Coalgebraic Modal Logic Beyond Sets

Klin, Bartek

doi:10.1016/j.entcs.2007.02.034

Cited by 38 publications

(56 citation statements)

References 26 publications

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“…The process of taking a finite set preserving functor and extending it to BA, and hence to Stone, is related to a construction in Worrell [56] where a set-functor is STRONGLY COMPLETE LOGICS FOR COALGEBRAS 5 lifted to complete ultrametric spaces. Klin [27] generalises the expressivity result of [51] working essentially with the same adjunction as in Diagram 1.2.…”

Section: Introductionmentioning

confidence: 99%

Strongly Complete Logics for Coalgebras

Kurz¹,

Rosický²

2012

Logical Methods in Computer Science

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Abstract. Coalgebras for a functor model different types of transition systems in a uniform way. This paper focuses on a uniform account of finitary logics for set-based coalgebras. In particular, a general construction of a logic from an arbitrary set-functor is given and proven to be strongly complete under additional assumptions. We proceed in three parts.Part I argues that sifted colimit preserving functors are those functors that preserve universal algebraic structure. Our main theorem here states that a functor preserves sifted colimits if and only if it has a finitary presentation by operations and equations. Moreover, the presentation of the category of algebras for the functor is obtained compositionally from the presentations of the underlying category and of the functor.Part II investigates algebras for a functor over ind-completions and extends the theorem of Jónsson and Tarski on canonical extensions of Boolean algebras with operators to this setting.Part III shows, based on Part I, how to associate a finitary logic to any finite-sets preserving functor T . Based on Part II we prove the logic to be strongly complete under a reasonable condition on T .

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Section: Introductionmentioning

confidence: 99%

Strongly Complete Logics for Coalgebras

Kurz¹,

Rosický²

2012

Logical Methods in Computer Science

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“…This is exemplified by the slogan in [36]: 'modal logic is to coalgebras what equational logic is to algebras'. Inspired by coalgebras on Stone spaces and the corresponding modal logic, recent developments [5,6,31,32,34,37,45] have identified the following situation as the essential mathematical structure underlying modal logics for coalgebras.…”

mentioning

confidence: 99%

Generic Trace Semantics via Coinduction

Hasuo¹,

Jacobs²,

Sokolova³

2007

Logical Methods in Computer Science

122

321

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Abstract. Trace semantics has been defined for various kinds of state-based systems, notably with different forms of branching such as non-determinism vs. probability. In this paper we claim to identify one underlying mathematical structure behind these "trace semantics," namely coinduction in a Kleisli category. This claim is based on our technical result that, under a suitably order-enriched setting, a final coalgebra in a Kleisli category is given by an initial algebra in the category Sets. Formerly the theory of coalgebras has been employed mostly in Sets where coinduction yields a finer process semantics of bisimilarity. Therefore this paper extends the application field of coalgebras, providing a new instance of the principle "process semantics via coinduction."

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“…However, while algebras and coalgebras do not completely constrain each other, and varying the algebras allows capturing the linear time/branching time spectrum, they were specified within the same category, with their semantical connection hardwired through distributivity requirements precluding, e.g., capturing the language hierarchies. On the other hand, semantic connections of algebras and coalgebras were studied in a wide variety of frameworks, and by a wide variety of techniques [25,21,23, to mention just a few], and it is possible that the language hierarchies, language acceptance, and computability and complexity concepts could have been captured in that framework if the community moved in that direction. The language acceptance relation and the trace equivalences have in fact been captured in [19], but by combining coalgebras and monads, which seem to indirectly capture the underlying semantical connection.…”

Section: Background: Semantic Connections Of Algebras and Coalgebrasmentioning

confidence: 99%

Smooth coalgebra: testing vector analysis

Pavlović

Fauser

2015

Math. Struct. Comp. Sci.

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Processes are often viewed as coalgebras, with the structure maps specifying the state transitions. In the simplest case, the state spaces are discrete, and the structure map simply takes each state to the next states. But the coalgebraic view is also quite effective for studying processes over structured state spaces, e.g. measurable, or continuous. In the present paper we consider coalgebras over manifolds. This means that the captured processes evolve over state spaces that are not just continuous, but also locally homeomorphic to normed vector spaces, and thus carry a differential structure. Both dynamical systems and differential forms arise as coalgebras over such state spaces, for two different endofunctors over manifolds. A duality induced by these two endofunctors provides a formal underpinning for the informal geometric intuitions linking differential forms and dynamical systems in the various practical applications, e.g. in physics. This joint functorial reconstruction of tangent bundles and cotangent bundles uncovers the universal properties and a high level view of these fundamental structures, which are implemented rather intricately in their standard form. The succinct coalgebraic presentation provides unexpected insights even about the situations as familiar as Newton's laws.

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Coalgebraic Modal Logic Beyond Sets

Cited by 38 publications

References 26 publications

Strongly Complete Logics for Coalgebras

Strongly Complete Logics for Coalgebras

Generic Trace Semantics via Coinduction

Smooth coalgebra: testing vector analysis

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