Unified Correspondence

Conradie, Willem; Ghilardi, Silvio; Palmigiano, Alessandra

doi:10.1007/978-3-319-06025-5_36

Cited by 68 publications

(112 citation statements)

References 31 publications

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“…9 Another standard topic would be algebraic analysis and extensions of modal algebras with operations that are monotonic in some arguments and distributive in others. See [12] and the references therein for generalizations of modal algebra in this direction.…”

Section: Further Directionsmentioning

confidence: 99%

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Instantial Neighbourhood Logic

Benthem

Bezhanishvili

Enqvist

et al. 2016

The Review of Symbolic Logic

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Abstract. This paper explores a new language of neighborhood structures where existential information can be given about what kind of worlds occur in a neighborhood of a current world. The resulting system of 'instantial neighborhood logic' INL has a non-trivial mix of features from relational semantics and from neighborhood semantics. We explore some basic model-theoretic behavior, provide a matching notion of bisimulation, and give a complete axiom system for which we prove completeness by a new normal form technique. In addition, we relate INL to other modal logics by means of translations, determine its precise SAT complexity, and formulate an adequate tableau calculus that can be used for automated deduction supporting INL inference. As for a broader setting, we discuss some motivations for INL in dynamic logics of evidence, consider some special cases in the realm of topology and of powers for players in games, and we point at general model-theoretic and especially, coalgebraic backgrounds for what is achieved in this paper. Many of these final themes suggest follow-up work of independent interest.

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Section: Further Directionsmentioning

confidence: 99%

“…For instance, neighborhood structures also arise naturally in the form of hypergraphs, that is, families of subsets of a domain, where each set is a "generalized arrow" [9,29]. 12 In this setting, once more, no obvious reducibilities arise to other modal logics, and the INL fragment of hypergraph theory may be well-worth exploring.…”

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confidence: 99%

Instantial Neighbourhood Logic

Benthem

Bezhanishvili

Enqvist

et al. 2016

The Review of Symbolic Logic

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“…However, they can be naturally encompassed within the existing algebraic approach to correspondence theory [5,9,10], and generalized to mu-calculi on a weakerthan-classical (and, particularly, intuitionistic) base.…”

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confidence: 99%

“…In Section 2.5 the limitations of these rules are discussed, which motivates the developments of Sections 4 and 5. In Section 3, the recursive mu-inequalities are defined in the same uniform style discussed and advocated in [5], which pivots on the order-theoretic properties of the algebraic interpretation of the logical connectives. This class is compared with other Sahlqvist-type classes in the literature.…”

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confidence: 99%

Algorithmic correspondence for intuitionistic modal mu-calculus

Conradie

Fomatati

Palmigiano

et al. 2015

Theoretical Computer Science

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In the present paper, the algorithmic correspondence theory developed in Conradie and Palmigiano [9] is extended to mu-calculi with a non-classical base. We focus in particular on the language of bi-intuitionistic modal mu-calculus. We enhance the algorithm ALBA introduced in Conradie and Palmigiano [9] so as to guarantee its success on the class of recursive mu-inequalities, which we introduce in this paper. Key to the soundness of this enhancement are the order-theoretic properties of the algebraic interpretation of the fixed point operators. We show that, when restricted to the Boolean setting, the recursive mu-inequalities coincide with the "Sahlqvist mu-formulas" defined in van Benthem, Bezhanishvili and Hodkinson [22]. IntroductionModal mu-calculus [17] is a logical framework combining simple modalities with fixed point operators, enriching the expressivity of modal logic so as to deal with infinite processes like recursion. It has a simple syntax, an easily given semantics, and is decidable. Modal mu-calculus has become a fundamental logical tool in theoretical computer science and has been extensively studied [3], and applied for instance in the context of temporal properties of systems, and of infinite properties of concurrent systems. Many expressive modal and temporal logics such as PDL, CTL, CTL * can be seen as fragments of the modal mu-calculus [3,15]. It provides a unifying framework connecting modal and temporal logics, automata theory and the theory of games, where fixed point constructions can be used to talk about the long term strategies of players, as discussed in [23].Correspondence theory studies the relationships between classical first-and second-order logic, and modal logic, interpreted on Kripke frames. A modal and a first-order formula correspond if they define the same class of structures. Specifically, Sahlqvist theory is concerned with the identification of syntactically specified classes of modal formulas which correspond to first-order formulas. Sahlqvist-style frame-correspondence theory for modal mu-calculus has recently been developed in [22]. Such analysis strengthens the general mathematical theory of the mu-calculus, facilitates the transfer of results ✩

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“…Parallel to the model theoretic approach to this type of result, there exists an algebraic-algorithmic approach (see e.g., [3,4,5]) which derives correspondence (and canonicity) results by means of 'calculi of correspondence' consisting of simple derivation rules which depend for their soundness on the order theoretic properties of the operations interpreting the logical connectives in the algebraic semantics. As indicated in Part 1 [2], these rules are divided into approximation and adjunction rules, together with the Ackermann rules used to eliminate propositional variables.…”

Section: Introductionmentioning

confidence: 99%

Algorithmic correspondence for intuitionistic modal mu-calculus, Part 2

Conradie¹,

Fomatati²,

Palmigiano³

et al.

EPiC Series in Computing

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Sahlqvist-style correspondence results remain a perennial theme and an active topic of research within modal logic. Recently there has been interest in extending classical results in this area to the modal mu-calculus. We show how the `calculus of correspondence' and the ALBA algorithm (Conradie and Palmigiano, 2012) can be extended to the intuitionistic mu-calculus, and be used to derive FO+LFP frame correspondents for formulas of that logic. We define the class of recursive mu-inequalities, which we compare it with related classes in the literature including the Sahlqvist mu-formulas of van Benthem, Bezhanishvili and Hodkinson. We show that the ALBA algorithm succeeds in reducing every recursive mu-inequality, and hence that every recursive mu-inequality has a frame correspondent in FO+LFP.

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Unified Correspondence

Cited by 68 publications

References 31 publications

Instantial Neighbourhood Logic

Instantial Neighbourhood Logic

Algorithmic correspondence for intuitionistic modal mu-calculus

Algorithmic correspondence for intuitionistic modal mu-calculus, Part 2

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