Duality for Logics of Transition Systems

Bonsangue, Marcello M.; Kurz, Alexander

doi:10.1007/978-3-540-31982-5_29

Cited by 43 publications

(56 citation statements)

References 24 publications

(27 reference statements)

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“…The Kripke polynomial functors on Set (including powerset) of [11] have duals on complete atomic Boolean algebras, which have a presentation. The description of the dual of the finite (or compact) powerspace on posets and sets in [7] also provides examples of presentations of functors.…”

Section: Definition 7 (Presented Functor)mentioning

confidence: 99%

“…We can then include the category of posets into the list of possible topological spaces and treat propositional logics without negation but with infinitary meets [6]. This approach was also used in [7].…”

Section: Top(x Sa) ∼ = Frm(a P X)mentioning

confidence: 99%

“…Let us emphasise that the last requirement is not essential [7]. But it simplifies the presentation considerably, as we can now take the initial algebra in Alg(L) as a canonical set of propositions.…”

Section: Abstract Logics For Coalgebrasmentioning

confidence: 99%

“…But, as shown in [7], this assumption is not necessary (neither always desirable: if T is the powerset functor, then Coalg(T ) does not have a final coalgebra and Alg(L) does not have an initial algebra). Theorem 27 then still holds (assuming, as in [7], that T weakly preserves limits of chains).…”

Section: Remark 28mentioning

confidence: 99%

“…This paper can also be seen as a sequel to [7], where we proposed a general framework for logics of coalgebras based on Stone duality. A general adequateness result was proved for what we call here abstract logics and then, studying the case of the compact powerspace P c , it was shown how to systematically obtain logics for coalgebras over different base categories by presenting the dual of P c by generators and relations.…”

Section: Introductionmentioning

confidence: 99%

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Presenting Functors by Operations and Equations

Bonsangue

Kurz

2006

Lecture Notes in Computer Science

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Abstract. We take the point of view that, if transition systems are coalgebras for a functor T, then an adequate logic for these transition systems should arise from the 'Stone dual' L of T. We show that such a functor always gives rise to an 'abstract' adequate logic for T-coalgebras and investigate under which circumstances it gives rise to a 'concrete' such logic, that is, a logic with an inductively defined syntax and proof system. We obtain a result that allows us to prove adequateness of logics uniformly for a large number of different types of transition systems and give some examples of its usefulness.

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Section: Definition 7 (Presented Functor)mentioning

confidence: 99%

Section: Top(x Sa) ∼ = Frm(a P X)mentioning

confidence: 99%

Section: Abstract Logics For Coalgebrasmentioning

confidence: 99%

Section: Remark 28mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Presenting Functors by Operations and Equations

Bonsangue

Kurz

2006

Lecture Notes in Computer Science

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Ultrafilter Extensions for Coalgebras

Kupke

Kurz

Pattinson

2005

Algebra and Coalgebra in Computer Science

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Abstract. This paper studies finitary modal logics as specification languages for Set-coalgebras (coalgebras on the category of sets) using Stone duality. It is wellknown that Set-coalgebras are not semantically adequate for finitary modal logics in the sense that bisimilarity does not in general coincide with logical equivalence. Stone-coalgebras (coalgebras over the category of Stone spaces), on the other hand, do provide an adequate semantics for finitary modal logics. This leads us to study the relationship of finitary modal logics and Set-coalgebras by uncovering the relationship between Set-coalgebras and Stone-coalgebras. This builds on a long tradition in modal logic, where one studies canonical extensions of modal algebras and ultrafilter extensions of Kripke frames to account for finitary logics. Our main contributions are the generalisations of two classical theorems in modal logic to coalgebras, namely the Jónsson-Tarski theorem giving a settheoretic representation for each modal algebra and the bisimulation-somewhereelse theorem stating that two states of a coalgebra have the same (finitary modal) theory iff they are bisimilar (or behaviourally equivalent) in the ultrafilter extension of the coalgebra.

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Testing Semantics: Connecting Processes and Process Logics

Pavlović

Mislove

Worrell

2006

Algebraic Methodology and Software Technology

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Abstract. We propose a methodology based on testing as a framework to capture the interactions of a machine represented in a denotational model and the data it manipulates. Using a duality that models machines on the one hand, and the data they manipulate on the other, testing is used to capture the interactions of each with the objects on the other side: just as the data that are input into a machine can be viewed as tests that the machine can be subjected to, the machine can be viewed as a test that can be used to distinguish data. While this approach is based on duality theories that now are common in semantics, it accomplishes much more than simply moving from one side of the duality to the other; it faithfully represents the interactions that embody what is happening as the computation proceeds. Our basic philosophy is that tests can be used as a basis for modeling interactions, as well as processes and the data on which they operate. In more abstract terms, tests can be viewed as formulas of process logics, and testing semantics connects processes and process logics, and assigns computational meanings to both.

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Duality for Logics of Transition Systems

Cited by 43 publications

References 24 publications

Presenting Functors by Operations and Equations

Presenting Functors by Operations and Equations

Ultrafilter Extensions for Coalgebras

Testing Semantics: Connecting Processes and Process Logics

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