heteroclinic connection breaks down and the stable and unstable manifold intersect transversely thus generating chaotic advection of passive particles.\\\\ \nIn this method, \n we seek for one of the simplest perturbations to create barriers around the broken separatrix\n of the integrable case. We choose a perturbation depending only on the position variables.\\\\\nWe will use the stream function (\\ref{1}) as the starting point to control or reduced the chaotic advection.\n\n\\section{Local control method}\n\\label{sec:control local} \nThe local control method has been extensively described in (Vittot, et al. (2005)) where the corresponding rigorous results were proved. We summerize here the results of this paper.\nFor a Hamiltonian system with $L$ degrees of freedom, the perturbed Hamiltonian is \n$$\nH({\\bf A},{\\bm\\theta})={\\bm \\omega}\\cdot {\\bf A}+ V({\\bf A},{\\bm\\theta}),$$ \nwhere $({\\bf A},{\\bm\\theta})\\in {\\mathbb R}^L\\times {\\mathbb T}^L$ are action-angle like variables and $\\bm\\omega$ is a non-resonant vector of ${\\mathbb R}^L$. \nWithout loss of generality, let us consider a region near ${\\bf A}={\\bf 0}$ (by translation), since the Hamiltonian is nearly integrable, the perturbation $V$ has constant and linear parts in actions of order $\\varepsilon$, i.e.\\ \n\\begin{equation}\n\\label{eqn:e4V}\nV({\\bf A},{\\bm\\theta})=\\varepsilon v({\\bm\\theta})+\\varepsilon {\\bf w}({\\bm\\theta})\\cdot {\\bf A}+Q({\\bf A},{\\bm\\theta}),\n\\end{equation} \nwhere $Q$ is of order $O(\\Vert {\\bf A}\\Vert ^2)$. Note that for $\\varepsilon=0$, the Hamiltonian $H$ has an invariant torus with frequency vector ${\\bm\\omega}$ at ${\\bf A}={\\bf 0}$ for any $Q$ not necessarily small.\nThe controlled Hamiltonian is then constructed as:\n\\begin{equation}\n\\label{eqn:gene}\nH_c({\\bf A},{\\bm\\theta})={\\bm \\omega}\\cdot {\\bf A}+ V({\\bf A},{\\bm\\theta})+ f({\\bm \\theta}).\n\\end{equation}\n In this situation, the control term $f$ only depends on the angle variables and is given by\n\\begin{equation}\n\\label{eqn:exf}\nf({\\bm\\theta})=V({\\bf 0},{\\bm\\theta})-V\\left( -\\Gamma \\partial_{\\bm\\theta} V({\\bf 0},{\\bm\\theta}),{\\bm\\theta}\\right),\n\\end{equation}\nwhere $\\Gamma$ is a linear operator defined as a pseudo-inverse of ${\\bm\\omega}\\cdot \\partial_{\\bm\\theta}$, i.e.\\ acting on $V=\\sum_{{\\bf k}}V_{{\\bf k}} {\\mathrm e}^{i{\\bm\\theta}\\cdot{\\bf k}}$ as\n\\begin{equation}\n\\label{gam}\n\\Gamma V=\\sum_{{\\bm\\omega}\\cdot{\\bf k}\\not= 0} \\frac{V_{{\\bf k}}}{i{\\bm\\omega}\\cdot{\\bf k}} {\\mathrm e}^{i{\\bm\\theta}\\cdot{\\bf k}}.\n\\end{equation}\nNote that $f$ is of order $\\varepsilon^2$. This can be seen from Eq.~(\\ref{eqn:e4V}) since $f$ can be rewritten as\n$$\nf({\\bm\\theta})=\\varepsilon^2 {\\bf w}({\\bm\\theta})\\cdot \\Gamma \\partial_{\\bm\\theta} v-Q\\left( -\\varepsilon \\Gamma \\partial_{\\bm\\theta} v, {\\bm\\theta} \\right),\n$$\nand $Q$ is quadratic in the actions.\nFor any perturbation $V$, Hamiltonian~(\\ref{eqn:gene}) has an invariant torus with frequency vector close to ${\\bm\\omega}$.\nThe equation of the torus which is constructed by the control is\n\\begin{equation}\n\\label{eqn:eto}\n\t{\\bf A}=-\\Gamma \\partial_{\\bm\\theta} V({\\bf 0},{\\bm\\theta}),\n\\end{equation}\nwhich is of order $\\varepsilon$ since $ V({\\bf 0},{\\bm\\theta})$ is of order $\\varepsilon$.\n\nIn the next section, we will see that the amplitude of the control term is small compared with the perturbation. \n\n\\subsection{Application to the suppression of chaotic advection}\n\n\nIn this section, we apply the method previously summerized to reduce chaotic transport of passive tracers. With this method, we create isolated barriers of transport. In particular, a barrier created between two convection rolls allows one to reduce the diffusion across these rolls. Let us first give some results in the integrable case.\\\\\nFor $\\epsilon=0$, the Hamiltonian is integrable, the trajectories of advected particles coincide with streamlines. The phase space, which is here the physical space, is characterized by a chain of rolls with separatrices localized at $x=m \\pi$, with $m \\in \\mathbb Z$. The fluid is limited by two invariant surfaces $y=\\pi$ and $y=0$ corresponding to the top and bottom roll boundaries.\\\\\nIn the integrable case, as mentionned in (Camassa and Wiggins (1991)), the flow is characterized by hyperbolic fixed points along the two invariant surfaces $y=\\pi$ and $y=0$ and localized at $x=m \\pi$, $m$ $\\in \\mathbb Z$. These points are joined by a vertical heteroclinic connection for which the stable and unstable manifolds coincide. \\\\\nIn order to built a barrier, we select a surface which is located around $x=0$. However we could also create a barrier at $x=\\pi$ (mod $2\\pi$)\\\\\nWe map the time-dependent stream function $\\Psi_{\\epsilon}$ given by Eq. ~(\\ref{1}) into an autonomous Hamiltonian written as $H(x,y,E,\\tau)=E+\\Psi_{\\epsilon}(x,y,\\tau)$. This corresponds to an extension of phase space\n $(x,y,E,\\tau)$, where ${\\bf A}=(x,E)$ and \n$\\boldsymbol{\\theta}=(y,\\tau)$ are momenta and positions. We assume that at $t=0$, $\\tau(0)=0$ for $H$.\nTherefore the equation of motion of $H$ are the same as the ones for $\\Psi_{\\epsilon}$ since $\\tau=t$.\\\\\nWe rewrite the autonomous Hamiltonian in the form:\n\\begin{eqnarray}\n\\label{34}\nH(x,y,E,\\tau)&= &E+ \\alpha \\sin(x + \\epsilon \\sin \\tau) \\sin y,\\nonumber \\\\\n&=& H_0(x,E)+ \\Psi_{\\epsilon}(x,y,\\tau),\n\\end{eqnarray}\nand develop the stream function in Taylor series around $x=0$\n\\begin{eqnarray}\nH(x,y,E,\\tau)&=& E+ \\epsilon v(y,\\tau)+ \\epsilon w(x,y,\\tau) x \\nonumber \\\\\n&+& Q(x,E,y,\\tau),\n\\end{eqnarray}\nwhere $H_0(x,E)=E$ is the integrable part, $\\epsilon v(y,\\tau)=\\Psi_{\\epsilon}(0,y,\\tau)$ and\n\\begin{eqnarray}\n\\label{35}\n\\epsilon w(x,y,\\tau)&=&\t\\partial_x \\Psi_{\\epsilon}(0,y,\\tau),\\nonumber\t\\\\\nQ(x,E,y,\\tau)&=&\\sum_{l=1}^\\infty \\frac{1}{(l+1)!} \\left[\\partial_x^{l+1} \\Psi_{\\epsilon} (0,y,\\tau)\\right] x^{l+1}.\t\\nonumber\n\\end{eqnarray}\nThe frequency vector $\\boldsymbol{\\omega}$ is given by\\\\ \n$\\boldsymbol{\\omega} = \\left({\\partial H_0}\/{\\partial x}, {\\partial H_0}\/{\\partial E}\\right)=\\left(0,1\\right)$ and is resonant.\\\\\n\nIn order to compute the operator $\\Gamma$ given by Eq.(\\ref{gam}), we rewrite $\\Psi_{\\epsilon}$ by expanding $\\sin (\\epsilon \\sin t)$ and $\\cos (\\epsilon \\sin t)$ as series of Bessel functions of first kind:\n\\begin{eqnarray}\n\\label{9}\n\\Psi_{\\epsilon}(x,y,t)&=&\n\\alpha\\sin y \\sin x (\\mathcal{J}_{0}^{\\epsilon}+ 2 \\sum_{n \\geq 1} \\mathcal{J}_{2n}^{ \\epsilon} \\cos 2 n t )\\nonumber \\\\\n &+& 2 \\alpha\\sin y \\cos x ( \\sum_{n \\geq 1} \\mathcal{J}_{2n+1}^{\\epsilon} \\sin (2n+1) t ),\\nonumber\n\\end{eqnarray}\nwhere $\\mathcal{J}_{l}^{\\epsilon}=\\mathcal{J}_{l}(\\epsilon)$ for $l \\in \\mathbb N$, and $\\mathcal{J}_{l}$ are Bessel functions of the first kind.\\\\\nThe control term is given by Eq.(\\ref{eqn:exf}):\n\\begin{eqnarray}\n\\label{37}\nf(y,t)= \\Psi_{\\epsilon} (0,y,t)-\\Psi_{\\epsilon}( -\\Gamma \\partial_{y} \\Psi_{\\epsilon},y,t).\\nonumber \\\\\n\\end{eqnarray}\nSince $\\Psi_{\\epsilon}$ does not depend on $E$, we only have to compute $\\Gamma \\partial_{y} \\Psi_{\\epsilon}(0,y,t)$ and then\n\\begin{eqnarray}\n\\label{30}\nf(y,t) &=&\\alpha\\sin( \\epsilon \\sin t )\t\\sin y \\nonumber \\\\\n&& - \\alpha\\sin \\left[-\\alpha\\cos y {C}_{\\epsilon}(t)+\\epsilon \\sin t\\right]\\sin y,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{serbessl}\nC_{\\epsilon}(t)&=&\\Gamma \\sin(\\epsilon \\sin t),\\nonumber \\\\\n&= & \\sum_{n \\geq 0} \\frac{-2}{2n+1}\\mathcal{J}_{2n+1}^{\\epsilon} \\cos (2n+1) t.\n\\end{eqnarray}\n\nThe equation of the invariant torus is given by Eq.~(\\ref{eqn:eto})\n\\begin{eqnarray}\n\\label{bar}\nx= \\alpha \\cos y {C}_{\\epsilon}(t).\n\\end{eqnarray}\nThe controlled stream function is given by:\n\n\\begin{eqnarray}\n\\label{streamcont}\n\\Psi_{c}(x,y,t)&=&\\alpha \\sin y \\left( \\sin(x+\\epsilon\\sin t)\n+ \\sin( \\epsilon \\sin t ) \\right.\\nonumber \\\\\n&& - \\left. \\sin \\left[-\\alpha \\cos y {C}_{\\epsilon}(t)+\\epsilon \\sin t\\right] \\right) .\n\\end{eqnarray}\n We notice that with a small modification of the stream function, provide that $\\alpha$ and $\\epsilon$ are such that $\\mid \\alpha \\epsilon \\mid < 1$ , there is an exact formula giving the equation of the torus. This invariant torus suppress the chaotic transport from roll to roll and then along the channel.\\\\\nTo illustrate numerically these results, we consider fixed values of the parameter $\\alpha$ and $\\epsilon$. Similar results hold for other values of the parameters $\\alpha$ and $\\epsilon$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[%\n width=3cm]{PSInc0.eps}\\includegraphics[%\n width=3cm]{PSInc3pis4.eps}\n \\end{center} \n \\begin{center}\n\\epsfig{file=RBnc.eps,width=6.2cm,height=3cm}\n \\end{center}\n\\caption{\\label{Fig1} Streamlines at $(a)$ $t=0$ and $(b)$ $t=3\\pi\/4$, and $(c)$ Poincar\\'{e} section \n of the stream function (\\ref{1}). The parameters are $\\alpha=0.6$ and $\\epsilon=0.63$.}\n\\end{figure} \n\nStreamlines of the streamfunction (\\ref{1}) are depicted on Fig.~\\ref{Fig1}$(a)$ and $(b)$ at two different times $t=0$ and $t=3\\pi\/4$ respectively and for $\\epsilon=0.63$ and $\\alpha=0.6$. We observe \n closed curves corresponding to lateral oscillations in the $x$-direction with a periodic displacement of $-\\epsilon\\sin t$.\nFor the same values of $\\epsilon$ and $\\alpha$, a Poincar\\'e section of the dynamics of the streamfunction (\\ref{1}) for initial conditions without control are represented in Fig.~\\ref{Fig1}$(c)$. It shows that the transport along the channel is greatly enhanced. \nThe phase space is characterized by regular (quasiperiodic) trajectories and a chaotic region around and between the rolls.\\\\ \nWhen we add the control term, the streamlines of the controlled stream function (\\ref{streamcont}) are then slightly modified (non-uniformly in $y$) as it can be seen on Fig.~\\ref{Fig2}$(a)$ and $(b)$ for $\\epsilon=0.63$ and $\\alpha=0.6$ at two different times $t=0$ and\n$t=3\\pi\/4$ respectively. Moreover, the displacement of these rolls remains parallel to the $x$-axis as it is for the stream function given by Eq.~(\\ref{1}). A Poincar\\'{e} section of the dynamics of passive tracers in a flow\ndescribed by the stream function $\\Psi_{c}$ given by Eq.~(\\ref{streamcont})\nis represented in Fig.~\\ref{Fig2}$(c)$. We notice first that there are invariant\nsurfaces which have been created around $x=0$ (mod $2\\pi$) (bold\ncurves) which prevent the diffusion of passive particles along the channel. Moreover we observe a regularisation of the dynamics whithin the cell bounded by the two barriers $x=0$ and $\\pi$ (mod $\\pi$). \\\\\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[%\n width=3cm]{PSIaffine0.eps}\\includegraphics[%\n width=3cm]{PSIaffine3pis4.eps}\n \\end{center} \n \\begin{center}\n \\epsfig{file=RBaffine.eps,width=6.2cm,height=3cm}\n \\end{center} \n \\caption{\\label{Fig2} Streamlines at $(a)$ $t=0$ and $(b)$ $t=3\\pi\/4$, and $(c)$ Poincar\\'{e} section \n of the stream function (\\ref{streamcont}). The parameters are $\\alpha=0.6$ and $\\epsilon=0.63$.} \n\\end{figure}\n\nIn Fig.~\\ref{Fig3}, we depict a numerical simulation of the dynamics of a dye in\nthe fluid. The left column shows the evolution of the tracers for\nthe stream function $\\Psi_{\\epsilon}$ given by Eq.~(\\ref{1}). The\nright column shows that the control term regularizes the dynamics and prevents any spread of a dye within a cell which is limited by two\nbarriers created by the stream function $\\Psi_{c}$ given by Eq.~(\\ref{streamcont}).\\\\\n\\begin{figure}\n\\includegraphics[%\n width=3cm]{Fig4a.eps}\\includegraphics[%\n width=3cm]{snap30.eps}\n\n\\includegraphics[%\n width=3cm]{Fig4c.eps}\\includegraphics[%\n width=3cm]{snap50.eps}\n\n\\includegraphics[%\n width=3cm]{Fig4e.eps}\\includegraphics[%\n width=3cm]{snap70.eps}\n\n\\includegraphics[%\n width=3cm]{Fig4g.eps}\\includegraphics[%\n width=3cm]{snap140.eps}\n\n\n\\caption{\\label{Fig3} Numerical simulation of the dynamics of a dye at $t=30$,\n$t=50$, $t=70$ and $t=140$ (from top to bottom)~: left column\nfor the stream function~(\\ref{1}) and right column for the\nstream function~(\\ref{streamcont}). The parameters are $\\alpha=0.6$\nand $\\epsilon=0.63$.}\n\\end{figure} \n\nFor $\\mid \\alpha \\epsilon\\mid << 1$, the control term (\\ref{streamcont}) can be simplified such as:\n\\begin{eqnarray}\n\\label{31}\nf_s(y,t)=-\\frac{\\alpha^2}{2} \\sin 2y\t\\cos( \\epsilon \\sin t ) {C}_{\\epsilon}(t).\n\\end{eqnarray}\n\nWe remark that the control term $f_s$ is of order $\\alpha^2 \\epsilon$ and does not depend on $x$ as it is expected from the method.\\\\\nIn order to test the robustness of the method and to try an experimentally\nmore tractable perturbation, we truncate the series given by Eq.~(\\ref{serbessl}) by considering only the first term of the series\n$C_{\\epsilon}(t)$ in Eq.~(\\ref{serbessl}) and $\\cos( \\epsilon \\sin t )$ which leads to the following\nstream function \n\\begin{eqnarray}\n\\Psi_{s}(x,y,t)&=&\\alpha\\sin(x+\\epsilon\\sin t)\\sin y \\nonumber \\\\\n &&-{\\alpha}^2 \\mathcal{J}_{1}^{\\epsilon}\\sin 2y \\cos t.\\label{psis}\\end{eqnarray}\nThe perturbation has a norm \n $\\sup_{x,y,t} \\mid V(x,y,t)\\mid=$\n $\\epsilon$ and the norm of the the simplified control term $f_s$ is \n\\begin{eqnarray}\n\\label{16}\n\\sup_{y,t} \\mid f_s(y,t)\\mid={\\alpha}^2\n{\\mathcal{J}_{1}^{\\epsilon}} .\t\n\\end{eqnarray}\n\nTherefore the ratio $r$ between the norm of the control term and the perturbation is given by $r \\approx {\\alpha} \\epsilon$.\\\\\n\nThe dynamics of the stream function $\\Psi_{s}$ is depicted in Fig.~\\ref{Fig4} for $\\epsilon=0.63$ and $\\alpha=0.6$. We see that invariant surface has been created around $x=0$ (mod $2 \\pi$) and that the control term \nstill reduces significantly the chaotic advection but there are some advected particles along the channel. We observe that \nan other invariant surface has been created around $x=\\pi$. This is due to the fact that the flow given by the stream function $\\Psi_{s}$ is invariant under the symmetry $$ t\\rightarrow t, x\\rightarrow x+\\pi, y\\rightarrow y+\\pi.$$ It results from this symmetry, a regularisation of the dynamics within the cell bounded by the two barriers $x=0$ and $\\pi$ (mod $2\\pi$). This symmetry is an approximate one ( up to order $\\mid \\alpha^2 \\epsilon \\mid$) for the stream function $\\Psi_{c}$. Thus the control term is able to regularize its dynamics also in this region.\n\\begin{figure}\n \\begin{center}\n \\epsfig{file=RBaffinsimp.eps,width=6.2cm,height=3cm}\n \\caption{\\label{Fig4} Poincar\\'{e} section of the stream function (\\ref{psis}) . The parameters are $\\alpha=0.6$ and $\\epsilon=0.63$.} \n \\end{center}\n\\end{figure}\n \nIn order to see more clearly the effect of the control term, we study the diffusion properties of the system.\nThe mean square displacement $$ of a distribution of $\\mathcal{M}$ particles (of order 3000) is computed as a function of time:\n\\begin{eqnarray}\n\t=\\frac{1}{\\mathcal{M}}\\sum_{i=1}^{\\mathcal{M}} \\left\\|{\\bf x}_i(t)-{\\bf x}_i(0)\\right\\|^{2},\n\\end{eqnarray}\nwhere ${\\bf x}_i(t)$, $i=1,...,\\mathcal{M}$ is the position of the $i$-th particle at time $t$ obtained by integrating Hamilton's equations with initial conditions ${\\bf x}_i(0)$. We find that $$ grows linearly for the considered time interval (see Fig.~\\ref{Fig5}$(a))$. The transport is then assumed to be described as normal diffusion and the corresponding diffusion coefficient can be determined from the slope of $$ versus $t$:\n\\begin{eqnarray}\nD=\\lim_{t\\rightarrow \\infty} \\frac{}{t}.\n\\end{eqnarray}\nFigure ~\\ref{Fig5}$(b)$ shows the values of $D$ as a function of $\\epsilon$ with and without control term (\\ref{psis}) determined from the mean square displacement for $ t > 1000$. We remark that the diffusion coefficient with the control term (\\ref{psis}) is significantly smaller than the uncontrolled case.\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=variance.eps,width=4cm,height=4cm} \\epsfig{file=coefdif1.eps,width=4cm,height=4cm}\n\\caption{\\label{Fig5} $(a)$ Mean square dispacement $$ versus time $t$ obtained for Hamiltonian (\\ref{1}) with three different values of $\\epsilon=0.12, 0.18, 0.2$ and $(b)$ Diffusion coefficient D versus $\\epsilon$ obtained for stream function (\\ref{1}) (circle) and stream function (\\ref{psis}) (square)}\n\\end{center}\n\\end{figure}\n\n\n\\section{Conclusion}\nWe presented in this paper an application of a recently\ndeveloped control technique to Rayleigh-B\\'enard convection.\nWe derived analytically the expansion of the control term required to reduce chaotic advection present\nin the original problem. \nUsing Poincar\\'e sections, and tracking the dynamics of a dye, we showed the efficiency of the method. The slightly deformed streamlines due to the control terms prevent any large spread through the channel but confine the material lines inside some deformed rolls.\nThe chaotic behavior of the flow was significantly reduced by taking only the first order of control term. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n The new data of the TOTEM Collaboration on the elastic differential cross sections\n at 13 TeV have two sets of data - at small momentum transfer \\cite{T66}\n and at middle and large momentum transfer \\cite{T67}.\n Recently, the first set of data has created a wide discussion of the determination of the total\n cross section and the value of $\\rho(t=0)$ (for example \\cite{MN,Khoze,CS-Diff}).\n A research of the structure of the elastic hadron scattering amplitude\n at superhigh energies and small momentum transfer - $t$\n can give a connection between\n the experimental knowledge and the basic asymptotic theorems,\n which are based on first principles \\cite{akm,fp,royt}.\n It gives information about the hadron interaction\n at large distances where the perturbative QCD does not work \\cite{Drem},\n and a new theory, as, for example, instantons or string theories,\n must be developed.\n\n\n\n\n\n\n Usually, a small region of the $t$ is taken into account\n for extraction of the sizes of $\\sigma_{tot}$ and $\\rho(t=0)$ (for example \\cite{T66,Protvino13}).\n Really, already in the analysis of the UA4\/2 data it was shown that the value of $\\rho(s,t)$\n has some phenomenological sense, as its determination requires some model assumptions \\cite{Sel-UA42}.\n The simple exponential approximation of the data gave $\\rho=0.24$ from the UA4 data and $\\rho=0.129$\n from UA4\/2 data (both at $\\sqrt{s} =540$ GeV. More complicated analyses gave\n $\\rho=0.19$ from the UA4 data and $\\rho=0.16$ from UA4\/2 data \\cite{Sel-UA42}.\n Hence it is do not the experimental problem,\n but the theoretical problem \\cite{CS-PRL09}.\n A phenomenological form of the scattering amplitude are determined for small $t$ can lead\n to very different differential cross sections at larger $t$.\n Especially, it is connected with the differential cross section at 13 TeV, as the diffraction minimum\n is located at a non-large $t$.\n\n\n\n\n Also the very important moment is related with the question how the experimental uncertainty,\n which usually are named as experimental errors, used in the our fitting procedure.\n In fact, the actual background rates and shapes of the measured distributions are sensitive\n to a number of experimental quantities, such as calibration constants, detector geometries,\n poorly known material budgets within experiments, particle identification efficiencies etc.\n A 'systematic error', referred to by a high energy physicist, usually corresponds\n to a 'nuisance parameter' by a statistician.\n\n Hence, the extraction of the main value of the elastic hadron interaction\n requires some model that can describe all the experimental data\n at the quantitative level with minimum free parameters.\n Now many groups of researchers have presented some physical models satisfying\n more or less these requirements. It is especially related with the HEGS\n (High Energy Generalized Structure) model \\cite{HEGS0,HEGS1}.\n As it takes into account two form factors (electromagnetic and gravitomagnetic),\n which are calculated from the GPDs function of nucleons, it has minimum free\n parameters and gives the quantitative description of the exiting experimental data\n in a wide energy region and momentum transfer.\n Analysis of new data of the TOTEM Collaboration at 13 TeV in the framework of the HEGS model\n discovered a new\n phenomenon in the hadron interaction - the oscillation term of the elastic scattering\n amplitude \\cite{Osc-13}. During this analysis only statistical errors of experimental\n data were taken into account in the fitting procedure. Systematic errors were taken as\n an additional coefficient of the normalization of the differential cross section,\n which is independent of the momentum transfer.\n\n Further careful analysis of the behavior of the differential cross sections\n in the framework of the HEGS model have shown additional unusual properties of the\n behavior of the elastic scattering amplitude at a very small momentum transfer.\n The effect is examined from different points of view in the present\n paper.\n\n In the second section of the paper, the new effect is analyzed in the framework of the HEGS model\n with taking into account experimental data both the sets of the TOTEM Collaboration\n obtained at 13 TeV and is compared with the results of some other models in the third section.\n In the fourth section, the existence of the new effect is examined in a simple phenomenological\n form of the scattering amplitude (as used most groups of researchers) and\n experimental data only first set at small momentum transfer are taken into account.\n The conclusions are given in the final section.\n\n\n\\section{ Description of the differential cross section in a wide region of momentum transfer}\n\n There are many different semi-phenomenological models which give a qualitatively\n description of the behavior of the differential cross sections of the elastic proton-proton\n scattering at $\\sqrt{s} = 13$ TeV (for example \\cite{Kohara,Gotsman,Koh-Bl18}.\n Some examples can be found in the review \\cite{Pakanoni}; hence, we do not give a deep analysis\n of those models. One of the common properties of practically all models is that they\n take into account statistic and systematic errors in a quadrature forms and in most part\n give only a qualitative description of the behavior of the differential cross section\n in a wide momentum transfer region.\n\n\n\n\n\n However, there are two essentially different ways of including statistical\n and systematic uncertainties in the fitting procedure, especially if we want to obtain\n a quantitative description of experimental data.\n The first one, mostly used in connection with the differential cross sections (for example \\cite{BSW,Bourrely-14,Gotsman,Kohara}),\n takes into account statistical and systematic errors in quadrature form:\n\n $\\sigma_{i(tot)}^2 = \\sigma_{i(stat)}^2+ \\sigma_{i(syst)}^2$.\n In this case, \n $\\chi^2$ can be simply written as\n\\begin{eqnarray}\n\\chi^{2}= \\sum_{i=1}^{n} \\frac{ ( \\hat{E}_{i} - F_{i}(\\vec{a}) )^2 }\n{\\sigma_{i(tot)}^{2}} .\n \\label{eq6}\n \\end{eqnarray}\n\n\n\n The second approach accounts for the basic property of systematic uncertainties,\n i.e. the fact that these errors have the same sign and size in proportion to the effect\n in one set of experimental data and, maybe, have a different sign and size in another set.\n To account for these properties, extra normalization coefficients for the measured data\n are introduced in the fit. For simplicity, this normalization is often transferred\n into the model parametrization, while it - in reality - accounts for the uncertainty of the normalization\n of experimental data. This method is often used by research collaborations to extract,\n for example, the parton distribution functions of nucleons\n \\cite{Stump01,exmp1-26,exmp1} and nuclei \\cite{EPPS16})\n in high energy accelerator experiments, or in astroparticle physics \\cite{Koh15}.\n In this case, $\\sigma_{i(tot)}^2 = \\sigma_{i(stat)}^2$ and\n the systematic uncertainty are taken into account as an additional normalization coefficient,\n $k$, and the size of this coefficient is assumed to have a standard systematic error,\n $k=1\\pm \\sigma $, \n \\begin{eqnarray}\n\\chi^{2} = \\sum_{j=1}^{m} [ \\sum_{i=1}^{n} \\frac{ ( f_j \\hat{E}_{ij} - \\bar{x} )^2 }{2 \\sigma^{2}_{ij(st.)} }\n + \\frac{(1-f_j)^2}{\\sigma^{2}_{j(sys)}} ].\n\\end{eqnarray}\n It should be noted that in the minimization procedure used in these two methods,\n different sizes of experimental errors were assumed. In the first case,\n we account for experimental errors in the quadrature of statistical and systematic errors\n and for experimental data with the normalization given by an experimental collaboration.\n In the second case, only statistical errors are considered as an experimental uncertainty.\n The systematic errors are accounted for as an additional normalization coefficient interpreted\n as a nuisance parameter applied to all experimental data of this separate data set.\n\n In the first case, the \"quadrature form\" of the experimental uncertainty gives a wide corridor\n in which different forms of the theoretical amplitude can exist. In the second case,\n the \"corridor of the possibility\" is essentially narrow, and it restricts the different forms\n of theoretic amplitudes.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n In this case, a problem appears - how to take into account systematic experimental errors.\n Most part of systematic errors come from the uncertainty of the Luminocity, which affects experimental data one way.\n For example, in the description of different errors in \\cite{T66,T67} the main part of the systematic\n errors is determined by the indefiniteness of the Luminocity.\n Hence, systematic errors can be represented as an additional\n normalization coefficient. Then, only statistical errors have to be taken into account\n in calculations of $\\chi^2$.\n\n\n\n\n\n\n However, to examine subtle effects in the behaviour of differential cross sections,\n it is needed to have the narrowest possible corridor for testing a theoretical function.