2pp

ACV-abstract-2026/sym-sim.tex

@@ -14,6 +14,7 @@
 \usepackage{microtype}
 \usepackage{stackengine}
 \usepackage{tikz-cd}
+\usepackage{subcaption}
 \usetikzlibrary{arrows.meta}
 \usetikzlibrary{decorations}           % Required for all decorations
 \usetikzlibrary{decorations.pathmorphing} % Specifically for 'zigzag'
@@ -206,8 +207,7 @@ These settings can be varied in at least two ways:
   nominal sets, to model systems with name management --- but we stick to $\BC=\Set$ in the sequel). 
 \end{itemize}
 %
-A convenient way to work with simulations and bisimulations is via the notion of 
+A convenient framing for simulations and bisimulations is the notion of \emph{relator}. 
-\emph{relator}. 
 
 \begin{definition}[Relators, Simulation, Bisimulation]
 Given a functor $F\c\Set\to\Set$, an \emph{$F$-relator} is a monotone map $\relar$
@@ -222,8 +222,9 @@ $R \;\subseteq\;  d^\op\comp\relar R\comp c$.
 %
 %The greatest $\relar$-simulation from $(X, c)$ to $(Y, d)$ (which exists by
 %Knaster-Tarski) is \emph{$\relar$-similarity}. 
-When $\relar$ is symmetric, $\relar$-simulations are called \emph{$\relar$-bisimulations} and $\relar$-similarity
+When $\relar$ is symmetric, $\relar$-simulations are called \emph{$\relar$-bisimulations}.
-is called \emph{$\relar$-bisimilarity}.
+% and $\relar$-similarity
+%is called \emph{$\relar$-bisimilarity}.
 \end{definition}
 %
 The canonical relator (usually) capturing bisimulation (like in \autoref{exa:pset})
@@ -262,8 +263,8 @@ abstract Howe's closure, for proving congruence of applicative bisimilarity
 in higher-oder mathematical semantics~\cite{UrbatTsampasEtAl23}. 
 %
 \begin{example}
-For $F=\PSet$, let $\appr$ be the subset inclusion relation. Then, all relators 
+For $F=\PSet$, let $\appr$ be the subset inclusion relation. Then, all the relators 
-\textsf{\bfseries\ref{it:ll-barr}}--\textsf{\bfseries\ref{it:ml-barr}} coincide and yield $\relar R = \{(U,V)\in\PSet X\times\PSet Y\mid \forall x\in U.\,\exists y\in V.\, (x,y)\in R\}$.
+\textsf{\bfseries\ref{it:ll-barr}}--\textsf{\bfseries\ref{it:ml-barr}} take the form $\relar R = \{(U,V)\in\PSet X\times\PSet Y\mid \forall x\in U.\,\exists y\in V.\, (x,y)\in R\}$.
 \end{example}
 %
 For any relator $\relar$, we can consider its \emph{symmetrization} $\relar^\leftrightarrow$, 
@@ -272,7 +273,7 @@ the symmetrization of any lax relator is the Barr relator. It is not clear thoug
 when this is true in general. The following is easy to verify:
 %
 \begin{proposition}
-If $\relar^\leftrightarrow\subseteq\bar F$ then any symmetric $\relar$-simulation
+If $\relar^\leftrightarrow\subseteq\bar F$ then a symmetric $\relar$-simulation
 $R\subseteq X\times X$ is a $\bar F$\dash bisimulation.
 \end{proposition}
 %
@@ -296,50 +297,56 @@ the premise ($\relar^\leftrightarrow\subseteq\bar F$) need not be true in genera
 which is indeed a bisimulation, as $\bar{F}$ is a symmetric relator. It is thus appropriate
 to regard $\bar{F}$-simulation as a generalization of Hermida and Jacobs' notion 
 to arbitrary relators.
-
-In $\Set$, with the axiom of choice, Hermida-Jacobs bisimulation coincides with 
-the \emph{Aczel-Mendler bisimulation}~\cite{Staton11}, i.e.\ such a relation $R\subseteq X\times Y$ that 
-the diagram 
 %
-\begin{equation}\label{eq:diag-sim}
-	\begin{tikzcd}[ampersand replacement=\&]
-		X \& R \& Y \\
-		FX \& FR \& FY
-		\arrow[" c"', 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["\sigma", dashed,from=1-2, to=2-2]
-		\arrow["d", 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}
 %
-commutes for some $\sigma$, where $(X,c)$ and $(Y,d)$ are given coalgebras. The 
+\begin{figure}[t]
-fact that $R$ is an $\bar F$-simulation thus comes with a \emph{witness} $\sigma\c R\to FR$.
+\begin{subfigure}[t]{0.48\textwidth}
-
+\centering
-\emph{Aczel-Mendler simulation}~\cite{Dubut25} adapts Aczel-Mendler bisimulation
+\begin{tikzcd}[ampersand replacement=\&]
-by replacing the strictly commuting diagram~\eqref{eq:diag-sim} with a laxly 
+	X \& R \& Y \\
-commuting one (\textsf{\bfseries\ref{it:ll-barr}--\textsf{\bfseries\ref{it:rl-barr}}} give rise to further variants):
+	FX \& FR \& FY
+	\arrow[" c"', 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["\sigma", dashed, from=1-2, to=2-2]
+	\arrow["d", 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}
+\caption{Aczel-Mendler bisimulation}
+\label{fig:diag-sim}
+\end{subfigure}
+\hfill
+\begin{subfigure}[t]{0.48\textwidth}
+\centering
+\begin{tikzcd}[ampersand replacement=\&]
+	X \& R \& X \\
+	FX \& FR \& FX
+	\arrow[" c"', from=1-1, to=2-1]
+	\arrow["\sqsubseteq"{marking, allow upside down}, draw=none, from=1-1, to=2-2]
+	\arrow["{p_1}"', from=1-2, to=1-1]
+	\arrow["{p_2}", from=1-2, to=1-3]
+	\arrow["\sigma", dashed, from=1-2, to=2-2]
+	\arrow[" c", from=1-3, to=2-3]
+	\arrow["\sqsubseteq"{marking, allow upside down}, draw=none, from=2-2, to=1-3]
+	\arrow["{{Fp_1}}", from=2-2, to=2-1]
+	\arrow["{{Fp_2}}"', from=2-2, to=2-3]
+\end{tikzcd}
+\caption{Aczel-Mendler simulation}
+\label{fig:diag-lax-sim}
+\end{subfigure}
+\end{figure}
 %
-\begin{equation}\label{eq:diag-lax-sim}
+In $\Set$, with the axiom of choice, Hermida-Jacobs bisimulation coincides with
-	\begin{tikzcd}[ampersand replacement=\&]
+the \emph{Aczel-Mendler bisimulation}~\cite{Staton11}: $R\subseteq X\times Y$
-		X \& R \& X \\
+for which \autoref{fig:diag-sim} commutes for some witness $\sigma\c R\to FR$.
-		FX \& FR \& FX
+\emph{Aczel-Mendler simulation}~\cite{Dubut25} replaces \autoref{fig:diag-sim} with
-		\arrow[" c"', from=1-1, to=2-1]
+the laxly commuting \autoref{fig:diag-lax-sim}
-		\arrow["\sqsubseteq"{marking, allow upside down}, draw=none, from=1-1, to=2-2]
+(\textsf{\bfseries\ref{it:ll-barr}}--\textsf{\bfseries\ref{it:rl-barr}} give rise to further variants).
-		\arrow["{p_1}"', from=1-2, to=1-1]
-		\arrow["{p_2}", from=1-2, to=1-3]
-		\arrow["\sigma", from=1-2, to=2-2]
-		\arrow[" c", from=1-3, to=2-3]
-		\arrow["\sqsubseteq"{marking, allow upside down}, draw=none, from=2-2, to=1-3]
-		\arrow["{{Fp_1}}", from=2-2, to=2-1]
-		\arrow["{{Fp_2}}"', from=2-2, to=2-3]
-	\end{tikzcd}
-\end{equation}
 %
 %(\textsf{\bfseries\ref{it:ll-barr}}--\textsf{\bfseries\ref{it:rl-barr}} give rise to 
 %further variants, obtained in the obvious way).
+%
 
