EXPTIME Tableaux for the Coalgebraic mu-Calculus

Cîrstea, Corina; Kupke, Clemens; Pattinson, Dirk

doi:10.2168/lmcs-7(3:3)2011

Cited by 28 publications

(65 citation statements)

References 30 publications

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“…Moreover, we considered decidability problems on regular trees. To this end, we developed an interpretation of a coalgebraic µ-calculus [CKP11] over regular trees that is parametric over the interpretation model and the syntax. Several formalizations or mechanizations of various modal logics in the context of proof assistants exists in the literature.…”

Section: Resultsmentioning

confidence: 99%

“…In this section, we introduce the syntax for a coalgebraic µ-calculus [CKP11] and define its semantics interpreted over an arbitrary T -coalgebra. The treatment of the underlying modal logic is parametric in the sense that modal operators are given by an arbitrary signature.…”

Section: Model-checking On Regular Treesmentioning

confidence: 99%

“…Instead of giving the syntax of the propositional µ-calculus, we follow the approach presented in [CKP11] which abstract away the set of modal operators into a signature Λ : Sig.…”

Section: Syntaxmentioning

confidence: 99%

“…Dans ce contexte, nous montrons comment un problème de terminaison peutêtre reformulé en problème de productivité. Dans un second temps, nous donnons une interprétation du µ-calcul coalgébrique [CKP11] sur les arbres réguliers. En particulier, nous prouvons la décidabilité de la satisfiabilité.…”

Section: Iii5 Transducteurs Descendantsà Observation Finieunclassified

“…Dans cette section, nous introduisons la syntaxe d'un µ-calcul coalgébrique [CKP11] et nous définissons sa sémantique interprétée sur une T -coalgèbre arbitraire. Le traitement de la logique modale sous-jacente est paramétrique, dans le sens où les opérateurs modaux sont donnés par une signature arbitraire.…”

Section: Iv2 Vérification De Modèles Sur Les Arbres Réguliersunclassified

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A Mechanized Theory of Regular Trees in Dependent Type Theory

Spadotti

2015

Interactive Theorem Proving

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Proof assistants are tools developed by computer scientists in order to ease formal reasoning. In this sense, they provide a framework to express statements and properties. Then, by using the proof rules of the underlying logic, theorems are proved and mechanically checked by the machine.Dependent type theory is a formalism which can serve as an alternative to set theory as a foundation for all mathematics. Type theory provides a unified framework for defining programs along with their data structures, and for expressing their properties. Concretely, this means that the same language is used to define programs, state their specifications, and express the proof of their soundness. Moreover, when the underlying logic is constructive, it is possible to extract a program from the proof term of its specification. Some type theories offer powerful reasoning principles such as induction for reasoning about finite objects, or coinduction for reasoning about infinite objects.Graphs are a ubiquitous data structure in computer science. They are used for giving semantics to various logic, for modeling computations, or for expressing relations between objects. The problem of representing graphs in dependent type theory can be quite challenging. Indeed, the main obstacle is that, in full generality, graphs can be circular structures. However, induction-the main reasoning principle in dependent type theory-fails to capture naturally such circularity. Indeed, inductive types are based around the notion of well-foundedness. Nevertheless, it is well-known that coalgebraic approaches are better-suited in order to reason about non well-founded structures, i.e., structures embedding some forms of circularity. As such, coinductive types may be used to define and reason about these infinite objects. The key idea is that circular structures can be thought of as infinite trees when cycles are unfolded infinitely.We are interested in the problem of mechanizing a theory of regular trees in dependent type theory. Informally, regular trees are characterized as the subset of infinite trees having the property that the set of their distinct subtrees is finite. As such, regular trees can be thought of as finite cyclic structures.In this thesis, we propose two formalizations of regular trees. The first one, based on coinduction, defines regular trees as a restriction of a coinductive type. The second one follows a syntactic approach, in the sense that regular trees are characterized as inductively defined cyclic terms, i.e., terms with back-pointers. We prove that these two representations are isomorphic.Then, we study the problem of defining transformations on trees that preserve the regularity property. To this end, we leverage the formalism of tree transducers as a tool to obtain a synctactic characterization of a subset of corecursive function definitions. Tree transducers are then interpreted back as tree morphisms preserving this regularity property.Finally, we study various decidability results through a mechanization of a coalge...

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Section: Resultsmentioning

confidence: 99%

Section: Model-checking On Regular Treesmentioning

confidence: 99%

“…Instead of giving the syntax of the propositional µ-calculus, we follow the approach presented in [CKP11] which abstract away the set of modal operators into a signature Λ : Sig.…”

Section: Syntaxmentioning

confidence: 99%

Section: Iii5 Transducteurs Descendantsà Observation Finieunclassified

Section: Iv2 Vérification De Modèles Sur Les Arbres Réguliersunclassified

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A Mechanized Theory of Regular Trees in Dependent Type Theory

Spadotti

2015

Interactive Theorem Proving

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Coalgebraic Logics & Duality

Kupke

2018

Coalgebraic Methods in Computer Science

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I will provide a brief introduction to coalgebraic modal logics and highlight a few central concepts concerning these logics. After that I will outline my current research in the area.This note is not a survey of coalgebraic logics such as [1,2]. Instead, I am going to highlight ideas that continue to be important for my research within the area. A leitmotif is the fundamental role played by duality theory.

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Optimal Satisfiability Checking for Arithmetic $$\mu $$-Calculi

Hausmann

Schröder

2019

Lecture Notes in Computer Science

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The coalgebraic µ-calculus provides a generic semantic framework for fixpoint logics with branching types beyond the standard relational setup, e.g. probabilistic, weighted, or game-based. Previous work on the coalgebraic µ-calculus includes an exponential time upper bound on satisfiability checking, which however requires a well-behaved set of tableau rules for the next-step modalities. Such rules are not available in all cases of interest, in particular ones involving either integer weights as in the graded µ-calculus, or real-valued weights in combination with non-linear arithmetic. In the present paper, we prove the same upper complexity bound under more general assumptions, specifically regarding the complexity of the (much simpler) satisfiability problem for the underlying so-called one-step logic, roughly described as the nesting-free next-step fragment of the logic. We also present a generic global caching algorithm that is suitable for practical use and supports on-the-fly satisfiability checking. Example applications include new exponential-time upper bounds for satisfiability checking in an extension of the graded µcalculus with Presburger arithmetic, as well as an extension of the (twovalued) probabilistic µ-calculus with polynomial inequalities. As a side result, we moreover obtain a new upper bound O(((nk)!) 2 ) on minimum model size for satisfiable formulas for all coalgebraic µ-calculi, where n is the size of the formula and k its alternation depth.

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EXPTIME Tableaux for the Coalgebraic mu-Calculus

Cited by 28 publications

References 30 publications

A Mechanized Theory of Regular Trees in Dependent Type Theory

A Mechanized Theory of Regular Trees in Dependent Type Theory

Coalgebraic Logics & Duality

Optimal Satisfiability Checking for Arithmetic $$\mu $$-Calculi

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