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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 10 (7 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.
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 9 (4 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.
Optimal Tableaux for Conditional Logics with Cautious Monotonicity
"... Conditional logics capture default entailment in a modal framework in which nonmonotonic implication is a firstclass citizen, and in particular can be negated and nested. There is a wide range of axiomatizations of conditionals in the literature, from weak systems such as the basic conditional lo ..."
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Cited by 6 (2 self)
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Conditional logics capture default entailment in a modal framework in which nonmonotonic implication is a firstclass citizen, and in particular can be negated and nested. There is a wide range of axiomatizations of conditionals in the literature, from weak systems such as the basic conditional logic CK, which allows only for equivalent exchange of conditional antecedents, to strong systems such as Burgess ’ system S, which imposes the full KrausLehmannMagidor properties of preferential logic. While tableaux systems implementing the actual complexity of the logic at hand have recently been developed for several weak systems, strong systems including in particular disjunction elimination or cautious monotonicity have so far eluded such efforts; previous results for strong systems are limited to semanticsbased decision procedures and completeness proofs for Hilbertstyle axiomatizations. Here, we present tableaux systems of optimal complexity PSPACE for several strong axiom systems in conditional logic, including system S; the arising decision procedure for system S is implemented in the generic reasoning tool CoLoSS.
Flat Coalgebraic Fixed Point Logics
"... Fixed point logics are widely used in computer science, in particular in artificial intelligence and concurrency. The most expressive logics of this type are the µcalculus and its relatives. However, popular fixed point logics tend to trade expressivity for simplicity and readability, and in fact ..."
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Fixed point logics are widely used in computer science, in particular in artificial intelligence and concurrency. The most expressive logics of this type are the µcalculus and its relatives. However, popular fixed point logics tend to trade expressivity for simplicity and readability, and in fact often live within the single variable fragment of the µcalculus. The family of such flat fixed point logics includes, e.g., CTL, the ∗nestingfree fragment of PDL, and the logic of common knowledge. Here, we extend this notion to the generic semantic framework of coalgebraic logic, thus covering a wide range of logics beyond the standard µcalculus including, e.g., flat fragments of the graded µcalculus and the alternatingtime µcalculus (such as ATL), as well as probabilistic and monotone fixed point logics. Our main results are completeness of the KozenPark axiomatization and a timedout tableaux method that matches EXPTIME upper bounds inherited from the coalgebraic µcalculus but avoids using automata.
The Logic of Exact Covers: Completeness and Uniform Interpolation
"... Abstract—We show that all (not necessarily normal or monotone) modal logics that can be axiomatised in rank1 have the interpolation property, and that in fact interpolation is uniform if the logics just have finitely many modal operators. As immediate applications, we obtain previously unknown inte ..."
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Abstract—We show that all (not necessarily normal or monotone) modal logics that can be axiomatised in rank1 have the interpolation property, and that in fact interpolation is uniform if the logics just have finitely many modal operators. As immediate applications, we obtain previously unknown interpolation theorems for a range of modal logics, containing probabilistic and graded modal logic, alternating temporal logic and some variants of conditional logic. Technically, this is achieved by translating to and from a new (coalgebraic) logic introduced in this paper, the logic of exact covers. It is interpreted over coalgebras for an endofunctor on the category of sets that also directly determines the syntax. Apart from closure under bisimulation quantifiers (and hence interpolation), we also provide a complete tableaux calculus and establish both the HennessyMilner and the small model property for this logic. I.
Coalgebraic Predicate Logic: Equipollence Results and Proof Theory
"... Abstract. The recently introduced Coalgebraic Predicate Logic (CPL) provides a general firstorder syntax together with extra modallike operators that are interpreted in a coalgebraic setting. The universality of the coalgebraic approach allows us to instantiate the framework to a wide variety of ..."
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Abstract. The recently introduced Coalgebraic Predicate Logic (CPL) provides a general firstorder syntax together with extra modallike operators that are interpreted in a coalgebraic setting. The universality of the coalgebraic approach allows us to instantiate the framework to a wide variety of situations, including probabilistic logic, coalition logic or the logic of neighbourhood frames. The last case generalises a logical setup proposed by C.C. Chang in early 1970’s. We provide further evidence of the naturality of this framework. We identify syntactically the fragments of CPL corresponding to extended modal formalisms and show that the full CPL is equipollent with coalgebraic hybrid logic with the downarrow binder and the universal modality. Furthermore, we initiate the study of structural proof theory for CPL by providing a sequent calculus and a cutelimination result. 1