\n In our paper \\cite{Osc-13}, it was shown that the new data of the TOTEM Collaboration at 13 TeV\n show the existence in the scattering amplitude of the oscillation term, which can be determined\n by the hadron potential at large distances. In the analysis of experimental data\n of both the sets of the TOTEM data the additional\n normalization was used. It size reaches sufficiently large values. In this case, a very small\n $\\chi^2_{dof}$ was obtained with taking into account only statistical errors and\n with a small number of free parameters in the scattering\n amplitude, which was obtained in our High Energy Generalized Structure (HEGS) model \\cite{HEGS0,HEGS1}.\n However, the additional normalization coefficient reaches a sufficiently large value,\n about $13\\%$. It can be in a large momentum transfer region but is very unusual for a small\n momentum transfer. However, both sets of experimental data (small and large region of $t$)\n overlap in some region and, hence, affect each other's normalization.\n It is to be noted, that the size of the normalization coefficient does not impact the size\n and properties of the oscillation term. We have examined many different variants of our\n model (including large and unity normalization coefficient) , but the parameters of\n the oscillation term have small variations.\n\n In the present work, the analysis of both sets of the TOTEM data at 13 TeV\n is carried out with additional normalization equal to unity and taking into account\n only statistical errors in experimental data.\n\n\n\n\n\\begin{figure}\n\\begin{center} \\vspace{-1.cm}\n\\includegraphics[width=.70\\textwidth]{ds134ca.ps}\n\\includegraphics[width=0.45\\textwidth]{ds13ss4.ps}\n \\includegraphics[width=0.45\\textwidth]{ds134m.ps}\n\\end{center}\n\\vspace{1.5cm}\n\\caption{The differential cross sections are calculated in the framework of the HEGS model\n with fixed additional normalization by $1.$ and with additional term eq.(7),\n a) [top] the full region of $t$ and the data \\cite{T66,T67};\n b) [bottom-left] the magnification of the region of the small momentum transfer of a);\n c) [bottom-right] the magnification of the region of the diffraction minimum.\n }\n\\label{Fig_1}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{ Model description of two sets at 13 TeV}\n\n Differential cross sections measured experimentally\n are described by the squared scattering amplitude\n\\begin{eqnarray}\nd\\sigma \/dt &=& \\pi \\ (F^2_C (t)+ (1 + \\rho^{2} (s,t)) \\ Im F^2_N(s,t)\n \\nonumber \\\\\n & & \\mp 2 (\\rho (s,t) +\\alpha \\varphi )) \\ F_C (t) Im F_N(s,t)). \\label{ds2}\n\\end{eqnarray}\nwhere $F_{C} = \\mp 2 \\alpha G^{2}\/|t|$ is the Coulomb amplitude;\n$\\alpha$ is the fine-structure constant,\n$\\varphi(s,t) $\n is the Coulomb hadron interference phase between the electromagnetic and strong\n interactions (in our case it takes from \\cite{selmp1,selmp2,selmp3}),\n and\n$Re\\ F_{N}(s,t)$ and $ Im\\ F_{N}(s,t)$ are the real and\nimaginary parts of the nuclear amplitude;\n$\\rho(s,t) = Re\\ F(s,t) \/ Im\\ F(s,t)$.\nJust this formula is used to fit experimental data\ndetermined by the Coulomb and hadron amplitudes and the Coulomb-hadron\nphase to obtain the value of $\\rho(s,t)$.\n\n\n\n As a basis, we take our high energy generalized structure (HEGS) model \\cite{HEGS0,HEGS1} which quantitatively describes, with only a few parameters, the differential cross section of $pp$ and $p\\bar{p}$\n from $\\sqrt{s} =9 $ GeV up to $13$ TeV, includes the Coulomb-hadron interference region and the high-$|t|$ region up to $|t|=15$ GeV$^2$\n and quantitatively well describes the energy dependence of the form of the diffraction minimum \\cite{HEGS-min}\n However, to avoid possible problems\n connected with the low-energy region, we consider here only the asymptotic variant of the model.\n The total elastic amplitude in general receives five helicity contributions, but at\n high energy it is enough to write it as $F(s,t) =\n F^{h}(s,t)+F^{\\rm em}(s,t) e^{\\varphi(s,t)} $\\,, where\n $F^{h}(s,t) $ comes from the strong interactions,\n $F^{\\rm em}(s,t) $ from the electromagnetic interactions.\n\n\n\n\n Note that all five spiral electromagnetic amplitudes are taken into account\n in the calculation of the differential cross sections.\n The Born term of the elastic hadron amplitude at large energy can be written as\n a sum of two pomeron and odderon contributions,\n \\begin{eqnarray\n F_{\\mathbb{P} }(s,t) & =& \\hat{s}^{\\epsilon_0}\\left(C_{\\mathbb{P}} F_1^2(t) \\ \\hat s^{\\alpha' \\ t} + C'_{\\mathbb{P}} A^2(t) \\ \\hat s^{\\alpha' t\\over 4} \\right) \\; , \\\\\n F_{\\mathbb{O} }(s,t) & =& i \\hat{s}^{\\epsilon_0+{\\alpha' t\\over 4}} \\left( C_{\\mathbb{O} }\n + C'_{\\mathbb{O}} \\ t \\right) A^2(t).\n \n \\end{eqnarray}\n All terms are supposed to have the same intercept $\\alpha_0=1+\\epsilon_0 = 1.11$, and the pomeron\n slope is fixed at $\\alpha'= 0.24$ GeV$^{-2}$.\n The model takes into account two hadron form factors $F_1(t)$ and $A(t)$, which correspond to the charge and matter\n distributions \\cite{GPD-PRD14}. Both form factors are calculated as the first and second moments of the same Generalized Parton Distributions (GPDs).\n The Born scattering amplitude has four free parameters (the constants $C$) at high energy:\ntwo for the two pomeron amplitudes and two for the odderon.\nThe real part of the hadronic elastic scattering amplitude is determined\n through the complexification $\\hat{s}=-i s$ to satisfy the dispersion relations.\n The oscillatory function was determined \\cite{Osc-13}\n\\vspace{-0.1 cm}\n\\begin{eqnarray}\n f_{osc}(t)=h_{osc} (i+\\rho_{osc}) J_{1}(\\tau))\/\\tau; \\ \\tau = \\pi \\ (\\phi_{0}-t)\/t_{0},\n\\end{eqnarray}\nhere $J_{1}(\\tau)$ is the Bessel function of the first order.\n This form has only a few additional fitting parameters and allows one to represent\n a wide range of possible oscillation functions.\n\n After the fitting procedure we obtain $\\chi^2\/n.d.f. =1.24$ (remember that we used only statistical errors).\n One should note that the last points of the second set above $-t=2.8$ GeV$^2$ show\n an essentially different slope, and we removed them. The total number of experimental points\n of both sets equals $415$. If we remove the oscillatory function, then\n $\\chi^2\/n.d.f. =2.7$, so an increase is more than two times. If we make a new fit without $f_{osc}$,\n then $\\chi^2\/n.d.f. =2.4$ decreases but remains large.\n\n\n\n However, such result was obtained with a sufficiently large addition coefficient of the normalization\n $n=1\/k=1.135$. It can be for a large momentum transfer, but unusual for the small region of $t$.\n\n Now let us put the additional normalization coefficient to unity and continue to take into account\n in our fitting procedure only statistical errors. Of course, we obtain the enormously huge\n $\\sum \\chi^2$. The new fit changes the basic parameters of the Pomeron and Odderon Born terms\n but does not lead to the reasonable size of $\\chi^2$.\n\n We find that the main part of $\\sum \\chi^2$ comes from the region of a very small momentum transfer.\n It requires the introduction of a new term which can help to describe the CNI region of $t$.\n This kind of term can be taken in different forms. In present paper, we examined\n two different forms. One is the simple exponential form\n \\begin{eqnarray}\n F_{d}(t)=h_{d} (i+\\rho_{d}) e^{-B_{d} |t|^{\\kappa} \\log{\\hat{s} }},\n \\label{fd-exp}\n\\end{eqnarray}\n and the other is the power form which has t-dependence similar to the squared the Coulomb amplitude.\n\\begin{eqnarray}\n F_{d}(t)=h_{d} (i+\\rho_{d})\/(1 + (r_{d} t)^2) \\ G_{el}^2 .\n \\label{fd-rb}\n\\end{eqnarray}\n where $ G_{el}^2 $ is the squared electromagnetic form factor of the proton.\n For simplicity, in a further fitting procedure the constant $\\rho_{osc}$ and the phase $\\phi_{0}$\n of the oscillatory term are taken as zero. Hence, the oscillatory term depends only on two parameters -\n $h_{osc}$ and $t_{0}$ period of oscillation. Also, to reduce the number of fitting parameters\n the correction to the main slope is taken in a simple form, we obtain the slope as\n \\begin{eqnarray}\n B(t)= \\alpha^{'} \\log{\\hat{s}} ( 1- t e^{ B_{ad} t}),\n \\label{slope-s}\n\\end{eqnarray}\n\n\n\n\n\\begin{figure}\n\\begin{flushright}\\vspace{-2.cm}\n\\includegraphics[width=.45\\textwidth]{fde.ps}\n\\includegraphics[width=0.45\\textwidth]{bfde.ps}\n\\end{flushright}\n\\vspace{0.5cm}\n\\caption{ The amplitude $F_{dop}(b)$ eq.(7) in the impact parameter representation\n a) [left] the real $F_{dop}(b)$- hard line and imaginary part $Im F_{dop}(b)$ -dashed line ;\n b) [right] overlapping function $b F_{dop}(b)$ (real part - hard line; imaginary part - dashed line) .\n }\n\\label{Fig_1}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\begin{flushright}\\vspace{-2.cm}\n\\includegraphics[width=.45\\textwidth]{fdrb.ps}\n\\includegraphics[width=0.45\\textwidth]{fdr2.ps}\n\\end{flushright}\n\\vspace{0.5cm}\n\\caption{ The amplitude $F_{dop}(b)$ eq.(8) in the impact parameter representation\n a) [left] the real $F_{dop}(b)$- hard line and imaginary part $Im F_{dop}(b)$ -dashed line ;\n b) [right] the same at large impact parameters.\n }\n\\label{Fig_3}\n\\end{figure}\n\n The fit of both sets of the TOTEM data simultaneously with taking into account only statistical errors and with additional normalization equal to unity with the additional term, eq.(7), gives\n a very reasonable $\\chi^2 = 551\/425=1.29$. The results are present for the full region\n of $t$ in Fig.1a, and with zoom of the region of small $t$ in Fig.1b, and zoom of the region\n of diffraction minimum in Fig.1c.\n\n The parameters of the additional term are well defined\n $ h_{d}=1.7 \\pm 0.01; \\ \\ \\ \\rho_{d}=-0.45 \\pm 0.06; $ \\\\\n $ B_{d} = 0.616 \\pm 0.026; \\ \\ \\ \\kappa=1.119 \\pm 0.024. $\n\n\n\n Using the second form of the additional term, eq.(8), we obtain\n practically the same picture with the same $\\chi^2=549\/425 =1.28$\n with the parameters of the additional term\n $ h_{d} = 1.067 \\pm 0.044; \\ \\ \\ \\rho_{d} = -0.53 \\pm 0.07 $ \\\\\n $ r_{d}= 7.62 \\pm 0.34 $. \\\\\n To check up the impact of the form of the CNI phase - $\\varphi (t)$ we made our calculations\n with the original Bethe phase $\\varphi = -( dLOG(Bsl\/2.*t)+0.577)$ as well.\n We found that $\\sum \\chi^2$\n changes by less than $0.2\\%$ and practically does not impact the parameters $F_d (t)$.\n Hence, our model calculations show two possibilities in the quantitative\n description of the two sets of the TOTEM data.\n One - take into account an additional normalization coefficient, which\n has a minimum size of about $13\\%$ ; the other - the introduction of the new anomalous term\n of the scattering amplitude which has a very large slope and gives the main contributions\n to the Coulomb-nuclear interference region.\n\n\n Of course, there are some other ways to obtain good descriptions of the\n new experimental data of the TOTEM Collaboration. One is to use some model\n with an essentially increasing number of the fitting parameters and many different\n parts of the scattering amplitude. Another is to use some polynomial model\n with many free parameters.\n In both cases, the physical value of such a description is doubtful.\n\n \n \n\n Let us examine the additional term in the impact parameter representation and\n use the fourier transform\n \\begin{eqnarray}\n F_{dop}(b) \\sim \\int_{0}^{\\infty} \\ dq \\ J_{0}(q b ) \\ F_{dop}(q^2) ,\n \\label{rep-qb}\n\\end{eqnarray}\n The results for the additional term in form eq.(7) is presented in Fig.2a.\n The Fig.2b show that the main contribution come from the non-large impact parameters.\n The maximum of $b F_{dop}(b)$ situated in the region of $r \\sim 1 $fm, slightly above the electromagnetic radius of proton.\n The Fig.3 show the impact parameter representation for the real and imaginary parts\n of $F_{dop}$ in form eq.(8).\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Other models}\n\n The above results were obtained in the framework of one specific model.\n Let us see what other models tell us. There are many different models with\n very different paradigms (for example, see reviews \\cite{J-S-rev,Pakanoni}) and we take only some\n of them as an example.\n One of the oldest models, which is based on the hadron structure\n \\cite{Islam}, says that the main result is\n \n \n \" The most striking feature of the preliminary $\\sqrt{s} = 13 $ TeV TOTEM data is that there are no oscillations in\n $d\\sigma\/dt$ beyond the\ninitial dip-bump structure. It shows a smooth falloff for large $|t|$, exactly as predicted by our model.\"\n The model gives, as many others, only a qualitative description of the differential cross section and does not feel\n the fine structure.\n\n\n\n\n Some other models, for example \\cite{Jnk-19}, developed the structure of the scattering\n amplitude, but in the analysis of the experimental data they do not include the specific properties\n of the hadron interaction at small momentum transfer -\n \" Note that, in this paper, we treat only the strong (nuclear) amplitude separated\n from Coulomb forces. The CNI effects modify the nuclear cross-section\nby less than $1 \\%$ for $|t| > 007$ GeV$^2$, thus, in the nuclear range, the CNI effects\ncan be ignored\"\nThe same specific bounds were taken by one of the famous model \\cite{DL-19}.\n In a recent paper they noted: \" This\npaper applies it in its simplest form to small- $t$ data from $13.76$ GeV to $13$ TeV for total cross sections\nand elastic scattering at small t, namely $|t| < 0.1$ GeV$^2$\"\nAs in many others paper in \\cite{DL-19} they speak only\nabout a good fit, which are shown in different Figures.\nHowever, they do not speak about the sizes\nof $\\chi^2$, especially in different regions of momentum transfer.\nIt is interesting that in \\cite{DL-19} they note \" ...but the slope for 7 TeV data lies between that for 8 and 13 TeV,\nwhich is surely anomalous\".\n Hence, all such models can not see some fine structure of hadron interactions which\n is discussed in our paper but note some anomalous behaviour of the slope.\n\n Some models include in the analysis the Coulomb-hadron interference region and\n note the importance of this region of $t$.\n However, in most part they are interested in the deviation of the differential cross sections\n from the exponential form, which leads to some \"break\" in the region of $-t \\sim 0.15$ GeV$^2$.\n For example in \\cite{Koh-Bl18} they note -\n\"The left plot shows the non-exponential behaviour of the differential cross\nsection for T8. ...\nThe plot\nin the RHS shows the ratio $(T2-R)\/T2$\n which exhibits information of a non-exponential\nbehaviour with advantages compared with the first plot, since is cancelled, and\nwith it most of the normalization systematic error.\"\n It is interesting that they show the importance of systematic errors.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIn a recent complicated work \\cite{Pakanoni},\nthe best description of experimental data at $|t| < 0.2$ GeV$^2$ is\n only the fourth variant of a complicated construction of the scattering amplitude.\n They note \"As we shall see, both the real and imaginary parts of the nuclear\namplitude are practically identical for $A_{2}$ and $A_{3}$, but\nthey are substantially different from $A_{0}$ and $A_{1}$. Moreover,\nthe predictions from the amplitudes $A_{2}$ and $A_{3}$ are in remarkable\naccord with the TOTEM data in the CNI region whereas\nthose from $A_{0}$ and $A_{1}$ are decidedly inferior.\"\nMoreover in that model \" cross sections very likely rise less than $L^{2}(s)$\".\nHowever, such a verdict contradict the definition of the total cross section\nby the TOTEM Collaboration.\n\n\n\n\n\n\nAnother interesting model \\cite{God} examines modern experimental data of the TOTEM Collaboration.\n The result is very interesting from our viewpoint. We present thier Fig. 2b in our Fig.4.\n The difference between the model results and the experimental data at small\n momentum transfer is remarkable . Very likely that it shows the necessity of additional normalization.\n Hence, this very different form our model also shows the problem of the normalization or\n the presence of some anomaly in the differential cross sections.\n\n\n\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.45\\textwidth]{13k.eps}\n\\end{center}\n\\caption{from paper \\cite{God}\n \"Figure 2: The predictions of the model [6] in the case $HP(0) - 1 = 0.32$ versus the TOTEM\n data at $\\sqrt{s}=13$ TeV [7]. The dashed line corresponds to the approximation $C(s,t) = 0$\" }\n\\label{Fig_6}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The fit of the differential cross section in the small momentum transfer region}\n\n Above, the examination of the new TOTEM experimental data at $\\sqrt{s}=13$ TeV carried out in a wide\n momentum transfer shows the existence of some anomaly in the behaviour of the differential cross sections\n at a very small momentum transfer. Of course, it has some dependence on the model structure.\n \n We cannot exclude a possibility of discovering a more complicated model that explains new features\n of hadron interactions at large distances.\n Hence, it is important also to see the phenomena of the new effect only in the small momentum transfer region and in the framework of the simplest form of the scattering amplitude .\n Now let us limit our examination to the small region of momentum transfer (up to $-t=0.069$\n which includes 79 experimental data of the first set of the TOTEM Collaboration \\cite{T66}.\n This region was examined by the TOTEM Collaboration\n and some other groups of researchers (for example \\cite{CS-Diff,Protvino13}).\n Unlike other groups, we will take into account only statistical errors and the\n additional normalization $k=1$. The new data of the TOTEM Collaboration have very small statistical\n errors, especially in the low momentum region. Hence, our fitting procedure will give the non-small\n $\\sum \\chi^2_{i}$; however, it imposes hard restrictions on different representations of the scattering amplitude.\n Firstly, let us examine the Born scattering amplitude using the standard eikonal representation,\n as was made in our model analysis of the whole region of the momentum transfer of experimental data.\n\n Let us take the hadronic Born scattering amplitude in the simple exponential form. Of course,\n after eikonalization such an amplitude is added the standard electromagnetic amplitude,\n as we made in the model analysis,\n\\begin{eqnarray\n F_{\\mathbb{P} }(s,t) = h\/(2 \\ 0.389 \\pi ) (i+\\rho) Exp( B\/2 t)\n \\end{eqnarray}\n As was made in a recent work \\cite{Protvino13}, we will made the fit\n \n in the different regions of $t$.\n Our results are given in Table 1. The maximum width of the examined region leads to the non-small\n $\\sum \\chi^2_{i}$. It is shown that a simple exponential form is not sufficient for our analysis.\n Of course, when we come to the small region of $t$, the description is improving more and more.\n It is to be noted that the size of the slope has small variation with decreasing $t$.\n In our analysis, the slope size is somewhat some less than was determined by the TOTEM group \\cite{T66} and\n by the Protvino group \\cite{Protvino13}. This may be the result of the slope determined by the Born\n scattering amplitude that is in further changed by the eikonalization procedure.\n However, we are interested in the possibility of the contribution of an additional\n rapidly decreasing term of the scattering amplitude.\n\n Let us add an additional term in form\n \\begin{eqnarray\n F_{ad }(s,t) = i \\ h_{d}\/(2 \\ 0.389 \\pi ) \\ e^{ D_{d} t}\n \\end{eqnarray}\n As a result, two additional parameters appear in the fitting procedure. We reduce the real part of the additional term as the increasing the number of the fitting parameters leads\n to large uncertainty in the results.\n It should be noted that if we add two additional parameters in the main Born amplitude\n as additional slopes - $B_{1} \\ \\sqrt(-t)$ and $B_{2} t^{2} $, this will not practically\n change the picture. The same result was obtained by the TOTEM Collaboration, too.\n\n The results of our new fitting are presented in Table 2.\n The $\\chi^2$ decreases essentially, especially for the complitely examined $t$ region.\n The constant of the additional term is determined sufficiently well.\n The slope of the additional term is large and also is determined with small errors.\n\n\n\\begin{figure}\n\\begin{center}\n\\vspace{-1.5cm}\n\\includegraphics[width=.45\\textwidth]{chi2fd0.ps}\n\\end{center}\n\\vspace{0.5cm}\n\\caption{ The $\\chi^{2}(t) $ in the case of taking into account the additional fast decreasing term\n (dashed line) and in the case of the absence of such a term (hard line).\n }\n\\label{Fig_5}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\includegraphics[width=.5\\textwidth]{chi2exp.ps}\n \\includegraphics[width=0.5\\textwidth]{chi2fdkb.ps}\n\\vspace{0.5cm}\n\\caption{ The dependence of $\\chi^2$ on the additional normalization coefficient\n of experimental data -\n a) [left] in the case of the simple exponential form of the scattering amplitude;\n b) [right] the same but with the additional fast decreasing term.\n\n }\n\\label{Fig_6}\n\\end{figure}\n\n\nThe $\\chi^{2}(t) $ are shown in Fig.5 in the case of taking into account the additional fast decreasing term\n (dashed line) and in the case of the absence of such a term (hard line).\n We can see that the largest difference comes from a very small region of momentum transfer.\n\n Let us include additional normalization (representing systematical errors) in our fitting\n procedure. The dependence of $\\sum_{i}^{N} \\chi^2$ on the additional coefficient\nof normalization of experimental data is shown in Fig. 6.\n The case of the simple exponential for the scattering amplitude is presented at the top of Figure 4;\n and with the additional fast decreasing term at the bottom.\nOne can see that $\\chi^2$ in the first case essentially depends on the normalization coefficient\n and has a sufficiently large value (remember that we used only statistical errors).\n Note that $k=1\/n$ is the coefficient by which we multiply our theoretical function\n to compare with experimental data in our fitting procedure. The minimum is reached when\n the additional normalization equals $13.5\\%$. This corresponds to the additional normalization\n which was used in our HEGS model calculations without the additional fast decreasing term.\n Contrary, in the case with the additional term the dependence on normalization is weak\n and the size of $\\chi^2$ has a reasonable value in a wide region of normalization.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n Now let us carry out analysis without eikonalization.\n In this case, the additional term will be represented in the power form (like a square of Coulomb amplitude)\n\n \\begin{eqnarray\n F_{ad }(s,t) = \\alpha_{el}^2 \\ h_{d}\/(2 \\ 0.389 \\pi ) \/[ \\epsilon + t^2]\n \\end{eqnarray}\n where $\\alpha_{el}=1\/137$ is the electromagnetic fine structure constant and $\\epsilon$ is free parameter\n order $\\alpha_{el}^2$. The comparison of $\\chi^2$ for a simple exponential term\n and with the added fast decreasing term are presented in Table 3. The difference is not large;\n however, it is about $10\\%$\n for every examined region of $t$. The constant $h_{d}$ is also determined well.\n\n\n\\begin{table}\n\\label{Table-1}\n\\caption{ The fit of $d\\sigma\/dt$ by the Born hard scattering amplitude\n in one exponential form using the eikonal representation. }\n\\vspace{.5cm}\n\\begin{tabular}{|c|c|c|c||c|c|} \\hline\n N&$-t_{max}(GeV^2)$&$\\sum \\chi^2_{i}$& $\\chi_{dof}$ &h (GeV$^{-1})$ & $B (GeV^{-2} )$ \\\\ \\hline\n 79 & 0.0699 & 162.1& 2.19 &$110.2 \\pm 0.3$ & $16.0 \\pm 0.03$ \\\\\n 70 & 0.0559 & 86.5 & 1.29 &$110.5 \\pm 0.4$ & $ 16.4 \\pm 0.04$ \\\\\n 65 & 0.0488 & 66.3 & 1.07 &$110.6 \\pm 0.6$ & $16.3 \\pm 0.05$ \\\\\n 60 & 0.0422 & 55.3 & 0.97 &$110.6 \\pm 1.4$ & $16.4 \\pm 0.06$ \\\\\n 55 & 0.0361 & 49.7 & 0.96 &$110.6 \\pm 1.7 $& $ 16.4 \\pm 0.08 $ \\\\\n 50 & 0.0305 & 47.8 & 1.02 &$110.6\\pm 1.4 $ & $ 16.4 \\pm 0.1 $ \\\\\n 40 & 0.0207 & 34.2 & 0.92 &$110.7 \\pm 0.6$ & $16.6 \\pm 0.2$ \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\\vspace{.5cm}\n\n\n\n\n\n\n\n\n\n\n\\begin{table}\n\\label{Table-2}\n\\vspace{.5cm}\n\\caption{The fit of $d\\sigma\/dt$ by the Born hard scattering amplitude in two exponential forms. }\n\\vspace{.5cm}\n\\begin{tabular}{|c|c|c|c||c|c|} \\hline\n N&$-t_{max}(GeV^2)$&$\\sum \\chi^2_{i}$& $\\chi_{dof}$ & h (GeV$^{-1})$ & $D_{d} (GeV^{-2} )$ \\\\ \\hline\n 79 & 0.0699 & 74 & 1.00& $3.26 \\pm 0.3$ & $41.2 \\pm 1.9$ \\\\\n 70 & 0.0559 & 62.2 & 0.93& $2.94 \\pm 0.4$ & $39.2 \\pm 4.1$ \\\\\n 65 & 0.0488 & 56.8 & 0.94& $2.26 \\pm 0.6$ & $31.6 \\pm 5.8$ \\\\\n 60 & 0.0422 & 53.0 & 0.96& $1.51 \\pm 1.4$ & $25.3 \\pm 7.7$ \\\\\n 55 & 0.0361 & 49.0 & 0.98& $1.56 \\pm 1.7 $ & $ 25.5 \\pm 11.4 $ \\\\\n 50 & 0.0305 & 44.4 & 0.98& $1.93 \\pm 1.4 $ & $ 29.7 \\pm 13.7 $ \\\\\n 40 & 0.0207 & 33.0 & 0.94& $2.4 \\pm 0.6$ & $29.7_{fixd}$ \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\\vspace{.5cm}\n\n\n\n\n\n\n\\begin{table}\n\\label{Table-3}\n\\vspace{.5cm}\n\\caption{The comparison of $\\sum \\chi^2_{i}$ from the fit of of $d\\sigma\/dt$\n by the hard scattering amplitude in the exponential and two exponential forms. }\n \\vspace{.5cm}\n\\begin{tabular}{|c|c|c|c|c|} \\hline\n N&$-t_{max}(GeV^2)$&$\\sum \\chi^2_{i}$ (Exp) & $\\sum \\chi^2_{i}$ (Exp+fd) & $h_{d}$ \\\\ \\hline\n 79 & 0.0699 & 67.61 & 62.87 & $ 1.95\\pm 0.35$ \\\\\n 70 & 0.0559 & 61.52 & 59.24 & $ 2.14\\pm 0.58$ \\\\\n 65 & 0.0488 & 57.14 & 55.32 & $ 2.25\\pm 0.73$ \\\\\n 60 & 0.0422 & 54.51 & 52.90 & $ 2.36\\pm 0.95$ \\\\\n 55 & 0.0361 & 50.26 & 48.39 & $ 2.03\\pm 0.82$ \\\\\n 50 & 0.0305 & 45.22 & 41.67 & $ 1.35\\pm 0.54$ \\\\\n 45 & 0.0254 & 38.03 & 34.58 & $ 2.33\\pm 1.35$ \\\\\n 40 & 0.0207 & 35.02 & 32.45 & $ 1.95\\pm 1.35$ \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\\vspace{.5cm}\n\n\n\n\n\\section{conclusion}\n\n Using only statistical errors and fixing additional normalization\n of differential cross sections equal to unity,\n we have limited the possible forms of the theoretical representation of the scattering\n amplitude.\n The phenomenological model - HEGSh model was used\n for examining the whole region of the momentum transfer\n of two sets of experimental data obtained by the TOTEM Collaboration at $13$ TeV.\n The simple exponential form of the scattering amplitude was used to examine only\n a small region of momentum transfer.\n In both cases, an additional fast decreasing term of\n the scattering amplitude was required for a quantitative description of the\n new experimental data.\n The large slope of this term can be connected with a large radius of the hadronic\n interaction and, hence, can be determined by the interaction potential at large distances.