Results 1  10
of
10
Categorical and Kripke Semantics for Constructive S4 Modal Logic
 In International Workshop on Computer Science Logic, CSL’01, L. Fribourg, Ed. Lecture Notes in Computer Science
, 2001
"... 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 m ..."
Abstract

Cited by 24 (1 self)
 Add to MetaCart
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 typetheoretic and categorytheoretic 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.
Categorical and Kripke Semantics for Constructive Modal Logics
, 2001
"... 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 studi ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
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 typetheoretic and categorytheoretic 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.
Topological Semantics and Bisimulations for Intuitionistic Modal Logics and Their Classical Companion Logics ⋆
"... Abstract. We take the wellknown intuitionistic modal logic of Fischer Servi with semantics in birelational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic preorder (or partialorder) with the modal ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. We take the wellknown intuitionistic modal logic of Fischer Servi with semantics in birelational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic preorder (or partialorder) with the modal accessibility relation generalise to the requirement that the relation and its inverse be lower semicontinuous 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 topologypreserving conditions are equivalent to the properties that the inverserelation 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 multimodal logics in the setting of the Gödel translation. After giving canonical topological Kripke models for the Hilbertstyle axiomatizations of the Fischer Servi logic and its classical multimodal 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
A First Order Modal Logic and its Sheaf Models
"... Abstract: We present a new way of formulating first order modal logic which circumvents the usual difficulties associated with variables changing their reference on moving between states. This formulation allows a very general notion of model (sheaf models). The key idea is the introduction of synta ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract: We present a new way of formulating first order modal logic which circumvents the usual difficulties associated with variables changing their reference on moving between states. This formulation allows a very general notion of model (sheaf models). The key idea is the introduction of syntax for describing relations between individuals in related states. This adds an extra degree of expressiveness to the logic, and also appears to provide a means of describing the dynamic behaviour of computational systems in a way somewhat different from traditional program logics. 1
On the Logical Content of Computational Type Theory: A Solution to Curry's Problem
 In Types for Proofs and Programs
, 2002
"... 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 fo ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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.
Relational sheaves and predicate intuitionistic modal logic
, 1999
"... 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, relationpreserving functions into the space, and the relational se ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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, relationpreserving functions into the space, and the relational sections of a relational bundle are defined as the relationpreserving 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 quasitoposes (thus modelling firstorder logic), and to have enough structure to model a certain firstorder modal logic described in a companion paper. 1
UMCS9961
, 1999
"... Abstract We prove soundness and adequacy for an intuitionistic modal sequent calculus with the modal Heyting algebra semantics presented in Hilken [7]. We produce a cutelimination for this calculus. For comparison a description of a corresponding classical modal logic in a sequent style is given al ..."
Abstract
 Add to MetaCart
Abstract We prove soundness and adequacy for an intuitionistic modal sequent calculus with the modal Heyting algebra semantics presented in Hilken [7]. We produce a cutelimination for this calculus. For comparison a description of a corresponding classical modal logic in a sequent style is given along with its semantics. 1 Introduction Modal logics have found many applications in computer science. In most cases the logics have been classical and have been used to descibe properties of relational structures. In other words the logics have been analysed relative to suitable Kripke relational semantics.
—Extended abstract—
, 2003
"... adjoint construction for topological models of intuitionistic modal logic ..."
Abstract
 Add to MetaCart
adjoint construction for topological models of intuitionistic modal logic
topological semantics
, 2008
"... On intuitionistic modal and tense logics and their classical companion logics: ..."
Abstract
 Add to MetaCart
On intuitionistic modal and tense logics and their classical companion logics: