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23
A Finite Model Construction For Coalgebraic Modal Logic
"... In recent years, a tight connection has emerged between modal logic on the one hand and coalgebras, understood as generic transition systems, on the other hand. Here, we prove that (finitary) coalgebraic modal logic has the finite model property. This fact not only reproves known completeness result ..."
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Cited by 24 (16 self)
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In recent years, a tight connection has emerged between modal logic on the one hand and coalgebras, understood as generic transition systems, on the other hand. Here, we prove that (finitary) coalgebraic modal logic has the finite model property. This fact not only reproves known completeness results for coalgebraic modal logic, which we push further by establishing that every coalgebraic modal logic admits a complete axiomatization of rank 1; it also enables us to establish a generic decidability result and a first complexity bound. Examples covered by these general results include, besides standard HennessyMilner logic, graded modal logic and probabilistic modal logic.
Admissibility of Cut in Coalgebraic Logics
 CMCS
, 2008
"... We study sequent calculi for propositional modal logics, interpreted over coalgebras, with admissibility of cut being the main result. As applications we present a new proof of the (already known) interpolation property for coalition logic and establish the interpolation property for the conditional ..."
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Cited by 8 (7 self)
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We study sequent calculi for propositional modal logics, interpreted over coalgebras, with admissibility of cut being the main result. As applications we present a new proof of the (already known) interpolation property for coalition logic and establish the interpolation property for the conditional logics CK and CK Id.
EXPTIME tableaux for the coalgebraic µ calculus
 Proc. CSL 2009, volume 5771 of Lecture Notes in Computer Science
, 2009
"... 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 µcalculus, an ext ..."
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Cited by 5 (3 self)
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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 µcalculus, an extension of the general (coalgebraic) framework with fixpoint operators. Our main results are completeness of the associated tableau calculus and EXPTIME decidability. Technically, this is achieved by reducing satisfiability to the existence of nonwellfounded 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 µcalculus and for an extension of coalition logic with fixpoints. 1
Generic modal cut elimination applied to conditional logics
 AUTOMATED REASONING WITH ANALYTIC TABLEAUX AND RELATED METHODS, TABLEAUX 2009, LECT. NOTES COMPUT. SCI
, 2009
"... We develop a general criterion for cut elimination in sequent calculi for propositional modal logics, which rests on absorption of cut, contraction, weakening and inversion by the purely modal part of the rule system. Our criterion applies also to a wide variety of logics outside the realm of normal ..."
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Cited by 4 (2 self)
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We develop a general criterion for cut elimination in sequent calculi for propositional modal logics, which rests on absorption of cut, contraction, weakening and inversion by the purely modal part of the rule system. Our criterion applies also to a wide variety of logics outside the realm of normal modal logic. We give extensive example instantiations of our framework to various conditional logics. For these, we obtain fully internalised calculi which are substantially simpler than those known in the literature, along with leaner proofs of cut elimination and complexity. In one case, conditional logic with modus ponens and conditional excluded middle, cut elimination and complexity are explicitly stated as open in the literature.
Cut Elimination in Coalgebraic Logics
"... We give two generic proofs for cut elimination in propositional modal logics, interpreted over coalgebras. We first investigate semantic coherence conditions between the axiomatisation of a particular logic and its coalgebraic semantics that guarantee that the cutrule is admissible in the ensuing s ..."
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Cited by 4 (4 self)
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We give two generic proofs for cut elimination in propositional modal logics, interpreted over coalgebras. We first investigate semantic coherence conditions between the axiomatisation of a particular logic and its coalgebraic semantics that guarantee that the cutrule is admissible in the ensuing sequent calculus. We then independently isolate a purely syntactic property of the set of modal rules that guarantees cut elimination. Apart from the fact that cut elimination holds, our main result is that the syntactic and semantic assumptions are equivalent in case the logic is amenable to coalgebraic semantics. As applications we present a new proof of the (already known) interpolation property for coalition logic and newly establish the interpolation property for the conditional logics CK and CK + ID.
Modal Logics are Coalgebraic
, 2008
"... Applications of modal logics are abundant in computer science, and a large number of structurally different modal logics have been successfully employed in a diverse spectrum of application contexts. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large vari ..."
