Use `SERE' consistently. Add more references.
* doc/tl/tl.tex: Replace all occurrences of ``rational [expression]'' by SERE. Add a couple of more notes and bibliographic references. * doc/tl/tl.bib: More entries.
This commit is contained in:
Alexandre Duret-Lutz
parent
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commit
fef1547bc3
2 changed files with 226 additions and 145 deletions
169
doc/tl/tl.bib
169
doc/tl/tl.bib
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@ -1,58 +1,18 @@
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@InProceedings{ somenzi.00.cav,
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author = {Fabio Somenzi and Roderick Bloem},
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title = {Efficient {B\"u}chi Automata for {LTL} Formul{\ae}},
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booktitle = {Proceedings of the 12th International Conference on
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Computer Aided Verification (CAV'00)},
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pages = {247--263},
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year = {2000},
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volume = {1855},
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@InProceedings{ beer.01.cav,
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author = {Ilan Beer and Shoham Ben-David and Cindy Eisner and Dana
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Fisman and Anna Gringauze and Yoav Rodeh},
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title = {The Temporal Logic Sugar},
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booktitle = {Proceedings of the 13th international conferance on
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Computer Aided Verification (CAV'01)},
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series = {Lecture Notes in Computer Science},
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address = {Chicago, Illinois, USA},
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publisher = {Springer-Verlag}
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}
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@InProceedings{ etessami.00.concur,
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author = {Kousha Etessami and Gerard J. Holzmann},
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title = {Optimizing {B\"u}chi Automata},
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booktitle = {Proceedings of the 11th International Conference on
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Concurrency Theory (Concur'00)},
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pages = {153--167},
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year = {2000},
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editor = {C. Palamidessi},
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volume = {1877},
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series = {Lecture Notes in Computer Science},
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address = {Pennsylvania, USA},
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publisher = {Springer-Verlag},
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note = {Beware of a typo in the version from the
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proceedings: $f \U g$ is purely eventual if both
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operands are purely eventual. The revision of the
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paper available at
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\url{http://www.bell-labs.com/project/TMP/} is
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fixed. We fixed the bug in Spot in 2005, thanks to
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LBTT. See also \url{http://arxiv.org/abs/1011.4214v2}
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for a discussion about this problem.}
|
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}
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@InProceedings{manna.87.podc,
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author = {Zohar Manna and Amir Pnueli},
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title = {A hierarchy of temporal properties},
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booktitle = {Proceedings of the sixth annual ACM Symposium on Principles of distributed computing (PODC'90)},
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year = {1990},
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location = {Quebec City, Canada},
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pages = {377--410},
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publisher = {ACM},
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address = {New York, NY, USA},
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}
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@InProceedings{ chang.92.icalp,
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author = {Edward Y. Chang and Zohar Manna and Amir Pnueli},
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title = {Characterization of Temporal Property Classes},
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booktitle = {Proceedings of the 19th International Colloquium on
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Automata, Languages and Programming (ICALP'92)},
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year = {1992},
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pages = {474--486},
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publisher = {Springer-Verlag},
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address = {London, UK}
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editor = {Berry, Gérard and Comon, Hubert and Finkel, Alain},
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publisher = {Springer},
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isbn = {978-3-540-42345-4},
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pages = {363--367},
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volume = {2102},
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year = {2001},
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month = jul
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}
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@InProceedings{ cerna.03.mfcs,
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@ -71,6 +31,94 @@
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publisher = {Springer-Verlag}
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}
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@InProceedings{ chang.92.icalp,
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author = {Edward Y. Chang and Zohar Manna and Amir Pnueli},
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title = {Characterization of Temporal Property Classes},
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booktitle = {Proceedings of the 19th International Colloquium on
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Automata, Languages and Programming (ICALP'92)},
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year = {1992},
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pages = {474--486},
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publisher = {Springer-Verlag},
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address = {London, UK}
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}
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@Article{ cimatti.08.tcad,
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author = {Alessandro Cimatti and Marco Roveri and Stefano Tonetta},
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journal = {IEEE Transactions on Computer Aided Design of Integrated
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Circuits and Systems},
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number = 10,
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pages = {1737--1750},
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title = {Symbolic Compilation of PSL},
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volume = 27,
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year = 2008,
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date = {2009-03-20},
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note = {\url{https://es.fbk.eu/people/tonetta/tests/tcad07/}}
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}
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@Book{ eisner.06.psl,
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author = {Cindy Eisner and Dana Fisman},
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title = {A Practical Introduction to {PSL}},
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publisher = {Springer},
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year = {2006},
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series = {Series on Integrated Circuits and Systems}
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}
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@InCollection{ eisner.08.hvc,
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author = {Cindy Eisner and Dana Fisman},
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title = {Structural Contradictions},
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booktitle = {Proceedings of the 4th International Haifa Verification
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Conference (HVC'2008)},
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series = {Lecture Notes in Computer Science},
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editor = {Hana Chockler and Alan Hu},
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publisher = {Springer},
