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The content of the talk: The content of the talk:
\begin{itemize} \begin{itemize}
\item Intuitive introduction to coalgebra and (bi)simulation \item Intuitive introduction to coalgebra and (bi)simulation
\item Reviewing basic definitions of coalgebra and (bi)simulation \item Reviewing definitions of coalgebra and (bi)simulation
\item Our motivation: To ease proving program equivalence \item Our motivation: To ease proving program equivalence
\item Relator-based notions \item Relator-based notions
\item Span-based notions \item Span-based notions
@@ -177,14 +177,14 @@
When can we say that two systems \emph{behave the same}? When can we say that two systems \emph{behave the same}?
\end{alertblock} \end{alertblock}
\vspace{-0.4cm} \vspace{-0.4cm}
This question is everywhere, often silently assumed:\vspace{-0.4cm} This question is everywhere, for example:\vspace{-0.4cm}
\begin{itemize} \begin{itemize}
\setlength{\itemsep}{6pt} \setlength{\itemsep}{6pt}
\item[\faServer] \textbf{Hardware redesign.} A chip redesigned for cheaper production must still compute exactly what the old one did. \item[\faServer] \textbf{Hardware redesign.} A chip redesigned for a cheaper price must still compute exactly what the old one did.
\item[\faCode] \textbf{Software upgrades.} A bank replaces its backend --- from the customer's view (balances, transactions), nothing should change. \item[\faCode] \textbf{Software upgrades.} A bank replaces its backend --- from the customer's view (balances, transactions), nothing should change.
\end{itemize} \end{itemize}
\vspace{0.2cm} \vspace{0.2cm}
\begin{block}{The catch} \begin{block}{The point}
``Behaving the same'' is intuitive, but making it \emph{precise} and \emph{checkable} is surprisingly subtle. ``Behaving the same'' is intuitive, but making it \emph{precise} and \emph{checkable} is surprisingly subtle.
\end{block} \end{block}
\end{frame} \end{frame}
@@ -453,8 +453,8 @@
-The method, applies a closure on the bisimilarity relation on $T$.\\ -The method, applies a closure on the bisimilarity relation on $T$.\\
-Then one should prove that the result is a simulation.\\ -Then one should prove that the result is a simulation.\\
-The closure preserves symmetry, and symmetric simulation is a bisimulation in traditional definitions.\\ -The closure preserves symmetry, and symmetric simulation is a bisimulation in traditional definitions.\\
-We have found out that it is not always the case.\\ -We need to know how to generally derive bisimulation from simulation.\\
-That is why we want to know when exactly a symmetric simulation is a bisimulation. -And what is the impact of choosing specific notion of simulation (span-based, relation-based)?
\end{frame} \end{frame}
\begin{frame}{Relators and Simulations} \begin{frame}{Relators and Simulations}
@@ -465,7 +465,7 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\column{.47\textwidth} \column{.47\textwidth}
To define coalgebraic simulation, relators are invented. Relators unify simulation and bisimulation.
\begin{alertblock}{Relator} \begin{alertblock}{Relator}
Assuming $F:\Set\to\Set$ is a functor, an $F$-relator a monotone map that sends a morphism of $\rel$ that is a relation $X\rto Y$ to a relation $FX\rto FY$. Assuming $F:\Set\to\Set$ is a functor, an $F$-relator a monotone map that sends a morphism of $\rel$ that is a relation $X\rto Y$ to a relation $FX\rto FY$.
\end{alertblock} \end{alertblock}
@@ -533,7 +533,7 @@
\begin{alertblock}{Symmetrization of a Relator} \begin{alertblock}{Symmetrization of a Relator}
Symmetrization of a relator $\relar$ is defined as: Symmetrization of a relator $\relar$ is defined as:
\begin{gather*} \begin{gather*}
\hat{\relar}=\relar r\cap (\relar r^{-1})^{-1} \hat{\relar}r=\relar r\cap (\relar r^{-1})^{-1}
\end{gather*} \end{gather*}
\end{alertblock} \end{alertblock}
\begin{block}{Symmetric Relators} \begin{block}{Symmetric Relators}
@@ -658,8 +658,8 @@
\vspace{-0.1cm} \vspace{-0.1cm}
\begin{block}{} \begin{block}{}
\begin{itemize} \begin{itemize}
\item Although, not every sound and complete relator is a Barr relator.\\ \item Although, not every sound and complete relator is a Barr relator!\\
\item We do not know when the symmetrizaion of a Barr relator is a Barr relator. \item We do not know when Clause 3 does not produce a Barr relator.
\end{itemize} \end{itemize}
\end{block} \end{block}
\end{frame} \end{frame}