91 lines
No EOL
2.9 KiB
TeX
91 lines
No EOL
2.9 KiB
TeX
\documentclass[99-notes-packed.tex]{subfiles}
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\begin{document}
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\paragraph*{CSP}
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Introduce the following operations:
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\begin{enumerate}
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\item {
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$0$ or $\mathsf{STOP}$ (inaction).
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Likewise CCS, a graph with 1 (initial) state, 0 transitions.
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}
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\item {
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$a.P$ or $a \rightarrow P$ (action prefix) for $\forall a \in A$.
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Likewise CCS.
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}
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\item {
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$P \square Q$ (external choice).
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Semantically it is similar to parallel composition without synchronization, where:
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\begin{itemize}
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\item {
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Prior to ``choice'', one of the two actions might happen between the processes.
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}
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\item {
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After one of the action happens, only the choiced process may occur at runtime.
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}
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\end{itemize}
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}
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\item {
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$P \sqcap Q$ (internal choice):
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\begin{equation*}
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\mathrm{CSP}[P \sqcap Q] \equiv \mathrm{CCS}[\tau.P + \tau.Q]
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\end{equation*}
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}
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\item {
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$P {||}_{S} Q$ (parallel composition) with enforced synchronization over $S \subseteq A$. Semantically speaking:
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\begin{itemize}
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\item $
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\mathsf{States}(P {||}_{S} Q) \coloneqq
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\mathsf{States}(P) \times \mathsf{States}(Q)
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$
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\item {
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$(s, t) \xrightarrow{a} (s^{'}, t)$ if
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$(a \notin S) \wedge (s \xrightarrow{a} s^{'} \in P)$
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}
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\item {
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$(s, t) \xrightarrow{a} (s, t^{'})$ if
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$(a \notin S) \wedge (t \xrightarrow{a} t^{'} \in Q)$
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}
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\item {
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$(s, t) \xrightarrow{a} (s^{'}, t^{'})$ if
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$(a \in S) \wedge (s \xrightarrow{a} s^{'} \in P) \wedge (t \xrightarrow{a} t^{'} \in Q)$
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}
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\end{itemize}
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}
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\item {
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$P/a$ (concealment).
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Like CCS, rename $a$ into $\tau$.
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}
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\item {
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$P[f]$ (renaming) for $f \in (A \rightarrow A)$
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Likewise CCS.
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}
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\end{enumerate}
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Weak and branching bisimulation are congurences for CSP.
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\paragraph*{GSOS}
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As a general form over languages, GSOS describes a transition rule of the following form: define
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\begin{itemize}
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\item {
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$\Sigma$ be the collection of function symbols wrt. a language.
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}
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\item {
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$\mathsf{arity}: \Sigma \rightarrow \mathbb{N}$ a function exposing the arity of the function symbol in question.
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}
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\end{itemize}
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then, under GSOS semantics, the language could be expressed as follows:
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\begin{equation*}
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\begin{prooftree}
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\hypo{x_i \xrightarrow{a} y_i, \dots\ (f \in \Sigma, i \in [1, \mathsf{arity}(f)], y_i \notin \mathsf{args}(f))}
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\infer1{f(x_1, \dots, x_{\mathsf{arity}(f)}) \xrightarrow{a} \mathsf{Expr}(x_1, \dots, x_{\mathsf{arity}(f)}, y_i, \dots)}
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\end{prooftree}
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\end{equation*}
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It is the generalization of SOS rules we have covered earlier.
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\end{document} |