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isbn = {978-3-642-01701-8},
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pages = {164--178},
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volume = {5394},
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year = {2009},
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month = oct
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}
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@InProceedings{ etessami.00.concur,
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author = {Kousha Etessami and Gerard J. Holzmann},
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title = {Optimizing {B\"u}chi Automata},
|
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booktitle = {Proceedings of the 11th International Conference on
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Concurrency Theory (Concur'00)},
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pages = {153--167},
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year = {2000},
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editor = {C. Palamidessi},
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volume = {1877},
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series = {Lecture Notes in Computer Science},
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address = {Pennsylvania, USA},
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publisher = {Springer-Verlag},
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note = {Beware of a typo in the version from the proceedings: $f
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\U g$ is purely eventual if both operands are purely
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eventual. The revision of the paper available at
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\url{http://www.bell-labs.com/project/TMP/} is fixed. We
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fixed the bug in Spot in 2005, thanks to LBTT. See also
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\url{http://arxiv.org/abs/1011.4214v2} for a discussion
|
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about this problem.}
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}
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@InProceedings{ manna.87.podc,
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author = {Zohar Manna and Amir Pnueli},
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title = {A hierarchy of temporal properties},
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booktitle = {Proceedings of the sixth annual ACM Symposium on
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Principles of distributed computing (PODC'90)},
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year = {1990},
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location = {Quebec City, Canada},
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pages = {377--410},
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publisher = {ACM},
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address = {New York, NY, USA}
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}
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@Book{ psl.04.lrm,
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title = {Property Specification Language Reference Manual v1.1},
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publisher = {Accellera},
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year = {2004},
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month = jun,
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note = {\url{http://www.eda.org/vfv/}}
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}
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@InProceedings{ schneider.01.lpar,
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author = {Klaus Schneider},
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title = {Improving Automata Generation for Linear Temporal Logic by
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@ -85,6 +133,19 @@
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publisher = {Springer-Verlag}
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}
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@InProceedings{ somenzi.00.cav,
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author = {Fabio Somenzi and Roderick Bloem},
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title = {Efficient {B\"u}chi Automata for {LTL} Formul{\ae}},
|
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booktitle = {Proceedings of the 12th International Conference on
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Computer Aided Verification (CAV'00)},
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pages = {247--263},
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year = {2000},
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volume = {1855},
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series = {Lecture Notes in Computer Science},
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address = {Chicago, Illinois, USA},
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publisher = {Springer-Verlag}
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}
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@TechReport{ tauriainen.03.a83,
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author = {Heikki Tauriainen},
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title = {On Translating Linear Temporal Logic into Alternating and
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202
doc/tl/tl.tex
202
doc/tl/tl.tex
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@ -58,6 +58,7 @@
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\newcommand{\OR}{\mathbin{\texttt{|}}}
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\newcommand{\ORALT}{\mathbin{\texttt{||}}}
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\newcommand{\ORALTT}{\mathbin{\texttt{\char`\\/}}}
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\newcommand{\ORALTTT}{\mathbin{\texttt{+}}}
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\newcommand{\AND}{\mathbin{\texttt{\&}}}
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\newcommand{\ANDALT}{\mathbin{\texttt{\&\&}}}
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\newcommand{\ANDALTT}{\mathbin{\texttt{/\char`\\}}}
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@ -80,9 +81,9 @@
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\newcommand{\AsuffixEQ}{\texttt{[]=>}}
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\newcommand{\AsuffixALTEQ}{\texttt{|=>}}
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\newcommand{\ratgroup}[1]{\texttt{\{}#1\texttt{\}}}
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\newcommand{\nratgroup}[1]{\texttt{!\{}#1\texttt{\}}}
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\newcommand{\ratgroupn}[1]{\texttt{\{}#1\texttt{\}!}}
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\newcommand{\sere}[1]{\texttt{\{}#1\texttt{\}}}
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\newcommand{\nsere}[1]{\texttt{!\{}#1\texttt{\}}}
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\newcommand{\seren}[1]{\texttt{\{}#1\texttt{\}!}}
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\def\limplies{\rightarrow}
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\def\simp{\rightrightharpoons}
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@ -337,12 +338,12 @@ Two temporal formul\ae{} $f$ and $g$ can be combined using the
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following Boolean operators:
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\begin{center}
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\begin{tabular}{cccc}
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\begin{tabular}{ccccc}
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& preferred & \multicolumn{2}{c}{other supported} \\
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operation & syntax & \multicolumn{2}{c}{syntaxes}\\
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\hline
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negation & $\NOT f$ & $\NOTALT f$ \\
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disjunction & $f\OR g$ & $f\ORALT g$ & $f\ORALTT g$ \\
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disjunction & $f\OR g$ & $f\ORALT g$ & $f\ORALTT g$ & $f\ORALTTT$ \\
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conjunction & $f\AND g$ & $f\ANDALT g$ & $f\ANDALTT g$ \\
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implication & $f\IMPLIES g$ & $f\IMPLIESALT g$ & $f\IMPLIESALTT g$\\
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exclusion & $f\XOR g$ & $f\XORALT g$ \\
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@ -574,12 +575,21 @@ section~\ref{sec:unabbbool} as well as the following two rewritings:
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selectively remove $\U$, $\R$, $\M$, or $\W$, using the equivalences
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from the previous section.}
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\section{Rational Operators}
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\section{SERE Operators}
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Any Boolean formula (section~\ref{def:boolform}) is a rational
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expression. Rational expression can be further combined with the
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following operators, where $f$ and $g$ denote arbitrary rational
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expressions and $b$ denotes a Boolean expression.