\n It can be some part of the hadronic potential responsible for the oscillation behavior\n of the elastic scattering amplitude \\cite{Osc-13}.\n\n The discovery of such anomaly in the behaviour of the differential\n cross section at very small momentum transfer\n in LHC experiments will give us important information about\n the behavior of the hadron interaction potential at large distances.\n It may be tightly connected with the problem of confinement.\n We have shown the existence of such anomaly\n \n \n \n at the statistical level and that some other models also\n revealed such unusual behaviour of the scattering amplitude.\n Very likely, such effects exist also in experimental data\n at essentially smaller energies \\cite{osc-conf}.\n However, the results of the TOTEM Collaboration have a unique\n unprecedentedly small statistical error and reach minimally small\n angles scattering with the largest number of experimental poits\n in this small region of the momentum transfer.\n \n \n \n \n The new effects can impact the determination of\n the sizes of the total cross sections, the ratio of\n the elastic to the total cross sections and the size of the\n $\\rho(s,t)$ - the ratio of the real to imaginary part of the elastic scattering\n amplitude.\n\n Now the results for the total cross sections and $\\rho(t=0)$ can be compared\n for the case with additional coefficient normalization $n$ and for the cases with\n an additional fast decreasing term and $n=1$.\n The results are presented in Table IV. The different variants with a large coefficient of the normalization\n give practically the same value, which is less than the total cross sections extracted by the TOTEM Collaboration\n - $\\sigma_{tot}(TOTEM) =110.6 \\pm 3.4$ mb in the analysis of only the small momentum transfer region\n \\cite{T-st}.\n The size of $\\rho(t=0)$ obtained in the model calculations essentially exceed\n the size of $\\rho(t=0)=0.1 \\pm 0.01$ extracted by the TOTEM Collaboration \\cite{T-rho}.\n On the contrary, the variants with an additional fast decreasing term\n in different forms give a\n large value of $\\sigma_{tot}(\\sqrt(s)=13$ TeV which exceed the $\\sigma_{tot}(TOTEM)$,\n and $\\rho(t=0)$ practically coincides with the predictions of the COMPETE Collaboration\n \\cite{COMPETE}.\n \n \n\\begin{table}\n\\label{Table-4}\n\\caption{The comparison of $\\sigma_{tot} (\\sqrt(s)=13 \\ TeV)$ and $\\rho(t=0,\\sqrt(s)=13 \\ TeV)$\nobtained in the different variants of the model calculations. }\n\\vspace{.5cm}\n\\begin{tabular}{|c|c|c|c|c|} \\hline\n n & model & $\\sum \\chi^2_{i}$ & $\\sigma_{tot}$ mb & $ \\rho(t=0)$ \\\\ \\hline\n 1.135 & & 525\/415 & 106.1 & $ 0.146 $ \\\\\n 1.135 &$f_{d}=0$ & 515\/425 & 106.2 & $ 0.142 $ \\\\\n 1.135 & & 527\/425 & 106.2 & $ 0.148 $ \\\\ \\hline\n 1. &$f_{d}(r_{d})$ & 539\/425 & 113.2 & $ 0.109 $ \\\\\n 1. &$f_{d}(r_{d})$ & 549\/425 & 113.1 & $ 0.113 $ \\\\\n 1. &$f_{d}(Exp)$ & 550\/425 & 112.6 & $ 0.115 $ \\\\ \\hline\n\\end{tabular}\n\\end{table}\n Of course, we can not excluded the case that the real experimental normalization\n reaches essentially large values than taken into account by the TOTEM\n Collaboration. However, for a small momentum transfer it is a very unlikely\n case, as practically in all existing experiments on the measure of the differential\n cross sections at the small momentum transfer systematic errors\n do not exceed a few percent.\n\n\n\n\n\n\\section{Reference}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\\addcontentsline{toc}{section}{Introduction}\n\\label{sec:intro}\n\\subsection*{Reversibility}\nBeing able to reverse a computation is an important feature of computing systems.\nReversibility is a key aspect in every system that needs to achieve distributed consensus~\\cite{Bouge1988} to escape local states where the consensus cannot be found.\nIn such problems, multiple computing agents have to reach a common solution.\nAllowing independent agents to backtrack and explore the solution space enables them to reach a globally accepted state if given enough time and if a common solution exists.\nFor example, the dining philophers problem~\\cite{Hoare1985} requires a backtracking mechanism to prevent deadlocks.\nRewinding a computation step by step is also a common way to debug programs.\nIn such settings the step by step approach is often more useful than restarting the program from an initial state.\n\nImportantly, the backtracking mechanism can be integrated to the operational semantics of a programming language, instead of adding a tailor-made implementation on top of each program.\nA formal model for reversible concurrent systems needs to address two challenges at the same time: (i) how to compute without forgetting and (ii) what is an optimal notion of legitimate backward moves.\nRoughly speaking, the first point is about syntax: processes need to carry a memory that keeps track of everything that has been done (and of the choices that have not been made).\nImportantly the needed information to backtrack is recorded in a \\emph{distributed} fashion instead of using a centralized store, which could be a bottleneck for the computation.\nThe second point is tied to the choice of the computation's semantics.\nIn a sequential program, one backtracks computations in the opposite order to the execution.\nHowever, in a concurrent setting, we do not want to undo the actions precisely in the opposite order than the one in which they were executed, as this order may not materialise.\nThe concurrency relation between actions has to be taken into account.\nIt can be argued that the most liberal notion of reversibility is the one that just respects causality: an action can be undone precisely after all the actions that causally depend on it have also been undone.\nThen an acceptable backward path is \\emph{causally consistent} with the forward computation.\n\nThere are different accounts of reversible operational semantics, RCCS~\\cite{Danos2004,Danos2007} and CCSK~\\cite{Phillips2007} being the two main propositions for a reversible CCS.\nIn these works, reversibility is embedded into a (classical) process calculus.\n\n\\subsection*{Causal models}\nIn interleaving models, the internal relations between different events cannot be observed.\nIn particular, causality is not treated as a primitive concept.\nOn the other hand, non-interleaving semantics have a primitive notion of \\emph{concurrency} between computation events.\nAs a consequence one can also derive a \\emph{causality} relation, generally defined as the complement of concurrency.\nThese models are therefore sometimes called \\emph{true-concurrent} or \\emph{causal} or, if causality is represented as a partial order on events, \\emph{partial order} semantics\\footnote{Event and configuration structures were introduced to define domains for concurrency~\\cite{Nielsen1979}.\n\tCausal models are thus often, but inaccurately, called \\emph{denotational}: a denotational interpretation is supposed to be invariant by reductions, a property that event structures do not have.}.\n\nA causal model is often an alternative representation of an existing interleaving semantics that helps in understanding the relations between computations in the latter.\nUsually in such models, sets of events are considered computational states.\nEach set, called a \\emph{configuration}, represents a reachable state in the run of the process.\nThe behaviour of a system is encoded as a collection of such sets.\nThe set inclusion relation between the configurations stands for the possible paths followed by the execution.\nConcurrency and causality are derivable from set inclusion.\nIn their generality, such models are called \\emph{configuration structures}~\\cite{Glabbeek2009}, they are a syntax-free and causal model that can interpret multiple calculi.\n\n\\emph{Stable families}~\\cite{Winskel1986} are configuration structures equipped with a set of axioms, that capture the intended behaviour of a CCS process.\nMorphisms of stable families capture sub-behaviours of processes and form a category of stable families.\nProcess combinators correspond then to universal constructions in this category.\nThe correspondence with CCS is established through an operational semantics defined on stable families, that we abusively name in that context configurations structures as well.\n\n\\subsection*{Behavioural equivalence}\nBehavioural equivalences are a major motivation in the study of formal semantics.\nFor instance, one wants to verify that the execution of a program satisfies its expected behaviour, or that binaries obtained from the same source code, but with different compilation techniques, behave the same.\nThus the interesting equivalences equate terms that behave the same.\nMoreover the equivalence should be a congruence: two processes are equivalent if they behave similarly in any context.\nLoosely speaking it aims at identifying process that have a common external behaviour in any environment.\n\nEquivalences defined on reduction semantics are often hard to prove.\nA proof technique in this case is to define a LTS-based equivalence that is equivalent with the reduction-based one and carry the proofs in LTS semantics.\n\nBehavioural equivalences are defined on the operational semantics and thus cannot access the structure of a term.\nThe observations one do during the execution of a process are called the \\emph{observables} of the relation.\nFor instance one observes whether the process terminates or whether it interacts with the environment~\\cite{Honda1995}.\n\n\\subsection*{Causality and reversibility}\nCausality and reversibility are tightly connected notions~\\cite{Nicola1990,Danos2004}.\nCausal consistency is a correctness criterion for reversible computations.\nTherefore whenever a reversible semantics is proposed, the calculus has to be equipped first with a causal semantics.\n\n\\emph{Prime} LTS are known~\\cite{Nielsen1981} to generates a prime event structure.\nSince a specific reversible LTS~\\cite{Phillips2007} is indeed prime, and moreover since the forward and backward reductions correspond to reductions in its causal representation, reversible models and causal ones are easily derivable from each other.\n\nNotably the connection between reversibility and causality is useful to define meaningful reversible equivalences.\nCausal equivalences are more discriminating than the traditional operational ones.\nHowever on a reversible operational semantics one define equivalences of the same expressivity.\nCausal equivalences have been extensively studied~\\cite{Phillips2012,Bednarczyk1991,Baldan2014,Vogler1993}.\nOf particular interest is the hereditary history preserving bisimulation, which was shown to correspond to a LTS-based equivalence for a reversible CCS~\\cite{Phillips2007}.\n\n\\subsection*{Equivalences on configuration structures}\nIn CCS equivalences are defined only on forward transitions and are therefore inappropriate to study reversible processes. %\n\nA reversible bisimulation~\\cite{Lanese2010} is more adapted but it is not contextual.\nWe introduce a contextual equivalence on RCCS by adapting the notions of contexts and barbs to the reversible setting.\nThe resulting relation, called \\emph{barbed back-and-forth congruence} is defined similarly to the barbed congruence of CCS except that the backwards reductions are also observed.\n\nConfiguration structures provide a causal semantics for CCS.\nEquivalences on configuration structures are more discriminating than the ones on the operational setting. %\nIt is possible to move up and down in the lattice, whereas in the operational semantics, only forward transitions have to be simulated.\nAs an example, consider the processes \\(a \\mid b\\) and \\(a.b+b.a\\) that are bisimilar in CCS but whose causal relations between events differ.\n\nIn particular we are interested in hereditary history preserving bisimulation (HHPB) in \\autoref{def:hhpb}, which equates configuration structures that simulate each others' forward and backward moves.\nPhillips and Ulidowski~\\cite{Phillips2012} showed that the back-and-forth bisimulation corresponds to HHPB, that can be defined in an operational setting thanks to reversibility.\nAllowing both forward and backward transitions gives to the operational world the discriminating power of causal models.\nWe show that HHPB also corresponds to a congruence on RCCS, the barbed back-and-forth congruence.\nIt is the a contextual characterisation of HHPB which implies a contextual equivalences in configuration structures.\n\n\\subsection*{Outline}\nWe begin by recalling notions on LTS and CCS, as well as their so-called reversible variants (\\autoref{sec:syntax}).\nRCCS (\\autoref{sec:rccs}) is then proven to be a conservative extension of CCS\u00a0over the traces: their is a strong bisimulation between a reversible process and a \\enquote{classical}, memory-less, process (\\autoref{lem:corresp_ccs_rccs}).\nLastly, we adapt the usual CCS notions of contexts, barbs, and barbed congruence to RCCS (\\autoref{sec:contextual-equiv-rccs}), thus introducing the back-and-forth barbed congruence (\\autoref{def:sbfc_rccs}).\n\nWe next introduce the interpretation of reversible process on configuration structures (\\autoref{sec:semantics}).\nWe recall the classical definitions (\\autoref{sec:st_fam}) as well as the encoding of CCS\u00a0terms in configuration structures (\\autoref{sec:causal_ccs}).\nEncoding of RCCS\u00a0terms is built on top of it (\\autoref{sec:stfam_rccs}), and an operational correspondence between reversible processes and their interpretations is proven (\\autoref{lem:operational_corresp}).\n\nFinally, we introduce a notion of context for configuration structures (\\autoref{sec:context_stfam}) and study the relation induced on configuration structures by the barbed back-and-forth congruence (\\autoref{sec:barbed_congruence_stfam}).\nIn \\autoref{sec:inductive_hhpb} we define the hereditary history preserving bisimilarity and provide a characterisation by inductive relations.\nLastly, we show in \\autoref{sec:contextual_hhpb} that HHPB is a congruence (\\autoref{prop:HHPB_congr}) and that whenever two configuration structures are barbed back-and-forth congruent, they also are hereditary history preserving bisimilar (\\autoref{main-thm}).\n\nOur main contribution is proving that barbed congruence in RCCS corresponds to hereditary history preserving bisimulation, which is defined on configuration structures\nAs a consequence, it provides a contextual characterization of equivalences defined in non-interleaving semantics.\n\n\\subsection*{Limitations}\nOur work is restrained to processes that forbid \\enquote{auto-concurrency} and \\enquote{auto-conflict} (\\autoref{rem-concur}).\nWe do not cover recursion, though a treatment of recursion in configuration structures exists~\\cite{Winskel1986}.\n\\enquote{Irreversible} action is a feature of RCCS~\\cite{Danos2004} that is absent of our work.\n\nWe tried to stick to canonical notations and to remind of common definitions.\nHowever, we consider the reader familiar with the syntax, congruence relation and reduction rules of CCS.\nIf not, a quick glance at a textbook~\\cite{Milner1989} or at lectures notes~\\cite{Amadio2014} should help the reader uneasy with them.\n\n\\section{Contextual equivalences in reversibility}\n\\label{sec:syntax}\nReversibility provides an implicit mechanism to undo computations.\nInterleaving semantics use a Labeled Transition System (LTS) to represent computations, henceforth refered to as the \\emph{forward} LTS.\nIn a reversible semantics a second LTS is defined that represents the \\emph{backward} moves (\\autoref{sec:lts_gen}).\n\nRCCS~\\cite{Danos2004, Danos2005, Krivine2006} (\\autoref{sec:rccs}) is a reversible variant of CCS, that allows computations to \\emph{backtrack}, hence introducing the notions of \\emph{forward} and \\emph{backward} transitions.\nMemories attached to processes store the relevant information to eventually do backward steps.\nWithout this memory, RCCS terms are essentially CCS terms (\\autoref{lem:corresp_ccs_rccs}), but their presence forces to be precise when defining contexts and contextual equivalence for the reversible case (\\autoref{sec:contextual-equiv-rccs}).\n\n\\subsection{(Reversible) labelled transition systems}\n\\label{sec:lts_gen}\nA labelled transition system is a multi-graph where the nodes are called \\emph{states} and the edges, \\emph{transitions}.\nTransitions are labelled by \\emph{actions} and may be fired non-deterministically.\n\n\\begin{definition}[Labelled Transition System]\n\tA \\emph{labelled transition system} is a tuple \\((\\to,S, \\textrm{Act})\\) made of a set \\(S\\) of \\emph{states}, a set \\(\\textrm{Act}\\) of \\emph{actions} (or labels) and a relation \\(\\to\\subseteq S\\times \\textrm{Act}\\times S\\).\n\n\tFor \\(s, s'\\in S\\) and \\(a, b \\in \\textrm{Act} \\) , we write \\(s\\redl{a}s'\\) for \\((s,a,s')\\in\\to\\) and \\(s\\to s'\\) if \\(s\\redl{a}s'\\) for some \\(a\\in \\textrm{Act}\\).\n\n\tElements \\(t:s\\redl{a}s'\\) of \\(\\mathrel{\\fw}\\) are called transitions.\n\tTwo transitions, \\(t\\) and \\(t'\\) are \\emph{composable}, written \\(t;t'\\), if the target of \\(t\\) is the source of \\(t'\\).\n\tThe empty trace is denoted \\(\\tempty\\).\n\\end{definition}\n\n\\begin{definition}[Trace]\n\t\\label{def:trace}\n\tA \\emph{trace}, denoted by \\(\\sigma:t_1;\\dots;t_n\\) is a sequence of composable transitions.\n\tExcept for the empty trace, all traces have a source and a target.\n\n\tDefine \\(\\mathrel{\\fw}^{\\star}\\subseteq S\\times \\textrm{Act}^{\\star}\\times S\\) the \\emph{reachability} relation as follows:\n\t\\[ s\\redl{\\alpha_1} \\dotsb \\redl{\\alpha_n}s'\\iff \\begin{multlined}[t]\n\t\t\\exists t_1,\\dots,t_n\\text{ and }s_1,\\dots,s_{n+1}\\text{ such that }\\\\\n\t\tt_i:s_i\\redl{\\alpha_i}s_{i+1}\\text{ and }s_1=s, s_{n+1} = s'.\n\t\t\\end{multlined}\n\t\\]\n\tWe say in that case that \\(s'\\) is \\emph{reachable} from \\(s\\), that \\(s'\\) is a \\emph{derivative} of \\(s\\), and that \\(s\\) is an \\emph{ancestor} of \\(s'\\).\n\\end{definition}\n\n\\begin{definition}[Reversible LTS]\n\t\\label{def:rlts}\n\tGiven \\((\\fw,S, \\textrm{Act})\\) and \\((\\bw,S, \\textrm{Act})\\) two labelled transition systems defined on the same set of states and actions, we define \\((\\fbw, S, \\textrm{Act})\\) a third LTS by taking \\(\\fbw = \\fw \\cup \\bw\\).\n\tBy convention, a transition \\(s \\fw t\\) is said to be \\emph{forward}, whereas a transition \\(t \\bw s\\) is said to be \\emph{backward}.\n\tIn \\(t \\bw s\\), \\(s\\) is an \\emph{ancestor} of \\(t\\).\n\\end{definition}\n\nA variety of semantically different backtracking mechanisms exists, for instance,\n\\begin{itemize}\n\t\\item taking \\(\\bw=\\emptyset\\) models a language with only irreversible moves,\n\t\\item in a sequential setting, if \\(\\fw\\) draws a tree, taking \\(\\bw = \\{(t, a, s) \\setst s \\redl{a} t\\}\\) forces the backward traces to follow exactly the forward execution.\n\\end{itemize}\n\nIn concurrency, backward traces are allowed if their source and target are respectively the target and source of a forward trace.\n\n\\subsection{Reversible CCS}\n\\label{sec:rccs}\nA RCCS term, also called a \\emph{monitored process}, is a CCS process equipped with a memory.\nA \\emph{thread} is a CCS term \\(P\\) guarded by a memory \\(m\\) and denoted \\(\\mproc{m}P\\).\nProcesses can be composed of multiple threads.\nThe memory acts as a stack for the previous computations.\nEach entry in the memory is called a \\emph{(memory) event} and has a unique identifier.\nThe forward transitions push events to the memories while the backward moves pop them out.\n\n\\begin{definition}[Names, labels and actions]\n\tWe define \\(\\mathsf{N}=\\{a,b,c,\\dots\\}\\) to be the set of \\emph{names} and \\(\\out{\\mathsf{N}}=\\{\\out{a},\\out{b},\\out{c},\\dots\\}\\) its \\emph{co-names}.\n\tThe complement of a (co-)name is given by a bijection \\(\\out{[\\cdot]}:\\mathsf{N} \\to \\out{\\mathsf{N}}\\), whose inverse is also denoted by \\(\\out{[\\cdot]}\\), so that \\(\\out{\\out{a}}=a\\).\n\n\tA \\emph{synchronisation} is a pair of names that complement each other, as \\((a,\\out{a})\\), and that is denoted with the special symbol \\(\\tau\\), whose complement is undefined.\n\n\tActions are labelled using the set \\(\\mathsf{L} = \\mathsf{N} \\cup \\out{\\mathsf{N}} \\cup \\{\\tau\\}\\) of (event) labels defined by the following grammar:\n\t\\begin{align}\n\t\t\\mathsf{N} \\cup \\out{\\mathsf{N}} : \\lambda, \\pi & \\coloneqq a \\enspace \\Arrowvert \\enspace \\out{a} \\enspace \\Arrowvert \\enspace \\hdots \\tag{CCS prefixes} \\\\\n\t\t\\mathsf{L} : \\alpha,\\beta & \\coloneqq \\tau \\enspace \\Arrowvert \\enspace a \\enspace \\Arrowvert \\enspace \\out{a} \\enspace \\Arrowvert \\enspace \\hdots \\tag{Event labels}\n\t\\end{align}\n\tAs it is common, we will sometimes use \\(a\\) and \\(b\\) to range over names, and call the set of names and co-names simply the set of names.\n\\end{definition}\n\nTransitions in both directions are decorated by the identifier of the associated event.\nIdentifiers on the (partial) events are used to remember their synchronisation partners.\nThus to combine into a \\(\\tau\\), the transitions need complementary labels and the same identifier.\n\n\\paragraph{Grammar}\nConsider the following \\emph{process constructors}, also called \\emph{combinators} or \\emph{operators}:\n\\begin{align}\n\te & \\coloneqq \\mem{i, \\alpha, P} & \\tag{memory events} \\\\\n\tm &\\coloneqq \\varnothing \\enspace \\Arrowvert \\enspace \\fork . m \\enspace \\Arrowvert \\enspace e. m \\tag{memory stacks}\\\\\n\tP,Q &\\coloneqq \\lambda.P \\enspace \\Arrowvert \\enspace P\\mid Q \\enspace \\Arrowvert \\enspace \\lambda.P+\\pi.Q \\enspace \\Arrowvert \\enspace P\\backslash a \\enspace \\Arrowvert \\enspace 0 \\tag{CCS processes}\\\\\n\tR, S & \\coloneqq m \\rhd P \\enspace \\Arrowvert \\enspace R \\mid S \\enspace \\Arrowvert \\enspace R\\backslash a \\tag{RCCS processes}\n\\end{align}\nA (memory) event \\(e=\\mem{i, \\alpha, P}\\) is made of:\n\\begin{itemize}\n\t\\item An event identifier \\(i \\in \\ids\\) that \\emph{tags} transitions.\n\t We may think of them as \\texttt{pid}, in the sense that they are a centrally distributed identifier attached to each transition.\n\t\\item A label \\(\\alpha\\) that marks which action has been fired (in the case of a forward transition), or what action should be restored (in the case of a backward move).\n\t\\item A backup of the whole process \\(P\\) that has been erased when firing a sum, or \\(0\\) otherwise.\n\\end{itemize}\n\nIn the memory stack, the fork symbol \\(\\fork\\) marks a parallel composition.\nThe memory is then copied in two local memories, as despicted in the congruence rule called \\enquote{\\ref{distrib-memory}} in \\autoref{def:congru_rccs}).\n\nLastly the null process, denoted \\(0\\), cannot perform any transition.\nWe will often omit it, so for example we write \\(a\\mid b\\) instead of \\(a.0\\mid b.0\\).\n\n\\begin{notations}\n\t\\label{notation:ids}\n\t\\begin{itemize}\n\t\t\\item We use \\(\\mathbb{N}\\) for the set of \\emph{event identifiers} \\(\\ids\\) and let \\(i,j,k\\) range over elements of \\(\\ids\\).\n\t\t Forward and backward transitions will be tagged with such identifiers, and so we write \\(\\fwlts{i}{\\alpha}\\) and \\(\\bwlts{i}{\\alpha}\\).\n\t\t We use \\(\\fbwlts{i}{\\alpha}\\) as a wildcard for \\(\\fwlts{i}{\\alpha}\\) or \\(\\bwlts{i}{\\alpha}\\), and if there are indices \\(i_1, \\hdots, i_n\\) and labels \\(\\alpha_1, \\hdots, \\alpha_n\\) such that \\(R_1 \\fbwlts{i_1}{\\alpha_1} \\dotsb \\fbwlts{i_n}{\\alpha_n} R_n\\), then we write \\(R_1 \\fbw^{\\star} R_n\\).\n\t\t We sometimes omit the identifier or the label in the transition.\n\t\t\\item For \\(R\\) a reversible process and \\(m\\) a memory, we denote \\(\\ids(m)\\) (resp.\\@\\xspace \\(\\ids(R)\\)) the set of identifiers occurring in \\(m\\) (resp.\\@\\xspace in \\(R\\)).\n\t\t\\item The sets \\(\\nm{R}\\) of names in \\(R\\), \\(\\fn{R}\\) of free names in \\(R\\) and \\(\\bn{R}=\\nm{R}\\setminus{\\fn{R}}\\) of bound (or private) names in \\(R\\) are defined by extending the definition of free names on CCS terms to memories and RCCS terms:\n\t\t \\begin{align*}\n\t\t \t &\n\t\t \t\\begin{aligned}\n\t\t \t\\fn{P\\backslash a} & =\\fn{P}\\setminus{\\{a\\}} \\\\\n\t\t \t\\fn{a.P} & =\\fn{\\out{a}.P}=\\{a\\}\\cup\\fn{P} \\\\\n\t\t \t\\fn{P\\mid Q} & =\\fn{P+Q}=\\fn{P}\\cup\\fn{Q} \\\\\n\t\t \t\\fn{0} & =\\emptyset\n\t\t \t\\end{aligned}\n\t\t \t\\tag{CCS rules}\\\\\n\t\t \t\\\\&\n\t\t \t\\begin{aligned}\n\t\t \t\\fn{R\\backslash a} & =\\fn{R}\\setminus{\\{a\\}} \\\\\n\t\t \t\\fn{R\\mid S} & =\\fn{R}\\cup\\fn{S} \\\\\n\t\t \t\\fn{m \\rhd P} & =\\fn{P}\n\t\t \t\\end{aligned}\n\t\t \t\\tag{RCCS rules}\n\t\t \\end{align*}\n\t\\end{itemize}\n\\end{notations}\n\n\\begin{remark}[On recording the past]\n\t\\label{rm:past}\n\tTo store the information needed to backtrack, RCCS attaches local memories to each thread.\n\tCCSK~\\cite{Phillips2007}, a variant of CCS, simulates reductions by movings a pointer in the term, that is left unchanged.\n\tReversible higher-order \\(\\pi\\)~\\cite{Lanese2010} uses a centralised, global memory to store the process before a reduction.\n\tKeys are associated to each reduction, thus reverting a transition with key \\(k\\) consists in restoring the process associated to \\(k\\) from the global memory.\n\tThe exact mechanism used for recording does not have an impact on the theory except for the structural rules, as we note in \\autoref{rk:congr}.\n\\end{remark}\n\nThe labelled transition system for RCCS is given by the rules of \\autoref{fig:lts_rccs}.