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37 lines
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37 lines
2.5 KiB
TeX
\subsubsection{Linear Time Properties (LTL)}
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We refer to Section~\ref{sec:ltl} for intuition, as these tend to mostly be intuition exercises.
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For doing verification of liveness / safety properties using \texttt{spin} and Promela,
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the operators are translated to Promela as follows:
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\verb+[] <> ! && || -> U+ for $\square \Diamond \neg \land \lor \Rightarrow U$, respectively.
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In addition, we have the equality operator as a primitive propositional formula. We have \texttt{p == true} for $p$ in the operators.
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Execute \texttt{spin -a file.pml}, followed by \texttt{gcc pan.c} and \texttt{./a.out}.
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As a reminder, these are the operators (details in Section~\ref{sec:ltl-details}):
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\begin{itemize}
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\item Now: $p$ states that the proposition is true ``\textit{now}''
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\item Holds until: $\Phi U \Psi$ states that $\Phi$ holds (with no other valid proposition holding between) until $\Psi$ holds.
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\item Next: $\bigcirc \Phi$ states that $\Phi$ holds for the ``next'' (if nothing else specified from start state)
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\item Eventually: $\Diamond \Phi$ states that $\Phi$ holds \textit{eventually}, $\Diamond \Phi \equiv (\texttt{true} U \Phi)$
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\item Always (from now on): $\square \Phi$ states that from now on, $\Phi$ will always hold, $\square \Phi \equiv \neg \Diamond \neg \Phi$
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\end{itemize}
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For the implication (where the left hand side is called the \textit{antecedent}, right hand side is called the \textit{consequent}),
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remember to show the case where the antecedent is true and state that for antecedent false, it is trivially true
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Another important remark is that you can't just write $\Diamond \neg s_1$, where $s_1$ is a state, you need to specify the propositions.
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We can also create statements, such as never as $\square \neg \Phi$ (always not $\Phi$), etc.
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Overall, be careful with parenthesis!
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\subparagraph{Liveness and Safety Properties}
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Proofs here run using the definitions directly, either by showing a counter example (for disproving) or showing that, in fact, the definition holds.
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Indirect proofs may also come in handy, because it is typically easier to show that something is not a safety property (or liveness property) that to show that it is.
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These two properties are \textit{mutually exclusive} (with one exception, \texttt{true}), to the extent that an LTL formula can't be both at the same time, but be a conjunct of both.
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In fact, every LTL formula is a either one of the two, or a conjunct of both.
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Finally, as a reminder, an LTL formula is for example $\square \Diamond a$, with $a$ a property.
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