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The ``SERE'' acronym will be translated to different word depending on
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the source. It can mean either ``\textit{Sequential Extended Regular
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Expression}''~\citep{eisner.06.psl,psl.04.lrm}, ``\textit{Sugar
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Extended Regular Expression}''~\citep{beer.01.cav}, or
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``\textit{Semi-Extended Regular Expression}''~\citep{eisner.08.hvc}.
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In any case, the intent is the same: regular expressions with
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traditional operations (union `$\OR$', concatenation `$\CONCAT$',
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Kleen star `$\STAR{}$') are extended with operators such as
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intersection `$\ANDALT$', and fusion `$\FUSION$'.
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Any Boolean formula (section~\ref{def:boolform}) is a SERE. SERE can
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be further combined with the following operators, where $f$ and $g$
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denote arbitrary SERE and $b$ denotes a Boolean formula.
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\begin{center}
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\begin{tabular}{ccccc}
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@ -587,7 +597,7 @@ expressions and $b$ denotes a Boolean expression.
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operation & syntax & \multicolumn{2}{c}{syntaxes}\\
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\hline
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empty word & $\eword$ \\
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union & $f\OR g$ & $f\ORALT g$ & $f\ORALTT g$ \\
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union & $f\OR g$ & $f\ORALT g$ & $f\ORALTT g$ & $f\ORALTTT g$ \\
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(synchronized) intersection & $f\ANDALT g$ & $f\ANDALTT g$ \\
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unsynchronized intersection & $f\AND g$ \\
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concatenation & $f\CONCAT g$ \\
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@ -619,10 +629,14 @@ expressions and $b$ denotes a Boolean expression.
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\end{tabular}
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\end{center}
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The character \samp{\$} can also be used as value for $\mvar{j}$ in
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the above operators to denote an unbounded range. For instance
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`$a\STAR{i,\texttt{\$}}$' and `$a\STAR{i..}$' represent the same SERE.
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\subsection{Semantics}
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The following semantics assume that $f$ and $g$ are two rational
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expressions, while $b$ is a Boolean formula.
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The following semantics assume that $f$ and $g$ are two SEREs, while
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$b$ is a Boolean formula.
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\begin{align*}
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\sigma\VDash \eword & \iff |\sigma| = 0 \\
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@ -666,8 +680,15 @@ expressions, while $b$ is a Boolean formula.
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\sigma\VDash\mathtt{\{\{}\NOT b\,\mathtt\}\STAR{0..}\CONCAT\ b\,\mathtt\}\STAR{\mvar{i}..}\\
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\end{align*}
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Note that the semantics of $\ANDALT$ and $\AND$ coincide if both
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Notes:
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\begin{itemize}
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\item The semantics of $\ANDALT$ and $\AND$ coincide if both
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operands are Boolean formul\ae.
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\item The SERE $f\FUSION g$ will never hold on $\eword$,
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regardless of the value of $f$ and $g$. For instance
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$a\STAR{}\FUSION b\STAR{}$ is actually equivalent to
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$a\STAR{}\CONCAT\sere{a\ANDALT b}\CONCAT b\STAR{}$.
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\end{itemize}
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\subsection{Syntactic Sugar}
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@ -702,8 +723,8 @@ output.
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\subsection{Trivial Identities (Occur Automatically)}
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The following identities also hold if $j$ or $l$ are missing (assuming
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they are then equal to $\infty$). $f$ can be any regular expression,
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while $b$, $b_1$, $b_2$ are assumed to be Boolean formul\ae.