\n\n\\begin{figure}\n\t\\begin{minipage}[b]{.99\\linewidth}\n\t\t\\centering\n\t\t\\begin{tabular}{c}\n\t\t\t \\\\[1em]\n\t\t\t\\begin{prooftree}\n\t\t\t\\Infer[left label = {\\drule{In\\(+\\)}}, right label = {\\(i\\notin\\ids(m)\\)}]{0}{\\mproc{m}{a.P + Q}\\fwlts{i}{a} \\mproc{\\mem{i,a,Q}.m}P}\n\t\t\t\\end{prooftree}\n\t\t\t \\\\[2em]\n\t\t\t\\begin{prooftree}\n\t\t\t\\Infer[left label = {\\drule{Out\\(+\\)}}, right label = {\\(i\\notin\\ids(m)\\)}]{0}{\\mproc{m}{\\out{a}.P + Q}\\fwlts{i}{\\out{a}} \\mproc{\\mem{i,\\out{a},Q}.m}P}\n\t\t\t\\end{prooftree}\n\t\t\t \\\\[2em]\n\t\t\t\\begin{prooftree}\n\t\t\t\\Infer[left label = {\\drule{In\\(-\\)}}, right label = {\\(i\\notin\\ids(m)\\)}]{0}{\\mproc{\\mem{i,a,Q}.m}P\\bwlts{i}{a} \\mproc{m}{a.P + Q}}\n\t\t\t\\end{prooftree}\n\t\t\t \\\\[2em]\n\t\t\t\\begin{prooftree}\n\t\t\t\\Infer[left label = {\\drule{Out\\(-\\)}}, right label = {\\(i\\notin\\ids(m)\\)}]{0}{\\mproc{\\mem{i,\\out{a},Q}.m}P\\bwlts{i}{\\out{a}} \\mproc{m}{\\out{a}.P + Q}}\n\t\t\t\\end{prooftree}\n\t\t\t \\\\[2em]\n\t\t\\end{tabular}\n\t\t\\subcaption{Prefix and sum rules}\\label{fig:lts_a}\n\t\\end{minipage}\n\n\t\\vspace{2em}\n\n\t\\begin{minipage}[b]{.99\\linewidth}\n\t\t\\centering\n\t\t\\begin{tabular}{c c}\n\t\t\t\\\\[1em]\n\t\t\t\\begin{prooftree}\n\t\t\t\\Hypo{R \\fwlts{i}{\\alpha} R' \\quad S \\fwlts{i}{\\out{\\alpha}} S'}\n\t\t\t\\Infer[left label = {\\drule{Com\\(+\\)}}]{1}{R \\mid S \\fwlts{i}{\\tau} R' \\mid S'}\n\t\t\t\\end{prooftree}\n\t\t\t &\n\t\t\t\\begin{prooftree}\n\t\t\t\\Hypo{R \\fbwlts{i}{\\alpha} R'}\n\t\t\t\\Infer[left label = {\\drule{ParL}}, right label = {\\(i\\notin\\ids(S)\\)}]{1}{R \\mid S \\fbwlts{i}{\\alpha} R' \\mid S}\n\t\t\t\\end{prooftree}\n\t\t\t\\\\[2em]\n\n\t\t\t\\begin{prooftree}\n\t\t\t\\Hypo{R \\bwlts{i}{\\alpha} R' \\quad S \\bwlts{i}{\\out{\\alpha}} S'}\n\t\t\t\\Infer[left label = {\\drule{Com\\(-\\)}}]{1}{R \\mid S \\bwlts{i}{\\tau} R' \\mid S'}\n\t\t\t\\end{prooftree}\n\t\t\t &\n\t\t\t\\begin{prooftree}\n\t\t\t\\Hypo{R \\fbwlts{i}{\\alpha} R'}\n\t\t\t\\Infer[left label = {\\drule{ParR}}, right label = {\\(i\\notin\\ids(S)\\)}]{1}{S \\mid R \\fbwlts{i}{\\alpha} S \\mid R'}\n\t\t\t\\end{prooftree}\n\t\t\t\\\\[2em]\n\t\t\\end{tabular}\n\t\t\\subcaption{Parallel constructions}\\label{fig:lts_c}\n\t\\end{minipage}\n\n\t\\vspace{2em}\n\n\t\\begin{minipage}[b]{.99\\linewidth}\n\t\t\\centering\n\t\t\\begin{tabular}{c c}\n\t\t\t\\\\[1em]\n\t\t\t\\begin{prooftree}\n\t\t\t\\Hypo{R \\fbwlts{i}{\\alpha} R'}\n\t\t\t\\Infer[left label = {\\drule{Hide}}, right label = {\\(a \\notin \\{\\alpha, \\out{\\alpha}\\}\\)}]{1}{ R\\backslash a \\fbwlts{i}{\\alpha} R'\\backslash a}\n\t\t\t\\end{prooftree}\n\t\t\t &\n\t\t\t\\begin{prooftree}\n\t\t\t\\Hypo{R \\congru R' \\fbwlts{i}{\\alpha} S' \\congru S}\n\t\t\t\\Infer[left label = {\\drule{Congr}}]{1}{R \\fbwlts{i}{\\alpha} S}\n\t\t\t\\end{prooftree}\n\t\t\t\\\\[2em]\n\t\t\\end{tabular}\n\t\t\\subcaption{Hiding and congruence}\\label{fig:lts_b}\n\t\\end{minipage}\n\t\\caption{Rules of the RCCS LTS}\n\t\\label{fig:lts_rccs}\n\\end{figure}\n\nThe prefix constructor \\(a.P\\) stands for sequential composition, the process interacts on \\(a\\) before continuing with \\(P\\).\nRules \\drule{In\\(+\\)} (for the input) and \\drule{Out\\(+\\)} (for the output) consumes a prefix by adding in the memory the corresponding event.\nThe backward moves, described by the rules \\drule{In\\(-\\)} and \\drule{Out\\(-\\)}, remove an event at the top of a memory and restores the prefix and the non-deterministic sum.\nThose rules are presented with a (guarded) sum, but we consider for instance \\(\\varnothing \\rhd a.P \\fwlts{1}{a} \\mem{1,a,0}. \\varnothing \\rhd P\\) to be a legal transition, taking \\(P + 0\\) (which is not syntactically correct) to be \\(P\\).\n\nParallel composition \\(P\\mid Q\\) employ the four rules of \\autoref{fig:lts_c} to derive a transition.\nRules \\drule{Com\\(+\\)} and \\drule{Com\\(-\\)} depicts two process agreeing to synchronize or to undo a synchronization by providing two dual prefixes\\footnote{Notice that since the complement of \\(\\tau\\) is not defined, only inputs and outputs synchronize.}, agreeing on the event identifier and triggering the transitions simultaneously.\nRules \\drule{ParL} and \\drule{ParR} allow respectively the left or the right process to compute independently of the rest of the process.\nIn those two later rules, the side condition \\(i \\notin \\ids(S)\\) ensures, in the forward direction, the uniqueness of the event identifiers and it prevents, in the backward direction, a part of a previous synchronisation to backtrack alone.\n\nOnce the name \\(a\\) is \\enquote{hidden in \\(P\\)}, that is, made private to the process \\(P\\), it cannot be used to interact with the environment.\nThis situation is denoted with \\(P\\backslash a\\) and illustrated in rule \\drule{Hide}.\nWhenever the private name \\(a\\) is encountered in the environment, \\(\\alpha\\)-renaming of \\(a\\) is done inside \\(P\\):\n\\[P\\backslash a =_{\\alpha} (P\\substs{b\/a})\\backslash b\\]\nwhere \\(P\\substs{b\/a}\\) stands for process \\(P\\) in which \\(b\\) substitutes \\(a\\).\nWe say that the hiding operator is a binder for the private name \\(a\\).\n\nThe structural congruence, whose definition follows, is applied on terms by the rule \\drule{Congr}.\nIt is built on top of some of the corresponding rules for CCS, and rewrites the terms under the memory or distributes it between two forking processes.\n\n\\begin{definition}[Structural congruence]\n\t\\label{def:congru_rccs}\n\tStructural congruence on monitored processes is the smallest equivalence relation up to uniform renaming of identifiers generated by the:\n\t\\begin{align}\n\t\tm \\rhd (P + Q) & \\congru m \\rhd (Q + P) \\\\\n\t\tm \\rhd ((P + Q) + R) & \\congru m \\rhd (P + (Q + R)) \\\\\n\t\t & \\frac{P =_\\alpha Q}{m \\rhd P \\congru m \\rhd Q} \\tag{\\(\\alpha\\)-conversion} \\\\\n\t\tm \\rhd (P \\mid Q) & \\congru (\\fork . m \\rhd P \\mid \\fork . m \\rhd Q)\\tag{distribution memory}\\label{distrib-memory} %\n\t\\end{align}\n\tAdding a \\emph{scope of restriction} rule \\(m \\rhd P\\backslash a \\congru (m \\rhd P)\\backslash a\\) could be done at the price of a cumbersome definition of what a free name in a memory is.\n\\end{definition}\n\n\\begin{remark}[On reduction semantics]\n\t\\label{rk:congr}\n\tCorrectness criteria for reversible semantics mostly relate it with its \\emph{only-forward} counterpart.\n\tHowever one may be interested in defining a reduction semantics for the LTS of \\autoref{fig:lts_rccs} if only to relate RCCS with other reversible semantics for CCS.\n\tOne, then needs a congruence relation on RCCS terms that has the monoid structure for parallel composition and the null process \\(0\\).\n\tHowever, due to the fork constructor, the associativity does not hold:\n\t\\[(R_1\\vert R_2)\\vert R_3\\not\\congru R_1\\vert (R_2\\vert R_3).\\]\n\tOther reversible calculi, in particular the reversible higher-order \\(\\pi\\)-calculus~\\cite{Lanese2010} fares better: its congruence relation respects associativity, thanks to a mechanism that uses bounded names for forking processes.\n\tThen \\(\\alpha\\)-renaming can be applied on these forking names.\n\n\tAlternatively, one could use \\enquote{at distance rewriting}~\\cite{Accatoli2013} to bypass the lack of flexibility of our structural congruence.\n\\end{remark}\n\nIn RCCS not all syntactically correct processes have an operational meaning.\nConsider for instance\n\\[\\fork . \\mem{i, \\alpha, 0} . \\varnothing \\triangleright P \\mid \\varnothing \\triangleright Q.\\]\nTo make a backward transition, one should first apply the congruence rule called \\enquote{\\ref{distrib-memory}} and then look for a rule of the LTS to apply.\nBut this is impossible, since the memory on the right-hand side of the parallel operator does not contain a fork symbol (\\(\\fork\\)) at its top.\nThe distributed memory does not agree on a common past, blocking the execution, but this term is correct from a syntactical point of view.\nIn the following, we will consider only the semantically correct terms, called \\emph{coherent}.\n\n\\begin{definition}[Coherent process]\n\t\\label{def:coherent_rccs}\n\tA RCCS process \\(R\\) is \\emph{coherent} if there exists a CCS process \\(P\\) such that \\(\\varnothing \\rhd P \\fw^{\\star} R\\).\n\\end{definition}\n\nCoherent terms are also called \\emph{reachable}, as they are obtained by a forward execution from a process with an empty memory.\nCoherence of terms can equivalently be defined in terms of coherence on memories~\\cite[Definition~1]{Danos2004}.\n\nBacktracking is non-deterministic because backtracking is possible on different threads.\nHowever, it is noetherian and confluent as backward synchronisations are deterministic~\\cite[Lemma~11]{Danos2005}.\n\n\\begin{lemma}[Unique origin]\n\tIf \\(R\\) is a coherent process, then \\(\\forall R'\\) such that \\(R \\congru R'\\) or \\(R \\fbw R'\\), then \\(R'\\) is also coherent.\n\tUp to structural congruence, there exists a unique process \\(P\\) such that \\(R \\bw^{\\star} \\emptyset \\triangleright P\\), we call it \\emph{the origin of \\(R\\)} and denote it \\(\\orig{R}\\).\n\\end{lemma}\n\nLastly, we recall a useful result, that asserts that every reversible trace can be rearranged as a sequence of only-backward moves followed by a sequence of only-forward moves.\n\n\\begin{lemma}[Parabolic traces, {\\cite[Lemma 10]{Danos2004}}]\n\t\\label{lem:rearrange_trace}\n\tIf $R \\fbw \\cdots \\fbw S$, then there exists \\(R'\\) such that \\(R \\bw^{\\star} R'\\fw^{\\star} S\\).\n\\end{lemma}\n\nIt is natural to wonder if our reversible syntax is a conservative extension of CCS.\nWe will make sure in the following that the forward rules in the reversible LTS correspond to the LTS of the natural semantics.\n\n\\begin{definition}[Map from RCCS to CCS]\n\t\\label{def:rccs_ccs}\n\tWe define inductively a map \\(\\erase{\\cdot}\\) from RCCS terms to CCS terms by erasing the memory:\n\t\\begin{align*}\n\t\t\\erase{m\\rhd P} & =P & & & \\erase{R\\vert S} & =\\erase{R}\\vert\\erase{S} & & & \\erase{R\\backslash a} & =(\\erase{R})\\backslash a\n\t\\end{align*}\n\\end{definition}\n\nIn the following lemma, we denote \\(\\redl{\\alpha}\\) the standard rewriting rule on CCS terms.\n\\begin{lemma}[Strong \\enquote{forward} bisimulation between \\(R\\) and \\(\\erase{R}\\)]\n\t\\label{lem:corresp_ccs_rccs}\n\n\tFor all \\(R\\) and \\(S\\), \\(R \\fwlts{i}{\\alpha} S\\) for some \\(i\\) iff \\( \\erase{R} \\redl{\\alpha} \\erase{S} \\).\n\\end{lemma}\n\n\\subsection{Contextual equivalences}\n\\label{sec:contextual-equiv-rccs}\n\nContextual equivalence for CCS terms~\\cite{Milner1992} is now standard, but its extension to RCCS is not straightforward, since contexts needs to be properly defined (\\autoref{def:rccs_context}).\nAs usual, reductions are part of the observables, but observing only them results in a too coarse relation, and adding termination is not relevant in concurrency.\n\\emph{Barbed congruence} (\\autoref{def:barbed_congr}) has proven to be the right notion for CCS, and we revisit it for RCCS terms.\nWe begin by recalling definitions of context and observables for CCS.\n\n\\begin{definition}[Context]\n\t\\label{def:context}\n\tA context is a process with a hole \\([\\cdot]\\) defined formally by the grammar:\n\t\\[C[\\cdot]\\coloneqq [\\cdot] \\enspace \\Arrowvert \\enspace \\lambda.C[\\cdot] \\enspace \\Arrowvert \\enspace P \\mid C[\\cdot] \\enspace \\Arrowvert \\enspace C[\\cdot]\\backslash a\\]\n\\end{definition}\n\n\\begin{definition}[Barbs]\n\t\\label{def:barb}\n\tWrite \\(P \\downarrow_\\alpha\\) if there exists \\(P'\\) such that \\(P \\redl{\\alpha} P'\\).\n\\end{definition}\n\nWe now define a contextual equivalence where reductions and barbs are the observables.\n\n\\begin{definition}[Barbed congruence]\n\t\\label{def:barbed_congr}\n\tThe \\emph{barbed bisimulation} is a symmetric relation \\({\\mathrel{\\mathcal{R}}}\\) on CCS processes such that whenever \\(P \\mathrel{\\mathcal{R}} Q\\) the following holds:\n\t\\begin{align}\n\t\tP\\mathrel{\\fw} P' & \\implies \\exists Q' \\text{ \\st~} Q\\mathrel{\\fw} Q'\\text{ and }Q\\mathrel{\\mathcal{R}} Q'\\tag{closed by reduction} \\\\\n\t\tP\\downarrow a & \\implies Q\\downarrow a\\tag{barb preserving}\n\t\\end{align}\n\tIf there exists a barbed bisimulation between \\(P\\) and \\(Q\\) we write \\(P \\overset{\\cdot}{\\bisim}^{\\tau} Q\\) and say that \\(P\\) and \\(Q\\) are \\emph{barbed bisimilar}.\n\n\tIf \\(\\forall C[\\cdot]\\), \\(C[P] \\overset{\\cdot}{\\bisim}^{\\tau} C[Q]\\), we write \\(P \\mathrel{\\bisim^{\\tau}} Q\\) and say that \\(P\\) and \\(Q\\) are \\emph{barbed congruent}.\n\\end{definition}\n\nAn interesting proposition allows to restrict the grammar of contexts in the following.\n\n\\begin{proposition}\n\t\\label{prop:context_that_counts}\n\t\\(\\forall a, P_1, P_2 , Q, \\lambda, a\\),\n\t\\[\n\t\tP_1 \\overset{\\cdot}{\\bisim}^{\\tau} P_2 \\implies\n\t\t\\begin{cases}\n\t\t\t\\lambda.P_1 \\overset{\\cdot}{\\bisim}^{\\tau} \\lambda.P_2 \\\\\n\t\t\tP_1 \\backslash a\\overset{\\cdot}{\\bisim}^{\\tau} P_2 \\backslash a \\\\\n\t\t\tP_1 + Q\\overset{\\cdot}{\\bisim}^{\\tau} P_2 + Q\n\t\t\\end{cases}\n\t\\]\n\\end{proposition}\n\n\\begin{proof}\n\t\\begin{enumerate}\n\t\t\\item \\(P_1 \\overset{\\cdot}{\\bisim}^{\\tau} P_2 \\implies \\lambda.P_1 \\overset{\\cdot}{\\bisim}^{\\tau} \\lambda.P_2\\).\n\t\t From CCS's grammar, \\(\\lambda\\neq\\tau\\), hence \\(\\nexists P_1', P_2'\\) such that \\(P \\redl{\\tau} P_1'\\) or \\(P_2 \\redl{\\tau} P_2'\\).\n\t\t The relation \\(\\{\\lambda.P_1,\\lambda.P_2\\}\\) is trivially a barbed bisimulation.\n\t\t\\item \\(P_1 \\overset{\\cdot}{\\bisim}^{\\tau} P_2 \\implies P_1 \\backslash a \\overset{\\cdot}{\\bisim}^{\\tau} P_2 \\backslash a\\).\n\t\t Let us denote \\({\\mathrel{\\mathcal{R}}_1}\\) the largest barbed bisimulation for \\(P_1\\) and \\(P_2\\).\n\t\t We show that the relation \\({\\mathrel{\\mathcal{R}}_2}=\\{P_1 \\backslash a, P_2 \\backslash a \\setst P_1 \\mathrel{\\mathcal{R}}_1 P_2\\}\\) is a barbed bisimulation.\n\t\t We have to show that:\n\t\t \\begin{itemize}\n\t\t \t\\item \\(\\forall b\\) such that \\(P_1\\backslash a\\downarrow \\beta\\) then \\(P_2\\backslash a\\downarrow \\beta\\).\n\t\t \t It follows from \\(P_1\\backslash a\\downarrow \\beta\\implies P_1 \\downarrow \\beta\\) and \\(\\beta\\neq a\\).\n\t\t \t\\item \\(P_1\\backslash a \\redl{\\tau} P_1'\\) implies that \\(P_2\\backslash a\\redl{\\tau} P_2'\\) and \\(P_1' \\mathrel{\\mathcal{R}}_2 P_2'\\).\n\t\t \t By structural induction on the transition \\(P_1\\backslash a \\redl{\\tau} P_1'\\) we have that \\(\\exists P_1''\\) such that\n\t\t \t \\(P_1\\redl{\\tau} P_1''\\) and \\(P_1''\\backslash a=P_1'\\).\n\t\t \t As \\(P_1 \\mathrel{\\mathcal{R}}_1 P_2\\) there exists \\(P_2''\\) such that \\(P_2 \\redl{\\tau}P_2''\\) and we apply the rule \\drule{Hide} we get \\(P_2\\backslash a \\redl{\\tau} P_2''\\backslash a\\).\n\t\t \t Thus there exists \\(P_2'=P_2''\\backslash a\\) and \\(P_1' \\mathrel{\\mathcal{R}}_2 P_2'\\).\n\t\t \\end{itemize}\n\t\t It follows similarly for the barbs and reductions on \\(P_2\\).\n\n\t\t\\item \\(P_1 \\overset{\\cdot}{\\bisim}^{\\tau} P_2 \\implies P_1 + Q \\overset{\\cdot}{\\bisim}^{\\tau} P_2 + Q\\).\n\t\t Let us denote \\({\\mathrel{\\mathcal{R}}_1}\\) the largest barbed bisimulation for \\(P_1\\) and \\(P_2\\).\n\t\t We show that the relation \\({\\mathrel{\\mathcal{R}}_2}=\\mathrel{\\mathcal{R}}_1\\cup\\{P_1, P_2\\}\\) is a barbed bisimulation.\n\t\t As above, we show that:\n\t\t \\begin{itemize}\n\t\t \t\\item \\(\\forall\\alpha\\) such that \\((P_1 + Q)\\downarrow \\alpha\\) then \\((P_2 + Q)\\downarrow \\alpha\\).\n\t\t \t It follows from the subcases \\(P_1 \\downarrow \\alpha\\) (hence \\(P_2 \\downarrow \\alpha\\)) or \\(Q \\downarrow \\alpha\\).\n\t\t \t\\item \\(P_1 + Q \\redl{\\tau} P_1'\\).\n\t\t \t From rules \\drule{SumL} and \\drule{SumR} either \\(Q \\redl{\\tau} P_1'\\) or \\(P_1 \\redl{\\tau} P_1'\\).\n\t\t \t In the first case we deduce, using rule \\drule{SumL} that \\(P_1 + Q\\redl{\\tau}P_1'\\), and therefore \\(P_1' \\mathrel{\\mathcal{R}}_2 P_1'\\).\n\t\t \t In the second case, we apply rule \\drule{SumR} and have that \\(P_2 \\redl{\\tau}P_2'\\) and \\(P_1' \\mathrel{\\mathcal{R}}_1 P_2'\\), which concludes our proof.\n\t\t \\end{itemize}\n\t\t It follows similarly for the barbs and reductions on \\(P_2\\).\\qedhere\n\t\\end{enumerate}\n\\end{proof}\n\n\\begin{corollary}\n\tIf a context \\(C[\\cdot]\\) does not contain the parallel operator, then for all \\(P\\), \\(Q\\), \\(C[P] \\not\\overset{\\cdot}{\\bisim}^{\\tau} C[Q]\\) implies \\(P \\not\\overset{\\cdot}{\\bisim}^{\\tau} Q\\)\n\\end{corollary}\n\nStated differently, this implies that discriminating contexts regarding barbed congruence involve parallel composition.\nAs we will focus on this relation, we will only consider in the following the contexts to be parallel compositions:\n\\[C[\\cdot]\\coloneqq [\\cdot] \\enspace \\Arrowvert \\enspace P \\mid [\\cdot] \\]\n\nThis is handy to define RCCS context, but some subtleties remain.\nA context has to become an executable process regardless of the process instantiated with it.\nWe say that a context has a coherent memory, it may backtrack up to the context with an empty memory (similar to the \\autoref{def:coherent_rccs} of coherent processes).\nWe distinguish three types of contexts, depending on their memories:\n\\begin{itemize}\n\t\\item Contexts with an empty memory.\n\t\\item Contexts with a non-empty memory that is coherent on its own\\footnote{Up to minor addition of \\(\\fork\\) symbols, as explained later on.}.\n\t The process that we instantiate with it can be\n\t \\begin{itemize}\n\t \t\\item incoherent\\footnotemark[\\value{footnote}], in which case we conjecture that the term obtained after instantiation is also incoherent,\n\t \t\\item coherent on their own\\footnotemark[\\value{footnote}], in which case it is possible to backtrack the memory of the context up to the empty memory.\n\t \\end{itemize}\n\\end{itemize}\nHence w.l.o.g.\\@\\xspace we consider contexts without memory and contexts with coherent memories to be equivalent.\nThese are the types of contexts that we use throughtout the article.\nHowever, a third case remains:\n\\begin{itemize}\n\t\\item Contexts that have a non-coherent memory.\n\t There exists incoherent terms whose instantiation with an incoherent context is coherent.\n\t For instance, \\(C=\\mem{1,a,0}.\\fork . \\varnothing \\rhd P\\mid [\\cdot]\\) and \\(R=\\mem{1,\\out{a},0}.\\fork . \\varnothing \\rhd P'\\) are incoherent individually, but \\(C[R]\\) is coherent and can backtrack to \\(\\fork . \\varnothing \\rhd P \\mid \\fork . \\varnothing \\rhd P'\\).\n\t We leave this case as future work.\n\\end{itemize}\n\nThe \\enquote{up to minor addition of \\(\\fork\\) symbols} comes from a simple consideration on the parallel composition in RCCS.\nA process with an empty memory compose with a RCCS term if fork symbols are added to reflect the parallel composition.\nFor instance, two processes with an empty memory \\(\\varnothing \\rhd P\\) and \\(\\varnothing \\rhd P'\\) compose and we obtain\n\\[\\fork.\\varnothing \\rhd P\\mid \\fork.\\varnothing \\rhd P'\\congru \\varnothing \\rhd (P\\mid P')\\]\ninstead of \\(\\varnothing \\rhd P\\mid\\varnothing \\rhd P'\\), an incoherent process.\n\nWe define a rewriting function on RCCS processes, that adds a fork symbol at the beginning of a memory.\nIt allows a process with a memory to compose with a context.\n\n\\begin{definition}[RCCS context]\n\t\\label{def:rccs_context}\n\tDefine \\(\\addfork(R)\\) the operator that adds a fork symbol at the beginning of the memory of each thread in \\(R\\):\n\t\\begin{align*}\n\t\t\\addfork(R_1\\mid R_2) & = \\addfork(R_1)\\mid\\addfork(R_2) \\\\\n\t\t\\addfork(R\\backslash a) & = (\\addfork(R))\\backslash a \\\\\n\t\t\\addfork(m\\rhd P) & = m'.\\fork.\\varnothing\\rhd P\\text{ where }m=m'.\\varnothing \\\\\n\t\t\\addfork(0) & = 0\n\t\\end{align*}\n\n\tDefine \\(\\context[R]\\) as follows\n\t\\[\n\t\t\\context[R] =\n\t\t\\begin{cases}\n\t\t\tR & \\text{if } C[\\cdot]=[\\cdot] \\\\\n\t\t\t\\fork.\\varnothing \\rhd P \\mid \\addfork(R) & \\text{if } C[\\cdot] = P \\mid [\\cdot]\n\t\t\\end{cases}\n\t\\]\n\\end{definition}\n\nRCCS context are basically CCS context with additional fork symbols in the memory of the context and in the memory of the process instantiated.\nWe now verify that \\(\\context[R]\\) is a coherent process, using the function \\(\\erase{\\cdot}\\) that erases the memories from a term (\\autoref{def:rccs_ccs}).\n\n\\begin{proposition}\n\tFor all \\(R\\) and \\(\\context[\\cdot]\\), \\(\\varnothing\\rhd C[\\erase{\\orig{R}}]\\fw^{\\star} \\context[R]\\).\n\\end{proposition}\n\\begin{proof}\n\tLet \\(C[\\cdot] = P \\mid [\\cdot]\\) and \\(\\orig{R}= \\varnothing\\rhd P'\\).\n\tBy definition and application of the congruence rules, we have that\n\t\\begin{align*}\n\t\t\\varnothing\\rhd C[\\erase{\\orig{R}}] & = \\varnothing\\rhd \\big(P\\mid \\erase{\\orig{R}}\\big) \\\\\n\t\t & = \\varnothing\\rhd \\big(P\\mid P'\\big) \\\\\n\t\t & \\congru (\\fork.\\varnothing\\rhd P) \\mid (\\fork.\\varnothing\\rhd P')\n\t\\end{align*}\n\tWe have from the trace \\(\\orig{R}\\mathrel{\\fw}^{\\star} R\\) that\n\t\\[\n\t\t(\\fork.\\varnothing\\rhd P) \\mid (\\fork.\\varnothing\\rhd P') \\mathrel{\\fw}^{\\star} (\\fork.\\varnothing\\rhd P) \\mid \\addfork (R) = \\context[R].\t\\qedhere\n\t\\]\n\\end{proof}\n\n\\begin{example}\n\tLet \\(R=\\fork.m . \\varnothing \\rhd P_1\\mid \\fork.m . \\varnothing \\rhd P_2\\) and \\(C[\\cdot] = P \\mid [\\cdot]\\).\n\tLet us rewind \\(R\\) to its origin:\n\t\\begin{align*}\n\t\tR & =\\fork.m . \\varnothing \\rhd P_1\\mid \\fork.m . \\varnothing \\rhd P_2 \\\\\n\t\t & \\qquad \\congru m . \\varnothing \\rhd (P_1\\mid P_2) \\\\\n\t\t & \\qquad \\qquad \\bw^{\\star} \\varnothing \\rhd P' \\\\\n\t\t & \\qquad \\qquad \\qquad = \\orig{R}\n\t\\end{align*}\n\tWe instantiate the context with \\(\\orig{R}\\) and redo the execution from the origin of \\(R\\) up to \\(R\\):\n\t\\begin{align*}\n\t\t\\context[\\orig{R}] & = (\\fork.\\varnothing\\rhd P)\\mid (\\fork.\\varnothing \\rhd P') \\\\\n\t\t & \\qquad \\fw^{\\star}(\\fork.\\varnothing\\rhd P)\\mid \\big( (m.\\fork.\\varnothing \\rhd P_1) \\mid (m.\\fork.\\varnothing\\rhd P_2)\\big) \\\\\n\t\t & \\qquad \\qquad =(\\fork.\\varnothing\\rhd P)\\mid\\addfork(R) \\\\\n\t\t & \\qquad \\qquad \\qquad= \\context[R]\n\t\\end{align*}\n\tHence we have that \\(\\context[\\orig{R}]\\fw^{\\star}\\context[R]\\).\n\\end{example}\n\nOnce this delicate notion of context for reversible process is settled, extending the CCS barbs (\\autoref{def:barb}) and barbed congruence (\\autoref{def:barbed_congr}) are straightforward.\n\n\\begin{definition}[RCCS barbs]\n\t\\label{def:barb_rccs}\n\tWe write \\(R \\downarrow_\\alpha\\) if there exists \\(i \\in I\\) and \\(R'\\) such that \\(R \\fwlts{i}{\\alpha} R'\\).\n\\end{definition}\n\n\\begin{definition}[Back-and-forth barbed congruence]\n\t\\label{def:sbfc_rccs}\n\tA \\emph{back-and-forth bisimulation} is a symmetric relation on coherent processes \\(\\mathrel{\\mathcal{R}}\\) such that if \\( R \\mathrel{\\mathcal{R}} S\\), then\n\t\\begin{align*}\n\t\tR \\bwlts{i}{\\tau} R' & \\implies \\exists S' \\text{ \\st~} S \\bwlts{i}{\\tau} S' \\text{ and } R' \\mathrel{\\mathcal{R}} S'; \\tag{back} \\label{arriere} \\\\\n\t\tR\\fwlts{i}{\\tau} R' & \\implies \\exists S' \\text{ \\st~} S \\fwlts{i}{\\tau} S'\\text{ and } R' \\mathrel{\\mathcal{R}} S'; \\tag{forth} \\label{avant}\n\t\t\\intertext{and it is a \\emph{back-and-forth \\emph{barbed} bisimulation} if, additionally,}\n\t\tR \\downarrow_{a} & \\implies S \\downarrow_{a}. \\tag{barbed} \\label{barbe}\n\t\\end{align*}\n\tWe write \\( R \\overset{\\cdot}{\\bisim}^{\\tau} S\\) if there exists \\(\\mathrel{\\mathcal{R}}\\) a back-and-forth barbed bisimulation such that \\( R\\mathrel{\\mathcal{R}} S\\).\n\n\tThe \\emph{back-and-forth barbed congruence}, denoted \\( R \\bisim^{\\tau} S\\), holds if for all context \\(\\context[\\cdot]\\), \\(\\context[\\orig{R}] \\overset{\\cdot}{\\bisim}^{\\tau} \\context[\\orig S]\\).\n\\end{definition}\n\nFrom the definition of \\(R \\bisim^{\\tau} S\\), the following lemma trivially holds.\n\\begin{lemma}\n\t\\label{lem-orig-sbfbc}\n\tFor all \\(R\\) and \\(S\\), \\(R \\mathrel{\\sim^{\\tau}} S \\implies \\orig{R} \\mathrel{\\sim^{\\tau}} \\orig{S}\\).\n\\end{lemma}\n\nHowever, the converse does not hold as \\(R\\) and \\(S\\) can be any derivative of the same origin, as illustrated below.\n\n\\begin{example}\n\tLet \\(R = \\mem{1, a, b.Q} . \\emptyset \\rhd P\\) and \\(S = \\mem{2, b, a.P} . \\emptyset \\rhd Q\\), with \\(P \\not\\overset{\\cdot}{\\bisim}^{\\tau} Q\\).\n\tWe have that \\(\\orig{R} \\bisim^{\\tau} \\orig{S}\\), as \\(\\orig{R} = \\orig{S}\\), but \\(R \\not\\bisim^{\\tau} S\\), as \\(P \\not\\overset{\\cdot}{\\bisim}^{\\tau} Q\\):\n\t\\begin{center}\n\t\t\\begin{tikzpicture} %\n\t\t\t\\tikz@path@overlay{node} (or) at (1,1) {\\(\\orig{R}\\mathrel{\\sim^{\\tau}} \\orig S\\)};\n\t\t\t\\tikz@path@overlay{node} (r) at (1, -0.5) {\\(\\mem{1, a, b.Q} . \\emptyset \\rhd P \\not\\mathrel{\\sim^{\\tau}} \\mem{2, b, a.P} . \\emptyset \\rhd Q\\)};\n\t\t\t\\path[draw] [->] (or) -- node[left]{\\small{\\(1:a\\)}~~} (-0.8,-0.3);\n\t\t\t\\path[draw] [->] (or) -- node[right]{~\\small{\\(2:b\\)}} (2.8,-0.3);\n\t\t\\end{tikzpicture}\n\t\\end{center}\n\\end{example}\n\nNote that even if the context is defined for any reversible process, we instantiate the context with processes with an empty memory in \\autoref{def:sbfc_rccs}.\nIf instead we had defined \\( R \\bisim^{\\tau} S\\) iff for all contexts, there exists \\(\\mathrel{\\mathcal{R}}\\) such that \\(\\context[R] \\mathrel{\\mathcal{R}} \\context[S]\\), then \\autoref{lem-orig-sbfbc} would not hold.\nWe highlight this in the following example.\n\n\\begin{example}\n\tLet us consider the processes \\(\\varnothing\\rhd a.P + Q\\) and \\(\\varnothing \\rhd a.P\\) which can do a transition on \\(a\\) to obtain \\(R=\\mem{1,a, Q}\\rhd P\\) and \\(S=\\mem{1,a}\\rhd P\\).\n\tWe have that \\(R\\) and \\(S\\) are back-and-forth barbed bisimular.\n\tAs we are using contexts without memory, there is no context able to backtrack on \\(a\\).\n\t\\begin{center}\n\t\t\\begin{tikzpicture} %\n\t\t\t\\tikz@path@overlay{node} (or) at (-1,1) {\\(\\orig{R}=\\varnothing\\rhd a.P+Q\\)};\n\t\t\t\\tikz@path@overlay{node} (rel1) at (1,1) {\\(\\not\\overset{\\cdot}{\\bisim}^{\\tau}\\)};\n\t\t\t\\tikz@path@overlay{node} (os) at (2.5,1) {\\(\\orig S =\\varnothing\\rhd a.P\\)};\n\t\t\t\\tikz@path@overlay{node} (r) at (-1,0) {\\(R =\\mem{1,a,Q}\\rhd P\\)};\n\t\t\t\\tikz@path@overlay{node} (rel2) at (1,0) {\\(\\mathrel{\\sim^{\\tau}}\\)};\n\t\t\t\\tikz@path@overlay{node} (s) at (2.5,0) {\\(S =\\mem{1,a}\\rhd P\\)};\n\t\t\t\\path[draw] [->] (or) -- node[right]{\\small{\\(1:a\\)}} (r);\n\t\t\t\\path[draw] [->] (os) -- node[right]{\\small{\\(1:a\\)}} (s);\n\t\t\\end{tikzpicture}\n\t\\end{center}\n\\end{example}\n\n\\begin{remark}[On backward barbs]\n\t\\label{rk:back_barb}\n\tLet us informally argue that backward barbs are not an interesting addition to a contextual equivalence.\n\tOne can always define ad-hoc barbs that potentially change the equivalence relations, however we end up with relations that have no practical meaning.\n\tWe consider below another definition of barb~\\cite[Definition 2.1.3]{Madiot2015}, which gives an intuitive reading and is not syntax-specific.\n\n\tLet the tick (\\(\\checkmark\\notin \\mathsf{N}\\)) be a special symbol denoting termination.