Categorical and Kripke Semantics for Constructive S4 Modal Logic

Abstract

Abstract. 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, etc. Both systems have so far been studied mainly from 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.

Cut Elimination in Nested Sequents for Intuitionistic Modal Logics

“…One can see this marking as a polarity assignment: • for input polarity, and • for output polarity. 1 Formally, nested sequents for IML are generated by the grammar (where n and k can both be zero):…”

“…Recently, researchers have also studied the variant which allows only k 1 and k 2 , and which is sometimes called constructive modal logic (e.g., [1,18]). Since this leads to a different proof theory, it will not be discussed here.…”

Maehara-style modal nested calculi

“…There exist other intuitionistic variants of K, e.g., [22][23][24][25], the most prominent being the one which has only the axioms k 1 and k 2 from (1). There is now consensus in the literature to call this variant constructive modal logic, e.g., [18,26,27].…”

Functional reactive types

“…In particular, we will show that FRP forms a category whose objects are functional reactive types, and whose morphisms are programs of type [As ⇒ Bs], where ∧ is product, ∨ is coproduct, is a comonad, and ♦ is a monad (and so form a model of constructive S4 modal logic [2]). The novelty here (compared to the author's previous work [11]) is that all the proof rules are defined just using the combinators in Figures 1 and 2, thus showing that it is sufficient to present streams as Before we can do this, however, we need to take a look at polymorphic constants such as the identity function.…”