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20
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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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.
Coalgebraic Correspondence Theory
"... Abstract. We lay the foundations of a firstorder correspondence theory for coalgebraic logics that makes the transition structure explicit in the firstorder modelling. In particular, we prove a coalgebraic version of the van Benthem/Rosen theorem stating that both over arbitrary structures and ove ..."
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Abstract. We lay the foundations of a firstorder correspondence theory for coalgebraic logics that makes the transition structure explicit in the firstorder modelling. In particular, we prove a coalgebraic version of the van Benthem/Rosen theorem stating that both over arbitrary structures and over finite structures, coalgebraic modal logic is precisely the bisimulation invariant fragment of firstorder logic.
UNIVERSAL COALGEBRAS AND THEIR LOGICS
, 2009
"... ABSTRACT. We survey coalgebras as models of state based systems together with their global and local logics. We convey some useful intuition regarding Setfunctors which leads naturally to coalgebraic modal logic where modalities are validity patterns for the successor object of a state. 1. ..."
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ABSTRACT. We survey coalgebras as models of state based systems together with their global and local logics. We convey some useful intuition regarding Setfunctors which leads naturally to coalgebraic modal logic where modalities are validity patterns for the successor object of a state. 1.
Flat Coalgebraic Fixed Point Logics
"... Abstract. 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 ..."
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Abstract. 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. 1
Hybrid Logic with the Difference Modality for Generalisations of Graphs
"... We discuss recent work generalising the basic hybrid logic with the difference modality to any reasonable notion of transition. This applies equally to both subrelational transitions such as monotone neighbourhood frames or selection function models as well as those with more structure such as Marko ..."
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We discuss recent work generalising the basic hybrid logic with the difference modality to any reasonable notion of transition. This applies equally to both subrelational transitions such as monotone neighbourhood frames or selection function models as well as those with more structure such as Markov chains and alternating temporal frames. We provide a generic canonical cutfree sequent system and a terminating proofsearch strategy for the fragment without the difference modality but including the global modality. Keywords: Global Modality, Difference Modality, Coalgebraic Semantics, Cutfree Sequent System
Coalgebras, Stone Duality, Modal Logic
, 2006
"... A brief outline of the topics of the course could be as follows. Coalgebras generalise transition systems. Modal logics are the natural logics for coalgebras. Stone duality provides the relationship between coalgebras and modal logic. Furthermore, some basic category theory is needed to understand c ..."
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A brief outline of the topics of the course could be as follows. Coalgebras generalise transition systems. Modal logics are the natural logics for coalgebras. Stone duality provides the relationship between coalgebras and modal logic. Furthermore, some basic category theory is needed to understand coalgebras as well as Stone duality. So we
Address for Correspondence:
"... 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 coalg ..."
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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. This is achieved by constructing for a given modal logic a canonical coalgebraic semantics, consisting of a signature functor and interpretations of modal operators, which turns out to be final among all such structures. The canonical semantics may be seen as a coalgebraic reconstruction of neighbourhood semantics, broadly construed. A finitary restriction of the canonical semantics yields a canonical weakly complete semantics which moreover enjoys the HennessyMilner property. As a consequence, the machinery of coalgebraic modal logic, in particular generic decision procedures and upper complexity bounds, becomes applicable to arbitrary rank 1 modal logics, without regard to their semantic status; we thus obtain purely syntactic versions of such results. As an extended example, we apply our framework to recently defined deontic logics. In particular, our methods lead to the new result that these logics are strongly complete.
Coalgebras and Their Logics 1
, 2006
"... Some comments about the last Logic Column, on nominal logic. Pierre Lescanne points out ..."
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Some comments about the last Logic Column, on nominal logic. Pierre Lescanne points out