\n\tA \\emph{barb} is an interaction with a context that can do a tick immediately after:\n\t\\[P\\downarrow_\\alpha \\iff P\\mid \\out{\\alpha}.\\checkmark\\redl{\\tau}Q\\mid\\checkmark \\text{ for some }Q.\\]\n\n\tNote that the definition above implies that (i) the barb is an interaction with a context that can terminate immediately after and (ii) the interaction \\emph{blocks} the termination on the context side, i.e.\\@\\xspace no further transition is possible on that side.\n\n\tIf we try to apply this definition to a backward barb then the tick has to be in the memory of the context and blocked by another action, i.e.\\@\\xspace the context has to be of the form \\(C[\\cdot] \\congru [\\cdot] \\mid (\\mem{i,\\out{\\alpha},0}.\\mem{\\checkmark}.\\varnothing \\rhd 0) \\).\n\tThis raises multiple problems:\n\t\\begin{enumerate}\n\t\t\\item Syntactically, \\(\\checkmark\\) becomes a prefix, rather than a \\enquote{terminal process}, i.e.\\@\\xspace terms of the form \\(\\checkmark . a . P\\) appear.\n\t\t This contradict the intuition that this symbol stands for termination.\n\t\t\\item In a situation where \\(C[R] \\bwlts{i}{\\tau} R' \\mid \\mem{\\checkmark}.\\varnothing \\rhd 0 \\), the \\(\\checkmark\\) symbol is not observable, and \\(R'\\) could continue its computation before \\(\\checkmark\\) is popped from the context's memory.\n\t\t So we would have to add the content of the memory to what is observable.\n\t\t But in that case, one might as well look directly in the process' memory if a label is present.\n\t\t\\item Lastly, defining the backward barb as the capability to do a backward step, and having immediately after the forward barb, seems to be equivalent to any reasonable definition of backward barb.\n\t\\end{enumerate}\n\n\tThus we argue that the backward barbs are a contrived notion.\n\\end{remark}\n\n\\section{Configuration structures as a model of reversibility}\n\\label{sec:semantics}\nCausal models take causality and concurrency between events as primitives.\nIn configuration structures, configurations stands for computational states and the set inclusion represents the executions, so that in each state one can infer a local order on the events based on the set inclusion.\nWe introduce them and their categorical representation modeling operations from process (\\autoref{sec:st_fam}).\n\nOne can also obtain a causal semantics of a process calculus, by decorating its LTS.\nIn \\autoref{sec:causal_ccs} we briefly show how to interpret CCS terms in configuration structures and how to decorate its LTS to derive causal information from the transitions.\n\nLastly, we introduce configuration structure for a restricted class of RCCS process, called \\emph{singly labelled} (\\autoref{def:singlylabelled}).\nThey are essentially an address in the configuration structure of the underlying, \\enquote{original} CCS term.\nWe then introduce the LTS of those configuration structures and prove their operational correspondence with the reversible syntax (\\autoref{lem:bisim_stfam_ccs}).\n\n\\subsection{Configuration structures as a causal model}\n\\label{sec:st_fam}\n\nLet \\(E\\) be a set, \\(\\subseteq\\) be the usual set inclusion and \\(C\\) be a family of subsets of \\(E\\).\nFor \\(X\\subseteq C\\), \\(X\\) is \\emph{compatible}, denoted \\(X\\compa\\), if \\(\\exists y\\in C\\) finite such that \\(\\forall x\\in X\\), \\(x\\subseteq y\\).\n\n\\begin{definition} [Configuration structures]\n\t\\label{def:conf_str}\n\tA \\emph{configuration structure} \\(\\mathcal{C}\\) is a triple \\((E,C,\\subseteq)\\) where \\(E\\) is a set of events, \\(\\subseteq\\) is the set inclusion and \\(C\\subseteq\\mathcal{P}(E)\\) is a set of subsets satisfying:\n\t\\begin{itemize}\n\t\t\\item \\emph{finiteness}:\n\t\t \\(\\forall x \\in C, \\forall e \\in x, \\exists z \\in C\\) finite such that \\( e\\in z\\) and \\(z \\subseteq x\\),\n\t\t\\item \\emph{coincidence freeness}:\n\t\t \\(\\forall x \\in C, \\forall e, e' \\in x, e\\neq e'\\Rightarrow \\big(\\exists z \\in C, z \\subseteq x\\text{ and }(e\\in z \\iff e'\\notin z)\\big)\\),\n\t\t\\item \\emph{finite completness}:\n\t\t \\( \\forall X \\subseteq C \\text{ if } X \\compa \\text{ then } \\bigcup X \\in C\\),\n\t\t\\item \\emph{stability}:\n\t\t \\(\\forall x,y\\in C \\text{ if }x\\cup y\\in C\\text{ then }x\\cap y\\in C\\).\n\t\\end{itemize}\n\tWe denote \\({\\mathbf{0}}\\) the configuration structure with \\(E = \\emptyset\\).\n\\end{definition}\n\nIntuitively, events are the actions occurring during the run of a process, while configurations represents computational states.\nThe first axiom, \\emph{finiteness}, guarantees that for each event the set of causes is finite. \\emph{Coincidence freeness} states that each computation step consists of a single event.\nAxioms \\emph{finite completness} and \\emph{stability} are more abstract and are better explained on some examples.\nConsider the structures in \\autoref{fig:counterex_stfam}: the structure \\ref{fig:counterex_stfam_a} does not satisfy the second axiom, as two events occur in a single step.\nThe structure \\ref{fig:counterex_stfam_b} does not satisfy finite completeness.\nIntuitively, configuration structures cannot capture \\enquote{pairwise} concurrence.\nFinally, the structure \\ref{fig:counterex_stfam_c} does not satisfies stability and the intuition is that the causes of event \\(e_3\\) are \\emph{either} \\(e_1\\) \\emph{or} \\(e_2\\), but not both.\n\n\\begin{figure}\n\t{\\centering\n\t\t\\begin{minipage}[b]{.1\\linewidth}\n\t\t\t\\begin{tikzpicture}[scale=0.8]\n\t\t\t\t\\tikz@path@overlay{node} (emptyset) at (0,-1) {\\(\\emptyset\\)};\n\t\t\t\t\\tikz@path@overlay{node} (ab) at (0,1) {\\(e_1,e_2\\)};\n\t\t\t\t\\path[draw] [->] (emptyset) -- (ab);\n\t\t\t\\end{tikzpicture}\n\t\t\t\\subcaption{}\\label{fig:counterex_stfam_a}\n\t\t\\end{minipage}\n\t\t\\begin{minipage}[b]{.45\\linewidth}\n\t\t\t\\begin{tikzpicture}[scale=0.8]\n\t\t\t\t\\tikz@path@overlay{node} (emptyset) at (0,-1) {\\(\\emptyset\\)};\n\t\t\t\t\\tikz@path@overlay{node} (a) at (-3,0) {\\(e_1\\)};\n\t\t\t\t\\tikz@path@overlay{node} (b) at (0,0) {\\(e_2\\)};\n\t\t\t\t\\tikz@path@overlay{node} (c) at (3,0) {\\(e_3\\)};\n\t\t\t\t\\tikz@path@overlay{node} (ab) at (-2, 1) {\\(e_1,e_2\\)};\n\t\t\t\t\\tikz@path@overlay{node} (ac) at (0,1) {\\(e_1,e_3\\)};\n\t\t\t\t\\tikz@path@overlay{node} (bc) at (2,1) {\\(e_2,e_3\\)};\n\t\t\t\t\\path[draw] [->] (emptyset) -- (a);\n\t\t\t\t\\path[draw] [->] (emptyset) -- (b);\n\t\t\t\t\\path[draw] [->] (emptyset) -- (c);\n\t\t\t\t\\path[draw] [->] (a) -- (ab);\n\t\t\t\t\\path[draw] [->] (a) -- (ac);\n\t\t\t\t\\path[draw] [->] (b) -- (bc);\n\t\t\t\t\\path[draw] [->] (b) -- (ab);\n\t\t\t\t\\path[draw] [->] (c) -- (bc);\n\t\t\t\t\\path[draw] [->] (c) -- (ac);\n\t\t\t\\end{tikzpicture}\n\t\t\t\\subcaption{}\\label{fig:counterex_stfam_b}\n\t\t\\end{minipage}\n\t\t\\begin{minipage}[b]{.36\\linewidth}\n\t\t\t\\begin{tikzpicture}[scale=0.8]\n\t\t\t\t\\tikz@path@overlay{node} (emptyset) at (0,-1) {\\(\\emptyset\\)};\n\t\t\t\t\\tikz@path@overlay{node} (b) at (-1,0) {\\(e_1\\)};\n\t\t\t\t\\tikz@path@overlay{node} (c) at (1,0) {\\(e_2\\)};\n\t\t\t\t\\tikz@path@overlay{node} (bcp) at (0, 1) {\\(e_1,e_2\\)};\n\t\t\t\t\\tikz@path@overlay{node} (ba) at (-2,1) {\\(e_1,e_3\\)};\n\t\t\t\t\\tikz@path@overlay{node} (ca) at (2,1) {\\(e_2,e_3\\)};\n\t\t\t\t\\tikz@path@overlay{node}[align=center] (bapc) at (0,3) {\\(e_1,e_2,e_3\\)};\n\t\t\t\t\\path[draw] [->] (emptyset) -- (b);\n\t\t\t\t\\path[draw] [->] (emptyset) -- (c);\n\t\t\t\t\\path[draw] [->] (b) -- (bcp);\n\t\t\t\t\\path[draw] [->] (c) -- (bcp);\n\t\t\t\t\\path[draw] [->] (b) -- (ba);\n\t\t\t\t\\path[draw] [->] (c) -- (ca);\n\t\t\t\t\\path[draw] [->] (ba) -- (bapc);\n\t\t\t\t\\path[draw] [->] (ca) -- (bapc);\n\t\t\t\t\\path[draw] [->] (bcp) -- (bapc);\n\t\t\t\\end{tikzpicture}\n\t\t\t\\subcaption{}\\label{fig:counterex_stfam_c}\n\t\t\\end{minipage}\n\t\t\\caption{Structures that are not coincidence free, finite complete and stable, respectively}\n\t\t\\label{fig:counterex_stfam}\n\t}\n\\end{figure}\n\n\\begin{notations}\n\tIn a configuration \\(\\mathcal{C}{C}\\), if \\(x, x' \\in C\\), \\(e\\in E \\), \\(e\\notin x \\) and \\(x' = x \\cup \\{e\\} \\), then we write \\(x\\redl{e}x'\\).\n\\end{notations}\n\n\\begin{definition}[Labelled configuration structure]\n\tA \\emph{labelled configuration structure} \\(\\mathcal{C}=(E,C,\\ell)\\) is a configuration structure endowed with a \\emph{labelling function} from events to labels \\(\\ell:E\\to\\mathsf{L}\\).\n\\end{definition}\nFrom now on, we will only consider configurations structures that are labelled, so we omit that adjective in the following.\n\nNow we define morphisms on configurations structures that permits to form a category whose objects are configuration structures.\nIntuitively, morphisms model the inclusion or refinement relations between processes.\nProcess algebras' operators are then extended to configuration structures, which makes it a modular model.\n\n\\begin{definition}[Category of configuration structures]\n\t\\label{def:st_fam_morph}\n\tA morphism of configurations structures \\(f:(E_1,C_1,\\ell_1)\\to(E_2,C_2,\\ell_2)\\) is a partial function on the underlying sets \\(f:E_1\\rightharpoonup E_2\\) that is:\n\t\\begin{itemize}\n\t\t\\item \\emph{configurations preserving}: \\(\\forall x\\in C_1, f(x)=\\{f(e) \\setst e\\in x\\}\\in C_2\\),\n\t\t\\item \\emph{local injective}: \\(\\forall x\\in C_1, \\forall e_1, e_2\\in x, f(e_1)=f(e_2) \\implies e_1=e_2\\),\n\t\t\\item \\emph{label preserving}: \\(\\forall x \\in C_1, \\forall e \\in x, \\ell_1(e)=\\ell_2(f(e))\\).\n\t\\end{itemize}\n\tAn isomorphism on configuration structures is denoted \\(\\iso\\).\n\\end{definition}\n\n\\begin{definition}[Operations on configuration structures]\n\t\\label{cat-op-def}\n\tLet \\(\\mathcal{C}_1=(E_1,C_1,\\ell_1)\\), \\(\\mathcal{C}_2=(E_2,C_2,\\ell_2)\\) be two configuration structures and set \\(E^\\star=E \\cup \\{\\star\\}\\).\n\t\\begin{description}\n\t\t\\item[Product]\n\t\tLet \\(\\star\\) denote \\emph{undefined} for a partial function.\n\t\tDefine \\emph{the product of \\(\\mathcal{C}_1\\) and \\(\\mathcal{C}_2\\)} as \\(\\mathcal{C}=\\mathcal{C}_1\\times\\mathcal{C}_2\\), for \\(\\mathcal{C}=(E,C,\\ell)\\), where \\(E=E_1\\times_{\\star} E_2\\) is the product in the category of sets and partial functions%\n\t\t\\footnote{The category of sets and partial functions has sets as objects and functions that can take the value \\(\\star\\) as morphisms~\\cite[Appendix A]{Winskel1982}.}:\n\t\t\\[\n\t\t\tE_1\\times_{\\star} E_2 =\n\t\t\t\\begin{multlined}[t]\n\t\t\t\t\\{(e_1,\\star)\\mid e_1\\in E_1\\}\\cup\\{(\\star,e_2)\\mid e_2\\in E_2\\}\\\\\n\t\t\t\t\\cup\\{(e_1,e_2)\\mid e_1\\in E_1, e_2\\in E_2\\}\n\t\t\t\\end{multlined}\n\t\t\\]\n\t\twith the projections \\(p_1:E\\to E_1\\cup\\{\\star\\}\\), \\(p_2:E\\to E_2\\cup\\{\\star\\}\\).\n\t\tDefine the projections \\(\\pi_1:(E,C)\\to (E_1,C_1)\\), \\(\\pi_2:(E,C)\\to (E_2,C_2)\\) such that the following holds, for \\(e\\in E\\) and \\(x\\in C\\):\n\t\t\\begin{itemize}\n\t\t\t\\item \\(\\pi_1(e)=p_1(e)\\) and \\(\\pi_2(e)=p_2(e)\\);\n\t\t\t\\item \\(\\pi_1(x)\\in C_1\\) and \\(\\pi_2(x)\\in C_2\\);\n\t\t\t\\item \\(\\forall e,e'\\in x\\), if \\(\\pi_1(e)=\\pi_1(e')\\neq\\star\\) or \\(\\pi_2(e)=\\pi_2(e')\\neq\\star\\) then \\(e=e'\\);\n\t\t\t\\item \\(\\forall e \\in x, \\exists z\\subseteq x\\) finite s.t. \\(\\pi_1(x) \\in C_1\\), \\(\\pi_2(x)\\in C_2\\) and \\(e\\in z\\);\n\t\t\t\\item \\(\\forall e, e' \\in x, e\\neq e'\\Rightarrow \\exists z\\subseteq x\\) s.t. \\(\\pi_1(z) \\in C_1\\), \\(\\pi_2(z)\\in C_2\\) and \\((e\\in z \\iff e'\\notin z)\\).\n\t\t\\end{itemize}\n\t\tThe labelling function \\(\\ell\\) is defined as follows:\n\t\t\\[\n\t\t\t\\ell(e) = \\begin{cases}\n\t\t\t\\ell_1(e_1) & \\text{ if }\\pi_1(e)=e_1,\\pi_2(e)=\\star \\\\\n\t\t\t\\ell_2(e_2) & \\text{ if }\\pi_1(e)=\\star,\\pi_2(e)=e_2 \\\\\n\t\t\t(\\ell_1(e_1),\\ell_2(e_2)) & \\text{ otherwise}\n\t\t\t\\end{cases}\n\t\t\\]\n\n\t\t\\item[Coproduct]\n\t\tDefine \\emph{the coproduct of \\(\\mathcal{C}_1\\) and \\(\\mathcal{C}_2\\)} as \\(\\mathcal{C}=\\mathcal{C}_1+\\mathcal{C}_2\\), for \\(\\mathcal{C}=(E,C,\\ell)\\), where \\(E=(\\{1\\}\\times E_1)\\cup(\\{2\\}\\times E_2)\\) and \\(C=\\{\\{1\\}\\times x \\setst x\\in C_1\\}\\cup\\{\\{2\\}\\times x \\setst x\\in C_2\\}\\).\n\t\tThe labelling function \\(\\ell\\) is defined as \\(\\ell(e)=\\ell_i(e_i)\\) when \\(e_i\\in E_i\\) and \\(\\pi_i(e_i)=e\\).\n\n\t\t\\item[Restriction]\n\t\tLet \\(E'\\subseteq E\\) and define \\emph{the restriction of a set of events} as \\((E,C,\\ell)\\upharpoonleft E'=(E',C',\\ell')\\) where \\(x\\in C'\\iff x\\in C\\) and \\(x\\subseteq E'\\). %\n\n\t\t\\emph{The restriction of a name} is then \\((E,C,\\ell)\\upharpoonleft a \\coloneqq (E,C,\\ell)\\upharpoonleft E_a\\) where \\(E_a=\\{e\\in E \\setst \\ell(e) \\in \\{a, \\out{a}\\}\\}\\).\n\t\tFor \\(a_1, \\hdots, a_n\\) a list of names, we define similarly \\(\\upharpoonleft \\cup_{1 \\leqslant i \\leqslant n}E_{a_i}\\).\n\n\t\t\\item[Prefix]Let \\(\\lambda\\) be the label of an event and define \\emph{the prefix operation on configuration structures} as \\(\\alpha.(E,C,\\ell)=(e\\cup E,C',\\ell')\\), for \\(e\\notin E\\) where \\(x'\\in C' \\iff \\exists x\\in C\\), \\(x'=x\\cup e\\) and \\(\\ell'(e) = \\alpha\\), and \\(\\forall e' \\neq e\\), \\(\\ell'(e') = \\ell(e')\\).\n\n\t\t\\item[Relabelling]\n\t\tDefine \\emph{the relabelling of a configuration structure} as \\(\\mathcal{C}_1\\circ\\ell=(E_1,C_1,\\ell)\\), where \\(\\mathcal{C}_1=(E_1,C_1,\\ell_1)\\) and \\(\\ell:E_1\\to\\mathsf{L}\\) is a labelling function.\n\n\t\t\\item[Parallel composition]\n\t\tDefine parallel composition \\(\\mathcal{C} = \\big((\\mathcal{C}_1\\times\\mathcal{C}_2)\\circ\\ell\\big)\\upharpoonleft E\\) as the application of product, relabelling and restriction, with \\(\\ell\\) and \\(E\\) defined below.\n\t\t\\begin{itemize}\n\t\t\t\\item First, \\(\\mathcal{C}_1\\times\\mathcal{C}_2 =\\mathcal{C}_3\\) is the product with \\(\\mathcal{C}_3=(E_3,C_3,\\ell_3)\\);\n\t\t\t\\item Then, \\(\\mathcal{C}'=\\mathcal{C}_3\\circ\\ell\\) with \\(\\ell\\) defined as follows:\n\t\t\t \\[\n\t\t\t \t\\ell(e) = \\begin{cases}\n\t\t\t \t\\ell_3(e) & \\text{ if }\\ell_3(e)\\in\\{a;\\out{a}\\} \\\\\n\t\t\t \t\\tau & \\text{ if }\\ell_3(e)\\in\\{(a,\\out{a}); (\\out{a},a)\\}\\\\\n\t\t\t \t0 & \\text{ otherwise }\n\t\t\t \t\\end{cases}\n\t\t\t \\]\n\t\t\t\\item Finally, \\(\\mathcal{C}= (E_1 \\times_{\\star} E_2,C_3,\\ell) \\upharpoonleft E\\) is the resulted configuration structure, where \\(E=\\{e\\in E_3 \\setst \\ell(e)\\neq 0\\}\\).\n\t\t\\end{itemize}\n\t\\end{description}\n\\end{definition}\n\nIn the definition of the product, the conditions guarantee that \\(\\mathcal{C}_1\\times\\mathcal{C}_2\\) is the product in the category of configuration structures and that the projections \\(\\pi_1\\), \\(\\pi_2\\) are morphisms.\nIn particular, the third condition ensures that the projections are local injective, the fourth and fifth enforce finiteness and coincidence-freeness axioms in the resulted configuration structure.\n\n\\begin{definition}[Causality]\n\t\\label{def:causality}\n\tLet \\(x\\in C\\) and \\(e_1,e_2\\in x\\) for \\((E,C,\\ell)\\) a configuration structure.\n\tThen we say that \\emph{\\(e_1\\) happens before \\(e_2\\) in \\(x\\)} or that \\emph{\\(e_1\\) causes \\(e_2\\) in \\(x\\)}, written \\(e_1\\leqslant_x e_2\\), iff \\(\\forall x_2\\in C, x_2\\subseteq x, e_2\\in x_2\\implies e_1\\in x_2\\).\n\\end{definition}\n\nConfigurations can also be interpreted as \\emph{temporal} observations~\\cite[Chap.~5]{Cristescu2015}, instead of causal orders present in the structure of a term.\nRefering to the order as \\emph{happens before} instead of causality highlights the observational nature of the order.\n\nMorphisms on configuration structures reflect causality.\nLet \\(f:\\mathcal{C}_1\\to\\mathcal{C}_2\\) a morphism and \\(x\\in C_1\\) a configuration.\nThen\n\\[ \\forall e_1,e_2\\in x,\\text{ if } f(e_1) \\leqslant_{f(x)} f(e_2)\\text{ then }e_1\\leqslant_x e_2.\\]\n\nHowever, morphisms do not preserve causality in general.\nIn the case of a product we can show that all \\emph{immediate} causalities are due to one of the two configurations structures.\nStated differenlty, a context can add but cannot remove causality in the process.\n\n\\begin{definition}[Immediate causality]\n\tLet \\(e,e'\\) be two events in a configuration \\(x\\) for a configuration structure \\((E,C,\\ell)\\).\n\tDenote \\(e\\to_x e'\\) if \\(e\\) is an \\emph{immediate cause} for \\(e'\\) in \\(x\\), that is \\(e<_xe'\\) and \\(\\nexists e''\\) such that \\(e<_x e''<_x e'\\).\n\\end{definition}\nNote that we overload the notation \\(e\\to_x e'\\) however as it is defined on events only, it is not ambiguous.\n\n\\begin{proposition}\n\t\\label{prop:cause_projection}\n\tLet \\(x\\in\\mathcal{C}_1\\times\\mathcal{C}_2\\).\n\tThen \\(e_1\\to_x e_2\\iff\\) either \\(\\pi_1(e_1)<_{\\pi_1(x)} \\pi_1(e_2)\\) or \\(\\pi_2(e_1)<_{\\pi_2(x)} \\pi_2(e_2)\\).\n\\end{proposition}\n\n\\begin{proof}\n\tThe proof~\\cite[Proposition 6]{Cristescu2015} follows by contradiction, using that if \\(x\\) is a configuration in \\(\\mathcal{C}\\) and if \\(e\\in x\\) is such that \\(\\forall e'\\in x\\), \\(e\\not <_x e'\\), then \\(x\\setminus e\\) is a configuration in \\(\\mathcal{C}\\).\n\\end{proof}\n\n\\subsection{Operational semantics, correspondence and equivalences}\n\\label{sec:causal_ccs}\nConfiguration structures are a causal model for CCS~\\cite{Winskel1995} in which the computational states are the configurations and the forward executions are dictated by set inclusion.\nTo show the correspondence with CCS (\\autoref{lem:bisim_stfam_ccs}), one defines an operational semantics on configurations structures (\\autoref{def:transition_stable}) that erases the part of the structure that is not needed in future computations.\nIn order to define a reversible semantics on configurations structures a second LTS is introduced (\\autoref{def:reverLTS}), that instead of being defined on configurations structures is defined on the configurations of a stable family.\nThus forward and backward moves are simply the set inclusion relation and its opposite, respectively.\n\nThe soundness of the model is proved by defining an operational semantics on configurations structures and showing an operational correspondence between the two worlds.\n\n\\begin{definition}[Encoding a CCS term]\n\t\\label{def:encoding-CCS-cf}\n\tGiven \\(P\\) a CCS term, its encoding \\(\\enc{P}\\) as a configuration structure is built inductively, using the operations of \\autoref{cat-op-def}:\n\t\\begin{align*}\n\t\t\\enc{a.P} & =a.\\enc{P} & & &\n\t\t\\enc{\\out{a}.P}&=\\out{a}.\\enc{P}\\\\\n\t\t\\enc{P\\mid Q} & =\\enc{P}\\mid\\enc{Q} & & &\n\t\t\\enc{P+Q}&=\\enc{P}+\\enc{Q}\\\\\n\t\t\\enc{P\\backslash a} & =\\enc{P}\\upharpoonleft E_a & & &\n\t\t\\enc{0}&={\\mathbf{0}}\n\t\\end{align*}\n\\end{definition}\n\n\\begin{definition}[LTS on configurations structures]\n\t\\label{def:transition_stable}\n\tLet $\\mathcal{C}=(E,C,\\ell)$ be a configuration structure.\n\tDefine $\\mathcal{C}\\setminus{e} = (E\\setminus{e},C',\\ell')$ where $\\ell'$ is the restriction of $\\ell$ to the set $E\\setminus{e}$ and\n\t\\[ x\\in C'\\iff x\\cup\\{e\\}\\in C.\\]\n\tWe easily make sure that $\\mathcal{C}\\setminus{e}$ is also a configurations structures.\n\n\tWe define a LTS on configurations structures thanks to the relation $\\mathcal{C}\\redl{e}\\mathcal{C}\\setminus{e}$, and we extend the notation to $\\mathcal{C}\\redl{\\ell(e)}\\mathcal{C}\\setminus{e}$ and to $\\mathcal{C}\\redl{x}\\mathcal{C}\\setminus{x}$, for $x$ a configuration.\n\\end{definition}\n\n\\begin{lemma}[Operational correspondence between a process $P$ and its encoding $\\enc{P}$]\n\t\\label{lem:bisim_stfam_ccs}\n\tLet $P$ a process and $\\enc{P}=(E,C,\\ell)$ its interpretation.\n\t\\begin{enumerate}\n\t\t\\item $\\forall \\alpha$, $P'$ such that $P\\redl{\\alpha}P'$, $\\exists e\\in E$ such that $\\ell(e)=\\alpha$ and $\\enc{P}\\setminus{e}\\iso\\enc{P'}$;\n\t\t\\item $\\forall e\\in E$, if $\\{e\\}\\in C$ then $\\exists P'$ such that $P\\redl{\\ell(e)}P'$ and $\\enc{P}\\setminus{e}\\iso\\enc{P'}$.\n\t\\end{enumerate}\n\\end{lemma}\nThe above lemma shows that \\emph{labelled} transitions are in correspondence, but labels are just a tool for compositionality.\nThe main result is that a process and its encoding simulate each others \\emph{reductions}.\n\n\\begin{theorem}[Operational correspondence with CCS]\n\tLet $P$ a process and $\\enc{P}=(E,C,\\ell)$ its encoding.\n\t\\begin{enumerate}\n\t\t\\item $\\forall P'$ such that $P\\redl{\\tau}P'$, $\\exists \\{e\\}\\in C$ closed such that $\\enc{P}\\setminus{e}\\iso\\enc{P'}$;\n\t\t\\item $\\forall e\\in E$, $\\{e\\}\\in C$ closed, $\\exists P'$ such that $P\\redl{\\tau}P'$ and $\\enc{P}\\setminus{e}\\iso\\enc{P'}$.\n\t\\end{enumerate}\n\\end{theorem}\n\nMultiple equivalence relations on configuration structures have been defined and studied~\\cite{Glabeek1989, Nicola1990,Phillips2012,Baldan2014,Vogler1993}.\nAmong them, hereditary history preserving bisimulation (HHPB)~\\cite{Bednarczyk1991} equates structures that simulate each others' forward and backward moves and thus connects configuration structures to reversibility.\nIt is considered a canonical equivalence on configuration structures as it respects the causality and concurrency relations between events and admits a categorical representation~\\cite{Joyal1996b}.\n\nThose connections between reversibility and causal models sheds a new light on what help are the meaningful equivalences on reversible processes.\nConsequently, one apply them in the operational setting.\nWe begin by modifying the definition of our LTS on configuration structures to include backward moves as well.\n\n\\begin{definition}[A reversible LTS on configuration structures]\n\t\\label{def:reverLTS}\n\tConsider \\((E, C,\\ell)\\) a configuration structure.\n\tFor \\(x\\in C\\), \\(e\\in E\\) define \\(x\\redl{e}x'\\) iff \\(x'=x\\cup\\{e\\}\\) and \\(x\\redl{\\alpha}x'\\) if additionally, \\(\\ell(e)=\\alpha\\).\n\tThe backward moves are defined as \\(x\\revredl{e}x'\\) and \\(x\\revredl{\\alpha}x'\\) if \\(x=x'\\cup\\{e\\}\\) and \\(\\ell(e)=\\alpha\\).\n\n\tDenote \\(x\\fbl{e}x'\\) when either \\(x\\redl{e}x'\\) or \\(x\\revredl{e}x'\\).\n\\end{definition}\n\nSuch a LTS naturally satisfies elementary criterion that one wold expect from a LTS~\\cite[Chap.~2]{Cristescu2015}.\n\n\\begin{definition}[HHPB {\\cite[Definition 1.4]{Bednarczyk1991}}]\n\t\\label{def:hhpb}\n\tA \\emph{hereditary history preserving bisimulation} on labelled configuration structures is a symmetric relation \\(\\mathrel{\\mathcal{R}}\\subseteq C_1\\times C_2\\times\\mathcal{P}(E_1\\times E_2)\\) such that \\((\\emptyset,\\emptyset,\\emptyset)\\in {\\mathrel{\\mathcal{R}}}\\) and if \\((x_1,x_2,f)\\in {\\mathrel{\\mathcal{R}}}\\), then\n\t\\begin{align*}\n\t\tf\\text{ is a label and order preserving bijection between }x_1\\text{ and }x_2\\notag \\\\\n\t\tx_1\\redl{e_1}x_1'\\implies \\exists x_2'\\in C_2 \\text{ \\st~} x_2\\redl{e_2}x_2' \\text{ and } f=f'\\upharpoonleft x_1, (x_1',x_2',f')\\in {\\mathrel{\\mathcal{R}}} \\notag \\\\\n\t\tx_1 \\revredl{e_1} x_1'\\implies \\exists x_2'\\in C_2 \\text{ \\st~} x_2 \\revredl{e_2} x_2'\\text{ and } f'=f\\upharpoonleft x_2, (x_1',x_2',f')\\in {\\mathrel{\\mathcal{R}}} \\notag\n\t\\end{align*}\n\\end{definition}\n\nIt is known~\\cite{Phillips2007} that hereditary history preserving bisimulation corresponds to the back-and-forth bisimulation (\\autoref{def:sbfc_rccs}), in the following sense: CCSK~\\cite{Phillips2007b} is proven to satisfy the \\enquote{the axioms of reversibility}~\\cite[Definition~4.2]{Glabbek1996}), so that its LTS is \\emph{prime}.\nThen, this LTS is represented as a process graph, on which the forward-reverse bisimulation~\\cite[Definition 5.1]{Phillips2007}---our back-and-forth bisimulation (\\autoref{def:sbfc_rccs})--- is defined.\nFinally, configuration graphs and hereditary history-preserving bisimulation are defined from configuration structures, and both relation are proven to coincide.~\\cite[Theorem~5.4, p.~105]{Phillips2007}.\n\n\\subsection{Configuration structures for RCCS }\n\\label{sec:stfam_rccs}\nAll the possible future behaviours of a process without memory are present in its encoding as a configuration structure.\nAll alike, we want our encoding of processes with memory to record both their past \\emph{and} their future, so that they can evolve in both directions, as process do.\nTo this end, we encode RCCS terms as a configuration in the configuration structure of their origins (\\autoref{def:encod_rccs}).\nThen, we show an operational correspondence between RCCS terms and their encoding.\n\nTo determine which configuration corresponds to the computational state of the term we are encoding, we need to uniquely identify a process from its past and future. %\nHowever, as the following example illustrates, this is not always possible: \\autoref{rem-concur} explains the limitations of the encoding we are going to develop.\n\n\\begin{figure}\n\t\\begin{minipage}[t]{.28\\linewidth}\n\t\t\\begin{tikzpicture}\n\t\t\t\\tikz@path@overlay{node} (emptyset) at (0, -1) {\\(\\emptyset\\)};\n\t\t\t\\tikz@path@overlay{node} (a) at (-1, 0) {\\(\\{e_1\\}\\)};\n\t\t\t\\tikz@path@overlay{node} (b) at (1, 0) {\\(\\{e_1''\\}\\)};\n\t\t\t\\tikz@path@overlay{node} (ab) at (-1, 1) {\\(\\{e_1, e'_1\\}\\)};\n\t\t\t\\path[draw] [->] (emptyset) -- (a);\n\t\t\t\\path[draw] [->] (emptyset) -- (b);\n\t\t\t\\path[draw] [->] (a) -- (ab);\n\t\t\t\\tikz@path@overlay{node}[align=center] (labels) at (0, -2) {\\(\\ell(e_1) = \\ell(e_1'') = a\\),\\\\ \\(\\ell(e'_1) = b\\)};\n\t\t\\end{tikzpicture}\n\t\t\\subcaption{}\\label{ex_unif_a}\n\t\\end{minipage}\n\t\\begin{minipage}[t]{.32\\linewidth}\n\t\t\\begin{tikzpicture}\n\t\t\t\\tikz@path@overlay{node} (emptyset) at (0, -1) {\\(\\emptyset\\)};\n\t\t\t\\tikz@path@overlay{node} (a) at (-1, 0) {\\(\\{e_2\\}\\)};\n\t\t\t\\tikz@path@overlay{node} (b) at (1, 0) {\\(\\{e''_2\\}\\)};\n\t\t\t\\tikz@path@overlay{node} (ab) at (-1, 1) {\\(\\{e_2, e'_2\\}\\)};\n\t\t\t\\tikz@path@overlay{node} (ba) at (1, 1) {\\(\\{e''_2, e'''_2\\}\\)};\n\t\t\t\\path[draw] [->] (emptyset) -- (a);\n\t\t\t\\path[draw] [->] (emptyset) -- (b);\n\t\t\t\\path[draw] [->] (a) -- (ab);\n\t\t\t\\path[draw] [->] (b) -- (ba);\n\t\t\t\\tikz@path@overlay{node}[align=center] (labels) at (0, -2) {\\(\\ell(e_2) = \\ell(e_2'') =a\\),\\\\ \\(\\ell(e_2') = \\ell(e_2''') =b\\)};\n\t\t\\end{tikzpicture}\n\t\t\\subcaption{}\\label{ex_unif_b}\n\t\\end{minipage}\n\t\\begin{minipage}[t]{.38\\linewidth}\n\t\t\\begin{tikzpicture}\n\t\t\t\\tikz@path@overlay{node} (emptyset) at (0, -1) {\\(\\emptyset\\)};\n\t\t\t\\tikz@path@overlay{node} (a) at (-1, 0) {\\(\\{e_3\\}\\)};\n\t\t\t\\tikz@path@overlay{node} (b) at (1, 0) {\\(\\{e_3'''\\}\\)};\n\t\t\t\\tikz@path@overlay{node} (aa1) at (-2, 1) {\\(\\{e_3, e_3'\\}\\)};\n\t\t\t\\tikz@path@overlay{node} (aa2) at (0, 1) {\\(\\{e_3, e_3''\\}\\)};\n\t\t\t\\tikz@path@overlay{node} (aaa) at (-1, 2) {\\(\\{e_3, e_3',e_3''\\}\\)};\n\t\t\t\\path[draw] [->] (emptyset) -- (a);\n\t\t\t\\path[draw] [->] (emptyset) -- (b);\n\t\t\t\\path[draw] [->] (a) -- (aa1);\n\t\t\t\\path[draw] [->] (a) -- (aa2);\n\t\t\t\\path[draw] [->] (aa1) -- (aaa);\n\t\t\t\\path[draw] [->] (aa2) -- (aaa);\n\t\t\t\\tikz@path@overlay{node}[align=center] (labels) at (0, -2) {\\(\\ell(e_3) = \\ell(e_3') =a, \\)\\\\ \\(\\ell(e_3'') = c, \\ell(e_3''') = b\\)};\n\t\t\\end{tikzpicture}\n\t\t\\subcaption{}\\label{ex_unif_c}\n\t\\end{minipage}\n\t\\caption{Encoding RCCS in configurations structures}\n\t\\label{ex_unif}\n\\end{figure}\n\n\\begin{example}\n\t\\label{ex:auto}\n\t\\begin{enumerate}\n\t\t\\item The process \\(P=a.b+a\\) is interpreted as the configuration structure in \\autoref{ex_unif_a}.