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Cited by 4 (0 self)
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Applications of modal logics are abundant in computer science, and a large number of structurally different modal logics have been successfully employed in a diverse spectrum of application contexts. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large variety of specific logics used in particular domains. The coalgebraic approach is generic and compositional: tools and techniques simultaneously apply to a large class of application areas and can moreover be combined in a modular way. In particular, this facilitates a pickandchoose approach to domain specific formalisms, applicable across the entire scope of application areas, leading to generic software tools that are easier to design, to implement, and to maintain. This paper substantiates the authors ’ firm belief that the systematic exploitation of the coalgebraic nature of modal logic will not only have impact on the field of modal logic itself but also lead to significant progress in a number of areas within computer science, such as knowledge representation and concurrency/mobility.
On modal logics of linear inequalities
 Proc. AiML 2010
, 2010
"... We consider probabilistic modal logic, graded modal logic and stochastic modal logic, where linear inequalities may be used to express numerical constraints between quantities. For each of the logics, we construct a cutfree sequent calculus and show soundness with respect to a natural class of mode ..."
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Cited by 4 (1 self)
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We consider probabilistic modal logic, graded modal logic and stochastic modal logic, where linear inequalities may be used to express numerical constraints between quantities. For each of the logics, we construct a cutfree sequent calculus and show soundness with respect to a natural class of models. The completeness of the associated sequent calculi is then established with the help of coalgebraic semantics which gives completeness over a (typically much smaller) class of models. With respect to either semantics, it follows that the satisfiability problem of each of these logics is decidable in polynomial space. Keywords: Probabilistic modal logic, graded modal logic, linear inequalities
Logic of Infons: the Propositional Case
"... Infons are statements viewed as containers of information (rather then representations of truth values). The logic of infons turns out to be a conservative extension of logic known as constructive or intuitionistic. Distributed Knowledge Authorization Language uses additional unary connectives “p sa ..."
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Cited by 3 (0 self)
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Infons are statements viewed as containers of information (rather then representations of truth values). The logic of infons turns out to be a conservative extension of logic known as constructive or intuitionistic. Distributed Knowledge Authorization Language uses additional unary connectives “p said ” and “p implied ” where p ranges over principals. Here we investigate infon logic and a narrow but useful primal fragment of it. In both cases, we develop model theory and analyze the derivability problem: Does the given query follow from the given hypotheses? Our more involved technical results are on primal infon logic. We construct an algorithm for the multiple derivability problem: Which of the given queries follow from the given hypotheses? Given a bound on the quotation depth of the hypotheses, the algorithm runs in linear time. We quickly discuss the significance of this result for access control.
ON THE COMPLEXITY OF ELEMENTARY MODAL LOGICS
"... Modal logics are widely used in computer science. The complexity of modal satisfiability problems has been investigated since the 1970s, usually proving results on a casebycase basis. We prove a very general classification for a wide class of relevant logics: Many important subclasses of modal lo ..."
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Cited by 3 (0 self)
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Modal logics are widely used in computer science. The complexity of modal satisfiability problems has been investigated since the 1970s, usually proving results on a casebycase basis. We prove a very general classification for a wide class of relevant logics: Many important subclasses of modal logics can be obtained by restricting the allowed models with firstorder Horn formulas. We show that the satisfiability problem for each of these logics is either NPcomplete or PSPACEhard, and exhibit a simple classification criterion. Further, we prove matching PSPACE upper bounds for many of the PSPACEhard logics.
Nominals for Everyone
"... It has been recognised that the expressivity of description logics benefits from the introduction of nonstandard modal operators beyond existential and number restrictions. Such operators support notions such as uncertainty, defaults, agency, obligation, or evidence, whose semantics often lies outs ..."
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Cited by 2 (2 self)
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It has been recognised that the expressivity of description logics benefits from the introduction of nonstandard modal operators beyond existential and number restrictions. Such operators support notions such as uncertainty, defaults, agency, obligation, or evidence, whose semantics often lies outside the realm of relational structures. Coalgebraic hybrid logic serves as a unified setting for logics that combine nonstandard modal operators and nominals, which allow reasoning about individuals. In this framework, we prove a generic EXPTIME upper bound for concept satisfiability over general TBoxes, which instantiates to novel upper bounds for many individual logics including probabilistic logic with nominals.