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they are then equal to $\infty$). $f$ can be any SERE, while $b$,
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$b_1$, $b_2$ are assumed to be Boolean formul\ae.
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\begin{align*}
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\0\STAR{0..\mvar{j}} &\equiv \eword &
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@ -758,9 +779,9 @@ The following rules are all valid with the two arguments swapped, except the one
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b:f &\stackrel{\dag}{\equiv} b\AND f\\
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\end{align*}
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\section{Rational-LTL Binding Operators}
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\section{SERE-LTL Binding Operators}
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The following operators combine a rational expression $r$ with a PSL
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The following operators combine a SERE $r$ with a PSL
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formula $f$ to form another PSL formula.
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\begin{center}
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@ -769,44 +790,44 @@ formula $f$ to form another PSL formula.
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operation & syntax & \multicolumn{2}{c}{syntaxes}\\
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\hline
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(universal) suffix implication
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& $\ratgroup{r}\Asuffix{} f$
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& $\ratgroup{r}\AsuffixALT{} f$
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& $\ratgroup{r}\texttt{(}f\texttt{)}$
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& $\sere{r}\Asuffix{} f$
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& $\sere{r}\AsuffixALT{} f$
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& $\sere{r}\texttt{(}f\texttt{)}$
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\\
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existential suffix implication
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& $\ratgroup{r}\Esuffix{} f$
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& $\sere{r}\Esuffix{} f$
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\\
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closure
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& $\ratgroup{r}$
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& $\sere{r}$
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\\
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negated closure
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& $\nratgroup{r}$
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& $\nsere{r}$
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\\
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\end{tabular}
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\end{center}
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For technical reasons, the negated closure is actually implemented as
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an operator, even if it is syntactically and semantically equal to the
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combination of $\NOT$ and $\ratgroup{r}$.
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combination of $\NOT$ and $\sere{r}$.
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\subsection{Semantics}
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The following semantics assume that $r$ is a rational expression,
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The following semantics assume that $r$ is a SERE,
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while $f$ is a PSL formula.
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\begin{align*}
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\sigma\vDash \ratgroup{r}\Esuffix f &\iff \exists k\ge 0, (\sigma^{0..k}\VDash r)\land(\sigma^{k..}\vDash f)\\
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\sigma\vDash \ratgroup{r}\Asuffix f &\iff \forall k\ge 0, (\sigma^{0..k}\VDash r)\limplies (\sigma^{k..}\vDash f)\\
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\sigma\vDash \ratgroup{r} & \iff (\exists k\ge 0, \sigma^{0..k}\VDash r)\lor(\forall k\ge 0,\,\exists\pi\in(\B^\AP)^\star,\, (\sigma^{0..k}\prec \pi)\land(\pi\VDash r))\\
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\sigma\vDash \nratgroup{r} & \iff (\forall k\ge 0, \sigma^{0..k}\nVDash r)\land(\exists k\ge 0,\,\forall\pi\in(\B^\AP)^\star,\, (\sigma^{0..k}\prec \pi)\limplies(\pi\nVDash r))\\
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\sigma\vDash \sere{r}\Esuffix f &\iff \exists k\ge 0, (\sigma^{0..k}\VDash r)\land(\sigma^{k..}\vDash f)\\
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\sigma\vDash \sere{r}\Asuffix f &\iff \forall k\ge 0, (\sigma^{0..k}\VDash r)\limplies (\sigma^{k..}\vDash f)\\
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\sigma\vDash \sere{r} & \iff (\exists k\ge 0, \sigma^{0..k}\VDash r)\lor(\forall k\ge 0,\,\exists\pi\in(\B^\AP)^\star,\, (\sigma^{0..k}\prec \pi)\land(\pi\VDash r))\\
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\sigma\vDash \nsere{r} & \iff (\forall k\ge 0, \sigma^{0..k}\nVDash r)\land(\exists k\ge 0,\,\forall\pi\in(\B^\AP)^\star,\, (\sigma^{0..k}\prec \pi)\limplies(\pi\nVDash r))\\
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\end{align*}
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The $\prec$ symbol should be read as ``\emph{is a prefix of}''. So
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the semantic for `$\sigma\vDash \ratgroup{r}$' is that either there is
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the semantic for `$\sigma\vDash \sere{r}$' is that either there is
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a (non-empty) finite prefix of $\sigma$ that is a model of $r$, or any
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prefix of $\sigma$ can be extended into a finite sequence $\pi$ that
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is a model of $r$. An infinite sequence $\texttt{a;a;a;a;a;}\ldots$
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is therefore a model of the formula \samp{$\ratgroup{a\PLUS{};\NOT
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is therefore a model of the formula \samp{$\sere{a\PLUS{};\NOT
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a}$} even though it never sees \samp{$\NOT a$}.
