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Rewrite "(Xc) M b" as "b & X(b U c)", plus three similar rules.

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* src/ltlvisit/simplify.hh (ltl_simplifier_options): New option
reduce_size_stricly.
* src/ltlvisit/simplify.cc (simplify_visitor): Implement these
rules.
* src/ltltest/reduc.cc: Check with reduce_size_strictly unset or
set, but only use the latter result to check sizes.
* src/ltltest/reduccmp.test: Test them.
* doc/tl/tl.tex: Document them.
This commit is contained in:
Alexandre Duret-Lutz 2012-02-13 17:58:57 +01:00
parent c9b34d684a
commit bb56c26d1c
5 changed files with 122 additions and 48 deletions
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doc/tl/tl.tex
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@ -96,6 +96,9 @@
\newcommand{\nsere}[1]{\texttt{!\{}#1\texttt{\}}}
\newcommand{\seren}[1]{\texttt{\{}#1\texttt{\}!}}
% rewriting rules that enlarge the formula
\newcommand{\equiV}{\stackrel{\star}{\equiv}}
\def\limplies{\rightarrow}
\def\simp{\rightrightharpoons}
\def\Simp{\stackrel{+}{\simp}}
@ -1088,11 +1091,11 @@ presented by~\citet{chang.92.icalp}, but other presentations have been
done including negation~\citep{cerna.03.mfcs} and weak
until~\citep{schneider.01.lpar}.
The following grammar rules describes extend the aforementioned
work slightly by dealing with PSL operators. These are the
rules used by Spot to decide upon
construction to which class a formula belongs (see the methods
\texttt{is\_syntactic\_safety()}, \texttt{is\_syntactic\_guarantee()},
The following grammar rules extend the aforementioned work slightly by
dealing with PSL operators. These are the rules used by Spot to
decide upon construction to which class a formula belongs (see the
methods \texttt{is\_syntactic\_safety()},
\texttt{is\_syntactic\_guarantee()},
\texttt{is\_syntactic\_obligation()},
\texttt{is\_syntactic\_recurrence()}, and
\texttt{is\_syntactic\_persistence()} listed on
@ -1231,12 +1234,12 @@ Section~\ref{sec:unabbbool}. Therefore it is never necessary to apply
`\verb|ltl_simplifier::negative_normal_form|`.
If the option `\verb|nenoform_stop_on_boolean|' is set, the above
recursive rewritings will not be applied to subformul\ae{} that are
Boolean formul\ae. For instance calling
`\verb|ltl_simplifier::negative_normal_form|` on $\NOT\F\G(a \XOR b)$
will produce $\G\F(((\NOT a)\AND(\NOT b))\OR(a\AND b))$ if
`\verb|nenoform_stop_on_boolean|' is unset, while it will produce
$\G\F(\NOT(a \XOR b))$ if `\verb|nenoform_stop_on_boolean|' is set.
recursive rewritings are not applied to Boolean subformul\ae{}. For
instance calling `\verb|ltl_simplifier::negative_normal_form|` on
$\NOT\F\G(a \XOR b)$ will produce $\G\F(((\NOT a)\AND(\NOT
b))\OR(a\AND b))$ if `\verb|nenoform_stop_on_boolean|' is unset, while
it will produce $\G\F(\NOT(a \XOR b))$ if
`\verb|nenoform_stop_on_boolean|' is set.
\section{Simplifications}
@ -1264,7 +1267,11 @@ The goals in most of these simplification are to:
\subsection{Basic Simplifications}
These simplifications are enabled with
\verb|ltl_simplifier_options::reduce_basics|'.
\verb|ltl_simplifier_options::reduce_basics|'. A couple of them may
enlarge the size of the formula: they are denoted using $\equiV$
instead of $\equiv$, and they can be disabled by setting the
\verb|ltl_simplifier_options::reduce_size_strictly|' option to
\texttt{true}.
\subsubsection{Basic Simplifications for Temporal Operators}
\label{sec:basic-simp-ltl}
@ -1280,25 +1287,29 @@ from left to right, as usual):
\G(f_1\OR\ldots\OR f_n \OR \G\F(g_1)\OR\ldots\OR \G\F(g_m)) & \equiv \G(f_1\OR\ldots\OR f_n)\OR \G\F(g_1\OR\ldots\OR g_m)
\end{align*}
Note that the latter rewriting rules for $\G$ has no dual:
rewriting $\F(f \AND \G\F g)$ to $\F(f) \AND \G\F(g)$ (instance as
suggested by~\citet{somenzi.00.cav}) goes against our goal of moving
the $\F$ operator in front of the formula. Conceptually, it is also
easier to understand $\F(f \AND \G\F g)$: has long as $f$ has not been
verified, there is no need to worry about the $\G\F g$ term.
