EXPTIME Tableaux for the Coalgebraic μ-Calculus

Cîrstea, Corina; Kupke, Clemens; Pattinson, Dirk

doi:10.1007/978-3-642-04027-6_15

Cited by 22 publications

(28 citation statements)

References 24 publications

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“…This assumption is generally made in the literature on the modal µ-calculus [15], but a recent paper [11] argues that in fact no algorithm is known that can perform the transformation in polynomial time. Therefore we formulate our EXPTIME-decidability result more restrictive than in [4]. We nevertheless conjecture that our tableau calculus can be used for proving EXPTIME-decidability for the full coalgebraic µ-calculus.…”

Section: Introductionmentioning

confidence: 95%

EXPTIME Tableaux for the Coalgebraic mu-Calculus

Cîrstea

Kupke

Pattinson

2011

Logical Methods in Computer Science

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Abstract. The coalgebraic approach to modal logic provides a uniform framework that captures the semantics of a large class of structurally different modal logics, including e.g. graded and probabilistic modal logics and coalition logic. In this paper, we introduce the coalgebraic µ-calculus, an extension of the general (coalgebraic) framework with fixpoint operators. Our main results are completeness of the associated tableau calculus and Exptime decidability for guarded formulas. Technically, this is achieved by reducing satisfiability to the existence of non-wellfounded tableaux, which is in turn equivalent to the existence of winning strategies in parity games. Our results are parametric in the underlying class of models and yield, as concrete applications, previously unknown complexity bounds for the probabilistic µ-calculus and for an extension of coalition logic with fixpoints.

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Section: Introductionmentioning

confidence: 95%

EXPTIME Tableaux for the Coalgebraic mu-Calculus

Cîrstea

Kupke

Pattinson

2011

Logical Methods in Computer Science

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“…However, all rule sets that we are aware of, in particular the rule sets that completely axiomatise the logics introduced in Example 1 satisfy an additional assumption: the set of conclusions of a rule can be polynomially encoded in terms of the size of the premise. This was used in [24] and applied to fixpoint logics in [4].…”

Section: Theorem 24 Suppose That (S R) Is a Finite Tableau System Amentioning

confidence: 99%

Optimal Tableau Algorithms for Coalgebraic Logics

Goré

Kupke

Pattinson

2010

Tools and Algorithms for the Construction and Analysis of Systems

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Abstract. Deciding whether a modal formula is satisfiable with respect to a given set of (global) assumptions is a question of fundamental importance in applications of logic in computer science. Tableau methods have proved extremely versatile for solving this problem for many different individual logics but they typically do not meet the known complexity bounds for the logics in question. Recently, it has been shown that optimality can be obtained for some logics while retaining practicality by using a technique called "global caching". Here, we show that global caching is applicable to all logics that can be equipped with coalgebraic semantics, for example, classical modal logic, graded modal logic, probabilistic modal logic and coalition logic. In particular, the coalgebraic approach also covers logics that combine these various features. We thus show that global caching is a widely applicable technique and also provide foundations for optimal tableau algorithms that uniformly apply to a large class of modal logics.

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“…This route is already well understood: it was noted in [6] that INL is a coalgebraic modal logic in a completely standard sense, and so the μ-calculus extension of INL is a coalgebraic modal μ-calculus as in [14,22]. Such coalgebraic μ-calculi have been quite extensively studied, with generic results on decidability and complexity, [11] and completeness [12,13]. But there are also other versions of modal fixpoint logics, often corresponding to fragments of μ-calculi.…”

Section: Introductionmentioning

confidence: 97%

A Propositional Dynamic Logic for Instantial Neighborhood Semantics

2018

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Abstract.We propose a new perspective on logics of computation by combining instantial neighborhood logic INL with bisimulation safe operations adapted from PDL. INL is a recent modal logic, based on an extended neighborhood semantics which permits quantification over individual neighborhoods plus their contents. This system has a natural interpretation as a logic of computation in open systems. Motivated by this interpretation, we show that a number of familiar program constructors can be adapted to instantial neighborhood semantics to preserve invariance for instantial neighborhood bisimulations, the appropriate bisimulation concept for INL. We also prove that our extended logic IPDL is a conservative extension of dual-free game logic, and its semantics generalizes the monotone neighborhood semantics of game logic. Finally, we provide a sound and complete system of axioms for IPDL, and establish its finite model property and decidability.

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EXPTIME Tableaux for the Coalgebraic μ-Calculus

Cited by 22 publications

References 24 publications

EXPTIME Tableaux for the Coalgebraic mu-Calculus

EXPTIME Tableaux for the Coalgebraic mu-Calculus

Optimal Tableau Algorithms for Coalgebraic Logics

A Propositional Dynamic Logic for Instantial Neighborhood Semantics

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