From 0813fec4f125a4f9fab992748651539cbbf538b5 Mon Sep 17 00:00:00 2001 From: Zhengyi Chen Date: Sun, 5 May 2024 22:46:08 +0100 Subject: [PATCH] ... --- 09-concurrency.tex | 72 ++++++++++++++++++++++++++++++++++++++++++-- 99-notes-packed.pdf | Bin 335471 -> 340866 bytes 2 files changed, 70 insertions(+), 2 deletions(-) diff --git a/09-concurrency.tex b/09-concurrency.tex index 0e3388e..7ed8a3a 100644 --- a/09-concurrency.tex +++ b/09-concurrency.tex @@ -32,7 +32,7 @@ A \textbf{pomset (partial-ordered multiset)} defines a $(E, <, l)$-tuple where: \begin{itemize} \item $E$ a set of ``events'' -- corresponding to each occurrence of action. - \item $<$ a partial-order of E -- $e_i$ happens before $e_j$, or incomparable, etc. + \item $<$ a partial-order of E such that $e_i$ happens before $e_j$, or incomparable, etc. \item $l: E \rightarrow A$ a mapping between events to their actions. \end{itemize} @@ -47,7 +47,75 @@ \end{itemize} \begin{definition}[Control \& Tokens] - Petri nets encode control at runtime. A \textbf{control} simply refers to the state at which the system is currently at. It is symbolized by a \textbf{token} for each concurrent state. + Petri nets encode control at runtime. A \textbf{control} simply refers to the state at which the system is currently at. It is represented as a assignment of \textbf{tokens} to each place in net i.e. a \textbf{marking}. +\end{definition} + +\paragraph*{Progress, Justness, Fairness} We define several properties on concurrent systems to make claim they display good behaviors. + +\begin{definition}[Safety] + A \textbf{safety property} defines that a ``bad'' predicate $\phi$ would never hold -- e.g., $P \models G(\neg \phi)$. +\end{definition} + +\begin{definition}[Liveness] + A \textbf{liveness property} defines that a ``good'' predicate $\phi$ would eventually hold -- e.g., $P \models F(\phi)$. +\end{definition} + +It's easy to see that safety and liveness properties are duo/convertible to each other -- $G(\neg \phi) \iff \neg F(\phi)$, after all. Hence we can speak of them in common contexts. + +Whether safety/liveness properties hold in a system depends often on whether we make appropriate progress and fairness assumptions (in increasing hierarchy): + +\begin{background}[Completeness Criteria] + LTL is a \textbf{linear-time logic} that specifies \textbf{linear-time properties} on paths in a Kripke structure. For example, $F(\phi)$ really means that, within each/a path, a state marked with proposition $\phi$ eventually occurs within the path string. + + We hence expand the satisfaction idea such that a process $P$ satisfies a LTL property $\varphi$ under a \textbf{completeness criterion} $C$: + \begin{equation*} + P \models^{C} \varphi \iff \forall\ \mathrm{path}\ \pi \in P: \pi \models C \implies \pi \models \varphi + \end{equation*} + + For example, $C$ might refer to the assumption that a path is infinite -- \textbf{deadlock-free}, as a finite path occurs only if we reach a state without outgoing transitions. This is often the \textbf{default completeness criterion} when unspecified -- in which case we simply use $\models$ without superscript. + + A completeness criterion $D$ is \textbf{stronger} than $C$ if: + \begin{equation*} + \{\pi\ |\ \pi \in \mathsf{path}(P), \pi \models D\} \subset \{\pi\ |\ \pi \in \mathsf{path}(P), \pi \models C\} + \end{equation*} +\end{background} + +We hence define the following completeness criteria in increasing strength (or reverse implication chain): + +\begin{definition}[Progress] + +\end{definition} + +\begin{definition}[Justness] + +\end{definition} + +\begin{definition}[Weak Fairness] + Define a \textbf{task} to be a set of transitions in a Kripke structure (you can think of it as a subprocedure in a program). Append the Kripke structure $(S, \rightarrowtriangle, I)$ with a novel structure $\tau$: + \begin{equation*} + \tau \coloneqq \textrm{Collection of tasks} + \end{equation*} + + A task $T$ is \textbf{enabled} in state $s \in S$ if there exists an outgoing transition from $s$ that is also in $T$. $T$ is \textbf{perpetually enabled} on path $\pi$ if it is enabled in every state of $\pi$ (i.e., every state in $\pi$ contains outgoing transition in $T$). + + Orthogonally, if a $T$-transition exists \underline{in} $\pi$ then $T$ \textbf{occurs} in $\pi$. + + A path $\pi$ is hence \textbf{weakly fair} if it satisfies the following LTL formula: + \begin{alignat*}{2} + \mathsf{WF}(T) + &\coloneqq &&\ G(\ G(\mathsf{enabled}(T)) \implies F(\mathsf{occurs(T)})\ ) \\ + &\Leftrightarrow &&\ F(G(\mathsf{enabled}(T))) \implies G(F(\mathsf{occurs}(T))) + \end{alignat*} + where predicates $\mathsf{enabled}$, $\mathsf{occurs}$ follow above definition -- a state $s$ is marked with $\mathsf{enabled}(T)$ iff $T$ is enabled at state $s$ i.e. exists outgoing transition from $s$ that is also in $T$. + + Hence, define $\models^{WF}$ as: + \begin{equation*} + P \models^{WF} \varphi \iff P \models [\wedge_{T \in \tau} {(\mathsf{WF}(T))}] \implies \varphi + \end{equation*} +\end{definition} + +\begin{definition}[Strong Fairness] + \end{definition} \end{document} \ No newline at end of file diff --git a/99-notes-packed.pdf b/99-notes-packed.pdf index cd3d941c168234fb707eec0702b41fa0d3e9a98a..d050d5a83f48d445683dee9300076751b668522b 100644 GIT binary patch delta 150922 zcmV(vKIk86x(gNg*J}wT`jd_nUYsaA(-ZYa>-~+ zl?}$^g4wt!-|erZShVHg{k2hxVt;$RRMsutmAir-RI4UIS1y@_kA*Q_UoTDO7OQ%{ zIvx(?{#w{Y3j=YzC{}G%@0--}N#>NBuAiM-zb_L*SjIDZZuuNjTy!1k>!ko9fkYw= zmeMbr$Q+ZvH%rbzFv7~gT&|&aTlvWx>th4!&aq)vUN1S2vy_4_n&S$eZ-2tWP0=RE zt}rk^tx-FMl#E&oid29&oL)cFMIxdFVM_hFMmYf;nkEaf4;%&TX(^ zM)VuJuVEjAG>dw_{eV|`v4W{n(_p{xK^5BLeq$@{w{LE0WN5sKEWZpIC1aVC<3%7z z5K0C~5K7?}@I6!^+JFiH*${>=ZjUWJC(_`RV%yYo)xXj_dUD~S41awZdg-|LkL<#W 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