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PSPACE bounds for rank 1 modal logics
 IN LICS’06
, 2006
"... For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank1 logics enjoy a sh ..."
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Cited by 37 (19 self)
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For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE. This leads to a unified derivation of tight PSPACEbounds for a number of logics including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant prooftheoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.
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 36 (17 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.
Modular algorithms for heterogeneous modal logics
 IN AUTOMATA, LANGUAGES AND PROGRAMMING, ICALP 07, VOL. 4596 OF LNCS
, 2007
"... Statebased systems and modal logics for reasoning about them often heterogeneously combine a number of features such as nondeterminism and probabilities. Here, we show that the combination of features can be reflected algorithmically and develop modular decision procedures for heterogeneous modal ..."
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Cited by 22 (15 self)
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Statebased systems and modal logics for reasoning about them often heterogeneously combine a number of features such as nondeterminism and probabilities. Here, we show that the combination of features can be reflected algorithmically and develop modular decision procedures for heterogeneous modal logics. The modularity is achieved by formalising the underlying statebased systems as multisorted coalgebras and associating both a logical and an algorithmic description to a number of basic building blocks. Our main result is that logics arising as combinations of these building blocks can be decided in polynomial space provided that this is the case for the components. By instantiating the general framework to concrete cases, we obtain PSPACE decision procedures for a wide variety of structurally different logics, describing e.g. Segala systems and games with uncertain information.
Rank1 modal logics are coalgebraic
 IN STACS 2007, 24TH ANNUAL SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE, PROCEEDINGS
, 2007
"... Coalgebras provide a unifying semantic framework for a wide variety of modal logics. It has previously been shown that the class of coalgebras for an endofunctor can always be axiomatised in rank 1. Here we establish the converse, i.e. every rank 1 modal logic has a sound and strongly complete coal ..."
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Cited by 21 (14 self)
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Coalgebras provide a unifying semantic framework for a wide variety of modal logics. It has previously been shown that the class of coalgebras for an endofunctor can always be axiomatised in rank 1. Here we establish the converse, i.e. every rank 1 modal logic has a sound and strongly complete coalgebraic semantics. As a consequence, recent results on coalgebraic modal logic, in particular generic decision procedures and upper complexity bounds, become applicable to arbitrary rank 1 modal logics, without regard to their semantic status; we thus obtain purely syntactic versions of these results. As an extended example, we apply our framework to recently defined deontic logics.
Coalgebraic hybrid logic
 IN FOUNDATIONS OF SOFTWARE SCIENCE AND COMPUTATION STRUCTURES, FOSSACS 09, VOLUME 5504 OF LNCS
, 2009
"... We introduce a generic framework for hybrid logics, i.e. modal logics additionally featuring nominals and satisfaction operators, thus providing the necessary facilities for reasoning about individual states in a model. This framework, coalgebraic hybrid logic, works at the same level of generality ..."
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Cited by 13 (10 self)
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We introduce a generic framework for hybrid logics, i.e. modal logics additionally featuring nominals and satisfaction operators, thus providing the necessary facilities for reasoning about individual states in a model. This framework, coalgebraic hybrid logic, works at the same level of generality as coalgebraic modal logic, and in particular subsumes, besides normal hybrid logics such as hybrid K, a wide variety of logics with nonnormal modal operators such as probabilistic, graded, or coalitional modalities and nonmonotonic conditionals. We prove a generic finite model property and an ensuing weak completeness result, and we give a semantic criterion for decidability in PSPACE. Moreover, we present a fully internalised PSPACE tableau calculus. These generic results are easily instantiated to particular hybrid logics and thus yield a wide range of new results, including e.g. decidability in PSPACE of probabilistic and graded hybrid logics.
Beyond rank 1: Algebraic semantics and finite models for coalgebraic logics
, 2008
"... Coalgebras provide a uniform framework for the semantics of a large class of (mostly nonnormal) modal logics, including e.g. monotone modal logic, probabilistic and graded modal logic, and coalition logic, as well as the usual Kripke semantics of modal logic. In earlier work, the finite model prop ..."
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Cited by 13 (9 self)
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Coalgebras provide a uniform framework for the semantics of a large class of (mostly nonnormal) modal logics, including e.g. monotone modal logic, probabilistic and graded modal logic, and coalition logic, as well as the usual Kripke semantics of modal logic. In earlier work, the finite model property for coalgebraic logics has been established w.r.t. the class of all structures appropriate for a given logic at hand; the corresponding modal logics are characterised by being axiomatised in rank 1, i.e. without nested modalities. Here, we extend the range of coalgebraic techniques to cover logics that impose global properties on their models, formulated as frame conditions with possibly nested modalities on the logical side (in generalisation of frame conditions such as symmetry or transitivity in the context of Kripke frames). We show that the finite model property for such logics follows from the finite algebra property of the associated class of complex algebras, and then investigate sufficient conditions for the finite algebra property to hold. Example applications include extensions of coalition logic and logics of uncertainty and knowledge.
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 11 (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.
CoLoSS: The Coalgebraic Logic Satisfiability Solver (System Description)
, 2007
"... CoLoSS, the Coalgebraic Logic Satisfiability Solver, decides satisfiability of modal formulas in a generic and compositional way. It implements a uniform polynomial space algorithm to decide satisfiability for modal logics that are amenable to coalgebraic semantics. This includes e.g. the logics K, ..."
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Cited by 9 (8 self)
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CoLoSS, the Coalgebraic Logic Satisfiability Solver, decides satisfiability of modal formulas in a generic and compositional way. It implements a uniform polynomial space algorithm to decide satisfiability for modal logics that are amenable to coalgebraic semantics. This includes e.g. the logics K, KD, Pauly’s coalition logic, graded modal logic, and probabilistic modal logic. Logics are easily integrated into CoLoSS by providing a complete axiomatisation of their coalgebraic semantics in a specific format. Moreover, CoLoSS is compositional: it synthesises decision procedures for modular combinations of logics that include the fusion of two modal logics as a special case. One thus automatically obtains reasoning support e.g. for logics interpreted over probabilistic automata that combine nondeterminism and probabilities in different ways.
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.
How Many Toes Do I Have? Parthood and Number Restrictions in Description Logics
"... The modelling of parthood relations in description logics via transitive roles often leads to undecidability when combined with number restrictions and role hierarchies. Here, we introduce the description logic PHQ that explicitly supports reasoning about parthood in the presence of qualified number ..."
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Cited by 6 (3 self)
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The modelling of parthood relations in description logics via transitive roles often leads to undecidability when combined with number restrictions and role hierarchies. Here, we introduce the description logic PHQ that explicitly supports reasoning about parthood in the presence of qualified number restrictions. Our main results are completeness and decidability in NEXPTIME. Conceptually, we argue that PHQ provides a better semantic fit for many applications: more often than not, parthoods occurring e.g. in biomedical ontologies are expected to be treelike. In such cases, PHQ supports stronger inferences than standard description logics. Technically this is achieved by explicitly excluding the merging of descendants, which, at the same time, eliminates the prime source of undecidability. We work in the general setting of coalgebraic modal logic, a generic semantic framework for notnecessarilynormal modal logics. This added generality allows the reuse of many of our results for other logics of sometimes quite different flavour.