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Left omega algebras and regular equations

Georg Struth
<span title="">2012</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/anq6kvwvqbegtipq5vfhvonzqq" style="color: black;">The Journal of Logic and Algebraic Programming</a> </i> &nbsp;
It turns out, for instance, that the omega operation captures precisely the empty word property in regular languages and wellfoundedness in relational models.  ...  The definability and solvability results are refined to concrete models, to languages, traces and relations.  ...  I would also like to apologise to my family for having worked between Christmas and New Year 2011 to meet the special issue deadline on December 31.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jlap.2012.05.004">doi:10.1016/j.jlap.2012.05.004</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/syfjyqp7hzgilg44q43zmhqttm">fatcat:syfjyqp7hzgilg44q43zmhqttm</a> </span>
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Classifying regular languages by a split game

Qiqi Yan
<span title="">2007</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
Namely we show that omega powers, together with boolean combinations and concatenations, are not sufficient to express the class of ω-regular languages.  ...  To gain insight into how the split game can be applied to attack the long-standing generalized star height 2 problem, we propose and solve the omega power problem, a similar but tractable problem in the  ...  Acknowledgments I thank Enshao Shen for pointing me to the star height 2 problem, as well as sharing his thoughts. I also thank Jean-Eric Pin and Wolfgang Thomas for their useful comments.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.tcs.2006.12.041">doi:10.1016/j.tcs.2006.12.041</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6jf4kttfrfgk7e4zhlq4cgr52q">fatcat:6jf4kttfrfgk7e4zhlq4cgr52q</a> </span>
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Logic and Rational Languages of Words Indexed by Linear Orderings

Nicolas Bedon, Alexis Bès, Olivier Carton, Chloé Rispal
<span title="2009-06-11">2009</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qj6jzeossfhihfshwf7nb7lbcq" style="color: black;">Theory of Computing Systems</a> </i> &nbsp;
We prove that every rational language of words indexed by linear orderings is definable in monadic second-order logic.  ...  As a corollary we prove that the inclusion problem for rational languages of words indexed by countable linear orderings is decidable.  ...  However, one can show that first-order definable languages of countable scattered linear orderings do not coincide any more with star-free and aperiodic ones [2, 22] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00224-009-9222-6">doi:10.1007/s00224-009-9222-6</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ql4zjihwwfb4hc4fbppyqflqmu">fatcat:ql4zjihwwfb4hc4fbppyqflqmu</a> </span>
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Omega Algebras and Regular Equations [chapter]

Michael R. Laurence, Georg Struth
<span title="">2011</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We instantiate these results in concrete models-languages, traces and relations-showing, for instance, that the omega captures precisely the empty word property in regular languages.  ...  We study a weak variant of omega algebra, where one of the usual star induction axioms is absent, in the context of recursive regular equations.  ...  To our knowledge, ours is the first proof that is entirely within first-order logic and which could be obtained by automated reasoning.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-21070-9_19">doi:10.1007/978-3-642-21070-9_19</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bswqjzgcyraspbt5623tkiszmi">fatcat:bswqjzgcyraspbt5623tkiszmi</a> </span>
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Algebraic notions of nontermination: Omega and divergence in idempotent semirings

Peter Höfner, Georg Struth
<span title="">2010</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/anq6kvwvqbegtipq5vfhvonzqq" style="color: black;">The Journal of Logic and Algebraic Programming</a> </i> &nbsp;
Two notions of nontermination are studied and compared in the setting of idempotent semirings: Cohen's omega operator and a divergence operator.  ...  It turns out that divergence yields a simple and natural way of modelling infinite behaviours of programs and discrete systems, whereas the omega operator shows some anomalies.  ...  on omega and divergence.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jlap.2010.07.016">doi:10.1016/j.jlap.2010.07.016</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/khxf6uhajraztonb4fl3m4eex4">fatcat:khxf6uhajraztonb4fl3m4eex4</a> </span>
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A Short Note on Infinite Union/Intersection of Omega Regular Languages [article]