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\subsection{Syntactic Sugar}
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@ -817,10 +838,10 @@ formula.\spottodo{It would be nice to have the opposite rewritings on
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output.}
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\begin{align*}
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\ratgroup{r}\EsuffixEQ f &\equiv \ratgroup{r\CONCAT\1}\Esuffix f &
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\ratgroup{r}\AsuffixEQ f &\equiv \ratgroup{r\CONCAT\1}\Asuffix f\\
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\ratgroupn{r} &\equiv \ratgroup{r}\Esuffix \1 &
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||||
\ratgroup{r}\AsuffixALTEQ f &\equiv \ratgroup{r\CONCAT\1}\Asuffix f\\
|
||||
\sere{r}\EsuffixEQ f &\equiv \sere{r\CONCAT\1}\Esuffix f &
|
||||
\sere{r}\AsuffixEQ f &\equiv \sere{r\CONCAT\1}\Asuffix f\\
|
||||
\seren{r} &\equiv \sere{r}\Esuffix \1 &
|
||||
\sere{r}\AsuffixALTEQ f &\equiv \sere{r\CONCAT\1}\Asuffix f\\
|
||||
\end{align*}
|
||||
|
||||
$\AsuffixEQ$ and $\AsuffixALTEQ$ are synonyms in the same way as
|
||||
|
|
@ -828,29 +849,29 @@ $\Asuffix$ and $\AsuffixALT$ are.
|
|||
|
||||
\subsection{Trivial Identities (Occur Automatically)}
|
||||
|
||||
For any PSL formula $f$, any rational expression $r$, and any Boolean
|
||||
For any PSL formula $f$, any SERE $r$, and any Boolean
|
||||
formula $b$, the following rewritings are systematically performed
|
||||
(from left to right).
|
||||
|
||||
\begin{align*}
|
||||
\ratgroup{\0}\Asuffix f &\equiv \1
|
||||
& \ratgroup{\0}\Esuffix f &\equiv \0
|
||||
& \ratgroup{\0} & \equiv \0
|
||||
& \nratgroup{\0} & \equiv \1 \\
|
||||
\ratgroup{\1}\Asuffix f &\equiv f
|
||||
& \ratgroup{\1}\Esuffix f &\equiv f
|
||||
& \ratgroup{\1} & \equiv \1
|
||||
& \nratgroup{\1} & \equiv \0 \\
|
||||
\ratgroup{\eword}\Asuffix f&\equiv \1
|
||||
& \ratgroup{\eword}\Esuffix f&\equiv \0
|
||||
& \ratgroup{\eword} & \equiv \0
|
||||
& \nratgroup{\eword} & \equiv \1 \\
|
||||
\ratgroup{b}\Asuffix f&\equiv (\NOT{b})\OR f
|
||||
& \ratgroup{b}\Esuffix f&\equiv b\AND f
|
||||
& \ratgroup{b} &\equiv b
|
||||
& \nratgroup{b} &\equiv \NOT b\\
|
||||
\ratgroup{r}\Asuffix \1&\equiv \1
|
||||
& \ratgroup{r}\Esuffix \0&\equiv \0 \\
|
||||
\sere{\0}\Asuffix f &\equiv \1
|
||||
& \sere{\0}\Esuffix f &\equiv \0
|
||||
& \sere{\0} & \equiv \0
|
||||
& \nsere{\0} & \equiv \1 \\
|
||||
\sere{\1}\Asuffix f &\equiv f
|
||||
& \sere{\1}\Esuffix f &\equiv f
|
||||
& \sere{\1} & \equiv \1
|
||||
& \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{b}\Asuffix f&\equiv (\NOT{b})\OR f
|
||||
& \sere{b}\Esuffix f&\equiv b\AND f
|
||||
& \sere{b} &\equiv b
|
||||
& \nsere{b} &\equiv \NOT b\\
|
||||
\sere{r}\Asuffix \1&\equiv \1
|
||||
& \sere{r}\Esuffix \0&\equiv \0 \\
|
||||
\end{align*}
|
||||
|
||||
\chapter{Grammar}
|
||||
|
|
@ -955,10 +976,11 @@ instance using the following methods:
|
|||
\\\texttt{is\_psl\_formula()}& Whether the formula uses only PSL
|
||||
operators. (Boolean and LTL operators are also allowed.)
|
||||
\\\texttt{is\_sere\_formula()}& Whether the formula uses only
|
||||
rational operators. (Boolean operators are also allowed, provided
|
||||
no rational operator is negated.)
|
||||
SERE operators. (Boolean operators are also allowed, provided
|
||||
no SERE operator is negated.)
