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16
ManySorted Coalgebraic Modal Logic: a Modeltheoretic Study
 Theoretical Informatics and Applications
, 2001
"... This paper gives a semantical underpinning for a manysorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in objectoriented languages. These systems will be described as coalgebras of socalled polynomial functors, built up from constants an ..."
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Cited by 53 (3 self)
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This paper gives a semantical underpinning for a manysorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in objectoriented languages. These systems will be described as coalgebras of socalled polynomial functors, built up from constants and identities, using products, coproducts and powersets. The semantical account involves Boolean algebras with operators indexed by polynomial functors, called MBAOs, for Manysorted Boolean Algebras with Operators, combining standard (categorical) models of modal logic and of manysorted predicate logic.
Behavioural Differential Equations: A Coinductive Calculus of Streams, Automata, and Power Series
, 2000
"... Streams, (automata and) languages, and formal power series are viewed coalgebraically. In summary, this amounts to supplying these sets with a deterministic automaton structure, which has the universal property of being final. Finality then forms the basis for both definitions and proofs by coinduct ..."
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Cited by 52 (17 self)
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Streams, (automata and) languages, and formal power series are viewed coalgebraically. In summary, this amounts to supplying these sets with a deterministic automaton structure, which has the universal property of being final. Finality then forms the basis for both definitions and proofs by coinduction, the coalgebraic counterpart of induction. Coinductive definitions take the shape of what we have called behavioural differential equations, after Brzozowski's notion of input derivative. A calculus is developed for coinductive reasoning about all of the afore mentioned structures, closely resembling (and at times generalising) calculus from classical analysis. 2000 Mathematics Subject Classification: 68Q10, 68Q55, 68Q85 1998 ACM Computing Classification System: F.1, F.3 Keywords & Phrases: Coalgebra, automaton, finality, coinduction, stream, formal language, formal power series, differential equation, input derivative, behaviour, semiring, maxplus algebra 1 Contents 1 Introductio...
Semantical Principles in the Modal Logic of Coalgebraic
"... Coalgebras for a functor on the category of sets subsume many formulations of the notion of transition system, including labelled transition systems, Kripke models, Kripke frames and many types of automata. This paper presents a multimodal language which is bisimulation invariant and (under a natur ..."
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Cited by 30 (6 self)
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Coalgebras for a functor on the category of sets subsume many formulations of the notion of transition system, including labelled transition systems, Kripke models, Kripke frames and many types of automata. This paper presents a multimodal language which is bisimulation invariant and (under a natural completeness condition) expressive enough to characterise elements of the underlying state space up to bisimulation. Like Moss' coalgebraic logic, the theory can be applied to an arbitrary signature functor on the category of sets. Also, an upper bound for the size of conjunctions and disjunctions needed to obtain characteristic formulas is given.
Expressive Logics for Coalgebras via Terminal Sequence Induction
 Notre Dame J. Formal Logic
, 2002
"... This paper introduces the proof principle of terminal sequence induction and shows, how terminal sequence induction can be used to obtain expressiveness results for logics, interpreted over coalgebras. ..."
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Cited by 26 (8 self)
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This paper introduces the proof principle of terminal sequence induction and shows, how terminal sequence induction can be used to obtain expressiveness results for logics, interpreted over coalgebras.
Simulations in Coalgebra
 THEOR. COMP. SCI
, 2003
"... A new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive. Such an order forms a basic building block for a "lax relation lifting", or "relator" as used by other authors. Simulations appear as coalgebras of this lifted functor, an ..."
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Cited by 17 (1 self)
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A new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive. Such an order forms a basic building block for a "lax relation lifting", or "relator" as used by other authors. Simulations appear as coalgebras of this lifted functor, and similarity as greatest simulation. Twoway similarity is then similarity in both directions. In general, it is different from bisimilarity (in the usual coalgebraic sense), but a su#cient condition is formulated (and illustrated) to ensure that bisimilarity and twoway similarity coincide. Also, suitable conditions are identified which ensures that similarity on a final coalgebra forms an (algebraic) dcpo structure. This involves a close investigation of the iterated applications F (#) and F (1) of a functor F with an order to the initial and final sets.
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 ..."
Coalgebras and Monads in the Semantics of Java
 Theoretical Computer Science
, 2002
"... This paper describes the basic structures in the denotational and axiomatic semantics of sequential Java, both from a monadic and a coalgebraic perspective. This semantics is an abstraction of the one used for the verification of (sequential) Java programs using proof tools in the LOOP project at th ..."
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Cited by 4 (0 self)
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This paper describes the basic structures in the denotational and axiomatic semantics of sequential Java, both from a monadic and a coalgebraic perspective. This semantics is an abstraction of the one used for the verification of (sequential) Java programs using proof tools in the LOOP project at the University of Nijmegen. It is shown how the monadic perspective gives rise to the relevant computational structure in Java (composition, extension and repetition), and how the coalgebraic perspective o#ers an associated program logic (with invariants, bisimulations, and Hoare logics) for reasoning about the computational structure provided by the monad.
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.