Algebraic modal correspondence: Sahlqvist and beyond

Conradie, Willem; Palmigiano, Alessandra; Sourabh, Sumit

doi:10.1016/j.jlamp.2016.10.006

Cited by 40 publications

(49 citation statements)

References 55 publications

(118 reference statements)

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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%

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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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“…Moreover, this analysis includes as a special case the correspondence theory for classical modal mu-calculus of [22]. Following the methodology developed in [10], the algebraic and order-theoretic principles underlying these results are isolated. This forms an intermediate level of analysis which is added to the model-theoretic analysis in [22].…”

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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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“…17 For any DLE-language L DLE , by a tense DLE-logic we understand any axiomatic extension of the basic tense bi-intuitionistic L DLE -logic in L * DLE . The algebraic semantics of L * DLE is given by the class of bi-intuitionistic 'tense' L DLE -algebras, defined as tuples A = (H, F * , G * ) such that H is a bi-Heyting algebra 18 and moreover, 1. for every f ∈ F s.t. n f ≥ 1, all a 1 , .…”

Section: Dlementioning

confidence: 99%

“…The logic L * DLE is the natural logic on the language L * DLE , however it is useful to introduce a specific notation for L * DLE , given that all the results holding for the minimal logic associated with an arbitrary DLE-language can be instantiated to the expanded language L * DLE and will then apply to L * DLE . 18 …”

Section: Dlementioning

confidence: 99%

Unified correspondence as a proof-theoretic tool

Greco

Palmigiano

et al. 2016

J Logic Computation

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The present paper aims at establishing formal connections between correspondence phenomena, well known from the area of modal logic, and the theory of display calculi, originated by Belnap.These connections have been seminally observed and exploited by Marcus Kracht, in the context of his characterization of the modal axioms (which he calls primitive formulas) which can be effectively transformed into 'analytic' structural rules of display calculi. In this context, a rule is 'analytic' if adding it to a display calculus preserves Belnap's cut-elimination theorem.In recent years, the state-of-the-art in correspondence theory has been uniformly extended from classical modal logic to diverse families of nonclassical logics, ranging from (bi-)intuitionistic (modal) logics, linear, relevant and other substructural logics, to hybrid logics and mu-calculi. This generalization has given rise to a theory called unified correspondence, the most important technical tools of which are the algorithm ALBA, and the syntactic characterization of Sahlqvisttype classes of formulas and inequalities which is uniform in the setting of normal DLE-logics (logics the algebraic semantics of which is based on bounded distributive lattices).We apply unified correspondence theory, with its tools and insights, to extend Kracht's results and prove his claims in the setting of DLE-logics. The results of the present paper characterize the space of properly displayable DLE-logics. Keywords: Display calculi, unified correspondence, distributive lattice expansions, properly displayable logics. Math. Subject Class. 03B45, 06D50, 06D10, 03G10, 06E15. , and mu-calculus [11,12]. The breadth of this work has stimulated many and varied applications. Some are closely related to the core concerns of the theory itself, such as the understanding of the relationship between different methodologies for obtaining canonicity results [40,15], or of the phenomenon of pseudocorrespondence [19]. Other, possibly surprising applications include the dual characterizations of classes of finite lattices [27]. Finally, the insights of unified correspondence theory have made it possible to determine the extent to which the Sahlqvist theory of classes of normal DLEs can be reduced to the Sahlqvist theory of normal Boolean expansions, by means of Gödel-type translations [20]. These and other results have given rise to a theory called unified correspondence [13]. Contents Tools of unified correspondence theory.The most important technical tools in unified correspondence are: (a) a very general syntactic definition of the class of Sahlqvist formulas, which applies uniformly to each logical signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives; (b) the algorithm ALBA, which effectively computes first-order correspondents of input term-inequalities, and is guaranteed to succeed on a wide class of inequalities (the so-called inductive inequalities) which, like the Sahlqvist class, can be defi...

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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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Algebraic modal correspondence: Sahlqvist and beyond

Cited by 40 publications

References 55 publications

Algorithmic correspondence for intuitionistic modal mu-calculus

Algorithmic correspondence for intuitionistic modal mu-calculus

Unified correspondence as a proof-theoretic tool

Algorithmic correspondence for intuitionistic modal mu-calculus, Part 2

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