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22
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.
Coalgebraic automata theory: Basic results
 Logical Methods in Computer Science
"... Vol. 4 (4:10) 2008, pp. 1–43 www.lmcsonline.org ..."
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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Cited by 8 (2 self)
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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.
Relentful Strategic Reasoning in AlternatingTime Temporal Logic
 In International Conference on Logic for Programming Artificial Intelligence and Reasoning’10, LNAI 6355
, 2010
"... Temporal logics are a well investigated formalism for the specification, verification, and synthesis of reactive systems. Within this family, alternating temporal logic, ATL*, has been introduced as a useful generalization of classical linear and branchingtime temporal logics by allowing temporal ..."
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Cited by 7 (3 self)
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Temporal logics are a well investigated formalism for the specification, verification, and synthesis of reactive systems. Within this family, alternating temporal logic, ATL*, 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 past 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*. 1
Proof systems for the coalgebraic cover modality
 Same volume. Clemens Kupke, Alexander Kurz and Yde Venema
, 2008
"... abstract. We investigate an alternative presentation of classical and positive modal logic where the coalgebraic cover modality is taken as primitive. For each logic, we present a sound and complete Hilbertstyle axiomatization. Moreover, we give a twosided sound and complete sequent calculus for t ..."
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Cited by 6 (3 self)
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abstract. We investigate an alternative presentation of classical and positive modal logic where the coalgebraic cover modality is taken as primitive. For each logic, we present a sound and complete Hilbertstyle axiomatization. Moreover, we give a twosided sound and complete sequent calculus for the negationfree language, and for the language with negation we provide a onesided sequent calculus which is sound, complete and cutfree.
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 6 (3 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.
Nabla Algebras and Chu Spaces
"... Abstract. This paper is a study into some properties and applications of Moss ’ coalgebraic or ‘cover ’ modality ∇. First we present two axiomatizations of this operator, and we prove these axiomatizations to be sound and complete with respect to basic modal and positive modal logic, respectively. M ..."
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Cited by 4 (3 self)
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Abstract. This paper is a study into some properties and applications of Moss ’ coalgebraic or ‘cover ’ modality ∇. First we present two axiomatizations of this operator, and we prove these axiomatizations to be sound and complete with respect to basic modal and positive modal logic, respectively. More precisely, we introduce the notions of a modal ∇algebra and of a positive modal ∇algebra. We establish a categorical isomorphism between the category of modal ∇algebra and that of modal algebras, and similarly for positive modal ∇algebras and positive modal algebras. We then turn to a presentation, in terms of relation lifting, of the Vietoris hyperspace in topology. The key ingredient is an Flifting construction, for an arbitrary set functor F, on the category Chu of twovalued Chu spaces and Chu transforms, based on relation lifting. As a case study, we show how to realize the Vietoris construction on Stone spaces as a special instance of this Chu construction for the (finite) power set functor. Finally, we establish a tight connection with the axiomatization of the modal ∇algebras.
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 4 (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.
Quotientbased Control Synthesis for NonDeterministic Plants with MuCalculus Specifications
 In 45th IEEE Conference on Decision and Control
, 2006
"... Abstract — We study the control of a nondeterministic discrete event system (DES) subject to a control specification expressed in the propositional mucalculus, under complete observation of events. Given a plant automaton model and a mucalculus specification we provide a set of rules that computes ..."
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Cited by 3 (1 self)
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Abstract — We study the control of a nondeterministic discrete event system (DES) subject to a control specification expressed in the propositional mucalculus, under complete observation of events. Given a plant automaton model and a mucalculus specification we provide a set of rules that computes the “quotient” of the specification against the plant, which is another mucalculus formula such that a supervisor exists if and only if the quotiented formula is satisfiable. Thus the control problem is reduced to one of mucalculus satisfiability. We also present a tableaubased satisfiability solving algorithm that identifies a model for the quotiented formula. The resulting model serves as a supervisor. The complexity of supervisor existence verification as well as model synthesis is single exponential in the size of the plant as well as the size of the specification formula. I.