\n\t\t Let us consider the execution \\(\\varnothing\\rhd P\\redl{a}R\\).\n\t\t To determine which of the configurations labelled \\(a\\) correspond to \\(R\\) we have to consider the future of \\(R\\) as well.\n\n\t\t\\item Hence we choose a configuration that respects the past and the future of \\(R\\), but such a configuration might not be unique.\n\t\t Let \\(Q=a.b + a.b\\) be a process whose configuration structure is in \\autoref{ex_unif_b}.\n\t\t For the trace \\(P \\redl{a} b\\) there is no way to choose between the two configurations labelled \\(a\\).\n\t\\end{enumerate}\n\\end{example}\n\nThe situation of \\autoref{ex:auto} is generalizable to any process whose reduction may lead to a process of the form \\(a.P+a.P\\) or \\(a.P\\mid a.P\\).\nWe consider then a restricted class of processes, as discussed in the following remark.\n\\begin{remark}[On auto-concurrency and others limitations]\n\t\\label{rem-concur}\n\tIn the following, we need to uniquely identify configurations based solely on the labels and orders of the \\emph{open} (i.e.\\@\\xspace non-synchronized) events.\n\tAs seen in \\autoref{ex:auto}, this is not possible in encoding of processes that may reduce to the form \\(a.P \\mid a.Q\\) or \\(a.P+a.Q\\).\n\n\tThe first kind of process is characterised by an \\emph{auto-concurrency} condition~\\cite[Definition 9.5]{Glabbeek2001}.\n\tWe need a stronger condition, a sort of \\emph{auto-conflict}, to forbid the second type of process.\n\\end{remark}\n\n\\begin{definition}[Singly labelled configuration structures and processes]\n\t\\label{def:singlylabelled}\n\tA configuration structure \\(\\mathcal{C}\\) is \\emph{singly labelled}, or \\emph{without auto-concurrency nor auto-conflict} if \\(\\forall x\\in\\mathcal{C}\\) and \\(\\forall e,e'\\notin x\\) we have that\n\t\\[\\big( x\\redl{e} y, x\\redl{e'}y'\\text{ and }\\ell(e)=\\ell(e')\\big)\\implies e = e'.\\]\n\n\tA process is singly labelled if its encoding as a configuration structure is.\n\\end{definition}\n\nRemark that being singly labelled does not mean that each label has to occurs only once in a process: whereas \\(a \\mid b.a \\) is not, since after firing \\(b\\) two transitions labelled \\(a\\) can be fired, \\(a.a\\) and \\(a.b + b\\) are singly labelled.\nHowever, a syntactical definition of this restriction cannot be inductively defined, since \\(P\\) and \\(Q\\) might be singly labelled, but not \\(P \\mid Q\\) nor \\(P + Q \\).\n\nThe following encoding, and all the results that use it, require the process to be singly labelled (on top of being coherent, if they are reversible).\nThis restriction could probably be removed at the price of a \\emph{tagging} of the occurrences of names, maybe in the spirit of the \\emph{localities}~\\cite{Boudol1998b}.\n\n\\begin{definition}[Encoding RCCS processes in configurations structures]\n\t\\label{def:encod_rccs}\n\tLet \\(R\\) be a singly labelled process and \\(\\mathcal{C} = \\encc{\\erase{\\orig{R}}}\\) the encoding (\\autoref{def:encoding-CCS-cf}) of its \\enquote{memory-less} origin (\\autoref{def:rccs_ccs}).\n\n\tWe first need the function \\(\\mathaddress_{\\mathcal{C}}\\), defined as:\n\t\\begin{align}\n\t\t\\funaddress{\\mathcal{C}}{x}{f,R_1\\redl{i:\\alpha}R_2\\mathrel{\\fw}^{\\star} R_3} & =\\funaddress{\\mathcal{C}}{x\\cup\\{e\\}}{f\\cup\\{e\\leftrightarrow i\\},R_2\\mathrel{\\fw}^{\\star} R_3} \\label{fun1} \\\\\n\t\t\\funaddress{\\mathcal{C}}{x}{f,R_2\\mathrel{\\fw}^{\\star}R_3} & = x \\text{ if } R_2 =R_3 \\label{fun2}\n\t\\end{align}\n\tWhere in \\eqref{fun1} \\(e\\) is such that\n\t\\begin{itemize}\n\t\t\\item \\(\\ell(e)=\\alpha\\);\n\t\t\\item \\(x\\cup\\{e\\}\\in \\mathcal{C}\\);\n\t\t\\item \\(j<_{R_2}i\\iff f(j)<_{x\\cup\\{e\\}}e\\);\n\t\t\\item and \\(\\enc{\\erase{R_2}}= \\big(\\mathcal{C}\\setminus (x\\cup\\{e\\})\\big)\\).\n\t\\end{itemize}\n\tNow we define the encoding of \\(R\\) in configuration structure by induction on the trace (\\autoref{def:trace}) \\(\\sigma: \\orig{R}\\mathrel{\\fw}^{\\star} R\\), as \\(\\encr{R}_{\\sigma} = (\\mathcal{C},\\funaddress{\\mathcal{C}}{\\emptyset}{\\emptyset,\\sigma})\\).\n\n\\end{definition}\nWe show in \\autoref{prop-soundness-rccs} that the function is well defined, i.e.\\@\\xspace for every singly labelled process \\(R\\) and for every trace \\(\\sigma: \\orig{R}\\mathrel{\\fw}^{\\star}R\\) there exists a unique configuration in \\(\\enc{\\erase{\\orig{R}}}_{\\sigma}\\) defined as above.\n\n\\begin{example}\n\tA first simple example is the encoding of a process with an empty memory.\n\tLet \\(S = \\varnothing \\rhd P\\), \\(\\erase{\\orig{S}} = P\\) and \\(\\enc{S}_{\\tempty}=(\\enc{P}, \\emptyset)\\).\n\n\tLet us show how to compute the encoding of the process\n\t\\[R=\\mem{2,a,0}.\\fork.\\mem{1,a,b}\\rhd 0\\mid \\fork.\\mem{1,a,b}\\rhd a.\\]\n\tWe backtrack to its origin and obtain \\(\\orig{R}=\\varnothing \\rhd a.(a \\mid c)+b\\).\n\tThe term is encoded in the configuration structure in \\autoref{ex_unif_c}.\n\tWe apply the function \\(\\funaddress{\\mathcal{C}}{\\emptyset}{\\orig{R}\\mathrel{\\fw}^{\\star} R}\\) on the trace\n\t\\begin{align*}\n\t\t\\varnothing\\rhd a.(a \\mid c)+b & \\fwlts{1}{a} \\mem{1,a,b}\\rhd (a\\mid c) \\\\\n\t\t & \\qquad \\congru (\\fork.\\mem{1,a,b}\\rhd a)\\mid (\\fork.\\mem{1,a,b}\\rhd c ) \\\\\n\t\t & \\qquad \\qquad \\fwlts{2}{a} \\mem{2,a,0}.\\fork.\\mem{1,a,b}\\rhd 0\\mid \\fork.\\mem{1,a,b}\\rhd c \\\\\n\t\t & \\qquad \\qquad \\qquad = R'\n\t\\end{align*}\n\n\tThe configuration corresponding to \\(R\\) is then \\(\\{e_3,e_3'\\}\\).\n\\end{example}\n\nLet us show that the encoding is correct, and in particular that the function \\(\\mathaddress_{\\mathcal{C}}\\) is well defined.\n\n\\begin{proposition}[Soundness of the RCCS encoding]\n\t\\label{prop-soundness-rccs}\n\tLet \\(P\\) be a singly labelled process and \\(\\mathcal{C}=\\encc{P}\\) its encoding.\n\tThen for any \\(R\\) reachable from \\(\\varnothing\\rhd P\\)\n\tthere exists a unique \\(x\\in\\mathcal{C}\\) such that \\(\\funaddress{\\mathcal{C}}{\\emptyset}{\\emptyset,\\varnothing\\rhd P\\mathrel{\\fw}^{\\star}R}=x\\).\n\\end{proposition}\n\n\\begin{proof}\n\tFrom \\autoref{lem:rearrange_trace}, we consider the trace \\(\\orig{R}\\mathrel{\\fw}^{\\star} R\\) to be only forward.\n\tWe proceed by induction on the trace \\(\\orig{R}\\mathrel{\\fw}^{\\star} R\\).\n\tFor the inductive case we have the trace \\(\\orig{R}\\mathrel{\\fw}^{\\star} R_n\\) %\n\tand\n\t\\(\\funaddress{\\mathcal{C}}{\\emptyset}{f_n,\\orig{R}\\mathrel{\\fw}^{\\star} R_n}=x_n\\), for \\(x_n\\in\\mathcal{C}\\), \\(f_n\\) a label and order preserving bijection between \\(x_n\\) and \\(R_n\\), and such that \\(\\encc{\\erase{R_n}}=\\mathcal{C}\\setminus x_n\\).\n\tWe have to show that for the trace \\(\\orig{R}\\mathrel{\\fw}^{\\star} R_n\\redl{i:a} R_{n+1}\\) %\n\tthere exists a unique configuration \\(x_{n+1}\\in \\mathcal{C}\\) such that\n\t\\begin{align*}\n\t\t\\funaddress{\\mathcal{C}}{\\emptyset}{\\emptyset,\\orig{R}\\mathrel{\\fw}^{\\star} R_n \\redl{i:\\alpha}R_{n+1}} =x_{n+1}\n\t\t\\shortintertext{ and }\n\t\t\\enc{\\erase{R_{n+1}}}=\\mathcal{C}\\setminus {x_{n+1}}.\n\t\\end{align*}\n\tWe have that\n\t\\[\n\t\t\\funaddress{\\mathcal{C}}{\\emptyset}{\\emptyset,\\orig{R}\\mathrel{\\fw}^{\\star} R_n\\redl{i:\\alpha}R_{n+1}}=\\funaddress{\\mathcal{C}}{x_n}{f_n,R_n\\redl{i:\\alpha}R_{n+1}}\n\t\\]\n\tHence we have that \\(x_{n+1}=x\\cup\\{e\\}\\), \\(f_{n+1}=f_n\\cup\\{e\\leftrightarrow i\\}\\) and we have to show that there exists a unique \\(e\\in\\mathcal{C}\\) such that \\(\\ell(e)=\\alpha\\) and\n\t\\[\\encc{\\erase{R_{n+1}}}=\\mathcal{C}\\setminus (x_n\\cup\\{e\\}).\\]\n\tHowever, if such an \\(e\\) exists then \\(e\\in\\encc{\\erase{R_n}}\\) and\n\t\\[\\mathcal{C}\\setminus (x_n\\cup\\{e\\})=\\encc{\\erase{R_n}}\\setminus{e}.\\]\n\tHence we reason on the transition \\(R_n\\redl{i:\\alpha}R_{n+1}\\) to show that there exists a unique \\(\\{e\\}\\in\\encc{\\erase{R_n}}\\) such that \\(\\encc{\\erase{R_{n+1}}}=\\encc{\\erase{R_n}}\\setminus{e}\\).\n\tWe consider only the case \\(\\alpha=a\\), the rest being similar.\n\tUsing structural congruence it is possible to rewrite \\(R_n\\) and \\(R_{n+1}\\) as follows\n\t\\[\n\t\tR_n\\congru(m_1\\rhd a.P_1 \\mid R_2)\\backslash (b_1\\dots b_n)\\qquad R_{n+1}\\congru(m_1\\rhd P_1 \\mid R_2)\\backslash (b_1\\dots b_n)\n\t\\]\n\tand hence, for \\(\\erase{R_2}=P_2\\),\n\t\\[\\erase{R_n}=(a.P_1 \\mid P_2)\\backslash (b_1\\dots b_n)\\qquad\\erase{R_{n+1}}=(P_1 \\mid P_2)\\backslash (b_1\\dots b_n).\\]\n\tWe have then to show that\n\t\\[ \\encc{(P_1 \\mid P_2)\\backslash (b_1\\dots b_n)}=\\encc{(a.P_1 \\mid P_2)\\backslash (b_1\\dots b_n)}\\setminus{e}.\\]\n\tFrom \\autoref{lem:bisim_stfam_ccs} such an event exists.\n\tTo show its uniqueness, consider \\(R_n\\congru m_1\\rhd a.P_1 \\mid \\big( m_2\\rhd a.P_2 \\mid R_2\\big)\\).\n\tEither \\(m_1=m_2\\) in which case the process exhibits auto-concurrency, or \\(m_1\\neq m_2\\) and in this case the condition \\(j<_{R_{n+1}}i\\iff f_n(j)<_{x_n\\cup\\{e\\}}e\\) from the definition of the encoding, points to either \\(m_1\\) or \\(m_2\\).\n\n\tLet us prove that \\(\\forall x\\in\\encc{(P_1 \\mid P_2)\\backslash (b_1\\dots b_n)}\\), \\(x\\in\\encc{(a.P_1 \\mid P_2)\\backslash (b_1\\dots b_n)}\\setminus{e}\\).\n\tThe other direction is similar.\n\tLet us unfold the encoding of \\autoref{def:encoding-CCS-cf} using the operations on configurations structures of \\autoref{cat-op-def}.\n\t\\begin{align*}\n\t\t\\encc{(P_1 \\mid P_2)\\backslash (b_1\\dots b_n)} & =(\\encc{P_1}\\times\\encc{P_2}) \\upharpoonleft \\cup_{1 \\leqslant i \\leqslant n}E_{b_i} \\\\\n\t\t\\encc{(a.P_1 \\mid P_2)\\backslash (b_1\\dots b_n)} & = (\\encc{a.P_1}\\times\\encc{P_2}) \\upharpoonleft \\cup_{1 \\leqslant i \\leqslant n}E_{b_i}\n\t\\end{align*}\n\tIf \\(x\\in (\\encc{P_1}\\times\\encc{P_2}) \\upharpoonleft \\cup_{1 \\leqslant i \\leqslant n}E_{b_i}\\) then\n\t\\begin{equation}\n\t\t\\label{eq:labl}\n\t\t\\forall e\\in x, \\ell(e)\\notin\\{b_i,\\out{b_i},0\\}.\n\t\\end{equation}\n\tHence \\(x\\in(\\encc{P_1}\\times\\encc{P_2})\\).\n\tLet \\(\\pi_1\\), \\(\\pi_2\\) be the two projections defined by the product.\n\tThen\n\t\\begin{equation}\n\t\t\\label{eq:proj}\n\t\t\\pi_1(x)\\in\\encc{P_1}\\text{ and }\\pi_2(x)\\in\\enc{P_2}.\n\t\\end{equation}\n\tAs \\(\\pi_1(x)\\in\\encc{P_1}\\), and from the definition of \\(\\encc{a.P_1}\\) we have that \\(\\exists e_1\\), \\(\\ell(e_1)=a\\) and such that \\(\\{e_1\\}\\cup \\pi_1(x)\\in a.\\encc{P_1}\\).\n\tFrom \\autoref{eq:proj} we have that \\(\\exists x_2\\in a.\\encc{P_1}\\times\\encc{P_2}\\) such that \\(\\pi_1(x_2)=\\{e_1\\}\\cup \\pi_1(x)\\) and \\(\\pi_2(x_2)=\\pi_2(x)\\).\n\tHence \\(\\exists !e\\) such that \\(\\pi_1(e)=e_1\\), \\(\\pi_2(e)=\\star\\) and \\(x_2=\\{e\\}\\cup x\\).\n\tFrom \\autoref{eq:labl} we have that \\(x_2\\in (a.\\encc{P_1}\\times\\encc{P_2})\\upharpoonleft \\cup_{1 \\leqslant i \\leqslant n}E_{b_i}\\).\n\tWe infer that if \\(x\\cup\\{e\\}\\in(b_1\\dots b_n)(a.\\encc{P_1}\\times\\encc{P_2})\\) then \\(x\\in\\encc{(b_1\\dots b_n)(a.P_1 \\mid P_2)}\\setminus{e}\\).\n\n\tFrom \\(\\encc{\\erase{R}}=\\mathcal{C}\\setminus x_n\\), we have that \\(\\forall y\\in \\encc{\\erase{R}}\\), \\(y\\cup x_n\\in\\mathcal{C}\\).\n\tIn particular \\(x_n\\cup\\{e\\}\\in\\mathcal{C}\\).\n\n\tLet us denote \\(x_{n+1}=x_n\\cup\\{e\\}\\).\n\tRemains to show that \\(j<_{R_{n+1}}i\\iff f(j)<_{x_{n+1}}e\\).\n\tWe show the implication \\(j<_{R_{n+1}}i\\implies f_n(j)<_{x_{n+1}}e\\) and consider the immediate order for \\(<_{R_{n+1}}\\), as the order is transitive.\n\tFrom \\(j<_{R_{n+1}}i\\), we have that \\(\\mem{i,a}.\\mem{j,b}\\in R_{n+1}\\), hence we retrieve a process \\(R_k\\) where \\(b.a.P'\\in R_k\\).\n\tHence the events \\(f_n(j)\\) and \\(f_n(i)\\) are causaly dependent in the configuration structure of \\(\\enc{\\erase{R_k}}\\), and therefore causally dependent in \\(\\mathcal{C}\\).\n\tFor the other direction \\(f_n (j)<_{x_{n+1}}e \\implies j<_{R_{n+1}}i\\) we show that \\(\\ell(f(j))\\) and \\(\\ell(e)\\) are causal in the origin process \\(P\\), hence they are causally dependent in the memory of \\(R_{n+1}\\).\n\n\tHence \\(\\funaddress{\\mathcal{C}}{\\emptyset}{\\orig{R}\\mathrel{\\fw}^{\\star} R_n\\redl{a}R_{n+1}}=x_n\\cup\\{e\\}\\) with \\(\\encc{\\erase{R_{n+1}}}=\\encc{\\orig{R_n}}\\setminus (x_n\\cup\\{e\\})\\).\n\\end{proof}\n\n\\begin{remark}[On encoding RCCS]\n\t\\label{rk:encode_rccs}\n\tAnother encoding exists~\\cite{Phillips2012}, but it is not compositional, since \\(\\encc{P_1 \\mid P_2}\\) is not defined as an operation on \\(\\encc{P_1}\\) and \\(\\encc{P_2}\\).\n\tCompositionality is important for the definition of contexts in configurations structures, in particular for the definition of congruence (\\autoref{bisim-cs}).\n\\end{remark}\n\nLet us now define a transition relation on configurations structures, useful in showing the operational correspondence between terms of RCCS and their encoding.\n\n\\begin{definition}[Reversible LTS in configurations structures]\n\t\\label{def:lts_stfam}\n\tDefine \\((\\encc{P}, x)\\redl{\\ell(e)}(\\encc{P}, x\\cup\\{e\\})\\) for \\(x\\cup\\{e\\}\\in\\enc{P}\\).\n\tSimilarly to \\autoref{def:rlts} we define \\((\\encc{P}, x)\\revredl{\\ell(e)}(\\encc{P}, x\\setminus{e})\\), for some \\(e\\) such that \\(x\\setminus{e}\\in\\encc{P}\\).\n\\end{definition}\n\nWe defined in \\autoref{def:encod_rccs} the encoding of a process parametrically on a trace.\nThe following proposition shows that any trace from \\(\\orig{R}\\) up to \\(R\\) leads to the same encoding.\n\\begin{proposition}\n\t\\label{prop:unique_conf_trace}\n\tFor all singly labelled processes \\(R\\) there exists \\(x\\) a configuration in \\(\\encc{\\erase{\\orig{R}}}\\) such that \\(\\forall \\sigma:\\orig{R}\\mathrel{\\fw}^{\\star} R \\), \\(\\encr{R}_{\\sigma}=x\\) holds.\n\\end{proposition}\n\n\\begin{proof}\n\tDenote \\(\\mathcal{C}=\\encc{\\erase{\\orig{R}}} = (E,C,\\ell)\\).\n\tFrom the definition of \\(\\encr{R}_{\\sigma}\\) it suffices to show that for any configurations \\(x,y\\in\\mathcal{C}\\) such that there exists a label and order preserving bijection between the two and such that \\(\\mathcal{C}\\setminus x=\\mathcal{C}\\setminus y\\), \\(x=y\\) holds.\n\n\tWe prove it by induction on the size of \\(x\\) and \\(y\\).\n\tSuppose that there exists two events \\(e,e'\\) such that \\(y=x\\cup\\{e\\}\\) and \\(z=x\\cup\\{e'\\}\\) are configurations of \\(\\mathcal{C}\\) as well, with \\(f:y\\leftrightarrow z\\).\n\tSince \\(R\\) is singly labelled (\\autoref{def:singlylabelled}), if \\(\\ell(e)=\\ell(e')\\), then \\(e=e'\\).\n\\end{proof}\n\n\\begin{example}\n\tConsider the configuration structure in \\autoref{ex_unif_c}, encoding the process \\(P=a.(a \\mid c)+b\\).\n\tThe process\n\t\\[S= \\big(\\mem{2,a,0}.\\fork.\\mem{1,a,b}\\rhd 0 \\big) \\mid \\big(\\mem{3,c,0}.\\fork.\\mem{1,a,b}\\rhd 0\\big)\\]\n\tcan be reached on the trace \\(\\sigma_1: \\varnothing \\rhd P\\fwlts{1}{a}\\fwlts{2}{a}\\fwlts{3}{c} S\\) or \\(\\sigma_1: \\varnothing \\rhd P\\fwlts{1}{a}\\fwlts{3}{c}\\fwlts{2}{a} S\\).\n\tHowever both traces lead to the same encoding of \\(S\\).\n\\end{example}\n\nHence we write \\(\\enc{R}\\) instead of \\(\\enc{R}_{\\sigma}\\).\nIt is an essential property to prove the existence of a bisimulation relation between a process and its encoding.\n\n\\begin{lemma}[Operational correspondence between a \\(R\\) and \\(\\encr{R}\\)]\n\t\\label{lem:operational_corresp}\n\tLet \\(R\\) a process and \\(\\enc{R}=(\\mathcal{C},x)\\) its interpretation.\n\t\\begin{enumerate}\n\t\t\\item \\(\\forall \\alpha\\), \\(S\\) and \\(i\\in\\ids\\) such that \\(R\\fwlts{i}{\\alpha}S\\) then \\(\\encr{R}\\redl{\\alpha}\\encr{S}\\);\n\t\t\\item \\(\\forall \\alpha\\), \\(S\\) and \\(i\\in\\ids\\) such that \\(R\\bwlts{i}{\\alpha}S\\) then \\(\\encr{R}\\revredl{\\alpha}\\encr{S}\\);\n\t\t\\item \\(\\forall e\\in E\\), \\((\\mathcal{C},x)\\redl{\\ell(e)}(\\mathcal{C}, x\\cup\\{e\\})\\) then \\(\\exists S\\), such that for some \\(i\\in\\ids\\), \\(R\\fwlts{i}{\\alpha}S\\) and \\(\\encr{S}=(\\mathcal{C},x\\cup\\{e\\})\\).\n\t\t\\item \\(\\forall e\\in E\\), \\((\\mathcal{C},x)\\revredl{\\ell(e)}(\\mathcal{C}, x\\setminus{e})\\) then \\(\\exists S\\), such that for some \\(i\\in\\ids\\), \\(R\\bwlts{i}{\\alpha}S\\) and \\(\\encr{S}=(\\mathcal{C},x\\setminus{e})\\).\n\t\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n\t\\begin{enumerate}\n\t\t\\item As \\(R\\fwlts{i}{\\alpha}S\\), \\(\\orig{R}=\\orig{S}\\), we have that \\(\\encr{S}=(\\mathcal{C},x_s)\\), where\n\t\t \\(x_S= \\funaddress{\\mathcal{C}}{\\emptyset}{\\orig{R}\\mathrel{\\fw}^\\star S}=\n\t\t \\funaddress{\\mathcal{C}}{\\emptyset}{\\orig{R}\\mathrel{\\fw}^\\star R\\redl{\\alpha} S}= x_R\\cup\\{e\\}\\) by \\autoref{prop-soundness-rccs}.\n\t\t As \\(\\encr{R}=(\\mathcal{C},x_R)\\) it follows that \\((\\mathcal{C},x_R)\\redl{\\alpha}(\\mathcal{C},x_S)\\).\n\t\t\\item The proof for the backward direction is similar except that it uses the trace up to \\(R\\).\n\t\t It uses \\autoref{prop:unique_conf_trace}, which allows us to backtrack on any path from the emptyset and leading to \\(x_R\\).\n\t\t\\item From \\((\\mathcal{C},x)\\redl{\\ell(e)}(\\mathcal{C}, x\\cup\\{e\\})\\) we have that \\(x\\cup\\{e\\}\\in\\mathcal{C}\\).\n\t\t Then \\(\\{e\\}\\in\\mathcal{C}\\setminus x\\).\n\t\t From \\(\\encr{R}=(\\mathcal{C},x)\\) we have that \\(\\mathcal{C}\\setminus x=\\encc{\\erase{R}}\\), hence \\(\\{e\\}\\in\\encc{\\erase{R}}\\).\n\t\t We use \\autoref{lem:bisim_stfam_ccs} and obtain that \\(\\exists P\\) such that \\(\\erase{R}\\redl{\\ell(e)}P\\).\n\t\t Then due to the strong bisimulation between a RCCS term and its corresponding CCS term in \\autoref{lem:corresp_ccs_rccs}, we have that, for some \\(i\\), \\(R\\fwlts{i}{\\alpha}S\\), where \\(\\erase{S}=P\\).\n\t\t That \\(\\encr{S}=(\\mathcal{C},x\\cup\\{e\\})\\) follows from a similar argument to above and from \\autoref{prop:unique_conf_trace}.\\qedhere\n\t\t\\item It is similar to the case above.\n\t\\end{enumerate}\n\\end{proof}\n\n\\section{Contextual equivalence on configuration structures}\n\\label{sec:def-bisim-cs}\n\nIn this section we introduce a notion of context for the configurations structures and then adapt the back-and-forth barbed bisimulation to configurations structures (\\autoref{bisim-cs}).\nWe define hereditary history preserving bisimulation and use two families of relations, denoted \\(F_i\\) and \\(B_i\\), to inductively approximate the bisimulation (\\autoref{Fn-Bn-and-h}).\nWe use these relations to show that two processes are barbed congruent whenever their denotations are in the HHPB relation (\\autoref{main-thm}).\nOnce the HHPB has been probed to be a congruence (\\autoref{prop:HHPB_congr}), one direction is straightforward , whereas the other is more technical and, as in CCS~\\cite{Milner1992}, follows by contradiction.\nIt uses the relations \\(\\Forw{i}\\) and \\(\\Backw{i}\\) (\\autoref{ForwBackwDef}) to build contexts that discriminate processes that are not bisimilar.\n\n\\subsection{Contexts for configurations structures}\n\\label{sec:context_stfam}\nContexts for configurations structures have never been defined as it is not clear what a configuration structure with a hole could be.\nHowever, if a structure \\(\\mathcal{C}\\) has an operational meaning, i.e.\\@\\xspace there exists \\(P\\) a process such that \\(\\mathcal{C}=\\enc{P}\\), we use a CCS context \\(C[\\cdot]\\) to build a configuration structure \\(\\enc{C[P]}\\).\n\nWhen analysing the reductions of a process in context, we need to know the contribution the process and the context have in the reduction.\nTo this aim we associate to the context \\(C[\\cdot]\\) instantiated by a process \\(P\\) a projection morphism \\(\\pi_{C, P}:\\enc{C[P]}\\to\\enc{P}\\) that retrieve in \\(\\enc{C[P]}\\) the parts of a configuration belonging to \\(\\enc{P}\\).\n\nFollowing \\autoref{prop:context_that_counts}, we continue to consider only context made of parallel compositions, but the following definition can be extended to arbitrary contexts~\\cite[Definition 46]{Cristescu2015}.\n\n\\begin{definition}\n\t\\label{def:conf_context}\n\tLet \\(C[\\cdot]\\) a context, and \\(P\\) a process.\n\tThe \\emph{projection} \\(\\pi_{C, P}:\\enc{C[P]}\\to\\enc{P}\\) is defined on the structure of \\(C\\) as follows:\n\t\\begin{itemize}\n\t\t\\item if \\(C[\\cdot]=C'[\\cdot]\\mid P'\\) then \\(\\pi_{C, P}:\\enc{C'[P]\\mid P'}\\to\\enc{P}\\) is defined as \\(\\pi_{C, P}(e)=\\pi_{C', P}(\\pi_1(e))\\), where \\(\\pi_1:\\enc{C'[P]\\mid P'}\\to\\enc{C'[P]}\\) is the projection morphism defined by the product in \\autoref{cat-op-def};\n\t\t\\item if \\(C[\\cdot]=[\\cdot]\\) then \\(\\pi_{C, P}:\\enc{C[P]}\\to\\enc{P}\\) is the identity.\n\t\\end{itemize}\n\\end{definition}\nWe naturally extend \\(\\pi_{C, P}\\) to configurations, and prove by case analysis that \\(\\pi_{C, P}:\\enc{C[P]}\\to\\enc{P}\\) is a morphism.\n\n\\subsection{Relation induced by barbed congruence on configurations structures}\n\\label{sec:barbed_congruence_stfam}\nWe define a relation on configurations structures that have an operational meaning %\nand we show it is the relation induced by the barbed congruence in RCCS (\\autoref{def:sbfc_rccs}).\nWe call the relation barbed back-and-forth congruence, to highlight its meaning, though it is not strictly speaking a congruence on configurations structures.\n\n\\begin{definition}[Back-and-forth barbed congruence on configurations structures]\n\t\\label{bisim-cs}\n\tA \\emph{back-and-forth barbed bisimulation on configurations structures} is a symmetric relation \\(\\mathrel{\\mathcal{R}}\\subseteq C_1\\times C_2\\) such that \\((\\emptyset,\\emptyset)\\in {\\mathrel{\\mathcal{R}}}\\), and if \\((x_1,x_2)\\in {\\mathrel{\\mathcal{R}}}\\), then\n\t\\begin{align*}\n\t\tx_1\\revredl{e_1}x_1' & \\implies \\begin{multlined}[t]\n\t\t\\exists x_2'\\in C_2\\text{ \\st~} x_2\\revredl{e_2}x_2',\\\\\n\t\t\\text{with }\\ell_1(e_1)=\\ell_2(e_2)=\\tau\\text{ and } (x_1',x_2')\\in {\\mathrel{\\mathcal{R}}};\n\t\t\\end{multlined}\n\t\t\\tag{back}\\\\\n\t\tx_1\\redl{e_1}x_1' & \\implies \\begin{multlined}[t] \\exists x_2'\\in C_2 \\text{ \\st~}x_2\\redl{e_2}x_2', \\\\\n\t\t\\text{with }\\ell_1(e_1)=\\ell_2(e_2)=\\tau \\text{ and }(x_1',x_2')\\in {\\mathrel{\\mathcal{R}}};\n\t\t\\end{multlined}\n\t\t\\tag{forth} \\\\\n\t\t\\text{if }\\exists e_1\\in E_1 & \\text{ \\st~}\\ell_1(e_1)\\neq\\tau \\begin{multlined}[t]\\text{ and }x_1\\redl{e_1}x_1' \\text{ then } \\exists x_2'\\in C_2 \\\\\n\t\t\\text{\\st~}x_2\\redl{e_2}x_2'\\text{, with }\\ell_1(e_1)=\\ell_2(e_2).\\end{multlined}\\tag{barbed}\n\t\\end{align*}\n\tLet \\(\\mathcal{C}_1\\overset{\\cdot}{\\bisim}^{\\tau}\\mathcal{C}_2\\) if and only if there exists a back-and-forth barbed bisimulation between \\(\\mathcal{C}_1\\) and \\(\\mathcal{C}_2\\).\n\n\tDefine \\(\\bisim^{\\tau}\\) the \\emph{back-and-forth barbed congruence} induced on configurations structures as a symmetric relation on configurations structures that have an operational meaning such that\n\t\\[\\enc{P_1}\\bisim^{\\tau}\\enc{P_2} \\iff\\forall C, \\enc{C[P_1]}\\overset{\\cdot}{\\bisim}^{\\tau}\\enc{C[P_2]}.\\]\n\\end{definition}\n\nWe now prove that this relation is the relation induced on the encoding of processes by the barbed back-and-forth congruence (\\autoref{def:sbfc_rccs}).\nWe begin by proving it in the non-contextual case.\nWe remind the reader that, as we are going to manipulate encoding of RCCS terms, some restrictions on the terms applies (\\autoref{rem-concur}).\n\\begin{proposition}\n\t\\label{prop:bfbisim_corresp}\n\tFor all \\(P\\) and \\(Q\\), \\(\\varnothing \\rhd P \\overset{\\cdot}{\\bisim}^{\\tau}\\varnothing \\rhd Q\\iff \\enc{P}\\overset{\\cdot}{\\bisim}^{\\tau}\\enc{Q}\\).\n\\end{proposition}\n\\begin{proof}\n\t\\begin{itemize}\n\t\t\\item \\(\\varnothing \\rhd P \\overset{\\cdot}{\\bisim}^{\\tau}\\varnothing \\rhd Q\\implies \\enc{P}\\overset{\\cdot}{\\bisim}^{\\tau}\\enc{Q}\\).\n\t\t Let \\({\\mathrel{\\mathcal{R}_{\\text{CCS}}}}\\) be a back-and-forth barbed bisimulation between \\(P\\) and \\(Q\\).\n\t\t We show that the following relation\n\t\t \\[\n\t\t \t{\\mathrel{\\mathcal{R}}} =\n\t\t \t\\begin{multlined}[t]\n\t\t \t\t\\{(x_1,x_2) \\setst x_1\\in\\enc{P}, x_2\\in\\enc{Q}, \\exists R, S\\text{ \\st~}\\orig{R}=P,\\\\\n\t\t \t\t\\orig S = Q, R \\mathrel{\\mathcal{R}_{\\text{CCS}}} S\\text{ and }\\enc{R}=(\\enc{P},x_1), \\enc{S}=(\\enc{Q},x_2)\\}\n\t\t \t\\end{multlined}\n\t\t \\]\n\t\t is a back-and-forth barbed bisimulation between \\(\\enc{P}\\) and \\(\\enc{Q}\\).\n\n\t\t We have that \\((\\emptyset,\\emptyset)\\in {\\mathrel{\\mathcal{R}}}\\), let \\((x_1,x_2)\\in {\\mathrel{\\mathcal{R}}}\\).\n\t\t We have to show that the conditions in \\autoref{bisim-cs} hold.\n\t\t Suppose that \\(x_1\\redl{e_1}x_1'=x_1\\cup\\{e_1\\}\\), for \\(\\ell(e_1)=\\tau\\).\n\t\t \\begin{align*}\n\t\t \tx_1\\redl{e_1}x_1' & \\implies (\\enc{P},x_1)\\redl{\\ell(e_1)}(\\enc{P}, x_1') \\tag{From \\autoref{def:lts_stfam}} \\\\\n\t\t \t & \\implies R\\fwlts{i}{\\ell(e_1)}R'\\text{ s.t. }\\enc{R'}=(\\enc{P}, x_1') \\tag{From \\autoref{lem:operational_corresp}} \\\\\n\t\t \t & \\implies S\\fwlts{i'}{\\tau}S' \\tag{From \\(R \\mathrel{\\mathcal{R}_{\\text{CCS}}} S\\)} \\\\\n\t\t \t & \\implies (\\enc{Q},x_2)\\redl{\\ell(e_2)}(\\enc{Q},x_2') \\tag{From \\autoref{lem:operational_corresp}}\n\t\t \\end{align*}\n\t\t with \\(\\ell(e_2)=\\tau\\).\n\t\t We have then \\((x_1',x_2')\\in {\\mathrel{\\mathcal{R}}}\\).\n\n\t\t We proceed in a similar manner to show that conditions on the backward transitions and on the barbs hold.