Wanwei Liu
<span title="2016-03-17">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We in this paper show that omega regular languages are not closed under infinite union and intersection.  ...  As an attempt, we propose to add step variables and quantifiers to temporal logics to enhance the expressiveness of the underlying logic.  ...  Indeed, L (P(2)) = k∈N L (X 2k p), and this indicates that star-free languages are not closed under infinite intersection.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1603.05426v1">arXiv:1603.05426v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2zs4wi4kvndavosjfp7q7e7jtq">fatcat:2zs4wi4kvndavosjfp7q7e7jtq</a> </span>
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Automated Engineering of Relational and Algebraic Methods in Isabelle/HOL [chapter]

Simon Foster, Georg Struth, Tjark Weber
<span title="">2011</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Class near-semiring = plus + mult + assumes mult-assoc: (x ·y)·z = x ·(y·z ) and add-assoc : (x +y)+z = x +(y+z ) and add-comm : x +y = y+x and distr : (x +y)·z = x ·z +y·z Class near-dioid = near-semiring  ...  We demonstrate further benefits of Isabelle for hypothesis learning, duality reasoning, theorem instantiation, and reasoning across models and theories.  ...  We acknowledge funding from EPSRC grants EP/F067909/1 and EP/G031711/1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-21070-9_5">doi:10.1007/978-3-642-21070-9_5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/oszkpd2pfrbsnovtsp7mvn3sha">fatcat:oszkpd2pfrbsnovtsp7mvn3sha</a> </span>
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Programming and automating mathematics in the Tarski–Kleene hierarchy

Alasdair Armstrong, Georg Struth, Tjark Weber
<span title="">2014</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/75odiq25o5gtzcy4h7mdcztb5y" style="color: black;">Journal of Logical and Algebraic Methods in Programming</a> </i> &nbsp;
For Kleene algebras we show how models can be programmed, including sets of traces and paths, languages, binary relations, max-plus and min-plus algebra, matrices, formal power series.  ...  For relation algebras we discuss primarily proof automation in a comprehensive library and present an advanced formalisation example.  ...  Hitoshi Furusawa and Yasuo Kawahara have helped to clarify the presentation of this paper through several discussions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jlap.2014.02.001">doi:10.1016/j.jlap.2014.02.001</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jxygndlhxrheba6rninswkp65m">fatcat:jxygndlhxrheba6rninswkp65m</a> </span>
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Page 6534 of Mathematical Reviews Vol. , Issue 2003i [page]

<span title="">2003</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
that the star-free regular languages admit a characterization as models of a certain kind of first-order sentence.  ...  F-PARIS7-ML; Paris) Borel hierarchy and omega context free languages.  ... 
<span class="external-identifiers"> </span>
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Automating Algebraic Methods in Isabelle [chapter]

Walter Guttmann, Georg Struth, Tjark Weber
<span title="">2011</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
They subsume computational logics such as dynamic and Hoare logics and form a basis for various software development methods.  ...  We also automate a modal correspondence result and soundness and relative completeness proofs of propositional Hoare logic.  ...  A benefit of Kleene algebra, which is slightly weaker, is that it replaces higher-order inductive reasoning about powers and unbounded suprema by equational reasoning in first-order logic.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-24559-6_41">doi:10.1007/978-3-642-24559-6_41</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5jpfr7dazveyddlinhogksywjq">fatcat:5jpfr7dazveyddlinhogksywjq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170810112040/http://user.it.uu.se/~tjawe125/publications/guttmann11automating.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/e1/3e/e13e791024eac893e63cf9add82b60cd0ad29f71.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-24559-6_41"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

First-Order logic and its Infinitary Quantifier Extensions over Countable Words [article]

