Categorical and Kripke Semantics for Constructive Modal Logics

We consider two systems of constructive modal logic which ar e computationally motivated. Their modalities admit several co mputational interpretations and are used to capture intensional features such as no tions of computation, constraints, concurrency design, etc. Both systems h ave so far been studied mainly from a type-theoretic and category-theoretic pe rs ctives, but Kripke models for similar systems were studied independently. Her e 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 c ategorical models for these logics. Submission Note:For the benefit of the referees we have included the proofs of our main results as an appendix to this paper. 1 Constructive Motivations This paper is about relating traditional Kripke-style sema ntics for constructive modal logics to their corresponding categorical semantics. Both f rms of semantics have important applications within computer science. Our aim is to persuade traditional modal logicians that categorical semantics is easy, fun and usefu l; just like Kripke semantics; and at the same time that categorical semantics can gen erat interesting new constructive modal logics, which differ somewhat from the trad itional diet of intuitionistic modal logics. ‘Categorical semantics’ can be used in two different senses : categorical model theory, where one does model theory but using categories instea d of sets; and categorical proof theory, where the main concern is with the proofs of ass ertions rather than the mere fact that the assertions are true. We will be using ‘cate gorical semantics’ in the second, proof-theoretic sense. Unlike traditional (Kripk e-style) model theory, we will not only provide meanings for the formulae of a logical syste m, but will also provide meanings for the proofs of these formulae. That is, derivati ons in the logic are studied as entities in their own right, and have their own semantic ob jects in the models. This study is useful: having a calculus of terms corresponding to derivations in the logic we obtain a direct correspondence between properties of proof s and properties of programs in the functional programming language based on these terms . Our primary goal in this paper is to show that Kripke and categ orical semantics interact as one might expect: from a categorical semantics f or formulae and proofs, one can recover both an algebraic and a Kripke semantics for form ulae. Many applications of modal logic to computer science rely on having a term calculus for natural deduction proofs in the logic. Such a term cal culus is a suitable variant of the -calculus, which is the prototypical functional programmi ng language. Reduction in the term-calculus should correspond to the process of nor malisation of proofs in the logic. Moreover, this term calculus should satisfy all the u sual syntactic properties of -calculi such as confluence, (strong) normalisation, subje ct reduction etc. This correspondence between natural deduction proofs and the appropr iate -calculus is called the Curry-Howard Isomorphism. Whereas the Curry-Howard isomorphism between intuitionistic propositional logic and the simply-typed -calculus has been known since the late 60s, establishing such isomorphisms for modal logics i s a more recent development. In this paper we develop a suitable categorical semantics an d associated -calculus for Constructive S4 , namedCS4, an intuitionistic modal logic with S4-like constructive 2 and3 modalities. We further highlight the special status of CS4 among non-classical modal logics and develop a suitable Kripke semantics and ass ociated Stone duality, which turns out to require a different approach compared to o ther more standard intuitionistic modal logics, in particular as regards the 3modality. This paper is organised as follows. Section 2 formally descr ibes the two constructive logics we investigate. Section 3 gives their Kripke-style s emantics and Section 4 uses traditional algebraic duality theory as a natural hinge to c onnect the Kripke semantics with the categorical semantics to be introduced in Section 6 . Before that, in Section 5 we explain some of the design criteria for our Kripke semanti cs. We finally conclude with some open questions and directions for further work. 2 The Constructive Modal SystemsCS4 and PLL In this paper we take a fresh look at two prominent constructi ve modal extensions to intuitionistic propositional logic ( IPL), which are particularly interesting because of their various applications in computer science. To give the reader a taste for these applications, we list a fe w. Davies and Pfenning [DP96] use the2-modality to give a -calculus for computation in stages. The idea is that a term2t represents a delayed computation. Ghani et al. [GdPR98] inv estigate refinements of this calculus which are suitable for the desig n of abstract machines. Similar ideas relating2with staged evaluation and the distinction between run-tim e and compile-time semantics have been developed by Moggi et.al. [BMTS99]. Despeyroux and Pfenning [DPS97] use a box modality to encode higher-ord er abstract syntax in theorem-provers like Elf and Isabelle. Still another use of the2 modality, to model the quote mechanism of Lisp, is proposed by Goubault-Larrecq [GL96]. A 3-style modality has been extensively used to distinguish a computa tion from its result in the -calculus: Moggi’s [Mog91] influential work on computation al monads describes the computational -calculus, which corresponds to an intuitionistic modal ty pe theory with a3-like modality (see [BBdP98]). Fairtlough and Mendler [Men 93,FMW97,Men00] use the same modality, which they call , in their work on lax logic for constraints and hardware verification. The calculus has also been used for de notational semantics of exception handling mechanisms, continuations, etc. On the syntactic side, it has been used, in the monadic-style of functional programming to add notion of ‘encapsulated state’ to functional languages. Despite their relevance for computer science these modal ex tensions ofIPL seem to be less well investigated as modal logics in their own righ t, perhaps because of the “unusual properties” of their associated modal operators.