some editting

ACV 2026/slides.tex

@@ -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
 			\[
-				x\mathrel{R}y,\;
+				\begin{tikzcd}[ampersand replacement=\&]
-				x\xrightarrow{a}x'
+					x \& y \\
-				\Rightarrow
+					{x'} \& {y'}
-				\exists\,y'.\;
+					\arrow["R"{description}, no head, from=1-1, to=1-2]
-				y\xrightarrow{a}y'
+					\arrow["a"', from=1-1, to=2-1]
-				\;\land\;
+					\arrow["a", dashed, from=1-2, to=2-2]
-				x'Ry'.
+					\arrow["R"{description}, dashed, no head, from=2-1, to=2-2]
-			\]
+				\end{tikzcd}
-			A simulation relation like $R$ is a bisimulation whenever
-			\[
-			x\mathrel{R}y,\;
-			y\xrightarrow{a}y'
-			\Rightarrow
-			\exists\,x'.\;
-			x\xrightarrow{a}x'
-			\;\land\;
-			x'Ry'.
 			\]
+%			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}
+			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
+%			\[
-			\iff
+%			x\sim y
-			\exists\,R.\;
+%			\iff
-			xRy,
+%			\exists\,R.\;
-			\]
+%			xRy,
-			where $R$ is a bisimulation.
+%			\]
+%			where $R$ is a bisimulation.
 		\end{alertblock}
-		\vspace{-0.2cm}
+%		\vspace{-0.2cm}
-			Ultimately, people care about similarity and bisimilarity, but simulation and bisimulation should be defined first.
+%			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}