Bharat Adsul, Saptarshi Sarkar, A.V. Sreejith
<span title="2021-07-03">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We contribute to the refined understanding of the language-logic-algebra interplay in the context of first-order properties of countable words.  ...  We also explicate its role in view of other natural and classical logical systems such as WMSO and FO[cut] - an extension of FO where quantification over Dedekind-cuts is allowed.  ...  ⊔ We claim that both first order logic with cuts (FO[cut]) and weak monadic second order logic (WMSO) can define the languages definable in FO[∞]. Theorem 6.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2107.01468v1">arXiv:2107.01468v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/k5zie3jeibgkrfapfgqzdoazhe">fatcat:k5zie3jeibgkrfapfgqzdoazhe</a> </span>
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A Complete Axiom System for Propositional Interval Temporal Logic with Infinite Time [article]

Ben Moszkowski
<span title="2012-07-16">2012</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
of omega automata and nontrivial techniques for complementing them.  ...  Our completeness proof uses a reduction to completeness for PITL with finite time and conventional propositional linear-time temporal logic.  ...  Shirley Craig provided outstanding library services and deserves special mention.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1207.3816v1">arXiv:1207.3816v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/z4he73ipszesjnwec2ncmebpje">fatcat:z4he73ipszesjnwec2ncmebpje</a> </span>
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Beyond omega-Regular Languages

Mikolaj Bojanczyk, Marc Herbstritt
<span title="2010-03-09">2010</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/npfxqbtrtjcmndol3ijtu4shem" style="color: black;">Symposium on Theoretical Aspects of Computer Science</a> </i> &nbsp;
It can be described using logic, namely by monadic second-order logic, or the equivalent weak monadic-second order logic.  ...  The paper presents some automata and logics on ω-words, which capture all ω-regular languages, and yet still have good closure and decidability properties.  ...  Above, we have described several classes of languages of ω-words, defined by: the logics with new quantifiers and automata with counters.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.stacs.2010.2440">doi:10.4230/lipics.stacs.2010.2440</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/stacs/Bojanczyk10.html">dblp:conf/stacs/Bojanczyk10</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6oaslsh6kndxzixfk6zzpwqj7m">fatcat:6oaslsh6kndxzixfk6zzpwqj7m</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180514074431/http://drops.dagstuhl.de/opus/volltexte/2010/2440/pdf/1001.BojanczikNikolaj.2440.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/57/cc/57ccf49451b7d325b22e980367d0ad5b5826427d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.stacs.2010.2440"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Modal Tools for Separation and Refinement

Georg Struth
<span title="">2008</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/uy5mv2ncw5eahkdx47hkrglxmm" style="color: black;">Electronical Notes in Theoretical Computer Science</a> </i> &nbsp;
Equational axioms for divergence are introduced, and the concept is refined for different contexts. In particular it is related to Löb's formula, which describes termination in modal logics.  ...  Kleene modules and modal Kleene algebras are applied to automatically verify refinement laws for infinite loops and separation of termination.  ...  Acknowledgement I am very grateful to Nachum Dershowitz, Jules Desharnais and Bernhard Möller for inspiring discussions and suggestions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.entcs.2008.06.005">doi:10.1016/j.entcs.2008.06.005</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gzvmozk35nhzncfs5p5arvwos4">fatcat:gzvmozk35nhzncfs5p5arvwos4</a> </span>
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Monadic partition logics and finite automata

Enshao Shen, Qijia Tian
<span title="">1996</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
By argumenting first-order logic with monadic partition quantifiers we get a new family of extended logics.  ...  They are strictly weaker in expressive power than monadic second-order logic, have some good semantic features and offer better specification formalisms for studying finite automata and forlnal languages  ...  Acknowledgements The first author would like to propose naming the partition quantifiers as Ebbinghaus quantifiers, for his first proposition of them and pushing forward their study anonymously.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0304-3975(95)00113-1">doi:10.1016/0304-3975(95)00113-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jf5c7duqb5bt3g4nvtj3dfpjnm">fatcat:jf5c7duqb5bt3g4nvtj3dfpjnm</a> </span>
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