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snf: Fix the handling of bounded repetition.

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star_normal_form() used to be called under bounded
repetitions like [*0..4], but some of these rewritings
are only correct for [*0..].  For instance
     (a*|1)[*]      can be rewritten to    1[*]
but  (a*|1)[*0..1]  cannot be rewritten to 1[*0..1]
it would be correct to rewrite the latter as (a[+]|1)[*0..1],
canceling the empty word in a*.

Also (a*;b*)[*]     can be rewritten to    (a|b)[*]
but  (a*;b*)[*0..1]  cannot be rewritten to (a|b)[*0..1]
and it cannot either be rewritten to (a[+]|b[+])[*0..1].

This patch introduces a new function to implement
rewritings under bounded repetition.

* src/ltlvisit/snf.hh, src/ltlvisit/snf.cc (star_normal_form_unbounded):
New function.
* src/ltlvisit/simplify.cc: Use it.
* src/ltltest/reduccmp.test: Add tests.
* doc/tl/tl.tex: Document the rewritings implemented.
This commit is contained in:
Alexandre Duret-Lutz 2014-05-15 20:06:43 +02:00
parent f431852af9
commit 139f7b49b4
5 changed files with 112 additions and 21 deletions
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44
doc/tl/tl.tex
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@ -1469,27 +1469,53 @@ SERE.
Starred subformul\ae{} are rewritten in Star Normal
Form~\cite{bruggeman.96.tcs} with:
\[r\STAR{\mvar{0}..\mvar{j}} \equiv r^\circ\STAR{\mvar{0}..\mvar{j}} \]
\[r\STAR{} \equiv r^\circ\STAR{} \]
where $r^\circ$ is recursively defined as follows:
\begin{align*}
r^\circ &= r\text{~if~} \varepsilon\not\VDash r \\
\eword^\circ &= \0 &
(r_1\CONCAT r_2)^\circ &= r_1^\circ\OR r_2^\circ \text{~if~} \varepsilon\VDash r_1\text{~and~}\varepsilon\VDash r_2\\
r\STAR{\mvar{0}..\mvar{j}}^\circ &= r^\circ &
r\STAR{\mvar{i}..\mvar{j}}^\circ &= r^\circ \text{~if~} i=0 \text{~or~} \varepsilon\VDash r&
(r_1\AND r_2)^\circ &= r_1^\circ\OR r_2^\circ \text{~if~} \varepsilon\VDash r_1\text{~and~}\varepsilon\VDash r_2\\
(r_1\OR r_2)^\circ &= r_1^\circ \OR r_2^\circ &
(r_1\ANDALT r_2)^\circ &= r_1 \ANDALT r_2
\end{align*}
Note: the original SNF definition~\cite{bruggeman.96.tcs} does not
include \samp{$\FUSION$}, \samp{$\AND$}, and \samp{$\ANDALT$}
operators, and it guarantees that $\forall r,\,\varepsilon\not\VDash
r^\circ$ because $r^\circ$ is stripping all the stars and empty words
that occur in $r$. For instance $\sere{a\STAR{}\CONCAT
b\STAR{}\CONCAT\sere{\1\OR c}}^\circ\STAR{} = \sere{a\OR b\OR
include \samp{$\AND$} and \samp{$\ANDALT$} operators, and it
guarantees that $\forall r,\,\varepsilon\not\VDash r^\circ$ because
$r^\circ$ is stripping all the stars and empty words that occur in
$r$. For instance $\sere{a\STAR{}\CONCAT
b\STAR{}\CONCAT\sere{\eword\OR c}}^\circ\STAR{} = \sere{a\OR b\OR
c}\STAR{}$. Our extended definition still respects this property in
presence of \samp{$\FUSION$} and \samp{$\AND$} operators, but
unfortunately not when the \samp{$\ANDALT$} operator is used.
presence of \samp{$\AND$} operators, but unfortunately not when the
\samp{$\ANDALT$} operator is used.
We extend the above definition to bounded repetitions with:
\begin{align*}
r\STAR{\mvar{i}..\mvar{j}} & \equiv r^\square\STAR{\0..\mvar{j}}\quad\text{if}\quad\varepsilon\VDash r\STAR{\mvar{i}..\mvar{j}}
\end{align*}
where $r^\square$ is recursively defined as follows:
\begin{align*}
r^\square &= r\text{~if~} \varepsilon\not\VDash r \\
\eword^\square &= \0 &
(r_1\CONCAT r_2)^\square &= r_1\CONCAT r_2\\
r\STAR{\mvar{i}..\mvar{j}}^\square &= r^\square\STAR{\mvar{\max(1,i)}..\mvar{j}} \text{~if~} i=0 \text{~or~} \varepsilon\VDash r &
(r_1\AND r_2)^\square &= r_1^\square\OR r_2^\square \text{~if~} \varepsilon\VDash r_1\text{~and~}\varepsilon\VDash r_2\\
(r_1\OR r_2)^\square &= r_1^\square \OR r_2^\square &
(r_1\ANDALT r_2)^\square &= r_1 \ANDALT r_2
\end{align*}
The differences between $^\square$ and $^\circ$ are in the handling
of $r\STAR{\mvar{i}..\mvar{j}}$ and in the handling of $r_1\CONCAT r_2$.
% Indeed $(c\STAR{}\OR\1)\STAR{0..1}$ is not equivalent to
% $(c\STAR{}\OR\1)^\circ\STAR{0..1}\equiv(c\OR\1)\STAR{0..1}\equiv
% 1\STAR{0..1}$ but to
% $(c\STAR{}\OR\1)^\square\STAR{0..1}\equiv(c\PLUS{}\OR\1)\STAR{0..1}$.
% Similarly $(a\STAR{}\CONCAT b\STAR{})\STAR{0..1})$ is definitely not
% equal to $(a\PLUS{}\OR b\PLUS{})\STAR{0..1}).
\subsubsection{Basic Simplifications SERE-LTL Binding Operators}