\n\n\t\t\\item \\(\\enc{P}\\overset{\\cdot}{\\bisim}^{\\tau}\\enc{Q}\\implies\\varnothing\\rhd P\\overset{\\cdot}{\\bisim}^{\\tau}\\varnothing\\rhd Q\\).\n\t\t Let \\({\\mathrel{\\mathcal{R}_{\\text{Conf}}}}\\) be a back-and-forth barbed bisimulation between \\(\\enc{P}\\) and \\(\\enc{Q}\\).\n\t\t We show that the following relation\n\t\t \\begin{align*}\n\t\t \t{\\mathrel{\\mathcal{R}}} =\n\t\t \t\\begin{multlined}[t]\n\t\t \t\\{(R,S) \\setst \\erase{\\orig{R}}=P, \\erase{\\orig{S}} = Q\\text{ and }\\enc{R}=(\\enc{P},x_1), \\\\\n\t\t \t\\enc{S}=(\\enc{Q},x_2),\\text{ with }(x_1,x_2)\\in {\\mathrel{\\mathcal{R}_{\\text{Conf}}}} \\}\n\t\t \t\\end{multlined}\n\t\t \\end{align*}\n\t\t is a back-and-forth barbed bisimulation between \\(P\\) and \\(Q\\).\n\t\t Let \\((R,S)\\in {\\mathrel{\\mathcal{R}}}\\), the following holds:\n\t\t \\begin{align*}\n\t\t \tR\\fwlts{i}{\\tau}R' & \\implies(\\enc{P},x_1)\\redl{\\ell(e_1)}(\\enc{P}, x_1') \\tag{From \\autoref{lem:operational_corresp}} \\\\\n\t\t \t & \\implies(\\enc{Q},x_2)\\redl{\\ell(e_2)}(\\enc{Q},x_2') \\tag{From \\((x_1,x_2)\\in {\\mathrel{\\mathcal{R}_{\\text{Conf}}}}\\)} \\\\\n\t\t \t & \\implies S\\fwlts{i'}{\\tau}S' \\tag{From \\autoref{lem:operational_corresp}}\n\t\t \\end{align*}\n\t\t where \\(x_1'=x_1\\cup\\{e_1\\}\\), \\(x_2'=x_2\\cup\\{e_2\\}\\) and \\(\\ell(e_1)=\\ell(e_2)=\\tau\\).\n\t\t We have that\n\t\t \\(\\orig {R'} = P\\), \\(\\orig {S'} = Q\\), \\(\\enc{R'}=(\\enc{P},x_1')\\), \\(\\enc{S'}=(\\enc{Q},x_2')\\) and \\((x_1',x_2')\\in {\\mathrel{\\mathcal{R}_{\\text{Conf}}}}\\).\n\t\t Hence \\((R',S')\\in {\\mathrel{\\mathcal{R}}}\\).\n\n\t\t To prove that the remaining conditions on the pair \\((R,S)\\) holds as well is similar.\n\t\\end{itemize}\n\\end{proof}\n\nThe contextual version of the proposition is straightforward.\n\n\\begin{lemma}\n\t\\label{soundness-bisim}\n\tFor all singly labelled processes \\(R\\) and \\(S\\), \\(\\orig{R}\\mathrel{\\sim^{\\tau}}\\orig{S}\\iff\\encc{\\erase{\\orig{R}}}\\mathrel{\\sim^{\\tau}}\\encc{\\erase{\\orig{S}}}\\).\n\\end{lemma}\n\\begin{proof}\n\t\\begin{align*}\n\t\t\\orig{R}\\mathrel{\\sim^{\\tau}}\\orig{S} & \\iff\\forall C[\\cdot], \\context[\\orig{R}]\\overset{\\cdot}{\\bisim}^{\\tau} \\context[\\orig{S}] \\tag{From \\autoref{def:sbfc_rccs}} \\\\\n\t\t & \\iff \\forall C[\\cdot],\\varnothing \\rhd C[\\erase{\\orig{R}}]\\overset{\\cdot}{\\bisim}^{\\tau} \\varnothing \\rhd C[\\erase{\\orig{S}}] \\tag{From \\autoref{def:rccs_context}} \\\\\n\t\t & \\iff \\forall C[\\cdot], \\enc{C[\\erase{\\orig{R}}]} \\overset{\\cdot}{\\bisim}^{\\tau} \\enc{C[\\erase{\\orig{S}}]} \\tag{From \\autoref{prop:bfbisim_corresp}} \\\\\n\t\t & \\iff \\encc{\\erase{\\orig{R}}}\\mathrel{\\sim^{\\tau}}\\encc{\\erase{\\orig{S}}} \\tag{From \\autoref{bisim-cs}}\n\t\\end{align*}\n\\end{proof}\n\n\\subsection{Inductive characterisation of HHPB}\n\\label{sec:inductive_hhpb}\nSimilarly to the proof in CCS, the correspondence between a contextual equivalence and a non-contextual one necessitates to approximate HHPB with (a family of) inductive relations defined on configuration structures.\nIf we are interested only in the forward direction (as in CCS), the inductive reasoning starts with the empty set, and constructs the bisimilarity relation by adding pairs of configurations reachable in the same manner from the empty set.\nHowever, to approximate HHPB, we need to have an inductive reasoning on the backward transition as well (\\autoref{ForwBackwDef}).\nThese relations are of major importance to prove our main theorem (\\autoref{main-thm}), as they re-introduce the possibility of an inductive reasoning thanks to a stratification of the HHPB relation.\n\n\\begin{definition} [Hereditary history preserving bisimilarity]\n\t\\label{sbisimilarity_h-def}\n\tThe hereditary history preserving bisimilarity, denoted \\(\\bisim\\), is the union of all HHPB relations (\\autoref{def:hhpb}).\n\\end{definition}\n\n\\begin{figure}\n\t\\centering\n\t\\begin{tikzpicture}\n\t\t\\tikz@path@overlay{node} (emptyset1) at (0, -1) {\\(\\emptyset\\)};\n\t\t\\tikz@path@overlay{node} (a1) at (-1, 0) {\\(\\{e_1\\}\\)};\n\t\t\\tikz@path@overlay{node} (b1) at (1, 0) {\\(\\{e_1'\\}\\)};\n\t\t\\path[draw] [->] (emptyset1) -- (a1);\n\t\t\\path[draw] [->] (emptyset1) -- (b1);\n\t\t\\tikz@path@overlay{node}[align=center] (labels) at (0, -3) {\\(\\ell_1(e_1) =\\ell_1(e'_1) = a\\)};\n\t\t\\tikz@path@overlay{node} (emptyset2) at (4, -1) {\\(\\emptyset\\)};\n\t\t\\tikz@path@overlay{node} (a2) at (3, 0) {\\(\\{e_2\\}\\)};\n\t\t\\tikz@path@overlay{node} (b2) at (5, 0) {\\(\\{e_2'\\}\\)};\n\t\t\\path[draw] [->] (emptyset2) -- (a2);\n\t\t\\path[draw] [->] (emptyset2) -- (b2);\n\t\t\\tikz@path@overlay{node}[align=center] (labels) at (4, -3) {\\(\\ell_2(e_2) = \\ell_2(e_2') =a\\)};\n\t\t\\path[draw][<->,colorp2, thick] (a1) .. controls (-1, -2.5) and (5, -2.5) .. (b2);\n\t\t\\path[draw][<->,colorp2, thick] (b1) .. controls (1, -2) and (3, -2) .. (a2);\n\t\t\\tikz@path@overlay{node}[colorp2] (f2) at (2, -1.8) {\\(f_2\\)};\n\t\t\\path[draw][<->,colorp3, thick] (a1) .. controls (-1, 2) and (3, 2) .. (a2);\n\t\t\\path[draw][<->,colorp3, thick] (b1) .. controls (1, 2) and (5, 2) .. (b2);\n\t\t\\tikz@path@overlay{node}[colorp3] (f1) at (2, 1.8) {\\(f_1\\)};\n\t\\end{tikzpicture}\n\t\\caption{Two possible hereditary history preserving bisimulations}\n\t\\label{fig:bisimilarity}\n\\end{figure}\n\n\\begin{remark}[On the uniqueness of hereditary history preserving bisimilarity]\n\tWriting \\(\\mathcal{C}_1\\bisim\\mathcal{C}_2\\) is an abuse of notation as hereditary history preserving \\emph{bisimulations} are defined on \\(C_1\\times C_2\\times\\mathcal{P}(E_1\\times E_2)\\).\n\tAlso the union of all bisimulations may contain triples that do not have \\enquote{compatible} bijections.\n\tFor instance, we have two possible bisimulations between the configurations structures of \\autoref{fig:bisimilarity}:\n\t\\[f_1 = \\{ e_1\\leftrightarrow e_2, e_1'\\leftrightarrow e_2' \\} \\qquad f_2 = \\{e_1\\leftrightarrow e_2', e_1'\\leftrightarrow e_2\\}\\]\n\tHowever, the bisimilarity relation contains both tuples \\( (\\{ e_1, e_2\\}, \\{ e_1', e_2'\\}, f_1) \\) and \\( (\\{ e_1, e_2\\}, \\{ e_1', e_2'\\}, f_2) \\).\n\n\\end{remark}\n\nWe give an inductive characterisation of HHPB by reasoning on the structures up to a level: we ignore the configurations that have greater cardinality then the considered level.\nHHPB is then the relation obtained when we reach the top level.\nHence we are able to detect, whenever two configurations structures are not HHPB, at which level the bisimulation does no longer hold.\n\nIn the following, denote \\(\\card(x)\\) the cardinality of a set \\(x\\).\n\\begin{definition}[Maximal and top configurations]\n\tA configuration \\(x\\in\\mathcal{C}\\) is \\emph{maximal} if there is no configuration \\(y\\in \\mathcal{C}\\) such that \\(x\\subsetneq y\\).\n\tIf moreover \\(\\forall y\\in\\mathcal{C}\\), \\(\\card(y)\\leqslant\\card(x)\\) then \\(x\\) is a \\emph{top configuration}.\n\\end{definition}\n\n\\begin{definition}[\\(\\Forw{i}\\), \\(\\Backw{i}\\)]\n\t\\label{ForwBackwDef}\n\tGiven \\(\\mathcal{C}_1\\), \\(\\mathcal{C}_2\\) two configurations structures\n\tdefine, for all \\(x_1 \\in C_1\\), \\(x_2 \\in C_2\\) and \\(f\\) a label and order-preserving function:\n\t\\begin{align*}\n\t\t(x_1, x_2, f)\\in \\Forw{i} & \\iff\n\t\t\\begin{cases}\n\t\t\\card(x_1) = \\card(x_2) = i, & \\text{if }x_1\\text{ and }x_2\\text{ are maximal } \\\\\n\t\t\\forall x'_1, \\exists x'_2, x_1 \\redl{e_1} x'_1, x_2 \\redl{e_1} x'_2 \\text{ and }\\\\\n\t\tf=f'\\upharpoonleft x_1\\text{ \\st~} (x'_1, x'_2, f') \\in \\Forw{i+1} & \\text{otherwise}\n\t\t\\end{cases}\\\\\n\t\t(x_1, x_2, f) \\in \\Backw{i} & \\iff\n\t\t\\begin{cases}\n\t\t(x_1, x_2, f) \\in \\Forw{i} & \\text{if \\(i = 0\\)} \\\\\n\t\t\\forall x'_1, \\exists x'_2, x_1 \\revredl{e_1} x'_1, x_2 \\revredl{e_1} x'_2 \\text{ and }\\\\\n\t\tf'=f\\upharpoonleft x_2\\text{ \\st~}(x'_1, x'_2, f') \\in \\Forw{i-1} \\cap \\Backw{i-1} & \\text{otherwise}\n\t\t\\end{cases}\n\t\\end{align*}\n\n\\end{definition}\nThe label and order-preserving function in the two relations helps to ensure that configurations that are in relation have the same labels and causal structure.\n\nThe relation \\(\\Backw{i}\\) is built on top of \\(\\Forw{i}\\): it tests for the backward steps all the couples that passed the forward test.\nIt should be remarked that, with this definition, \\(\\Backw{i} \\subseteq \\Forw{i}\\), but, at the price of slight modifications, one could have defined \\(\\Forw{i}\\) on top of \\(\\Backw{i}\\).\n\n\\begin{figure}\n\t\\begin{minipage}[b]{.45\\linewidth}\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\t\\tikz@path@overlay{node} (emptyset) at (0, -1) {\\(\\emptyset\\)};\n\t\t\t\\tikz@path@overlay{node} (a) at (-1, 0) {\\(\\{e_1\\}\\)};\n\t\t\t\\tikz@path@overlay{node} (b) at (1, 0) {\\(\\{e_1'\\}\\)};\n\t\t\t\\tikz@path@overlay{node} (ab) at (0, 1) {\\(\\{e_1, e_1'\\}\\)};\n\t\t\t\\path[draw] [->] (emptyset) -- (a);\n\t\t\t\\path[draw] [->] (emptyset) -- (b);\n\t\t\t\\path[draw] [->] (a) -- (ab);\n\t\t\t\\path[draw] [->] (b) -- (ab);\n\t\t\t\\tikz@path@overlay{node}[align=center] (labels) at (0, -2) {\\(\\ell_1(e_1) = a\\),\\\\ \\(\\ell_1(e'_1) = b\\)};\n\t\t\\end{tikzpicture}\n\t\t\\subcaption{\\(a \\mid b\\)}\\label{fig:ex_bf_a}\n\t\\end{minipage}\n\t\\begin{minipage}[b]{.45\\linewidth}\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\t\\tikz@path@overlay{node} (emptyset) at (0, -1) {\\(\\emptyset\\)};\n\t\t\t\\tikz@path@overlay{node} (a) at (-1, 0) {\\(\\{e_2\\}\\)};\n\t\t\t\\tikz@path@overlay{node} (b) at (1, 0) {\\(\\{e''_2\\}\\)};\n\t\t\t\\tikz@path@overlay{node} (ab) at (-1, 1) {\\(\\{e_2, e'_2\\}\\)};\n\t\t\t\\tikz@path@overlay{node} (ba) at (1, 1) {\\(\\{e''_2, e'''_2\\}\\)};\n\t\t\t\\path[draw] [->] (emptyset) -- (a);\n\t\t\t\\path[draw] [->] (emptyset) -- (b);\n\t\t\t\\path[draw] [->] (a) -- (ab);\n\t\t\t\\path[draw] [->] (b) -- (ba);\n\t\t\t\\tikz@path@overlay{node}[align=center] (labels) at (0, -2) {\\(\\ell_2(e_2) = \\ell_2(e_2''') =a\\),\\\\ \\(\\ell_2(e_2') = \\ell_2(e_2'') =b\\)};\n\t\t\\end{tikzpicture}\n\t\t\\subcaption{\\(a.b+b.a\\)}\\label{fig:ex_bf_b}\n\t\\end{minipage}\n\t\\caption{Encoding parallel and sum in configurations structures}\n\t\\label{fig:ex_bf}\n\\end{figure}\n\n\\begin{example}\n\t\\label{example2}\n\tConsider the configurations structures in Figures \\ref{ex_unif_b} and \\ref{ex_unif_c}, the relations \\(F_n\\) are enough to discriminate them:\n\t\\begin{align*}\n\t\t & F_2 = \\big(\\{e_1, e_1'\\}, \\{e_2, e_2'\\}\\big); \\big(\\{e_1, e_1'\\}, \\{e_2'',e_2'''\\}\\big) \\\\\n\t\t & \\qquad F_1 = \\big(\\{e_1\\}, \\{e_2\\}\\big); \\big(\\{e_1\\}, \\{e_2''\\}\\big) \\\\\n\t\t & \\qquad \\qquad F_0 = \\emptyset\n\t\\end{align*}\n\n\tThis intuitively is due to the fact that forward transitions are enough to discriminate \\(a+a.b\\) and \\(a.b+a.b\\).\n\tHowever for comparing the processes \\(a \\mid b\\) and \\(a.b+b.a\\) whose configurations are in Figures \\ref{fig:ex_bf_a} and \\ref{fig:ex_bf_b}, we need the backward moves as well.\n\tLet us first build the \\(F_n\\) relations:\n\t\\begin{align*}\n\t\t & F_2 = \\big(\\{e_1,e_1'\\},\\{e_2,e_2'\\}\\big); \\big(\\{e_1,e_1'\\};\\{e_2'',e_2'''\\}\\big) \\\\\n\t\t & \\qquad\tF_1 = \\big(\\{e_1\\},\\{e_2\\}\\big); \\big(\\{e_1'\\};\\{e_2''\\}\\big) \\\\\n\t\t & \\qquad \\qquad F_0 = \\big(\\emptyset,\\emptyset\\big)\n\t\\end{align*}\n\n\tWe first construct the \\(B_0\\) relation and then move up in the structures.\n\tIn our example, the \\(B_2\\) relation breaks the HHPB.\n\t\\begin{align*}\n\t\t & \\qquad \\qquad B_0 =F_0=\\big(\\emptyset,\\emptyset\\big) \\\\\n\t\t & \\qquad B_1 = \\big(\\{e_1\\},\\{e_2\\}\\big); \\big(\\{e_1'\\};\\{e_2''\\}\\big) \\\\\n\t\t & B_2 = \\emptyset\n\t\\end{align*}\n\n\\end{example}\nThe following proposition states that pairs of configurations are in a bisimulation relation if they have the same cardinality.\nIt follows from the fact that any configuration is reachable from the empty set and that they have to mimick each other's step in the backward direction.\n\\begin{proposition}\n\t\\label{prop:size_config}\n\tLet \\(\\mathcal{C}_1\\bisim\\mathcal{C}_2\\) be two configuration structures in a hereditary history preserving bisimulation and \\(x_1\\in\\mathcal{C}_1\\), \\(x_2\\in\\mathcal{C}_2\\) be two configurations.\n\n\tIf \\(\\exists f\\) such that \\((x_1,x_2,f)\\in \\{\\bisim\\}\\) then \\(\\card(x_1)=\\card(x_2)\\).\n\\end{proposition}\n\\begin{proof}\n\tIt follows by induction on the trace \\(\\emptyset\\mathrel{\\fw}^\\star x_1\\).\n\tFor every event in \\(x_1\\), we have to add an event in \\(x_2\\) in order to obtain that the pair \\(x_1\\) and \\(x_2\\) are in a HHPB relation.\n\\end{proof}\n\nThe following lemma, that will be handy to prove \\autoref{main-thm}, implies that if for all \\(n \\leqslant k\\) the maximum cardinal considered, \\(\\Forw{n} \\cap \\Backw{n} \\neq \\emptyset\\), then \\(\\cup_{n \\leqslant k} (\\Forw{n} \\cap \\Backw{n})\\) is a bisimulation.\n\n\\begin{lemma}\n\t\\label{Fn-Bn-and-h}\n\tFor all \\(\\mathcal{C}_1\\), \\(\\mathcal{C}_2\\), if \\(\\mathcal{C}_1 \\bisim \\mathcal{C}_2\\) , then \\(\\forall x_1 \\in \\mathcal{C}_1 (\\exists x_2 \\in \\mathcal{C}_2, \\exists f, (x_1, x_2, f) \\in \\Forw{n} \\cap \\Backw{n}) \\iff (\\exists x_2 \\in \\mathcal{C}_2, \\exists f, (x_1, x_2, f) \\in\\bisim )\\), where \\(\\card(x_1)=n\\).\n\\end{lemma}\n\n\\begin{proof}\n\tLet us denote \\(\\mathrel{\\mathcal{R}}\\) the relation \\(\\bisim\\).\n\tOne should first remark that \\(\\mathcal{C}_1 \\bisim \\mathcal{C}_2\\) implies that \\(\\forall x_1 \\in \\mathcal{C}_1, \\exists x_2 \\in \\mathcal{C}_2\\), and \\(\\exists f\\) such that \\( (x_1, x_2, f) \\in {\\mathrel{\\mathcal{R}}}\\), as \\((\\emptyset,\\emptyset,\\emptyset)\\in {\\mathrel{\\mathcal{R}}}\\) and all configurations are reachable from the empty set.\n\tThe reader should notice that the \\(x_2 \\in \\mathcal{C}_2\\) and \\(f\\) on both sides of the \\(\\iff\\) symbols may be different.\n\n\tWe prove that statement by induction on the cardinal of \\(x_1\\).\n\t\\paragraph{\\(\\card(x_1) = 0\\)}\n\t\\begin{itemize}\n\t\t\\item[\\(\\Rightarrow\\)]\n\t\t \\(x_2 \\in \\mathcal{C}_2\\) \\st \\((\\emptyset, x_2, f) \\in {\\mathrel{\\mathcal{R}}}\\) follows by the definition of the bisimulation from \\(x_2 = \\emptyset\\) and \\(f = \\emptyset\\).\n\t\t\\item[\\(\\Leftarrow\\)]\n\t\t By definition, \\(\\Forw{0} \\cap \\Backw{0} = \\Forw{0}\\).\n\t\t Since there exists \\(x_2 \\in \\mathcal{C}_2\\) such that \\((\\emptyset, x_2, f) \\in {\\mathrel{\\mathcal{R}}}\\), we know that any forward transition made by \\(\\emptyset\\) can be simulated by a forward transition from \\(x_2\\), and that the elements obtained are in the relation \\({\\mathrel{\\mathcal{R}}}\\).\n\t\t By an iterated use of this notion, we find top configurations \\(x_1^m \\in \\mathcal{C}_1\\) and \\(x_2^m \\in \\mathcal{C}_2\\) (that is, elements of maximum cardinality, \\(k\\)) such that \\((x_1^m, x_2^m, f^m) \\in {\\mathrel{\\mathcal{R}}}\\).\n\t\t By \\autoref{prop:size_config}, \\(x_1^m\\) and \\(x_2^m\\) have the same cardinality, and \\((x_1^m, x_2^m, f^m) \\in \\Forw{k}\\).\n\t\t By just {reversing the trace}, we go backward and stay in relation \\(\\Forw{i}\\) until \\(i = 0\\), hence we found the \\(x_2\\) and \\(f\\) we were looking for.\n\t\\end{itemize}\n\t\\paragraph{\\(\\card(x_1) = k +1\\)}\n\tAs \\(\\card(x_1) > 0\\), we know there exists \\(x'_1\\) such that \\(x_1 \\revredl{e_1} x'_1\\).\n\t\\begin{itemize}\n\t\t\\item[\\(\\Rightarrow\\)]\n\t\t Let \\(x_2\\) and \\(f\\) such that \\((x_1, x_2, f) \\in \\Forw{k+1} \\cap \\Backw{k+1}\\).\n\t\t We know that\n\t\t \\begin{align*}\n\t\t \t\\forall x'_1, \\exists x'_2\\text{ and } f', x_1 \\revredl{e_1} x'_1, x_2 \\revredl{e_1} x'_2 \\text{ and } (x'_1, x_2', f') \\in \\Backw{k} \\tag{By Definition of \\(\\Backw{k} \\)} \\\\\n\t\t \t\\exists x''_2,f'', (x'_1, x_2'', f'') \\in {\\mathrel{\\mathcal{R}}} \\tag{By induction hypothesis}\n\t\t \\end{align*}\n\t\t And as \\(x'_1 \\redl{e_1} x_1\\), there exists \\(x'''_2\\) and \\(f'''\\) such that \\((x_1, x'''_2, f''') \\in {\\mathrel{\\mathcal{R}}}\\).\n\t\t\\item[\\(\\Leftarrow\\)]\n\t\t We prove it by contraposition: suppose that \\(\\exists x_2,f\\) such that \\((x_1, x_2, f) \\in {\\mathrel{\\mathcal{R}}}\\), we prove that \\(\\forall x_2\\), \\((x_1, x_2, f) \\notin \\Forw{k+1} \\cap \\Backw{k+1}\\) leads to a contradiction.\n\n\t\t As \\((x_1, x_2, f) \\in {\\mathrel{\\mathcal{R}}}\\), we know that there exists \\(x'_1\\) and \\(x'_2,f'\\) such that \\(x_1 \\revredl{e_1} x'_1\\), \\(x_2 \\revredl{e_1} x'_2\\) and \\((x'_1, x'_2, f') \\in {\\mathrel{\\mathcal{R}}}\\).\n\t\t By induction hypothesis, \\(\\exists x''_2\\) and \\(\\exists f''\\) such that \\((x'_1, x''_2, f'') \\in \\Forw{k} \\cap \\Backw{k}\\).\n\t\t As \\(x'_1 \\redl{e_1} x_1\\), \\(\\exists x'''_2\\) and \\(\\exists f'''\\) such that \\(x''_2 \\redl{e_1} x'''_2\\) and \\((x_1, x'''_2, f''') \\in \\Forw{k+1}\\), by definition of \\(\\Forw{k}\\).\n\n\t\t So \\((x_1, x'''_2, f''') \\notin \\Backw{k+1}\\), but as \\(x_1 \\revredl{e_1} x'_1\\) and \\(x'''_2 \\revredl{e_1} x''_2\\), and as moreover \\((x'_1, x''_2, f'') \\in \\Forw{k} \\cap \\Backw{k}\\), we have that \\((x_1, x'''_2, f''') \\in \\Backw{k+1}\\).\n\n\t\t From this contradiction we know that we found the right element (\\(x'''_2\\)) that is in relation with \\(x_1\\) according to \\(\\Forw{k+1} \\cap \\Backw{k+1}\\).\n\t\t \\qedhere\n\t\\end{itemize}\n\\end{proof}\n\n\\begin{figure}\n\t\\begin{tikzpicture}[scale=0.85]\n\t\t\\path[fill][colorp1] (0,0) ellipse (1cm and 2cm) node[above=2cm, black]{\\(\\enc{P_1} = \\mem{E_1, C_1, \\ell_1}\\)};\n\t\t\\path[draw] (0, -1) node[below]{\\(x_1\\)} node (x1) {\\(\\bullet\\)};\n\t\t\\path[draw] (0, 1) node (x'1) {\\(\\bullet\\)};\n\t\t\\path[draw] [->] (x1) -- node[right]{\\(e''_1\\)} (x'1);\n\t\t\\path[fill][colorp1] (6, 0) ellipse (1cm and 2cm) node[above=2cm, black]{\\(\\enc{P_1 \\mid Q} = \\mem{E'_1, C'_1, \\ell'_1} = (\\enc{P_1} \\times \\enc{Q}) \\upharpoonleft E_1 \\)};\n\t\t\\path[fill][colorp1] (6.7, 0) ellipse (1cm and 2cm);\n\t\t\\path[draw] (6.35, -1) node[below]{\\(y_1\\)} node (y1) {\\(\\bullet\\)};\n\t\t\\path[draw] (6.35, 1) node[above]{\\(y'_1\\)} node (y'1) {\\(\\bullet\\)};\n\t\t\\path[draw] [->] (y1) -- node[right]{\\(e'' = (e''_1, e''_q)\\)}(y'1);\n\t\t\\begin{scope}[yshift=-4.3cm]\n\t\t\t\\path[fill][colorp2] (0,0) ellipse (1cm and 2cm) node[below=2cm, black]{\\(\\enc{P_2} = \\mem{E_2, C_2, \\ell_2}\\)};\n\t\t\t\\path[draw] (0, -1) node[below]{\\(x_2\\)} node (x2) {\\(\\bullet\\)};\n\t\t\t\\path[draw] (0, 1) node[above]{\\(x''_2\\)} node (x'2) {\\(\\bullet\\)};\n\t\t\t\\path[draw] [->] (x2) -- node[right]{\\(e''_2\\)} (x'2);\n\t\t\t\\path[fill][colorp2] (6, 0) ellipse (1cm and 2cm) node[below=2cm, black]{\\(\\enc{P_2 \\mid Q} = \\mem{E'_2, C'_2, \\ell'_2} = (\\enc{P_2} \\times \\enc{Q}) \\upharpoonleft E_2\\)};\n\t\t\t\\path[fill][colorp2] (6.7, 0) ellipse (1cm and 2cm);\n\t\t\t\\path[draw] (6.35, -1) node[below]{\\(y_2\\)} node (y2) {\\(\\bullet\\)};\n\t\t\t\\path[draw] (6.35, 1) node[above]{\\(y'_2\\)} node (y'2) {\\(\\bullet\\)};\n\t\t\t\\path[draw] [->] (y2) -- node[right]{\\(e'_2\\)}(y'2);\n\t\t\\end{scope}\n\t\t\\path[draw] [<->, double] (y2) to [bend left] node[midway, left] {$f_c$} ($(y1)-(.3, 0)$);\n\t\t\\path[draw] [<->, double] (x2) to [bend left] node[midway, left] {$f$} ($(x1)-(.3, 0)$);\n\t\t\\path[draw] [->, double] (y2) to [bend right] node[midway, below ,sloped] {$\\pi_2$} (x2);\n\t\t\\path[draw] [->, double] (y1) to [bend right] node[midway, below ,sloped] {$\\pi_1$} (x1);\n\t\t\\path[draw] [->, double] (y'1) to [bend right] node[midway, below ,sloped] {$\\pi_1$} (x'1);\n\t\t\\tikz@path@overlay{node}[rectangle callout,draw,inner sep=2pt,fill=colorp1,\n\t\t\tcallout absolute pointer=(y1.east),\n\t\tbelow right= 25pt and 35pt of y1.north east]\n\t\t{\\begin{tabular}{c c c c} \\(e\\) & \\( = \\)& \\(e_1,\\) & \\(e_q\\) \\\\ \\rotatebox{-90}{\\(\\leqslant\\)} & & \\rotatebox{-90}{\\(\\leqslant\\)}\\(_{\\pi_1}\\) & \\rotatebox{-90}{\\(\\leqslant\\)}\\(_{\\pi_2}\\) \\\\ \\(e'\\) & \\( =\\) & \\(e'_1,\\) & \\(e'_q\\)\\end{tabular}};\n\t\\end{tikzpicture}\n\t\\caption{Configurations Structures by the end of the proof of \\autoref{prop:HHPB_congr}}\n\t\\label{fig-HHPB_congr}\n\\end{figure}\n\n\\subsection{Contextual characterisation of HHPB}\n\\label{sec:contextual_hhpb}\n\n\\begin{proposition}[Hereditary history preserving bisimulation is a congruence]\n\t\\label{prop:HHPB_congr}\n\tFor all singly labelled \\(P_1\\), \\(P_2\\), \\(\\enc{P_1} \\bisim \\enc{P_2} \\implies \\forall C, \\enc{C[P_1]} \\bisim \\enc{C[P_2]}.\\)\n\\end{proposition}\n\n\\begin{proof}\n\tThe proof amounts to carefully build a relation between \\(\\enc{C[P_1]}\\) and \\(\\enc{C[P_2]}\\) that reflects the known bisimulation between \\(\\enc{P_1}\\) and \\(\\enc{P_2}\\).\n\tIts uses that causality in a product is the result of the entanglement of the causality of its elements (\\autoref{prop:cause_projection}).\n\n\tDue to the restriction on the contexts we consider, we only have to prove that\n\t\\[\n\t\t\\forall P_1, P_2, \\enc{P_1}\\bisim\\enc{P_2}\\implies\\forall Q, \\enc{P_1\\vert Q}\\bisim\\enc{P_2\\vert Q}\n\t\\]\n\n\tAs \\(\\enc{P_1}\\bisim\\enc{P_2}\\), there exists \\({\\mathrel{\\mathcal{R}}}\\) a hereditary history preserving bisimulation (HHPB) between \\(\\enc{P_1}\\) and \\(\\enc{P_2}\\).\n\t\\autoref{fig-HHPB_congr} introduces the variables names and types.\n\n\tDefine \\(\\mathrel{\\mathcal{R}}_c\\subseteq C'_1\\times C'_2\\times\\mathcal{P}(E'_1\\times E'_2)\\) as follows:\n\t\\[\n\t\t(y_1, y_2, f_c) \\in {\\mathrel{\\mathcal{R}}}_c \\iff\n\t\t\\begin{cases}\n\t\t\t(\\pi_1(y_1),\\pi_2(y_2),\\pi_1\\circ f) \\in {\\mathrel{\\mathcal{R}}} \\\\\n\t\t\tf_c(e)=(\\pi_1 \\circ f(e)),\\pi_2(e))\\in y_2\\text{ for all }e \\in y_1\n\t\t\\end{cases}\n\t\\]\n\n\tInformally \\((y_1, y_2, f_c)\\) is in the relation \\(\\mathrel{\\mathcal{R}}_c\\) if there is \\((x_1, x_2, f)\\) in \\(\\mathrel{\\mathcal{R}}\\) such that \\(x_i\\) is the first projection of \\(y_i\\) and such that \\(f_c\\) satisfies the property: for \\((e_1, e_{q})\\in E'_1\\), \\(f_c(e_1,e_{q})=(f(e_1),e_{q})\\) and \\((f(e_1),e_{q})\\in E'_2\\).\n\n\tLet us show that \\(\\mathrel{\\mathcal{R}}_c\\) is a HHPB between \\(\\mem{E'_1, C'_1, \\ell'_1}\\) and \\(\\mem{E'_2, C'_2, \\ell'_2}\\).\n\t\\begin{itemize}\n\t\t\\item \\((\\emptyset,\\emptyset,\\emptyset)\\in {\\mathrel{\\mathcal{R}}}_c\\);\n\t\t\\item For \\((y_1, y_2, f_c)\\in {\\mathrel{\\mathcal{R}}}\\) we show that \\(f_c\\) is label and order preserving bijection.\n\t\t We have that \\(f_c\\) is defined as \\(f_c(e)=(\\pi_1 \\circ f(e)),\\pi_2(e))\\), for some \\(f\\) label and order preserving bijection such that \\((\\pi_1(y_1),\\pi_2(y_2),\\pi_1\\circ f)\\in {\\mathrel{\\mathcal{R}}}\\).\n\n\t\t That \\(f_c\\) is a bijection follows from \\(f\\) being a bijection.\n\n\t\t Let \\(e \\in y_1\\) with \\(\\pi_1(e)=e_1\\), \\(\\pi_2(e)=e_{q}\\), then \\(f_c(e)=(f(e_1),e_{q})\\) for some \\(f_c\\) \\st \\((\\pi(y_1), \\pi_2(y_2), f)\\in {\\mathrel{\\mathcal{R}}}\\).\n\t\t We have that \\(\\ell'_1(e)=(\\ell_1(e_1),\\ell_{Q}(e_{q}))\\) and\n\t\t \\[\n\t\t \t\\ell'_2(f_c(e))=\\ell'_2(f(e_1),e_{q})=\\big(\\ell_2(f(e_1)),\\ell_{Q}(e_{q})\\big)\n\t\t \\]\n\t\t As \\(f\\) is label preserving we get \\(\\ell'_2(f_c(e))=(\\ell_1(e_1),\\ell_{Q}(e_{q}))\\), hence \\(\\ell'_1(e)=\\ell'_2(f_c(e))\\).\n\n\t\t Let us now show that for \\(e, e' \\in y_1\\), if \\(e\\to_{y_1} e'\\) then \\(f_c(e)\\leqslant_{y_2} f_c(e')\\).\n\t\t We denote \\(\\pi_1(e)=e_1\\), \\(\\pi_2(e)=e_{q}\\) and \\(\\pi_1(e')=e_1'\\), \\(\\pi_2(e')=e_{q}'\\).\n\t\t Then from \\autoref{prop:cause_projection}\n\t\t \\begin{equation*}\n\t\t \te\\to_{y_1} e'\\implies e_1\\leqslant_{\\pi_1(y_1)} e_1'\\text{ or }e_{q}\\leqslant_{\\pi_2(y_1)} e_{q}'\n\t\t \\end{equation*}\n\t\t We consider the case where \\(e_1\\leqslant_{\\pi_1(y_1)} e_1'\\).\n\t\t As \\(f\\) is order preserving we have that \\(f(e_1)\\leqslant_{\\pi_1(y_2)}f(e_1')\\).\n\t\t Then \\((f(e_1),e_{q})\\leqslant_{x_2}(f(e_1'),e_{q}')\\), as the projections are order reflecting.\n\n\t\t\\item Let \\((y_1, y_2,f_c)\\in {\\mathrel{\\mathcal{R}}}_c\\) and \\(y_1 \\redl{e''} y_1'\\), \\(y_1'=y_1\\cup\\{e''\\}\\).\n\t\t We consider only the case when \\(\\pi_1(e'') = e''_1\\neq\\star\\), \\(\\pi_2(e'')= e''_{q}\\neq\\star\\) as the rest is similar.\n\t\t From the definition of the projections \\(\\pi_1(y_1)\\), \\(\\pi_1(y_1')\\in C_1'\\) and as \\(\\pi_1(e'')=e''_1\\neq\\star\\), we have that \\(\\pi_1(y_1')=\\pi_1(y_1)\\cup\\{e''_1\\}\\).\n\t\t We reason similarly on \\(\\pi_2(y_1)\\) and get\n\t\t \\begin{equation}\n\t\t \t\\label{eq1}\n\t\t \t\\pi_1(y_1)\\redl{e''_1}\\pi_1(y_1')\\text{ and }\\pi_2(y_1)\\redl{e''_{q}}\\pi_2(y_1').\n\t\t \\end{equation}\n\t\t From \\autoref{eq1} and as \\((\\pi_1(y_1), \\pi_2(y_2),f)\\in {\\mathrel{\\mathcal{R}}}\\), by definition of \\(\\mathrel{\\mathcal{R}}_c\\), we have that\n\t\t \\begin{equation}\n\t\t \t\\label{eq2}\n\t\t \t\\exists x_2'\\text{ \\st~}\\pi_1(y_2)\\redl{e''_2}x_2'=x_2\\cup\\{e''_2\\}\n\t\t \\end{equation}\n\t\t and\n\t\t \\begin{equation}\\label{eq3}\n\t\t \tf'=f\\cup\\{e_1''\\leftrightarrow e_2''\\}\n\t\t \\end{equation}\n\t\t such that \\((x_1',x_2',f')\\in {\\mathrel{\\mathcal{R}}}\\).\n\n\t\t Let us show that \\(\\exists y'_2 \\in (\\enc{P_2}\\times\\enc{P_{Q}})\\) with \\(y_2'=y_2\\cup\\{e_2'\\}\\) and \\(\\pi_1(e'_2)=e''_2\\), \\(\\pi_2(e'_2)=e_{q}''\\).\n\t\t From \\autoref{eq1} and \\autoref{eq2} we have that the projections are defines with \\(\\pi_1(y'_2)=x'_2\\), \\(\\pi_2(y'_2)=\\pi_2(y_1')\\).\n\t\t The axioms of finiteness and coincidence freeness on \\(y_2'\\) follows from \\(y_2\\) being a configuration in \\((\\enc{P_2}\\times\\enc{P_{Q}})\\).\n\n\t\t Let us show that \\(y_2'\\notin X_2\\).\n\t\t We have that \\(y_1'\\notin X_1\\).\n\t\t As \\(\\ell(e_1'')\\) and \\(\\ell(e_q'')\\) are compatible, then so are \\(\\ell(e_2'')\\) and \\(\\ell(e_q'')\\), hence \\(y_2\\cup\\{(e_2'',e_q'')\\}\\notin X_2\\).\n\n\t\t Remains to show \\((y_1',y_2',f_c')\\in {\\mathrel{\\mathcal{R}}}\\), where \\(f_c'=f_c\\cup\\{e''_1 \\leftrightarrow e''_2\\}\\).