 Curiously, if we switch to this notion of simulation, it is no longer granted 
 that symmetric simulation is a bisimulation even for $F=\PSet$.
@@ -353,7 +360,7 @@ Take $R=\{(1,2),(2,1),(1,3),(3,1)\}$, and $X=\{1,2,3\}$, and $ c(x)=X$ for every
 		R\setminus\{(1,2)\} & w=(1,3)
 	\end{cases}
 \end{gather*}
-Then $\sigma$ witnesses $R$ to be a simulation but not a bisimulation: $c(p_2(1,3))=c(3)=X\neq\mathcal{P}p_2(\sigma(1,3))=X\setminus\{2\}$.
+Then $\sigma$ is a witnesses of simulation but not a bisimulation: $c(p_2(1,3))=c(3)=X\neq\mathcal{P}p_2(\sigma(1,3))=X\setminus\{2\}$.
 \end{example}
 
 %Inspired by Hermida-Jacobs bisimulation\cite{HermidaJacobs98} we modify the definition by setting $\BC$ to be a regular category, and changing the span $(FR,Fp_1,Fp_2)$ in~\eqref{eq:diag-lax-sim} with the span $((FR)^\dagger,(Fp_1)^\dagger,(Fp_2)^\dagger)$, where $(FR)^\dagger$ is the image of $\brks{Fp_1,Fp_2}$, and $\brks{(Fp_1)^\dagger,(Fp_2)^\dagger}\c(FR)^\dagger\to FX\times FX$ is monic, so $(FR)^\dagger$ is a relation. By the symmetry of $R$ there exists $s\c R\to R$ that we call \emph{swap}, where $p_1\comp s=p_2$ and $p_2\comp s=p_1$. The necessary and sufficient condition for $\sigma$ to be a witness for $R$ to be a bisimulation is that