|
||||
\\\texttt{is\_finite()}& Whether a SERE describes a finite
|
||||
language, or an LTL formula uses no temporal operator but $\X$.
|
||||
language (no unbounded stars), or an LTL formula uses no
|
||||
temporal operator but $\X$.
|
||||
\\\texttt{is\_eventual()}& Whether the formula is a pure eventuality.
|
||||
\\\texttt{is\_universal()}& Whether the formula is purely universal.
|
||||
\\\texttt{is\_syntactic\_safety()}& Whether the formula is a syntactic
|
||||
|
|
@ -973,8 +995,8 @@ instance using the following methods:
|
|||
persistence property.
|
||||
\\\texttt{is\_marked()}& Whether the formula contains a special
|
||||
``marked'' version of the $\Esuffix$ operator.\footnotemark
|
||||
\\\texttt{accepts\_eword()}& Whether the formula accept
|
||||
$\eword$. (This can only be true for a rational formula.)
|
||||
\\\texttt{accepts\_eword()}& Whether the formula accepts
|
||||
$\eword$. (This can only be true for a SERE formula.)
|
||||
\end{tabulary}
|
||||
\footnotetext{This special operator is used when translating recurring
|
||||
$\Esuffix$, it is rendered as $\EsuffixMarked$ and it obeys the same
|
||||
|
|
@ -1049,9 +1071,8 @@ and weak until~\citep{schneider.01.lpar}.
|
|||
In the following grammar rules, the symbols $\varphi_G$, $\varphi_S$,
|
||||
$\varphi_O$, $\varphi_P$, $\varphi_R$ designate any formula belonging
|
||||
respectively to the Guarantee, Safety, Obligation, Persistence, or
|
||||
Recurrence classes. $v$ designates any variable, $r$ any rational
|
||||
property, $r_F$ a finite rational property (no loops), and $r_I$ an
|
||||
infinite rational property.
|
||||
Recurrence classes. $v$ designates any variable, $r$ any SERE, $r_F$
|
||||
a bounded SERE (no loops), and $r_I$ an unbounded SERE.
|
||||
|
||||
|
||||
\begin{align*}
|
||||
|
|
@ -1059,36 +1080,36 @@ infinite rational property.
|
|||
\varphi_G\AND \varphi_G\mid (\varphi_G\OR \varphi_G)\mid
|
||||
\X\varphi_G \mid \F\varphi_G\mid
|
||||
\varphi_G\U\varphi_G\mid \varphi_G\M\varphi_G\\
|
||||
\mid&\, \ratgroup{r}\Esuffix \varphi_G\mid
|
||||
\ratgroup{r_F}\Asuffix \varphi_G\\
|
||||
\mid&\, \sere{r}\Esuffix \varphi_G\mid
|
||||
\sere{r_F}\Asuffix \varphi_G\\
|
||||
\varphi_S ::=&\, \0\mid\1\mid v\mid \NOT\varphi_G\mid
|
||||
\varphi_S\AND \varphi_S\mid (\varphi_S\OR \varphi_S)\mid
|
||||
\X\varphi_S \mid \G\varphi_S\mid
|
||||
\varphi_S\R\varphi_S\mid \varphi_S\W\varphi_S\\
|
||||
\mid&\, \ratgroup{r_F}\Esuffix \varphi_S\mid
|
||||
\ratgroup{r}\Asuffix \varphi_S\\
|
||||
\mid&\, \sere{r_F}\Esuffix \varphi_S\mid
|
||||
\sere{r}\Asuffix \varphi_S\\
|
||||
\varphi_O ::=&\, \varphi_G \mid \varphi_S\mid \NOT\varphi_O\mid
|
||||
\varphi_O\AND \varphi_O\mid (\varphi_O\OR \varphi_O)\mid
|
||||
\X\varphi_O \mid
|
||||
\varphi_O\U\varphi_G\mid\varphi_O\R\varphi_S \mid
|
||||
\varphi_S\W\varphi_O\mid \varphi_G\M\varphi_O\\
|
||||
\mid&\, \ratgroup{r_F}\Esuffix \varphi_O \mid \ratgroup{r_I}\Esuffix \varphi_G\mid
|
||||
\ratgroup{r_F}\Asuffix \varphi_O\mid
|
||||
\ratgroup{r_I}\Asuffix \varphi_S\\
