From c7ea851920caffa33d0b517feab7b61fa327d7a3 Mon Sep 17 00:00:00 2001 From: partowp Date: Wed, 17 Jun 2026 17:05:46 +0100 Subject: [PATCH] 4 defs! --- draft/draft.tex | 157 ++++++++++++++++++++++++++++-------------------- 1 file changed, 92 insertions(+), 65 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index 5437601..4e67328 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -387,45 +387,51 @@ Pouya Partow\inst{1}\orcidID{0009-0003-9652-9469}} % \section{Coalgebraic Bisimulation}%\label{sec:} - -\begin{definition}[Relation Lifting] - Assuming $F\c\Set\to\Set$ is a functor, then we call $\rel(F)\c\rel\to\rel$ a relation lifting of $F$, where the following diagram commutes: - \begin{equation*} - \begin{tikzcd}[ampersand replacement=\&] - \rel \&\& \rel \\ - {\Set\times\Set} \&\& {\Set\times\Set} - \arrow["{\rel(F)}", from=1-1, to=1-3] - \arrow[from=1-1, to=2-1] - \arrow[from=1-3, to=2-3] - \arrow["{F\times F}"', from=2-1, to=2-3] - \end{tikzcd} - \end{equation*} +In this section, by $\spa(\BC)$ we refer to spans in a category $\BC$ that at least has products, and by $\rel(\BC)$ we refer to the relatios in $\BC$, and by relation we are referring to a span $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ such that the morphism $\brks{p_1,p_2}$ is a mono. For the time being, we limit the discussion to the case $\BC=\Set$. For simplicity, by $\rel$ and $\spa$ we mean $\rel(\Set)$ and $\spa(\Set)$, accordingly. +%\begin{definition}[Relation Lifting] +% Assuming $F\c\BC\to\BC$ is a functor, then we call $\rel(F)\c\rel(\BC)\to\rel(\BC)$ a relation lifting of $F$, where the following diagram commutes: +% \begin{equation*} +% \begin{tikzcd}[ampersand replacement=\&] +% \rel(\BC) \&\& \rel(\BC) \\ +% {\BC\times\BC} \&\& {\BC\times\BC} +% \arrow["{\rel(F)}", from=1-1, to=1-3] +% \arrow[from=1-1, to=2-1] +% \arrow[from=1-3, to=2-3] +% \arrow["{F\times F}"', from=2-1, to=2-3] +% \end{tikzcd} +% \end{equation*} +%\end{definition} +%We have notions of bisimulation that may involve relation lifting. An example of relation lifting is to use the image factorization provided in regular categories. +\begin{definition}[$F$-Relator] + For a set functor $F$, and for sets $X$ and $Y$, an $F$-relator $\relar$ is a map that takes every relation on $X\times Y$ to a relation on $FX\times FY$, and it is monotone with respect to inclusion. \end{definition} +%Also, for the time being, we limit the discussion to the case $\BC=\Set$. For simplicity, by $\rel$ we mean $\rel(\BC)$. % -We take $\rel(F)\c\rel\to\rel$ to be the functor that for an arbitrary functor $F$ takes a relation $R$, where $R\in\obj(\rel)$ and $R\subseteq X_1\times X_2$, and gives the relation that is the image of the function $\brks{Fp_1,Fp_2}\c FR\to FX\times FY$. -\begin{definition}[Bisimulation] - For a functor $F\c\Set\to\Set$, a bisimulation is a $\rel(F)$-coalgebra in $\rel$. -\end{definition} +%We take $\rel(F)\c\rel(\BC)\to\rel(\BC)$ to be the functor that for an arbitrary functor $F$ takes a relation $R$, where $R\in\obj(\rel)$ and $R\subseteq X_1\times X_2$, and gives the relation that is the image of the function $\brks{Fp_1,Fp_2}\c FR\to FX\times FY$. +%\begin{definition}[Bisimulation] +% For a functor $F\c\BC\to\BC$, a bisimulation is a $\rel(F)$-coalgebra in $\rel$. +%\end{definition} % -\begin{prop} - Assuming that $(R,\alpha)$ is a $\rel(F)$-coalgebra, where $\alpha=\beta_1\times\beta_2$ in $\Set\times\Set$, then the following diagram commutes, and vice-versa: - \begin{equation*} - \begin{tikzcd}[ampersand replacement=\&] - {X_1} \& R \& {X_2} \\ - {FX_1} \& FR \& {FX_2} - \arrow["{\beta_1}"', from=1-1, to=2-1] - \arrow["{p_1}"', from=1-2, to=1-1] - \arrow["{p_2}", from=1-2, to=1-3] - \arrow["\beta", from=1-2, to=2-2] - \arrow["{\beta_2}", from=1-3, to=2-3] - \arrow["{Fp_1}", from=2-2, to=2-1] - \arrow["{Fp_2}"', from=2-2, to=2-3] - \end{tikzcd} - \end{equation*} -\end{prop} +%\begin{prop} +% Assuming that $(R,\alpha)$ is a $\rel(F)$-coalgebra, where $\alpha=\beta_1\times\beta_2$ in $\BC\times\BC$, then the following diagram commutes, and vice-versa: +% \begin{equation*} +% \begin{tikzcd}[ampersand replacement=\&] +% {X_1} \& R \& {X_2} \\ +% {FX_1} \& FR \& {FX_2} +% \arrow["{\beta_1}"', from=1-1, to=2-1] +% \arrow["{p_1}"', from=1-2, to=1-1] +% \arrow["{p_2}", from=1-2, to=1-3] +% \arrow["\beta", from=1-2, to=2-2] +% \arrow["{\beta_2}", from=1-3, to=2-3] +% \arrow["{Fp_1}", from=2-2, to=2-1] +% \arrow["{Fp_2}"', from=2-2, to=2-3] +% \end{tikzcd} +% \end{equation*} +%\end{prop} \begin{definition}[Aczel-Mendler Bisimulation] - \begin{equation*} + A relation $R\subseteq X\times Y$ is an \emph{Aczel-Mendler bisimulation} from an $F$-coalgebra $(X,\alpha)$ to an $F$-coalgebra $(Y,\beta)$ whenever there is a morphism $\gamma\c R\to FR$ called witness that commutes in the following diagram: + \begin{equation*}\label{eq:acz-mend-diag} \begin{tikzcd}[ampersand replacement=\&] {X} \& R \& {Y} \\ {FX} \& FR \& {FY} @@ -440,51 +446,72 @@ We take $\rel(F)\c\rel\to\rel$ to be the functor that for an arbitrary functor $ \end{equation*} \end{definition} -\begin{definition}[Hermida-Jacobs Bisimulation] +\begin{definition}[Witnessless $\relar$-Bisimulation] + For a relator $\relar$, a relation $R\subseteq X\times Y$ is a \emph{witnessless $\relar$-bisimulation} from an $F$-coalgebra $(X,\alpha)$ to an $F$-coalgebra $(Y,\beta)$ whenever for every $x\in X$ and $y\in Y$, we have $x\mathrel{R} y\Rightarrow \alpha(x)\mathrel{(\relar R)}\beta(y)$. +% \begin{equation*} +% \begin{tikzcd}[ampersand replacement=\&] +% {X} \& R \& {Y} \\ +% {FX} \& \rel(F)R \& {FY} +% \arrow["{\alpha}"', from=1-1, to=2-1] +% \arrow["{p_1}"', from=1-2, to=1-1] +% \arrow["{p_2}", from=1-2, to=1-3] +% \arrow["{\beta}", from=1-3, to=2-3] +% \arrow["{q_1}", from=2-2, to=2-1] +% \arrow["{q_2}"', from=2-2, to=2-3] +% \end{tikzcd} +% \end{equation*} +\end{definition} + +\begin{definition}[Witnessful $\relar$-Bisimulation] + For a relator $\relar$, a relation $R\subseteq X\times Y$ is a \emph{witnessful $\relar$-bisimulation} from an $F$-coalgebra $(X,\alpha)$ to an $F$-coalgebra $(Y,\beta)$ whenever there is a morphism $\gamma\c R\to \relar R$ called witness that commutes in the following diagram: \begin{equation*} \begin{tikzcd}[ampersand replacement=\&] {X} \& R \& {Y} \\ - {FX} \& (FR)^\dagger \& {FY} + {FX} \& \relar R \& {FY} \arrow["{\alpha}"', from=1-1, to=2-1] \arrow["{p_1}"', from=1-2, to=1-1] \arrow["{p_2}", from=1-2, to=1-3] \arrow["\gamma", from=1-2, to=2-2] \arrow["{\beta}", from=1-3, to=2-3] - \arrow["{(Fp_1)^\dagger}", from=2-2, to=2-1] - \arrow["{(Fp_2)^\dagger}"', from=2-2, to=2-3] - \end{tikzcd} - \end{equation*} -\end{definition} - -\begin{definition}[Hughes-Jacobs Bisimulation] - \begin{equation*} - \begin{tikzcd}[ampersand replacement=\&] - {X} \& R \& {Y} \\ - {FX} \& (FR)^\dagger \& {FY} - \arrow["{\alpha}"', from=1-1, to=2-1] - \arrow["{p_1}"', from=1-2, to=1-1] - \arrow["{p_2}", from=1-2, to=1-3] - \arrow["{\beta}", from=1-3, to=2-3] - \arrow["{(Fp_1)^\dagger}", from=2-2, to=2-1] - \arrow["{(Fp_2)^\dagger}"', from=2-2, to=2-3] + \arrow["{q_1}", from=2-2, to=2-1] + \arrow["{q_2}"', from=2-2, to=2-3] \end{tikzcd} \end{equation*} \end{definition} \begin{definition}[Vanilla Bisimulation] - \begin{equation*} - \begin{tikzcd}[ampersand replacement=\&] - {X} \& R \& {Y} \\ - {FX} \& FR \& {FY} - \arrow["{\alpha}"', from=1-1, to=2-1] - \arrow["{p_1}"', from=1-2, to=1-1] - \arrow["{p_2}", from=1-2, to=1-3] - \arrow["{\beta}", from=1-3, to=2-3] - \arrow["{Fp_1}", from=2-2, to=2-1] - \arrow["{Fp_2}"', from=2-2, to=2-3] - \end{tikzcd} - \end{equation*} + A relation $R\subseteq X\times Y$ is a \emph{vanilla bisimulation} from an $F$-coalgebra $(X,\alpha)$ to an $F$-coalgebra $(Y,\beta)$ whenever for every $x\in X$ and $y\in Y$, we have $x\mathrel{R} y\Rightarrow \alpha(x)\mathrel{(FR)}\beta(y)$. +% \begin{equation*} +% \begin{tikzcd}[ampersand replacement=\&] +% {X} \& R \& {Y} \\ +% {FX} \& FR \& {FY} +% \arrow["{\alpha}"', from=1-1, to=2-1] +% \arrow["{p_1}"', from=1-2, to=1-1] +% \arrow["{p_2}", from=1-2, to=1-3] +% \arrow["{\beta}", from=1-3, to=2-3] +% \arrow["{Fp_1}", from=2-2, to=2-1] +% \arrow["{Fp_2}"', from=2-2, to=2-3] +% \end{tikzcd} +% \end{equation*} \end{definition} +\begin{prop} + The following propositions hold: + \begin{enumerate}[label=(\Roman*), ref=(\Roman*)] + \item Every Aczel-Mendler bisimulation is a vanilla bisimulation. + \item Every vanilla bisimulation is an Aczel-Mendler bisimulation. + \item Every Aczel-Mendler bisimulation is a witnessful bisimulation. + \item Assuming the axiom of choice, every witnessful bisimulation is an Aczel-Mendler bisimulation. + \item Every witnessless bisimulation is a witnessful bisimulation. + \item Assuming the axiom of choice, every witnessless bisimulation is a witnessful bisimulation. + \end{enumerate} +\end{prop} +\begin{proof} + (I): Assuming $x\mathrel{R}y$, then given by~\eqref{eq:acz-mend-diag} we have $\gamma(x,y)\in FR$, $Fp_1\comp\gamma(x,y)=\alpha(x)$, and $Fp_2\comp\gamma(x,y)=\beta(y)$ that means $\alpha(x)\mathrel{(FR)}\beta(y)$. + + (II): Since $R$ is a vanilla bisimulation, for every $(x,y)\in R$ we have $\alpha(x)\mathrel{(FR)}\beta(y)$, so we can define $\gamma\c R\to FR$ as $\gamma(x,y)=(\alpha(x),\beta(y))$, and then $\gamma$ commutes in~\eqref{eq:acz-mend-diag}. + + (III): +\end{proof} \todo{Discuss 4 versions of bisimulation (with witness/without witness, for relations/for spans). Which are equivalent? Which do not make sense?} \todo{In next section run a similar analysis for simulation: relator-based vs. Aczel-Mendler.}