Categorical and Kripke Semantics for Constructive Modal Logics

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 design, etc. Both systems have so far been studied mainly from a 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. 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 semantics for constructive modal logics to their corresponding categorical semantics. Both forms of semantics have important applications within computer science. Our aim is to persuade traditional modal logicians that categorical semantics is easy, fun and useful; just like Kripke semantics; and at the same time that categorical semantics can generate interesting new constructive modal logics, which differ somewhat from the traditional 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 instead of sets; and categorical proof theory, where the main concern is with the proofs of assertions rather than the mere fact that the assertions are true. We will be using ‘categorical semantics’ in the second, proof-theoretic sense. Unlike traditional (Kripke-style) model theory, we will not only provide meanings for the formulae of a logical system, but will also provide meanings for the proofs of these formulae. That is, derivations in the logic are studied as entities in their own right, and have their own semantic objects 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 proofs 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 categorical semantics interact as one might expect: from a categorical semantics for formulae and proofs, one can recover both an algebraic and a Kripke semantics for formulae. Many applications of modal logic to computer science rely on having a term calculus for natural deduction proofs in the logic. Such a term calculus is a suitable variant of the -calculus, which is the prototypical functional programming language. Reduction in the term-calculus should correspond to the process of normalisation of proofs in the logic. Moreover, this term calculus should satisfy all the usual syntactic properties of -calculi such as confluence, (strong) normalisation, subject reduction etc. This correspondence between natural deduction proofs and the appropriate -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 is a more recent development. In this paper we develop a suitable categorical semantics and associated -calculus for Constructive S4, named CS4, an intuitionistic modal logic with S4-like constructive 2 and 3 modalities. We further highlight the special status of CS4 among non-classical modal logics and develop a suitable Kripke semantics and associated Stone duality, which turns out to require a different approach compared to other more standard intuitionistic modal logics, in particular as regards the 3 modality. This paper is organised as follows. Section 2 formally describes the two constructive logics we investigate. Section 3 gives their Kripke-style semantics and Section 4 uses traditional algebraic duality theory as a natural hinge to connect 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 semantics. We finally conclude with some open questions and directions for further work. 2 The Constructive Modal Systems CS4 and PLL In this paper we take a fresh look at two prominent constructive 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 few. Davies and Pfenning [DP96] use the 2-modality to give a -calculus for computation in stages. The idea is that a term 2t represents a delayed computation. Ghani et al. [GdPR98] investigate refinements of this calculus which are suitable for the design of abstract machines. Similar ideas relating2with staged evaluation and the distinction between run-time and compile-time semantics have been developed by Moggi et.al. [BMTS99]. Despeyroux and Pfenning [DPS97] use a box modality to encode higher-order abstract syntax in theorem-provers like Elf and Isabelle. Still another use of the 2 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 computation from its result in the -calculus: Moggi’s [Mog91] influential work on computational monads describes the computational -calculus, which corresponds to an intuitionistic modal type theory with a 3-like modality (see [BBdP98]). Fairtlough and Mendler [Men93,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 denotational semantics of exception handling mechanisms, continuations, etc. On the syntactic side, it has been used, in the monadic-style of functional programming to add a notion of ‘encapsulated state’ to functional languages. Despite their relevance for computer science these modal extensions of IPL seem to be less well investigated as modal logics in their own right, perhaps because of the “unusual properties” of their associated modal operators.