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Implement basic rewriting rules for {r} and !{r}.

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* src/ltlvisit/simplify.cc: Here.
* src/ltltest/reduccmp.test: Test them.
* doc/tl/tl.tex: Document them.
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Alexandre Duret-Lutz 2012-04-27 00:18:08 +02:00
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doc/tl/tl.tex
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@ -759,16 +759,16 @@ formula $f$ to form another PSL formula.
existential suffix implication
& $\sere{r}\Esuffix{} f$
\\
closure
weak closure
& $\sere{r}$
\\
negated closure
negated weak closure
& $\nsere{r}$
\\
\end{tabular}
\end{center}
For technical reasons, the negated closure is actually implemented as
For technical reasons, the negated weak closure is actually implemented as
an operator, even if it is syntactically and semantically equal to the
combination of $\NOT$ and $\sere{r}$.
@ -809,6 +809,8 @@ rewritings are also preformed on output to ease reading.
$\AsuffixEQ$ and $\AsuffixALTEQ$ are synonyms in the same way as
$\Asuffix$ and $\AsuffixALT$ are.
The $\seren{r}$ operator is a \emph{strong closure} operator.
\subsection{Trivial Identities (Occur Automatically)}
For any PSL formula $f$, any SERE $r$, and any Boolean
@ -826,8 +828,8 @@ formula $b$, the following rewritings are systematically performed
& \nsere{\1} & \equiv \0 \\
\sere{\eword}\Asuffix f&\equiv \1
& \sere{\eword}\Esuffix f&\equiv \0
& \sere{\eword} & \equiv \0
& \nsere{\eword} & \equiv \1 \\
& \sere{\eword} & \equiv \1
& \nsere{\eword} & \equiv \0 \\
\sere{b}\Asuffix f&\equiv (\NOT{b})\OR f
& \sere{b}\Esuffix f&\equiv b\AND f
& \sere{b} &\equiv b
@ -1205,7 +1207,7 @@ rewriting arrange any PSL formula into negative normal form.
\NOT(f \M g) & \equiv (\NOT f) \W (\NOT g)&
\NOT(\sere{r} \Esuffix f) &\equiv \sere{r} \Asuffix \NOT f
\end{align*}
\noindent Recall the that negated closure $\nsere{r}$ is actually
\noindent Recall the that negated weak closure $\nsere{r}$ is actually
implemented as a specific operator, so it not actually prefixed by the
$\NOT$ operator.
\begin{align*}
@ -1474,6 +1476,22 @@ denoted with $\equiV$ can be disabled by setting the
\sere{r_1\OR r_2}\Esuffix f &\equiV (\sere{r_1}\Esuffix f)\OR(\sere{r_2}\Esuffix f)
\end{align*}
Here are basic the rewritings for the weak closure and its negation:
\begin{align*}
\sere{r}&\equiv \1\text{~if~}\varepsilon\VDash r&
\nsere{r}&\equiv \0\text{~if~}\varepsilon\VDash r\\
\sere{r_1;r_2}&\equiv \sere{r_1}\text{~if~}\varepsilon\VDash r_2&
\nsere{r_1;r_2}&\equiv \nsere{r_1}\text{~if~}\varepsilon\VDash r_2\\
\sere{b;r}&\equiV b\AND\X\sere{r}&
\nsere{b;r}&\equiV (\NOT b)\OR\X\nsere{r}\\
\sere{b\STAR{};r}&\equiv b\W\sere{r}&
\nsere{b\STAR{};r}&\equiv (\NOT b)\M\nsere{r}\\
\sere{b\STAR{\mvar{i}..\mvar{j}};r}&\equiV \underbrace{b\AND \X(b\ldots}_{\mathclap{i\text{~occurences of~}b}}\AND\X\sere{b\STAR{\mvar{0}..\mvar{j-i}}\CONCAT r})&
\nsere{b\STAR{\mvar{i}..\mvar{j}};r}&\equiV \underbrace{(\NOT b)\OR \X((\NOT b)\ldots}_{\mathclap{i\text{~occurences of~}\NOT b}}\OR\X\nsere{b\STAR{\mvar{0}..\mvar{j-i}}\CONCAT r}) \\
\sere{b\STAR{\mvar{i}..\mvar{j}}}&\equiV \underbrace{b\AND \X(b\AND \X(\ldots b))}_{i\text{~occurences of~}b}&
\nsere{b\STAR{\mvar{i}..\mvar{j}}}&\equiV \underbrace{(\NOT b)\OR \X((\NOT b)\OR \X(\ldots(\NOT b)))}_{i\text{~occurences of~}\NOT b}
\end{align*}
\subsection{Simplifications for Eventual and Universal Formul\ae}
\label{sec:eventunivrew}