|
||||
\mid&\, \sere{r_F}\Esuffix \varphi_O \mid \sere{r_I}\Esuffix \varphi_G\mid
|
||||
\sere{r_F}\Asuffix \varphi_O\mid
|
||||
\sere{r_I}\Asuffix \varphi_S\\
|
||||
\varphi_P ::=&\, \varphi_O \mid \NOT\varphi_R\mid
|
||||
\varphi_P\AND \varphi_P\mid (\varphi_P\OR \varphi_P)\mid
|
||||
\X\varphi_P \mid \F\varphi_P \mid
|
||||
\varphi_P\U\varphi_P\mid\varphi_P\R\varphi_S\mid
|
||||
\varphi_S\W\varphi_P\mid\varphi_P\M\varphi_P\\
|
||||
\mid&\, \ratgroup{r}\Esuffix \varphi_P\mid
|
||||
\ratgroup{r_F}\Asuffix \varphi_P\mid
|
||||
\ratgroup{r_I}\Asuffix \varphi_S\\
|
||||
\mid&\, \sere{r}\Esuffix \varphi_P\mid
|
||||
\sere{r_F}\Asuffix \varphi_P\mid
|
||||
\sere{r_I}\Asuffix \varphi_S\\
|
||||
\varphi_R ::=&\, \varphi_O \mid \NOT\varphi_P\mid
|
||||
\varphi_R\AND \varphi_R\mid (\varphi_R\OR \varphi_R)\mid
|
||||
\X\varphi_R \mid \G\varphi_R \mid
|
||||
\varphi_R\U\varphi_G\mid\varphi_R\R\varphi_R\mid
|
||||
\varphi_R\W\varphi_R\mid\varphi_G\M\varphi_R\\
|
||||
\mid&\, \ratgroup{r_F}\Esuffix \varphi_R \mid \ratgroup{r_I}\Esuffix \varphi_G\mid \ratgroup{r}\Asuffix \varphi_P\\
|
||||
\mid&\, \sere{r_F}\Esuffix \varphi_R \mid \sere{r_I}\Esuffix \varphi_G\mid \sere{r}\Asuffix \varphi_P\\
|
||||
\end{align*}
|
||||
|
||||
\chapter{Rewritings}
|
||||
|
|
@ -1130,13 +1151,13 @@ rewriting arrange any PSL formula into negative normal form.
|
|||
\\
|
||||
\NOT\G f & \equiv \F\NOT f &
|
||||
\NOT(f \W g) & \equiv (\NOT f) \M (\NOT g) &
|
||||
\NOT(\ratgroup{r} \Asuffix f) &\equiv \ratgroup{r} \Esuffix \NOT f
|
||||
\NOT(\sere{r} \Asuffix f) &\equiv \sere{r} \Esuffix \NOT f
|
||||
\\
|
||||
\NOT(\ratgroup{r})&\nratgroup{r}&
|
||||
\NOT(\sere{r})&\equiv \nsere{r}&
|
||||
\NOT(f \M g) & \equiv (\NOT f) \W (\NOT g)&
|
||||
\NOT(\ratgroup{r} \Esuffix f) &\equiv \ratgroup{r} \Asuffix \NOT f
|
||||
\NOT(\sere{r} \Esuffix f) &\equiv \sere{r} \Asuffix \NOT f
|
||||
\end{align*}
|
||||
\noindent Recall the that negated closure $\nratgroup{r}$ is actually
|
||||
\noindent Recall the that negated closure $\nsere{r}$ is actually
|
||||
implemented as a specific operator, so it not actually prefixed by the
|
||||
$\NOT$ operator.
|
||||
\begin{align*}
|
||||
|
|
@ -1157,7 +1178,7 @@ 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 rewriting will not be applied to subformul\ae{} that are
|
||||
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
|
||||
|
|
@ -1267,9 +1288,9 @@ in the OR arguments:
|
|||
|
||||
\subsubsection{Basic Simplifications for SERE Operators}
|
||||
|
||||
\spottodo[inline]{These rules, mostly taken from ``Symbolic
|
||||
computation of PSL'' (Cimatti, Roveri, and Tonetta) are not complete
|
||||
yet.}
|
||||
\spottodo[inline]{The following rules, mostly taken
|
||||
from~\citet{cimatti.08.tcad} are not complete yet. We only show
|
||||
those that are implemented.}
|
||||
|
||||
The following simplification rules are used for the $n$-ary operators
|
||||
$\ANDALT$, $\AND$, and $\OR$. The patterns are of course commutative.
|
||||
|
|
@ -1292,17 +1313,17 @@ SERE.
