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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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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.
ALGORITHMIC CORRESPONDENCE AND COMPLETENESS IN MODAL LOGIC. I. The Core Algorithm SQEMA
 CONSIDERED FOR PUBLICATION IN LOGICAL METHODS IN COMPUTER SCIENCE
, 2006
"... In terms of frame validity, modal formulae express universal monadic secondorder properties, but in many important cases these have firstorder equivalents. This is important for both logical and computational reasons, since firstorder logic is much better studied and behaved than monadic second ..."
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Cited by 20 (2 self)
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In terms of frame validity, modal formulae express universal monadic secondorder properties, but in many important cases these have firstorder equivalents. This is important for both logical and computational reasons, since firstorder logic is much better studied and behaved than monadic secondorder logic. Furthermore, firstorder definability often goes together with canonicity, which in turn implies framecompleteness of logics axiomatized with such formulae. Sahlqvist’s theorem is a general result on firstorder definability and canonicity of a large syntactic class of modal formulae. Sahlqvist’s approach was paralleled and further developed by van Benthem into the substitution method. Establishing firstorder definability of modal formulae amounts to elimination of secondorder quantifiers. Two algorithms have been developed and implemented for elimination of predicate quantifiers in secondorder logic: SCAN, based on a constraint resolution procedure, and DLS, based on a logical equivalence established by Ackermann. In this paper we introduce a new algorithm, SQEMA, for computing firstorder equivalents and proving canonicity of modal formulae. Like DLS, it uses (a modal version of) Ackermann’s lemma, but unlike both SCAN and DLS it works directly on modal formulae,
Coalgebraic automata theory: Basic results
 Logical Methods in Computer Science
"... Vol. 4 (4:10) 2008, pp. 1–43 www.lmcsonline.org ..."
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Closure properties of coalgebra automata
 in: LICS 2005 [1
"... We generalize some of the central results in automata theory to the abstraction level of coalgebras. In particular, we show that for any standard, weak pullback preserving functor F, the class of recognizable languages of Fcoalgebras is closed under taking unions, intersections and projections. ..."
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We generalize some of the central results in automata theory to the abstraction level of coalgebras. In particular, we show that for any standard, weak pullback preserving functor F, the class of recognizable languages of Fcoalgebras is closed under taking unions, intersections and projections. Our main technical result concerns a construction which transforms a given alternating Fautomaton into an equivalent nondeterministic one. 1.
Relentful Strategic Reasoning in AlternatingTime Temporal Logic
, 2012
"... Temporal logics are a well investigated formalism for the specification, verification, and synthesis of reactive systems. Within this family, AlternatingTime Temporal Logic (ATL ∗ , for short) has been introduced as a useful generalization of classical linear and branchingtime temporal logics, by ..."
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Temporal logics are a well investigated formalism for the specification, verification, and synthesis of reactive systems. Within this family, AlternatingTime Temporal Logic (ATL ∗ , for short) has been introduced as a useful generalization of classical linear and branchingtime temporal logics, by allowing temporal operators to be indexed by coalitions of agents. Classically, temporal logics are memoryless: once a path in the computation tree is quantified at a given node, the computation that has led to that node is forgotten. Recently, mCTL ∗ has been defined as a memoryful variant of CTL ∗ , where path quantification is memoryful. In the context of multiagent planning, memoryful quantification enables agents to “relent ” and change their goals and strategies depending on their history. In this paper, we define mATL ∗ , a memoryful extension of ATL ∗ , in which a formula is satisfied at a certain node of a path by taking into account both the future and the past. We study the expressive power of mATL ∗, its succinctness, as well as related decision problems. We also investigate the relationship between memoryful quantification and past modalities and show their equivalence. We show that both the memoryful and the past extensions come without any computational price; indeed, we prove that both the satisfiability and the modelchecking problems are 2EXPTIMECOMPLETE, as they are for ATL ∗.
Completions of µalgebras
 In Proceedings of the Twentieth Annual IEEE Symposium on Logic in Computer Science (LICS 2005
, 2005
"... A µalgebra is a model of a first order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms (f, µx.f) where µx.f is axiomatized as the least prefixed point of f, whose axioms are equations or equational implications. Standard µalgebras are complete meaning ..."
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A µalgebra is a model of a first order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms (f, µx.f) where µx.f is axiomatized as the least prefixed point of f, whose axioms are equations or equational implications. Standard µalgebras are complete meaning that their lattice reduct is a complete lattice. We prove that any non trivial quasivariety of µalgebras contains a µalgebra that has no embedding into a complete µalgebra. We focus then on modal µalgebras, i.e. algebraic models of the propositional modal µcalculus. We prove that free modal µalgebras satisfy a condition – reminiscent of Whitman’s condition for free lattices – which allows us to prove that (i) modal operators are adjoints on free modal µalgebras, (ii) least prefixed points of Σ1operations satisfy the constructive relation µx.f = W n≥0 f n (⊥). These properties imply the following statement: the MacNeilleDedekind completion of a free modal µalgebra is a complete modal µalgebra and moreover the canonical embedding preserves all the operations in the class Comp(Σ1, Π1) of the fixed point alternation hierarchy.
Improved Model Checking of Hierarchical Systems
, 2009
"... We present a unified gamebased approach for branchingtime model checking of hierarchical systems. Such systems are exponentially more succinct than standard statetransition graphs, as repeated subsystems are described only once. Early work on model checking of hierarchical systems shows that one ..."
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Cited by 11 (7 self)
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We present a unified gamebased approach for branchingtime model checking of hierarchical systems. Such systems are exponentially more succinct than standard statetransition graphs, as repeated subsystems are described only once. Early work on model checking of hierarchical systems shows that one can do better than a naive algorithm that “flattens ” the system and removes the hierarchy. Given a hierarchical system S and a branchingtime specification ψ for it, we reduce the modelchecking problem (does S satisfy ψ?) to the problem of solving a hierarchical game obtained by taking the product of S with an alternating tree automaton Aψ for ψ. Our approach leads to clean, uniform, and improved modelchecking algorithms for a variety of branchingtime temporal logics. In particular, by improving the algorithm for solving hierarchical parity games, we are able to solve the modelchecking problem for the µcalculus in Pspace and time complexity that is only polynomial in the depth of the hierarchy. Our approach also leads to an abstractionrefinement paradigm for hierarchical systems. The abstraction maintains the hierarchy, and is obtained by merging both states and subsystems into abstract states.
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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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.
Completeness of the finitary Moss logic
 In Areces and Goldblatt [3
"... abstract. We give a sound and complete derivation system for the valid formulas in the finitary version of Moss ’ coalgebraic logic, for coalgebras of arbitrary type. ..."
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abstract. We give a sound and complete derivation system for the valid formulas in the finitary version of Moss ’ coalgebraic logic, for coalgebras of arbitrary type.