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Register a new userIntuitionistic Non-Normal Modal Logics: A general framework
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Added by
Tiziano Dalmonte
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We define a family of intuitionistic non-normal modal logics; they can bee seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only one between Necessity and Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. For all logics we provide both a Hilbert axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then give a semantic characterisation of our logics in terms of neighbourhood models. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K (CK) and the propositional fragment of Wijesekeras Constructive Concurrent Dynamic Logic.
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This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics. Given a set Gamma of modal formulas of the form gamma(x, p1, . . ., pn), where x occurs only positively in gamma, the language Lsharp (Gamma) is obtained by adding to the language of polymodal logic a connective sharp_gamma for each gamma epsilon. The term sharp_gamma (varphi1, . . ., varphin) is meant to be interpreted as the least fixed point of the functional interpretation of the term gamma(x, varphi 1, . . ., varphi n). We consider the following problem: given Gamma, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language Lsharp (Gamma) on Kripke frames. We prove two results that solve this problem. First, let Ksharp (Gamma) be the logic obtained from the basic polymodal K by adding a Kozen-Park style fixpoint axiom and a least fixpoint rule, for each fixpoint connective sharp_gamma. Provided that each indexing formula gamma satisfies the syntactic criterion of being untied in x, we prove this axiom system to be complete. Second, addressing the general case, we prove the soundness and completeness of an extension K+ (Gamma) of K_sharp (Gamma). This extension is obtained via an effective procedure that, given an indexing formula gamma as input, returns a finite set of axioms and derivation rules for sharp_gamma, of size bounded by the length of gamma. Thus the axiom system K+ (Gamma) is finite whenever Gamma is finite.
Open bisimilarity is defined for open process terms in which free variables may appear. The insight is, in order to characterise open bisimilarity, we move to the setting of intuitionistic modal logics. The intuitionistic modal logic introduced, called $mathcal{OM}$, is such that modalities are closed under substitutions, which induces a property known as intuitionistic hereditary. Intuitionistic hereditary reflects in logic the lazy instantiation of free variables performed when checking open bisimilarity. The soundness proof for open bisimilarity with respect to our intuitionistic modal logic is mechanised in Abella. The constructive content of the completeness proof provides an algorithm for generating distinguishing formulae, which we have implemented. We draw attention to the fact that there is a spectrum of bisimilarity congruences that can be characterised by intuitionistic modal logics.
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