Note that the latter three rewriting rules for $\G$ have no dual:
rewriting $\F(f \AND \G\F g)$ to $\F(f) \AND \G\F(g)$ (as suggested
by~\citet{somenzi.00.cav}) goes against our goal of moving the $\F$
operator in front of the formula. Conceptually, it is also easier to
understand $\F(f \AND \G\F g)$: has long as $f$ has not been verified,
there is no need to worry about the $\G\F g$ term.
Here are the basic rewriting rules for binary operators (excluding
$\OR$ and $\AND$ which are considered in Spot as $n$-ary operators).
$b$ denotes a Boolean formula.
Here are the basic rewriting rules for binary operators (excluding $\OR$ and
$\AND$ which are considered in Spot as $n$-ary operators):
\begin{align*}
\1 \U f & \equiv \F f & f \W \0 & \equiv \G f \\
f \M \1 & \equiv \F f & \0 \R f & \equiv \G f \\
(\X f)\U (\X g) & \equiv \X(f\U g) & (\X f)\W(\X g) & \equiv \X(f\W g) \\
(\X f)\M (\X g) & \equiv \X(f\M g) & (\X f)\R(\X g) & \equiv \X(f\R g) \\
f \U(\G f) & \equiv \G f & f \W(\G f) & \equiv \G f \\
f \M(\F f) & \equiv \F f & f \R(\F f) & \equiv \F f \\
f \U (g \OR \G(f)) & \equiv f\W g & f \W (g \OR \G(f)) & \equiv f\W g \\
f \M (g \AND \F(f)) & \equiv f\M g & f \R (g \AND \F(f)) & \equiv f\M g \\
f \U (g \AND f) & \equiv g\M f & f \W (g \AND f) & \equiv g\R f
\1 \U f & \equiv \F f & f \W \0 & \equiv \G f \\
f \M \1 & \equiv \F f & \0 \R f & \equiv \G f \\
(\X f)\U (\X g) & \equiv \X(f\U g) & (\X f)\W(\X g) & \equiv \X(f\W g) \\
(\X f)\M (\X g) & \equiv \X(f\M g) & (\X f)\R(\X g) & \equiv \X(f\R g) \\
(\X f)\U b & \equiV b\OR \X(b\M f) & (\X f)\W b & \equiV b\OR \X(f\R b) \\
(\X f)\M b & \equiV b\AND \X(b\U f) & (\X f)\R b & \equiV b\AND \X(f\W b) \\
f \U(\G f) & \equiv \G f & f \W(\G f) & \equiv \G f \\
f \M(\F f) & \equiv \F f & f \R(\F f) & \equiv \F f \\
f \U (g \OR \G(f)) & \equiv f\W g & f \W (g \OR \G(f)) & \equiv f\W g \\
f \M (g \AND \F(f)) & \equiv f\M g & f \R (g \AND \F(f)) & \equiv f\M g \\
f \U (g \AND f) & \equiv g\M f & f \W (g \AND f) & \equiv g\R f
\end{align*}
Here are the basic rewriting rules for $n$-ary operators ($\AND$ and
@ -1341,7 +1352,7 @@ The above rules are applied even if more terms are presents in the
operator's arguments. For instance $\F\G(a)\AND \G(b) \AND \F\G(c) \AND
\X(d)$ will be rewritten as $\X(d \AND \F\G(a\AND c))\AND \G(b)$.
The following more complicated rules are generalization of $f\AND
The following more complicated rules are generalizations of $f\AND
\X\G f\equiv \G f$ and $f\OR \X\F f\equiv \F f$:
\begin{align*}
f\AND \X(\G(f\AND g\ldots)\AND h\ldots) &\equiv \G(f) \AND \X(\G(g\ldots)\AND h\ldots) \\
@ -1351,7 +1362,7 @@ The latter rule for $f\OR \X(\F(f)\OR h\ldots)$ is only applied if all
$\F$-formul\ae{} can be removed from the argument of $\X$ with the
rewriting. For instance $a \OR b \OR c\OR \X(\F(a\OR b)\OR \F(c)\OR \G d)$
will be rewritten to $\F(a \OR b \OR c) \OR \X\G d$ but
$b \OR c\OR \X(\F(a\OR b)\OR \F(c)\OR \G d)$ would only become
$b \OR c\OR \X(\F(a\OR b)\OR \F(c)\OR \G d)$ will only become
$b \OR c\OR \X(\F(a\OR b\OR c)\OR \G d)$.
Finally the following rule is applied only when no other terms are present
@ -1503,6 +1514,7 @@ sometimes generalized to support operators such as $\M$ and $\W$.
\appendix
\chapter{Defining LTL with only one of $\U$, $\W$, $\R$, or $\M$}
\label{sec:ltl-equiv}
The operators \samp{F}, \samp{G}, \samp{U}, \samp{R}, \samp{M}, and
\samp{W} can all be defined using only Boolean operators, \samp{X},