|
|||
b \ANDALT r &\text{if~} i\le 1\le j\\
|
||||
\0 &\text{else}\\
|
||||
\end{cases}\\
|
||||
b \ANDALT \ratgroup{r_1 \FUSION \ldots \FUSION r_n} &\equiv b \ANDALT r_1 \ANDALT \ldots \ANDALT r_n \\
|
||||
b \ANDALT \ratgroup{r_1 \CONCAT \ldots \CONCAT r_n} &\equiv
|
||||
b \ANDALT \sere{r_1 \FUSION \ldots \FUSION r_n} &\equiv b \ANDALT r_1 \ANDALT \ldots \ANDALT r_n \\
|
||||
b \ANDALT \sere{r_1 \CONCAT \ldots \CONCAT r_n} &\equiv
|
||||
\begin{cases}
|
||||
b \ANDALT r_i & \text{if~}\exists!i,\,\varepsilon\not\VDash r_i\\
|
||||
b \ANDALT (r_1 \OR \ldots \OR r_n) & \text{if~}\forall i,\, \varepsilon\VDash r_i\\
|
||||
\0 &\text{else}\\
|
||||
\end{cases}\\
|
||||
\ratgroup{b_1\CONCAT r_1}\ANDALT\ratgroup{b_2\CONCAT r_2} &\equiv \ratgroup{b_1\ANDALT b_2}\CONCAT\ratgroup{r_1\ANDALT r_2} \\
|
||||
\ratgroup{b_1\FUSION r_1}\ANDALT\ratgroup{b_2\FUSION r_2} &\equiv \ratgroup{b_1\ANDALT b_2}\FUSION\ratgroup{r_1\ANDALT r_2} \\
|
||||
\ratgroup{r_1\CONCAT b_1}\ANDALT\ratgroup{r_2\CONCAT b_2} &\equiv \ratgroup{r_1\ANDALT r_2}\CONCAT\ratgroup{b_1\ANDALT b_2} \\
|
||||
\ratgroup{r_1\FUSION b_1}\ANDALT\ratgroup{r_2\FUSION b_2} &\equiv \ratgroup{r_1\ANDALT r_2}\FUSION\ratgroup{b_1\ANDALT b_2} \\
|
||||
\sere{b_1\CONCAT r_1}\ANDALT\sere{b_2\CONCAT r_2} &\equiv \sere{b_1\ANDALT b_2}\CONCAT\sere{r_1\ANDALT r_2} \\
|
||||
\sere{b_1\FUSION r_1}\ANDALT\sere{b_2\FUSION r_2} &\equiv \sere{b_1\ANDALT b_2}\FUSION\sere{r_1\ANDALT r_2} \\
|
||||
\sere{r_1\CONCAT b_1}\ANDALT\sere{r_2\CONCAT b_2} &\equiv \sere{r_1\ANDALT r_2}\CONCAT\sere{b_1\ANDALT b_2} \\
|
||||
\sere{r_1\FUSION b_1}\ANDALT\sere{r_2\FUSION b_2} &\equiv \sere{r_1\ANDALT r_2}\FUSION\sere{b_1\ANDALT b_2} \\
|
||||
\end{align*}
|
||||
|
||||
\subsection{Simplifications for Eventual and Universal Formul\ae}
|
||||
|
|
@ -1401,7 +1422,7 @@ first presented by~\citet{somenzi.00.cav}.
|
|||
|
||||
A few words about implication first. For two PSL formul\ae{} $f$ and
|
||||
$g$, we say that $f\implies g$ if $\forall\sigma,\,(\sigma\vDash
|
||||
f)\implies(\sigma\vDash g)$. For two rational formul\ae{} $f$ and
|
||||
f)\implies(\sigma\vDash g)$. For two SERE $f$ and
|
||||
$g$, we say that $f\implies g$ if $\forall\pi,\,(\pi\VDash
|
||||
f)\implies(\pi\VDash g)$.
|
||||
|
||||
|
|
@ -1446,7 +1467,7 @@ $f_b \simp g_b$ & $\mathrm{BDD}(f_b)\land \mathrm{BDD}(g_b) =
|
|||
\def\band#1#2#3#4{$\begin{aligned}#1 &\simp #2 \\{}\land #3 &\simp #4\end{aligned}$}
|
||||
\def\banD#1#2#3#4#5#6{$\begin{aligned}#1 &\simp #2 \\{}\land #3 &\simp #4\\{}\land #5 &\simp #6\end{aligned}$}
|
||||
|
||||
\begin{center}
|
||||
\centering
|
||||
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|}
|
||||
\hline
|
||||
$\simp$ & $g$ & $g_E$ & $\X g$ & $g_1\U g_2$ & $g_1\W g_2$ & $g_1 \R g_2$ & $g_1\M g_2$ & $\F g$ & $\G g$ & $g_1\OR g_2$ & $g_1 \AND g_2$ \\
|
||||
|
|
@ -1475,7 +1496,6 @@ $f_1\OR f_2$ & \band{f_1}{g}{f_2}{g} & & &
|
|||
$f_1\AND f_2$ & \bor{f_1}{g}{f_2}{g} & & & & & & & & & & \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\caption{Recursive rules for syntactic implication.\label{tab:syntimpl}}
|
||||
\end{sidewaystable}
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue