mirror of
https://github.com/janishutz/eth-summaries.git
synced 2026-07-28 03:39:08 +02:00
28 lines
2.0 KiB
TeX
28 lines
2.0 KiB
TeX
\subsubsection{Linear Time Properties (LTL)}
|
|
We refer to Section~\ref{sec:ltl} for intuition, as these tend to mostly be intuition exercises.
|
|
|
|
For doing verification of liveness / safety properties using \texttt{spin} and Promela,
|
|
the operators are translated to Promela as follows:
|
|
\verb+[] <> ! && || -> U+ for $\square \Diamond \neg \land \lor \Rightarrow U$, respectively.
|
|
In addition, we have the equality operator as a primitive propositional formula. We have \texttt{p == true} for $p$ in the operators.
|
|
|
|
Execute \texttt{spin -a file.pml}, followed by \texttt{gcc pan.c} and \texttt{./a.out}.
|
|
|
|
As a reminder, these are the operators (details in Section~\ref{sec:ltl-details}):
|
|
\begin{itemize}
|
|
\item Now: $p$ states that the proposition is true ``\textit{now}''
|
|
\item Holds until: $\Phi U \Psi$ states that $\Phi$ holds (with no other valid proposition holding between) until $\Psi$ holds.
|
|
\item Next: $\bigcirc \Phi$ states that $\Phi$ holds for the ``next''
|
|
\item Eventually: $\Diamond \Phi$ states that $\Phi$ holds \textit{eventually}, $\Diamond \Phi \equiv (\texttt{true} U \Phi)$
|
|
\item Always (from now on): $\square \Phi$ states that from now on, $\Phi$ will always hold, $\square \Phi \equiv \neg \Diamond \neg \Phi$
|
|
\end{itemize}
|
|
For the implication (where the left hand side is called the \textit{antecedent}, right hand side is called the \textit{consequent}),
|
|
remember to show the case where the antecedent is true and state that for antecedent false, it is trivially true
|
|
|
|
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.
|
|
|
|
|
|
\subparagraph{Liveness and Safety Properties}
|
|
Proofs here run using the definitions directly, either by showing a counter example (for disproving) or showing that, in fact, the definition holds.
|
|
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.
|