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23
Coalgebraic modal logic: Soundness, completeness and decidability of local consequence
 Theoret. Comput. Sci
, 2002
"... This paper studies finitary modal logics, interpreted over coalgebras for an endofunctor, and establishes soundness, completeness and decidability results. The logics are studied within the abstract framework of coalgebraic modal logic, which can be instantiated with arbitrary endofunctors on the ca ..."
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Cited by 51 (24 self)
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This paper studies finitary modal logics, interpreted over coalgebras for an endofunctor, and establishes soundness, completeness and decidability results. The logics are studied within the abstract framework of coalgebraic modal logic, which can be instantiated with arbitrary endofunctors on the category of sets. This is achieved through the use of predicate liftings, which generalise atomic propositions and modal operators from Kripke models to arbitrary coalgebras. Predicate liftings also allow us to use induction along the terminal sequence of the underlying endofunctor as a proof principle. This induction principle is systematically exploited to establish soundness, completeness and decidability of the logics. We believe that this induction principle also opens new ways for reasoning about modal logics: Our proof of completeness does not rely on a canonical model construction, and the proof of the finite model property does not use filtrations. 1
Expressivity of coalgebraic modal logic: The limits and beyond
 IN FOUNDATIONS OF SOFTWARE SCIENCE AND COMPUTATION STRUCTURES, VOLUME 3441 OF LNCS
, 2005
"... Modal logic has a good claim to being the logic of choice for describing the reactive behaviour of systems modeled as coalgebras. Logics with modal operators obtained from socalled predicate liftings have been shown to be invariant under behavioral equivalence. Expressivity results stating that, c ..."
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Cited by 39 (13 self)
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Modal logic has a good claim to being the logic of choice for describing the reactive behaviour of systems modeled as coalgebras. Logics with modal operators obtained from socalled predicate liftings have been shown to be invariant under behavioral equivalence. Expressivity results stating that, conversely, logically indistinguishable states are behaviorally equivalent depend on the existence of separating sets of predicate liftings for the signature functor at hand. Here, we provide a classification result for predicate liftings which leads to an easy criterion for the existence of such separating sets, and we give simple examples of functors that fail to admit expressive normal or monotone modal logics, respectively, or in fact an expressive (unary) modal logic at all. We then move on to polyadic modal logic, where modal operators may take more than one argument formula. We show that every accessible functor admits an expressive polyadic modal logic. Moreover, expressive polyadic modal logics are, unlike unary modal logics, compositional.
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 25 (15 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 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.
Automata and fixed point logics: a coalgebraic perspective
 Electronic Notes in Theoretical Computer Science
, 2004
"... This paper generalizes existing connections between automata and logic to a coalgebraic level. Let F: Set → Set be a standard functor that preserves weak pullbacks. We introduce various notions of Fautomata, devices that operate on pointed Fcoalgebras. The criterion under which such an automaton a ..."
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Cited by 15 (7 self)
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This paper generalizes existing connections between automata and logic to a coalgebraic level. Let F: Set → Set be a standard functor that preserves weak pullbacks. We introduce various notions of Fautomata, devices that operate on pointed Fcoalgebras. The criterion under which such an automaton accepts or rejects a pointed coalgebra is formulated in terms of an infinite twoplayer graph game. We also introduce a language of coalgebraic fixed point logic for Fcoalgebras, and we provide a game semantics for this language. Finally we show that any formula p of the language can be transformed into an Fautomaton Ap which is equivalent to p in the sense that Ap accepts precisely those pointed Fcoalgebras in which p holds.
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 14 (11 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.
NonDeterministic Kleene Coalgebras
"... In this paper, we present a systematic way of deriving (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of systems. This generalizes both the results of Kleene (on regular languages and deterministic finite automata) and Miln ..."
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Cited by 13 (4 self)
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In this paper, we present a systematic way of deriving (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of systems. This generalizes both the results of Kleene (on regular languages and deterministic finite automata) and Milner (on regular behaviours and finite labelled transition systems), and includes many other systems such as Mealy and Moore machines.
Coalgebraic modal logic beyond Sets
 In MFPS XXIII
, 2007
"... Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be ..."
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Cited by 10 (3 self)
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Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be
A Kleene theorem for polynomial coalgebras
 In Foundations of Software Science and Computational Structures, 12th International Conference, FOSSACS 2009, volume 5504 of LNCS
, 2009
"... Abstract. For polynomial functors G, we show how to generalize the classical notion of regular expression to Gcoalgebras. We introduce a language of expressions for describing elements of the final Gcoalgebra and, analogously to Kleene’s theorem, we show the correspondence between expressions and ..."
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Cited by 10 (3 self)
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Abstract. For polynomial functors G, we show how to generalize the classical notion of regular expression to Gcoalgebras. We introduce a language of expressions for describing elements of the final Gcoalgebra and, analogously to Kleene’s theorem, we show the correspondence between expressions and finite Gcoalgebras. 1