A Coalgebraic Perspective on Monotone Modal Logic

Hansen, Helle Hvid; Kupke, Clemens

doi:10.1016/j.entcs.2004.02.028

Cited by 50 publications

(45 citation statements)

References 9 publications

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“…More generally, reversing the original viewpoint that modal logic serves as a specification language for coalgebras, our results show that coalgebra constitutes a good semantic framework also for non-normal and even non-monotone modal systems (for non-normal systems cf. also [7]). …”

Section: Resultsmentioning

confidence: 99%

Expressivity of Coalgebraic Modal Logic: The Limits and Beyond

Schröder

2005

Foundations of Software Science and Computational Structures

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Abstract. Modal logic has a good claim to being the logic of choice for describing the reactive behaviour of systems modeled as coalgebras. Logics with modal operators obtained from so-called predicate liftings have been shown to be invariant under behavioral equivalence. Expressivity results stating that, conversely, logically indistinguishable states are behaviorally equivalent depend on the existence of separating sets of predicate liftings for the signature functor at hand. Here, we provide a classification result for predicate liftings which leads to an easy criterion for the existence of such separating sets, and we give simple examples of functors that fail to admit expressive normal or monotone modal logics, respectively, or in fact an expressive (unary) modal logic at all. We then move on to polyadic modal logic, where modal operators may take more than one argument formula. We show that every accessible functor admits an expressive polyadic modal logic. Moreover, expressive polyadic modal logics are, unlike unary modal logics, compositional.

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Section: Resultsmentioning

confidence: 99%

Expressivity of Coalgebraic Modal Logic: The Limits and Beyond

Schröder

2005

Foundations of Software Science and Computational Structures

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“…Probabilistic transition system as coalgebras go back to Rutten and de Vink [21]. Coalgebras for the double contravariant powerset functor are investigated in Kupke and Hansen [28].…”

Section: Notesmentioning

confidence: 99%

Coalgebras and their logics

Kurz

2006

SIGACT News

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Some comments about the last Logic Column, on nominal logic. Pierre Lescanne points out that the terminology "de Bruijn levels" was introduced in the paper Explicit Substitutions with de Bruijn's Levels, by Pierre Lescanne and Jocelyne Rouyer-Degli, presented at the 1995 RTA conference. He also points out that Stoy diagrams were probably invented by Stoy, but appear in work by Bourbaki as early as 1939 (published in 1954). Merci, Pierre.That article also initiated what is bound to be an interesting discussion. The critique of higherorder abstract syntax in that article prompted Karl Crary and Robert Harper to prepare a response to the leveled criticisms. The response should appear in an upcoming Column.In this issue, Alexander Kurz describes recent work on the topic of specifying properties of transition systems. It turns out that by giving a suitably abstract description of transition systems as coalgebras, we can derive logics for capturing properties of these transition systems in a rather elegant way. I will let you read the details below.

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“…Similarly, for the predicate lifting that interprets the monotonic neighbourhood modality. M-coalgebras are known in modal logic as monotonic neighbourhood frames [3,8,9]. We will refer to M-coalgebras as monotonic neighbourhood functions.…”

Section: Coalgebraic Modal Logicmentioning

confidence: 99%

“…The modal language of GL is obtained by extending the program operations of PDL with the game operation dual ( d ) which corresponds to a role switch of the two players. Game Logic semantics is given by multi-modal monotone neighbourhood models [3,8,9] in which each game γ is interpreted as a monotone neighbourhood function E γ : X → M(X) (we formally define M later in Example 1(iii)) which assigns to each state x ∈ X the collection of all subsets U ⊆ X for which player 1 has a strategy in γ starting in x to ensure an outcome in U . As a deductive system, GL is defined to be the least monotone multi-modal logic containing the following axioms and rule:…”

Section: Pdl and Glmentioning

confidence: 99%

Strong Completeness for Iteration-Free Coalgebraic Dynamic Logics

Hansen

Kupke

Leal

2014

Advanced Information Systems Engineering

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Abstract. We present a (co)algebraic treatment of iteration-free dynamic modal logics such as Propositional Dynamic Logic (PDL) and Game Logic (GL), both without star. The main observation is that the program/game constructs of PDL/GL arise from monad structure, and the axioms of these logics correspond to certain compatibilty requirements between the modalities and this monad structure. Our main contribution is a general soundness and strong completeness result for PDL-like logics for T -coalgebras where T is a monad and the "program" constructs are given by sequential composition, test, and pointwise extensions of operations of T .

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A Coalgebraic Perspective on Monotone Modal Logic

Cited by 50 publications

References 9 publications

Expressivity of Coalgebraic Modal Logic: The Limits and Beyond

Expressivity of Coalgebraic Modal Logic: The Limits and Beyond

Coalgebras and their logics

Strong Completeness for Iteration-Free Coalgebraic Dynamic Logics

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