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.

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.

In this paper we relate the lax modality O to Intuitionistic Propositional Logic (IPL) and give a complete characterisation of inhabitation in Computational Type Theory (CTT) as a logic of constraint contexts. This solves a problem open since the 1940's, when Curry was the first to suggest a formal syntactic interpretation of O in terms of contexts.

This paper generalises and adapts the theory of sheaves on a topological space to sheaves on a relational space: a topological space with a binary relation. The relational bundles on a relational space are defined as the continuous, relation-preserving functions into the space, and the relational sections of a relational bundle are defined as the relation-preserving partial sections. This defines a functor to the category of presheaves on the space, which has a left adjoint. The presheaves which arise as the relational sections of a relational bundle are characterised by separation and patching conditions similar to those of a sheaf: we call them the relational sheaves. The relational bundles which arise from presheaves are characterised by local homeomorphism conditions: we call them the local relational homeomorphisms. The adjunction restricts to an equivalence between the categories of relational sheaves and local relational homeomorphisms. The paper goes on to investigate the structure of these equivalent categories. They are shown to be quasi-toposes (thus modelling firstorder logic), and to have enough structure to model a certain firstorder modal logic described in a companion paper. 1

Abstract. We take the well-known intuitionistic modal logic of Fischer Servi with semantics in bi-relational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic pre-order (or partial-order) with the modal accessibility relation generalise to the requirement that the relation and its inverse be lower semi-continuous with respect to the topology. We then investigate the notion of topological bisimulation relations between topological Kripke frames, as introduced by Aiello and van Benthem, and show that their topology-preserving conditions are equivalent to the properties that the inverse-relation and the relation are lower semicontinuous with respect to the topologies on the two models. Our first main result is that this notion of topological bisimulation yields semantic preservation w.r.t. topological Kripke models for both intuitionistic tense logics, and for their classical companion multi-modal logics in the setting of the Gödel translation. After giving canonical topological Kripke models for the Hilbert-style axiomatizations of the Fischer Servi logic and its classical multi-modal companion logic, we show that the syntactic Gödel translation induces a natural semantic map from the intuitionistic canonical model into the canonical model of the classical companion logic, and this map is itself a topological bisimulation. 1