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ACV 2026/slides.tex
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@@ -166,9 +166,9 @@
\begin{itemize}
\item Intuitive introduction to coalgebra and (bi)simulation
\item Reviewing basic definitions of coalgebra and (bi)simulation
\item Our motivation: To ease proving program equivalence
\item Relator-based notions
\item Span-based notions
\item Motivation! Howe's method for categorical operational semantics
\end{itemize}
\end{frame}
@@ -200,10 +200,9 @@
%\end{frame}
\begin{frame}{From ``System'' to Mathematics}
The word ``system'' needs to be made precise.
% \vspace{-0.6cm}
\begin{block}{}
We model systems as mathematical notions like \textbf{labeled transition systems}: a set of states, together with a transition relation between them.
We model systems as mathematical notions like \textbf{labeled transition systems}: a set of states, together with a labeled transition relation between them.
\end{block}
\begin{alertblock}{Example: a vending machine}
% \begin{columns}
@@ -266,7 +265,8 @@
node[below,font=\tiny] {select tea} (tea);
\draw[->] (active) to[bend left=10] node[above,font=\tiny,yshift=2mm] {select coffee} (tea);
\draw[->] (tea) -- node[above,font=\tiny] {take tea} (idle);
\draw[->] (tea) to[bend right=10] node[below,font=\tiny] {take tea} (idle);
\draw[->] (tea) to[bend left=10] node[above,font=\tiny] {take coffee} (idle);
\end{tikzpicture}
\end{alertblock}
@@ -317,28 +317,43 @@
A relation $R\subseteq X\times Y$ is a simulation from $(X,A,T)$ to $(Y,A,S)$ whenever
\[
x\mathrel{R}y,\;
x\xrightarrow{a}x'
\Rightarrow
\exists\,y'.\;
y\xrightarrow{a}y'
\;\land\;
x'Ry'.
\]
A simulation relation like $R$ is a bisimulation whenever
\[
x\mathrel{R}y,\;
y\xrightarrow{a}y'
\Rightarrow
\exists\,x'.\;
x\xrightarrow{a}x'
\;\land\;
x'Ry'.
\begin{tikzcd}[ampersand replacement=\&]
x \& y \\
{x'} \& {y'}
\arrow["R"{description}, no head, from=1-1, to=1-2]
\arrow["a"', from=1-1, to=2-1]
\arrow["a", dashed, from=1-2, to=2-2]
\arrow["R"{description}, dashed, no head, from=2-1, to=2-2]
\end{tikzcd}
\]
% A simulation relation like $R$ is a bisimulation whenever
% \[
% x\mathrel{R}y,\;
% y\xrightarrow{a}y'
% \Rightarrow
% \exists\,x'.\;
% x\xrightarrow{a}x'
% \;\land\;
% x'Ry'.
% \]
\end{block}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\column{.48\textwidth}
\begin{block}{Bisimulation}
A simulation relation like $R$ is a bisimulation whenever:\vspace{-0.5cm}
\[
\begin{tikzcd}[ampersand replacement=\&]
x \& y \\
{x'} \& {y'}
\arrow["R"{description}, no head, from=1-1, to=1-2]
\arrow["a", dashed, from=1-1, to=2-1]
\arrow["a"', from=1-2, to=2-2]
\arrow["R"{description}, dashed, no head, from=2-1, to=2-2]
\end{tikzcd}
\]
\end{block}
\vspace{-0.2cm}
\begin{alertblock}{Similarity and Bisimilaritiy}
The \emph{similarity relation} $\precsim\subseteq X\times Y$ is the greatest simulation relation. Equivalently
\vspace{-0.3cm}
@@ -349,17 +364,18 @@
xRy,
\]
where $R$ is a simulation.\\
Similarly, the \emph{bisimilarity relation} $\sim\subseteq X\times Y$ is the greatest bisimulation relation. Equivalently\vspace{-0.3cm}
\[
x\sim y
\iff
\exists\,R.\;
xRy,
\]
where $R$ is a bisimulation.
Similarity is defined similarly, where $R$ is a simulation, and it is shown with $\precsim$.
% Similarly, the \emph{bisimilarity relation} $\sim\subseteq X\times Y$ is the greatest bisimulation relation. Equivalently\vspace{-0.3cm}
% \[
% x\sim y
% \iff
% \exists\,R.\;
% xRy,
% \]
% where $R$ is a bisimulation.
\end{alertblock}
\vspace{-0.2cm}
Ultimately, people care about similarity and bisimilarity, but simulation and bisimulation should be defined first.
% \vspace{-0.2cm}
% Ultimately, people care about similarity and bisimilarity, but simulation and bisimulation should be defined first.
\end{columns}
\end{frame}
@@ -421,6 +437,27 @@
\end{frame}
\begin{frame}{Towards Program Equivalence}
\footnotesize
We have developed a coalgbera $(T,\gamma)$ that gives the big-step semantics for an arbitrary programming language in a specified family of them, called \emph{abstract HO-GSOS}.
% \begin{block}{}
% -$\mS$ is representing terms of a programming language,\\
% -and $\gamma$ is giving their big-step reduction.
% \end{block}
\begin{alertblock}{Main Goal}
We want to prove that for terms $t$ and $s$, and a context $C$:\vspace{-0.3cm}
\begin{gather*}
t\sim s\Rightarrow C[t]\sim C[s]
\end{gather*}
\end{alertblock}\vspace{-0.5cm}
-In the literature, this is often proved using \emph{Howe's method}. \\
-The method, applies a closure on the bisimilarity relation on $\mS$.\\
-Then one should prove that the result is a simulation.\\
-The closure preserves symmetry, and symmetric simulation is a bisimulation in traditional definitions.\\
-We have found out that it is not always the case.\\
-That is why we want to know when exactly a symmetric simulation is a bisimulation.
\end{frame}
\begin{frame}{Relators, Simulations and Bisimulations}
\footnotesize
@@ -621,27 +658,6 @@
\end{columns}
\end{frame}
\begin{frame}{Motivation!}
\footnotesize
We have a coalgbera $(\mS,\gamma)$ in an arbitrary category $\mathbb{C}$.
\begin{block}{}
-$\mS$ is representing terms of a programming language,\\
-and $\gamma$ is giving their big-step reduction.
\end{block}
\begin{alertblock}{Main Goal}
We want to prove that for terms $t$ and $s$, and a context $C$:\vspace{-0.3cm}
\begin{gather*}
t\sim s\Rightarrow C[t]\sim C[s]
\end{gather*}
\end{alertblock}\vspace{-0.5cm}
-In the literature, this is often proved using \emph{Howe's method}. \\
-The method, applies a closure on the bisimilarity relation on $\mS$.\\
-Then one should prove that the result is a simulation.\\
-The closure preserves symmetry, and symmetric simulation is a bisimulation in traditional definitions.\\
-We have found out that it is not always the case.\\
-That is why we want to know when exactly a symmetric simulation is a bisimulation.
\end{frame}
\bibliographystyle{apalike}
\bibliography{references}
\addtocounter{framenumber}{-1}