A Framework for Intuitionistic Modal Logics

Plotkin, Gordon; Stirling, Colin

doi:10.1016/b978-0-934613-04-0.50032-6

Cited by 91 publications

(100 citation statements)

References 3 publications

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“…This variant of IK that contains all 5 axioms k 1 -k 5 has first been studied in [20,21] and investigated in detail in [9]. There exist other intuitionistic variants of K, e.g., [22][23][24][25], the most prominent being the one which has only the axioms k 1 and k 2 from (1).…”

Section: Syntax and Semantics Of Intuitionistic Modal Logicsmentioning

confidence: 99%

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Maehara-style modal nested calculi

Kuznets

Straßburger

2018

Arch. Math. Logic

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We develop multi-conclusion nested sequent calculi for the fifteen logics of the intuitionistic modal cube between IK and IS5. The proof of cut-free completeness for all logics is provided both syntactically via a Maehara-style translation and semantically by constructing an infinite birelational countermodel from a failed proof search. Interestingly, the Maehara-style translation for proving soundness syntactically fails due to the hierarchical structure of nested sequents. Consequently, we only provide the semantic proof of soundness. The countermodel construction used to prove completeness required a completely novel approach to deal with two independent sources of nontermination in the proof search present in the case of transitive and Euclidean logics.

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Section: Syntax and Semantics Of Intuitionistic Modal Logicsmentioning

confidence: 99%

“…Let us now recall the birelational models [21,29] for intuitionistic modal logics, which are a combination of the Kripke semantics for propositional intuitionistic logic and for classical modal logic. …”

Section: Syntax and Semantics Of Intuitionistic Modal Logicsmentioning

confidence: 99%

Maehara-style modal nested calculi

Kuznets

Straßburger

2018

Arch. Math. Logic

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“…It is not only the fallible worlds but also the extension by sets Θ, capturing hereditary refutation information, that distinguishes the representation of constructive modal logic, such as CS4, from that for standard intuitionistic modal logics, such as those of [PS86,FS80,Ewa86]. Indeed, if the axioms ¬3⊥ and 3(φ ∨ ψ) → 3φ ∨ 3ψ are adopted the sets Θ and fallible worlds become redundant.…”

Section: Theorem 4 (Representation For Pll) Letmentioning

confidence: 99%

“…While it is easy to agree that a Kripke model of constructive modal logic should consist of a set of worlds W and two accessibility relations, one intuitionistic ≤ and the other modal R, it is not so clear how these relations should interact (frame conditions) and just how they should be used to interpret specifically the 3 modality. The mainstream approach as exemplified by Ewald [Ewa86], Fischer-Servi [FS80], Plotkin and Stirling [PS86], Simpson [Sim94] is based on the analogy of 2 with ∀ and of 3 with ∃-quantification over the modal accessibility R. Reading these quantifiers intuitionistically, relative to ≤, one arrives at the semantic interpretation w |= 2A iff ∀v. w ≤ v ⇒ ∀u.…”

Section: Kripke Modelsmentioning

confidence: 99%

Categorical and Kripke Semantics for Constructive S4 Modal Logic

Alechina

Mendler

Paiva

et al. 2001

Lecture Notes in Computer Science

116

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Abstract. We consider two systems of constructive modal logic which are computationally motivated. Their modalities admit several computational interpretations and are used to capture intensional features such as notions of computation, constraints, concurrency, etc. Both systems have so far been studied mainly from type-theoretic and category-theoretic perspectives, but Kripke models for similar systems were studied independently. Here we bring these threads together and prove duality results which show how to relate Kripke models to algebraic models and these in turn to the appropriate categorical models for these logics.

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“…We shall define the formulae of modal μ-calculus by extending those of intuitionistic modal logic [21] with the least and greatest fixpoints. Here, we do not follow the standard definition of modal μ-formulae, since we will be giving an intuitionistic interpretation in which (♦ϕ) ⇔ (¬ ¬ϕ) and (ϕ ⇒ ψ) ⇔ (¬ϕ ∨ ψ) do not necessarily hold, and we must, therefore, have ⇒ and ♦ as primitives.…”

Section: Complete Heyting Algebramentioning

confidence: 99%

Multi-Valued Modal Fixed Point Logics for Model Checking

Nishizawa

2010

IEICE Trans. Inf. & Syst.

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SUMMARYIn this paper, I will show how multi-valued logics are used for model checking. Model checking is an automatic technique to analyze correctness of hardware and software systems. A model checker is based on a temporal logic or a modal fixed point logic. That is to say, a system to be checked is formalized as a Kripke model, a property to be satisfied by the system is formalized as a temporal formula or a modal formula, and the model checker checks that the Kripke model satisfies the formula. Although most existing model checkers are based on 2-valued logics, recently new attempts have been made to extend the underlying logics of model checkers to multi-valued logics. I will summarize these new results. key words: fixed point logic, De Morgan algebra, Heyting algebra, minplus algebra IntroductionIn order to discover bugs of hardware or software, various formal methods have been studied. Model checking [6] is the most successful as the technique to automatically verify transition systems modelling hardware and software against temporal formulas expressing specifications. The specifications for model checking are described in modal logics, temporal logics, and modal fixed point logics [4], [16], [17], [22], [27]. Until the 1990s, the logics used for model checking were based on only two truth values, i.e., 'true' and 'false'. However, in the 2000s, multi-valued logics for model checking are studied extensively. This paper introduces their logics.First, we introduce the ordinary logics for model checking. The system to be verified is formalized as a Kripke structure. A Kripke structure consists of a set S, a subset → of S × S, and a subset ρ of S × Atom, where Atom is called the set of atomic propositions. The elements of S are called states and → are called the transition relation. We write s → t for (s, t) ∈ →.To formalize the specification for verified systems, various modal logics and temporal logics are used. However, most of them are included in modal μ-calculus [16] which is a powerful logic equipped with modalities and the least fixpoints. Therefore, this paper introduces only modal μ-calculus (both of 2-valued and multi-valued). The formulae of modal μ-calculus are generated by the following grammar. Here, P is an atomic proposition. ¬ and ∧ are the negation operator and the conjunction operator. So, modal μ-calculus includes the propositional logic. We write ϕ ∧ ψ for ¬(¬ϕ ∨ ¬ψ), ϕ ⇒ ψ for ¬ϕ ∨ ψ, ϕ for ¬♦¬ϕ, and νX.ϕ for ¬μX.¬ϕ[¬X/X]. ♦ is a modal operator interpreted by using the transition relation of a Kripke structure. A state s satisfies ♦ϕ if and only if there exists a state t satisfying ϕ and s → t. μ is called least fixed point operator. μX.ϕ represents the strongest formula ψ which is equal to the result of replacing X in ϕ with ψ. For example, μX.P ∨ ♦X is interpreted as follows.Therefore, a state s satisfies μX.P ∨ ♦X if and only if there exists a path from s to a state t satisfying P. The grammar of formulae of modal μ-calculus is subject to the side condition that μX.ϕ has no free negat...

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A Framework for Intuitionistic Modal Logics

Cited by 91 publications

References 3 publications

Maehara-style modal nested calculi

Maehara-style modal nested calculi

Categorical and Kripke Semantics for Constructive S4 Modal Logic

Multi-Valued Modal Fixed Point Logics for Model Checking

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