\n\t\t We have that \\((\\pi_1(y_1'),\\pi_1(y_2'),f')\\in {\\mathrel{\\mathcal{R}}}_c\\) and from \\autoref{eq3} that \\(\\pi_1\\circ f_c'=f'\\). \\qedhere\n\t\\end{itemize}\n\\end{proof}\n\n\\newcommand{\\horiz}{4.6} %\n\\pgfmathsetmacro{\\horizhoriz}{2*\\horiz}\n\n\\begin{figure}\n\tFor \\(i \\in \\{1, 2\\}\\), we have:\n\t\\begin{tikzpicture}\n\t\t\\path[fill][colorp1] (0,0) ellipse (1cm and 2cm) node[below=1.5cm, left=.7cm, black]{\\(\\enc{P_i}\\)};\n\t\t\\path[draw] (0, -1) node[below]{\\(x_i\\)} node (x1) {\\(\\bullet\\)};\n\t\t\\path[draw] (0, 1) node[above]{\\(x'_i\\)} node (x'1) {\\(\\bullet\\)};\n\t\t\\path[draw] [->] (x1) -- (x'1);\n\t\t\\path[fill][colorp1] (\\horiz,0) ellipse (1cm and 2cm) node[below=1.5cm, left=.7cm, black]{\\(\\enc{C[P_i]}\\)};\n\t\t\\path[draw] (\\horiz, -1) node[below]{\\(y_i\\)} node (y1) {\\(\\bullet\\)};\n\t\t\\path[draw] (\\horiz, 1) node[above]{\\(y'_i\\)} node (y'1) {\\(\\bullet\\)};\n\t\t\\path[draw] [->] (y1) -- (y'1);\n\t\t\\path[fill][colorp1] (\\horizhoriz,0) ellipse (1cm and 2cm) node[below=1.5cm, left=.7cm, black]{\\(\\enc{C'[P_i]}\\)};\n\t\t\\path[draw] (\\horizhoriz, 1) node[above]{\\(z'_i\\)} node (z'1) {\\(\\bullet\\)};\n\t\t\\path[draw] [->, double] (y1) to [bend right] node[midway,below,sloped] {\\(\\pi_{C, P_i}\\)} (x1);\n\t\t\\path[draw] [->, double] (z'1) to [bend right] node[midway,below,sloped] {\\(\\pi_{C', C[P_i]}\\)} (y'1);\n\t\t\\path[draw] [->, double] (y'1) to [bend right] node[midway,below,sloped] {\\(\\pi_{C, P_i}\\)} (x'1);\n\t\\end{tikzpicture}\n\n\tWe start with \\(y_1 \\mathrel{\\sim^{\\tau}} y_2\\), then prove that \\(z'_1 \\mathrel{\\sim^{\\tau}} z'_2\\), to end up with \\((x'_1, x'_2, f) \\in \\Forw{n} \\cap \\Backw{n}\\).\n\n\t\\caption{Configurations Structures by the end of the proof of \\autoref{main-thm}}\n\t\\label{fig-main-thm}\n\\end{figure}\n\n\\begin{theorem}\n\t\\label{main-thm}\n\tFor all singly labelled \\(P_1\\) and \\(P_2\\), \\(\\enc{P_1} \\bisim \\enc{P_2} \\iff \\enc{P_1} \\mathrel{\\sim^{\\tau}} \\enc{P_2}\\).\n\\end{theorem}\n\n\\begin{proof}\n\tThe left-to-right direction follows from the definition of \\(\\bisim\\) (\\autoref{def:hhpb}) and from \\autoref{prop:HHPB_congr}.\n\n\tWe prove the other direction %\n\tby contraposition: let us suppose that \\(\\enc{P_1}\\mathrel{\\sim^{\\tau}}\\enc{P_2}\\) and \\(\\enc{P_1} \\not\\bisim \\enc{P_2}\\), we will find a contradiction.\n\t\\autoref{fig-main-thm} presents the general shape of the configurations at the end of the proof.\n\n\tAs \\(\\enc{P_1} \\not\\bisim \\enc{P_2}\\), by \\autoref{Fn-Bn-and-h}, there exists \\(x_1\\in\\encc{P_1}\\) such that \\(\\forall x_2\\in\\encc{P_2}\\), \\((x_1,x_2,f)\\notin F_n\\cap B_n\\) holds.\n\tLet us consider the largest such \\(x_1\\).\n\tNote that we consider only \\(x_2\\) such that \\(\\card(x_1)=\\card(x_2)=n\\), and that we use the projections \\(\\pi_{C, P}\\) (\\autoref{def:conf_context}) to separate the events of the process \\(P\\) from the events of the context \\(C\\).\n\n\tFor any \\(x_1\\) we define\n\t\\( C[\\cdot] \\coloneqq \\prod_{e_i \\in x_i} (\\overline{\\ell(e_i)} + c_{e_i}) \\mid [\\cdot] \\)\n\twhere \\(c_{e_i}\\notin \\nm{P_1}\\cup\\nm{P_2}\\), such that the following holds\n\t\\begin{itemize}\n\t\t\\item \\(\\exists y_1 \\in \\enc{C[P_1]}\\) such that \\(y_1\\) is closed, \\(\\pi_{C, P_1} (y_1) = x_1\\) and \\(y_1 \\not\\downarrow_{c_{e_i}}\\) for all \\(e_i \\in x_1\\);\n\t\t\\item We supposed that \\(\\enc{P_1}\\mathrel{\\sim^{\\tau}} \\enc{P_2}\\), so \\(\\enc{C[P_1]}{\\overset{\\cdot}{\\bisim}^{\\tau}}\\enc{C[P_2]}\\).\n\t\t Hence \\(\\exists\\mathrel{\\mathcal{R}}\\) a back-and-forth barbed bisimulation and \\(\\exists y_2 \\in \\enc{C[P_2]}\\) such that \\((y_1, y_2) \\in {\\mathrel{\\mathcal{R}}}\\) and \\(y_2 \\not\\downarrow_{c_{e_i}}\\) for all \\(e_i \\in x_1\\).\n\t\\end{itemize}\n\tWe proceed as follows:\n\t\\begin{itemize}\n\t\t\\item we show that there exists \\(f\\) a label and order preserving bijection between \\(x_1\\) and \\(\\pi_{C, P_1}(y_2)\\);\n\t\t\\item then we show that \\((x_1,\\pi_{C, P_1}(y_2),f)\\in F_n\\) for \\(f\\) defined above;\n\t\t\\item similarly we show that \\((x_1,\\pi_{C, P_1}(y_2),f)\\in B_n\\).\n\t\\end{itemize}\n\n\tWe denote \\(\\pi_{C, P_1}(y_2)\\) with \\(x_2\\).\n\tWe have by induction on the trace \\(\\emptyset\\mathrel{\\fw}^{\\star} y_1\\) that if \\(y_1\\) is closed then \\(y_2\\) is closed as well.\n\tMoreover we define a bijection \\(g:y_1\\to y_2\\) that is order and label preserving.\n\tIt follows again from an induction on the trace \\(\\emptyset\\mathrel{\\fw}^{\\star} y_1\\) and from \\(y_2 \\not\\downarrow_{c_{e_i}}\\) for all \\(e_i \\in x_1\\).\n\n\tWe have that \\(\\forall e_1, e'_1 \\in x_1\\), and \\(e_2 \\in x_2\\),\n\t\\begin{equation}\n\t\t\\label{reason2}\n\t\te_2 \\in x_2 \\iff e_1 \\in x_2\\text{ and } \\ell(e_1) = \\ell (e_2)\n\t\\end{equation}\n\t\\begin{align}\n\t\te_1 <_{x_1} e'_1 & \\Longrightarrow \\pi_{C, P_1}^{-1} (e_1) <_{y_1} \\pi_{C, P_1}^{-1} (e'_1) \\label{reason} \\\\\n\t\t & \\Longrightarrow g(\\pi_{C, P_1}^{-1} (e_1)) <_{y_2} g(\\pi_{C, P_1}^{-1} (e'_1)) \\label{reason3}\n\t\\end{align}\n\n\tRemark that \\eqref{reason2} follows from \\(y_2 \\not\\downarrow_{c_{e_i}}\\) and from the fact that if \\(y_1\\) is closed we can show by contradiction that \\(y_2\\) is closed as well.\n\tSecondly, \\eqref{reason} follows from the morphisms reflecting causality.\n\tLastly, \\eqref{reason3} follows from \\(g\\) being an order preserving bijection between \\(y_1\\) and \\(y_2\\).\n\n\tFor every events in \\(e_1'',e_2''\\in y_2\\) such that \\(e_1''\\to_{y_2} e_2''\\) from \\autoref{prop:cause_projection}, either \\(\\pi_{C, P_2}(e_1'')\\leqslant_{\\pi_{C, P_2}(y_2)} \\pi_{C, P_2}(e_2'')\\) or the projection of the two events are causal dependent in the context.\n\tHowever, the context does not induce any causality between the events.\n\tAs \\(\\pi_{C, P_2}(e_1'')\\leqslant_{\\pi_{C, P_2}(y_2)} \\pi_{C, P_2}(e_2'')\\), we have that there exists \\(f\\) a label and order preserving bijection between \\(x_1\\) and \\(\\pi_{C, P_1}(y_2)\\).\n\n\tLet us now prove that \\((x_1, x_2, f) \\in \\Forw{n+1}\\).\n\tThere are two cases:\n\t\\begin{align}\n\t\t\\not\\exists x'_1, x_1 \\redl{e_1} x'_1, \\exists x'_2, x_2 \\redl{e_2} x'_2 \\label{case-no-transition} \\\\\n\t\t\\exists x'_1, x_1 \\redl{e_1} x'_1, \\forall x'_2, x_2 \\redl{e_2} x'_2 \\text{ and } (x'_1, x'_2, f') \\notin \\Forw{k}\\label{case-no-extension}\n\t\\end{align}\n\n\tThe implication \\eqref{case-no-transition} is easier: if \\(\\exists x'_2, x_2 \\redl{e_2} x'_2\\), then, as a context cannot remove transitions from the original process, \\(\\exists y'_2, y_2 \\redl{(e_2, \\star)} y'_2\\).\n\tAs \\(\\enc{C[P_2]} \\overset{\\cdot}{\\bisim}^{\\tau} \\enc{C[P_1]}\\), \\(\\exists y'_1, y_1 \\redl{(e_1, \\star)} y'_1\\), and a similar argument on the context shows that \\(\\exists x'_1, x_1 \\redl{e_1} x'_1\\).\n\tHence a contradiction.\n\n\tTo prove \\eqref{case-no-extension} requires more work.\n\tFirst, let\n\t\\(C'[\\cdot] \\coloneqq C[\\cdot] \\mid (\\overline{\\ell(e_1)} + c_{e_1})\\).\n\tBy induction hypothesis, there exists \\(z'_1 \\in \\enc{C'[P_1]}\\) such that \\(z'_1\\) is closed, \\(\\pi_{C', C[P_1]} (z'_1) = y'_1\\) and \\(z'_1 \\not\\downarrow_{c_{e_i}}\\) and \\(z'_1 \\not\\downarrow_{c_{e_1}}\\) for all \\(e_i \\in x_1\\).\n\n\tBy hypothesis, \\(\\enc{P_1}\\mathrel{\\sim^{\\tau}} \\enc{P_2}\\), hence there exists \\(\\mathrel{\\mathcal{R}}'\\) a back-and-forth barbed bisimulation between \\(\\enc{C'[P_1]}\\) and \\(\\enc{C'[P_2]}\\).\n\tIt implies that \\(\\exists z_2'\\) such that \\(z_2 \\in \\enc{C'[P_2]}\\) and \\(z'_2 \\not\\downarrow_{c_{e_i}}\\) and \\(z'_2 \\not\\downarrow_{c_{e_1}}\\) for all \\(e_i \\in x_1\\).\n\n\tUsing a similar argument to above we have that \\(z_2'\\) is closed and that there exists a bijection \\(h\\) between \\(z_1'\\) and \\(z_2'\\).\n\n\tLet us denote the projection \\(\\pi_{C', P_2} (z'_2)\\) as \\(x_2'\\).\n\tWe infer using the fact that \\(z_2'\\) is closed and that \\(z'_2 \\not\\downarrow_{c_{e_1}}\\) that \\(\\exists e_2'\\in x_2'\\) such that \\(\\ell(e_2')=\\ell(e_1)\\).\n\n\tAs there exists a label and order preserving bijection \\(h'\\) between \\(z_1'\\) and \\(z_2'\\), and as we forbid auto concurrency and ambiguous non-deterministic sum (\\autoref{rem-concur}), we conclude that\n\t\\(x_2'\\setminus\\{e_2'\\}=x_2\\), for \\(\\pi_{C', P_2} (z'_2)=x_2'\\).\n\n\tThen we have \\(\\pi_{C', P_1}(z'_1) = x'_1, \\pi_{C', P_2}(z'_2) = x'_2\\) and \\(f\\cup\\{e_1'\\leftrightarrow e_2'\\}\\) a bijection between the two.\n\tAs we supposed that \\(x_1\\) is the largest configuration for which the HHPB breaks we get that \\(\\exists x_2''\\) such that \\((x_1',x_2'',f'')\\in\\Forw{n+1}\\).\n\tBut such an \\(x_2''\\) is unique since \\(P_2\\) is singly labelled.\n\tThus we conclude that \\( (x'_1, x'_2, f\\cup\\{e_1'\\leftrightarrow e_2'\\}) \\in \\Forw{n+1}\\).\n\n\tThe proof that \\((x_1, x_2, f) \\in \\Backw{n}\\) goes along the line of (and uses) the proof that \\((x_1, x_2, f) \\in \\Forw{n}\\).\n\\end{proof}\n\n\\begin{remark}[On \\autoref{main-thm}]\n\tNote that \\autoref{main-thm} is a result on RCCS processes that have \\emph{an empty memory}.\n\tIt is a consequence of HHPB and the back-and-forth barbed congruence on configuration structure (\\autoref{bisim-cs}) being defined on configurations structures, and not on the tuples of configurations structures and configurations.\n\tHowever, we need the reversible setting to simulate the back-and-forth behaviour that we acquire when moving to configurations structures.\n\tThe result above then should be read as: \\emph{reversible process with \\emph{an empty memory} are barbed congurent if and only if their encodings in configurations structures are in a HHPB relation}.\n\n\tTo make the result more general and include any reversible process we need to reformulate it as follows.\n\n\t\\begin{conjecture}\n\t\tIf \\(R\\mathrel{\\sim^{\\tau}} S\\) such that \\(\\encc{R}=(\\mathcal{C}_R, x_R)\\) and \\(\\encc{S}=(\\mathcal{C}_S, x_S)\\) then there exists \\(\\mathrel{\\mathcal{R}}\\) a HHPB between \\(\\mathcal{C}_R\\) and \\(\\mathcal{C}_S\\) with \\((x_R,x_S,f)\\in {\\mathrel{\\mathcal{R}}}\\), for some \\(f\\).\n\t\\end{conjecture}\n\tWe leave this as future work.\n\\end{remark}\n\n\\section*{Conclusions and future work}\n\\addcontentsline{toc}{section}{Conclusions and future work}\nWe showed that, for a restricted class of RCCS processes (without recursion, auto-concurrency nor auto-conflict (\\autoref{rem-concur})) hereditary history preserving bisimilarity has a contextual characterisation in CCS.\nWe used the barbed congruence defined on RCCS as the congruence of reference, adapted it to configurations structures and then showed a correspondence with HHPB.\nAs a proof tool, we defined two inductively relations that approximate HHPB.\nConsequently we have that adding reversibility into the syntax helps in retrieving some of the discriminating power of configurations structures.\n\nNote that one could prove the main result of the paper by showing that the bisimulation defined on the LTS of RCCS and the barbed congruence (\\autoref{def:sbfc_rccs}) equate the same terms.\nWe chose to use configurations structures instead, as we plan to investigate other equivalences on reversible process algebra and their interpretations in configurations structures give interesting insights.\n\n\\paragraph{Weak equivalences}\nThis work follows notable efforts~\\cite{Phillips2007,Lanese2010} to understand equivalences for reversible processes.\nThere are numerous interesting continuations.\nA first one is to move to weak equivalences, which ignores silent moves \\(\\tau\\) and focus on the observable part of a process.\nThis is arguably a more interesting relation than the strong one, in which processes have to mimick \\emph{exactly} each other's silent moves.\nEven if such a relation on configurations structures exists~\\cite{Vogler1993,Fiore1999} one still has to show that this is indeed the relation we expect.\n\nIn configuration structures, the adjective \\emph{weak} has sometimes~\\cite{Phillips2012,Glabeek1989} a different meaning: it stands for the ability to change the label and order preserving bijection as the relation grows, to modify choices that were made before this step.\nIt would be interesting to understand what \\enquote{weak} relations in this sense represent for reversible processes.\n\n\\paragraph{Insensitiveness to the direction of the transitions and irreversibility}\nThe relations defined so far simulate forward (resp.\\@\\xspace backward) transitions only with forward (resp.\\@\\xspace backward) transitions, and only consider \\emph{forward} barb.\nIgnoring the direction of the transitions could introduce some fruitful liberality in the way processes simulate each other.\nDepending on the answer, \\(a + \\tau . b\\) and \\(a + b\\) would be weakly bisimilar or not.\nA weak bisimulation that ignores the direction of transitions~\\cite{Lanese2010} already exists, but it equates a reversible process with all its derivatives.\nIrreversible moves could play an important role in such equivalences and would help to understand what are the meaningful equivalences in the setting of transactions~\\cite{Danos2005}.\n\nReversibility is commonly used in transactional systems, i.e.\\@\\xspace participative computations where a commitment phase is reached whenever a consensus occurs.\nThis has two effects: it forbids the further exploration of the solution space, and prevents all the participants to complete if a participant cancels the transaction~\\cite{Danos2007}.\nCommitment is modelled as an \\emph{irreversible} action: such a feature is present in RCCS~\\cite{Danos2004}, but absent from our work.\nIt could probably be implemented by adding a mechanism to \\enquote{update} the origin of a term, and by \\enquote{cutting} the configuration structure after an irreversible transition (in the spirit of the LTS of \\autoref{def:transition_stable}).\nHowever, it remains to prove that those two actions would be equivalent.\n\n\\paragraph{Removing the limitations}\nContext---which plays a key role---raise questions on the memory handling of RCCS : what about a context that could fix the memory of an incoherent process?\n\nMaybe of less interest but important for the generality of these results, one should include infinite processes as well.\nThis needs a rework of the relations in \\autoref{ForwBackwDef} used to approximate the HHPB.\nIn configurations structures however one usually handles the recursive case by unfolding the process up to a finite level.\n\nOne way to retrieve the class of processes with auto-conflict and auto-concurrence could be to define bisimulations that take into account tagged labels.\nAt the price of a verbose syntax, one could imagine being able to discriminate between configurations reached after firing events with the same labels, thus allowing to define configurations structures for arbitrary RCCS terms.\nAre relations taking into account those \\enquote{localities}~\\cite{Boudol1988a}, which uniquely determine occurrences of a label, more discriminating the traditional bisimulations?\n\nLastly, we conjecture that HHPB is equivalent to a congruence relation on terms that do not exhibit auto-conflict.\nMore precisely, we could imagine that congruent processes have isomorphic event structures, and that configurations structures are isomorphic if and only if they are in HHPB relation.\n\n\\section*{Acknowledgement}\nWe would like to warmly thank D.~Varacca and J.~Krivine for the useful discussions as well as the referees of an earlier version~\\cite{Aubert2015d} for their helpful remarks.\n\n\\section*{References}\n\\bibliographystyle{elsarticle-num.bst}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nConsider the Cauchy problem of the following two-dimensional\ngeneralized magnetohydrodynamic equations:\n\\begin{eqnarray}\n\\left\\{\n \\begin{array}{llll}\\label{eq}\n u_t + u \\cdot \\nabla u = - \\nabla p + b \\cdot \\nabla b - \\nu \\Lambda^{\n 2\\alpha} u, \\\\\n b_t + u \\cdot \\nabla b = b \\cdot \\nabla u - \\kappa \\Lambda^{2\\beta} b,\\\\\n \\nabla \\cdot u = \\nabla \\cdot b = 0, \\\\\n u\\left(x,0\\right)=u_0\\left(x\\right),\\,\\,\\, b\\left(x,0\\right)=b_0\\left(x\\right)\n \\end{array}\\right.\n\\end{eqnarray}\nfor $x\\in \\mathbb{R}^2$ and $t>0$, where $ u=u\\left(x,t\\right) $ is\nthe velocity, $ b =b\\left(x,t\\right) $ is the magnetic,\n$ p =p\\left(x,t\\right) $ is the pressure, and $\nu_0\\left(x\\right),\\,b_0\\left(x\\right) $ with $\\mathrm{div}\nu_0\\left(x\\right)=\\mathrm{div} b_0\\left(x\\right)=0$ are the initial\nvelocity and magnetic, respectively. Here $\\nu, \\kappa,\n\\alpha, \\beta \\ge 0$ are nonnegative constants and $\\Lambda$\nis defined by\n\\begin{equation*}\n {\\widehat{\\Lambda f}}(\\xi) = \\left| \\xi \\right| \\widehat{f}(\\xi),\n\\end{equation*}\nwhere $\\wedge$ denotes the Fourier transform. In the following\nsections, we will use the inverse Fourier transform $\\vee$.\n\nThe global regularity of the 2D GMHD equations (\\ref{eq}) has\nattracted a lot of attention and there have been extensive studies\n(see\\cite{FNZ2013}-\\cite{ZF2011}).\n It follows from \\cite{W2011} that the\nproblem (\\ref{eq}) has a unique global regular solution if\n\\begin{displaymath}\n \\alpha \\geqslant 1 , \\ \\ \\beta > 0,\n \\ \\ \\alpha + \\beta \\geqslant 2.\n\\end{displaymath}\nTran, Yu and Zhai \\cite{TYZ2013} got a global regular solution under\nassumptions that\n\\begin{equation*}\n \\alpha \\geqslant 1 \/ 2,\\,\\, \\beta \\geqslant 1 \\hspace{2em}\\mbox{or}\\hspace{2em}\n 0 \\leqslant \\alpha < 1 \/ 2,\\,\\, 2 \\alpha + \\beta > 2 \\hspace{2em}\\mbox{or}\\hspace{2em}\n \\alpha \\geqslant 2, \\,\\,\\beta = 0.\n\\end{equation*}\nRecently, it was shown in {\\cite{JZ2013}} that if $ 0\\leqslant\\alpha\n< 1 \/ 2,\\,\\, \\beta \\geqslant 1, \\,\\,3\\alpha + 2\\beta >3 $, then the\nsolution is global regular. In particular, when $\\alpha=0,\n\\beta>\\frac 32$, the solution is global regular. This was proved\nindependently in \\cite{Y2013-2}\\cite{YB2013}. Meanwhile, Fan,\nNakamura and Zhou {\\cite{FNZ2013}} used properties of the heat\nequation and presented a global regular solution when\n$0<\\alpha<\\frac{1}{2}, \\beta=1$.\n\nIn this paper, we aim at getting the global regular solution of\n\\eqref{eq} when $\\nu=0, \\kappa>0$ and $\\beta>1$. For simplicity, we\nlet $\\kappa=1$. That is, we consider\n\\begin{eqnarray}\n\\left\\{\n \\begin{array}{llll}\\label{eq2}\n u_t + u \\cdot \\nabla u = - \\nabla p + b \\cdot \\nabla b, \\\\\n b_t + u \\cdot \\nabla b = b \\cdot \\nabla u - \\kappa \\Lambda^{2\\beta} b,\\\\\n \\nabla \\cdot u = \\nabla \\cdot b = 0, \\\\\n u\\left(x,0\\right)=u_0\\left(x\\right),\\,\\,\\,\n b\\left(x,0\\right)=b_0\\left(x\\right).\n \\end{array}\\right.\n\\end{eqnarray}\n\n\n\n Let $\\omega = -\n\\partial_2 u_1 +\\partial_1 u_2$ and\n $j = - \\partial_2 b_1 + \\partial_1 b_2$ represent the vorticity and the current respectively.\n We will prove that $\\omega, j\\in L^2(0,T;L^\\infty)$\n and obtain the global regularity of the solution by the BKM type criterion in \\cite{CKS1997}.\nTo this end, we will take advantage of the approaches used in\n{\\cite{FNZ2013}} and \\cite{JZ2013} to deal with the higher\nregularity estimates of $j$. More precisely, using the equation\nsatisfied by the current $j$, we will obtain the estimates of\n$\\|\\Lambda^r j\\|_{L^2}^2 \\left( t \\right) + \\int_0^t \\|\n\\Lambda^{\\beta+r} j\\|_{L^2}^2\\le C$ with $r=\\beta-1$. Using the\nsingular integral representation of $\\Lambda^\\delta j$ with some\n$\\delta>0$, we will obtain the estimate $\\|\\nabla\nj\\|_{L^2(0,T;L^\\infty(\\mathbb{R}^2))}$. Then we get the estimates of\n$\\|\\omega\\|_{L^2(0,T;L^\\infty(\\mathbb{R}^2))}$ using the particle\ntrajectory method. It should be noted that after the paper is\nfinished, at the almost same time, Cao, Wu and Yuan obtain the\nsimilar result independently using a different method\n(see\\cite{CWY-2013}). In comparison with result obtained in\n\\cite{CWY-2013}, it is not required that $\\|\\nabla\nj_0\\|_{L^\\infty}<\\infty$ in our result.\n\nThe main result of this paper is stated as follows.\n\\begin{theorem}\n Let $\\beta>1$ and assume that $\\left( u_0, b_0 \\right)\\in H^\\rho$ with $\\rho >\\max\\{2,\\beta\\}$. Then for any $T>0$, the Cauchy problem \\eqref{eq2} has a unique regular solution\n\\begin{equation*}\n(u,b)\\in L^\\infty([0,T];H^\\rho(\\mathbb R^2)) \\,\\,\\mbox{and}\\,\\, b\\in L^2([0,T];H^{\\rho+\\beta}(\\mathbb R^2)).\n\\end{equation*}\n\\end{theorem}\n\\begin{remark}\nWhen $\\alpha=0, \\beta>\\frac{3}{2}, \\rho>2$, the result has been\nobtained in \\cite{JZ2013}, \\cite{Y2013-2}and \\cite{YB2013}.\n\\end{remark}\n\n\\section{Preliminaries}\\label{sec:the preparations}\nLet us first consider the following equation\n\\begin{eqnarray*}\n \\left\\{\n \\begin{array}{llll}\n v_t +\\Lambda^{2\\beta}v = f\n \\\\\n v\\left(x,0\\right)=v_0(x).\n \\end{array}\\right.\n\\end{eqnarray*}\nSimilar to the heat equation, we can get\n\\begin{equation}\\label{heq}\nv\\left(x,t\\right)=\\int_{\\mathbb{R}^2} t^{-\\frac{1}{\\beta}}h\\left(\\frac{x-y}{t^\\frac{1}{2\\beta}}\\right)v_0(y) \\mathrm dy+\n\\int_0^t \\int_{\\mathbb{R}^2} (t-s)^{-\\frac{1}{\\beta}}h\\left(\\frac{x-y}{(t-s)^\\frac{1}{2\\beta}}\\right)f\\left(y,s\\right) \\mathrm dy\\mathrm d s,\n\\end{equation}\nwhere $h(x)=\\left(e^{-|\\cdot|^{2\\beta}}\\right)^\\vee(x)$ and it has the similar properties as the heat kernel.\n\\begin{lemma}Let $l$ be a nonnegative integer and $\\eta\\geqslant0$, then\n\\begin{eqnarray}\n\\left\\|\\nabla^{l} h\\right\\|_{L^1}+\\left\\|\\Lambda^{\\eta}h\\right\\|_{L^1}\\leqslant C.\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nFirst, we give the proof of the estimates of $\\nabla^{l}h$.\n\\begin{eqnarray*}\n\\left\\|\\nabla^{l} h\\right\\|_{L^1}&=&C\\sup_{|\\gamma|=l}\\int_{\\mathbb R^2}\\left|\\int_{\\mathbb R^2}\\xi^\\gamma e^{-\\left|\\xi\\right|^{2\\beta}}e^{ix\\cdot \\xi}\\mathrm d\\xi\\right|\\mathrm dx\\\\\n&=&C\\sup_{|\\gamma|=l} \\int_{\\left|x\\right|\\leqslant 1}\\left|\\int_{\\mathbb R^2}\\xi^\\gamma e^{-\\left|\\xi\\right|^{2\\beta}}e^{ix\\cdot \\xi}\\mathrm d\\xi\\right|\\mathrm dx+\nC\\int_{\\left|x\\right|\\geqslant 1}\\left|\\int_{\\mathbb R^2}\\xi^\\gamma e^{-\\left|\\xi\\right|^{2\\beta}}e^{ix\\cdot \\xi}\\mathrm d\\xi\\right|\\mathrm dx\\\\\n&\\leqslant & C+ C\\sup_{|\\gamma|=l}\\int_{\\left|x\\right|\\geqslant 1}(1+\\left|x\\right|^2)^{-2}\\left|\\int_{\\mathbb R^2}\\xi^\\gamma e^{-\\left|\\xi\\right|^{2\\beta}}(1-\\Delta_\\xi)^2e^{ix\\cdot \\xi}\\mathrm d\\xi\\right|\\mathrm dx\\\\\n&\\leqslant & C+ C\\sup_{|\\gamma|=l}\\int_{\\left|x\\right|\\geqslant 1}(1+\\left|x\\right|^2)^{-2}\\left|\\int_{\\mathbb R^2}(1-\\Delta_\\xi)^2(\\xi^\\gamma e^{-\\left|\\xi\\right|^{2\\beta}})e^{ix\\cdot \\xi}\\mathrm d\\xi\\right|\\mathrm dx\\\\\n&\\leqslant & C.\n\\end{eqnarray*}\nNext, we start to estimate $\\Lambda^\\eta h$ and let $l>\\eta$.\n\\begin{eqnarray*}\n\\left\\|\\Lambda^{\\eta} h\\right\\|_{L^1}&=&\\left\\|\\sum_{k\\geqslant-1}\\Delta_{k}\\Lambda^{\\eta} h\\right\\|_{L^1}\\\\\n&\\leqslant&\\left\\|\\Delta_{-1}\\Lambda^{\\eta} h\\right\\|_{L^1}+\\sum_{k\\geqslant 0}\\left\\|\\Delta_{k}\\Lambda^{\\eta} h\\right\\|_{L^1}\\\\\n&\\leqslant&C\\left\\| h\\right\\|_{L^1}+ C\\sum_{k\\geqslant 0}2^{k(-l+\\eta)}\\left\\|\\Delta_{k}\\nabla^{l} h\\right\\|_{L^1}\\\\\n&\\leqslant&C+C\\sum_{k\\geqslant 0}2^{k(-l+\\eta)}\\left\\|\\nabla^{l} h\\right\\|_{L^1}\\\\\n&\\leqslant&C,\n\\end{eqnarray*}\nwhere we use the nonhomogeneous Littlewood-Paley decompositions $Id=\\sum_k \\Delta_k$ and Bernstein-Type inequalities (see\\cite{BCD2011}).\n\n\\end{proof}\n\nNow, let $\\omega = \\nabla^{\\bot} \\cdot u = - \\partial_2 u_1 + \\partial_1 u_2$\n and $j = \\nabla^{\\bot} \\cdot b = - \\partial_2 b_1 + \\partial_1 b_2$,\nand applying $\\nabla^{\\bot} \\cdot$ to the equations (\\ref{eq}), we obtain the following equations for\n $\\omega$ and $j$:\n \\begin{eqnarray}\n \\omega_t + u \\cdot \\nabla \\omega & = & b \\cdot \\nabla j, \\label{eq:omega-L2}\\\\\n j_t + u \\cdot \\nabla j & = & b \\cdot \\nabla \\omega + T \\left( \\nabla u,\n \\nabla b \\right) - \\Lambda^{2 \\beta} j, \\label{eq:j-L2}\n \\end{eqnarray}\n where\n \\begin{equation*}\n T \\left( \\nabla u, \\nabla b \\right) = {\\color{black} 2 \\partial_1 b_1\n \\left( \\partial_1 u_2 + \\partial_2 u_1 \\right) + 2 \\partial_2 u_2 \\left(\n \\partial_1 b_2 + \\partial_2 b_1 \\right)} .\n \\end{equation*}\nThe estimates for $\\omega, j$ are obtained in \\cite{TYZ2013} and \\cite{JZ2013}, which is presented\nin the following lemma.\n\\begin{lemma}\n Assume that $\\alpha = 0,\\,\\,\\beta > 1,\\, r=\\beta-1,\\,\\,\\mbox{and}\\,\\, k \\geqslant \\beta$.\n Let $u_0, b_0 \\in H^k$. For any $T > 0$, we have\n \\begin{eqnarray}\n \\left\\| \\omega \\right\\|_{L^2}^2 \\left( t \\right) + \\left\\| j\n \\right\\|_{L^2}^2 \\left( t \\right) + \\int_0^t \\left\\| \\Lambda^{\\beta} j\n \\right\\|_{L^2}^2 \\mathrm d\\tau \\leqslant C \\left( T\n \\right), \\\\\n \\left\\|\\Lambda^r j\\right\\|_{L^2}^2 \\left( t \\right) + \\int_0^t \\left\\| \\Lambda^{\\beta+r} j\n \\right\\|_{L^2}^2 \\mathrm d\\tau \\leqslant C \\left(T\n \\right).\n \\end{eqnarray}\n\\end{lemma}\n\n\\section{The Proof of Theorem 1.1}\\label{sec:the proof of theorem 1.1}\nIn this section, we will prove our main result Theorem 1.1. The\nproof is divided into three steps.\n\nStep 1: $\\omega\\in L^\\infty(0,T;L^p(\\mathbb R^2)), j\\in L^p(0,T;\\mathbb R^2)$ for any $22)$, and\nintegrating with respect to $x$, we get\n\\begin{eqnarray*}\n \\frac{1}{p}\\frac{\\mathrm d}{\\mathrm dt} \\left\\| \\omega \\right\\|_{L^p}^p\n & \\leqslant & \\int_{\\mathbb{R}^2} \\left| b \\right| \\left| \\nabla j \\right|\n \\left| \\omega \\right|^{p - 1} \\mathrm dx,\\\\\n & \\leqslant & \\left\\| b \\right\\|_{L^{\\infty}}\\left\\| \\nabla j \\right\\|_{L^p}\\left\\| \\omega \\right\\|_{L^p}^{p-1}\n\\end{eqnarray*}\nThus, we have\n\\begin{eqnarray*}\n \\frac{1}{2}\\frac{\\mathrm d}{\\mathrm dt} \\left\\| \\omega \\right\\|_{L^p}^2\n & \\leqslant & \\left\\| b \\right\\|_{L^{\\infty}}\\left\\| \\nabla j \\right\\|_{L^p}\\left\\| \\omega \\right\\|_{L^p}\n\\end{eqnarray*}and\n\\begin{eqnarray*}\n\\left\\| \\omega \\right\\|_{L^p}^2 &\\leqslant& C\\left\\| \\omega(x,0) \\right\\|_{L^p}^2+ C\\int_0^t (\\left\\| \\nabla j \\right\\|_{L^p}^2+\\left\\| \\omega \\right\\|_{L^p}^2)\\mathrm ds\\\\\n& \\leqslant & C + C\\int_0^t (\\left\\| \\nabla^2 b \\right\\|_{L^p}^2+\\left\\| \\omega \\right\\|_{L^p}^2)\\mathrm ds\\\\\n& \\stackrel{(\\ref{in1})}{\\leqslant} & C + C\\int_0^t (\\left\\| b\\cdot\\nabla u-u\\cdot\\nabla b \\right\\|_{L^p}^2+\\left\\| \\omega \\right\\|_{L^p}^2)\\mathrm ds\\\\\n& \\leqslant & C + C\\int_0^t (\\left\\| \\nabla b \\right\\|_{L^p}^2 \\left\\| u \\right\\|_{L^\\infty}^2+\\left\\| \\omega \\right\\|_{L^p}^2)\\mathrm ds\\\\\n& \\leqslant & C + C\\int_0^t (1+ \\left\\| \\nabla b \\right\\|_{L^p}^2 )\\left\\| \\omega \\right\\|_{L^p}^2\\mathrm ds.\n\\end{eqnarray*}\nThis, combining with the Gronwall's inequality, leads to $\\omega\\in\nL^\\infty(0,T;L^p(\\mathbb R^2))$ for any $2