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The ProofTheory and Semantics of Intuitionistic Modal Logic
, 1994
"... Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpret ..."
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Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic metatheory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are selfjustifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic firstorder logic. It is also established that, in many cases, the natural deduction systems induce wellknown intuitionistic modal logics, previously given by Hilbertstyle axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their
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 ..."
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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.
Knowledge Sharing among Ideal Agents
 School of Computer Science, University of Birmingham
, 1999
"... Multiagent systems operating in complex domains crucially require agents to interact with each other. An important result of this interaction is that some of the private knowledge of the agents is being shared in the group of agents. This thesis investigates the theme of knowledge sharing from a th ..."
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Cited by 30 (10 self)
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Multiagent systems operating in complex domains crucially require agents to interact with each other. An important result of this interaction is that some of the private knowledge of the agents is being shared in the group of agents. This thesis investigates the theme of knowledge sharing from a theoretical point of view by means of the formal tools provided by modal logic. More specifically this thesis addresses the following three points. First, the case of hypercube systems, a special class of interpreted systems as defined by Halpern and colleagues, is analysed in full detail. It is here proven that the logic S5WD constitutes a sound and complete axiomatisation for hypercube systems. This logic, an extension of the modal system S5 commonly used to represent knowledge of a multiagent system, regulates how knowledge is being shared among agents modelled by hypercube systems. The logic S5WD is proven to be decidable. Hypercube systems are proven to be synchronous agents with perfect recall that communicate only by broadcasting, in separate work jointly with Ron van der Meyden not fully reported in this thesis.
Extended CurryHoward Correspondence for a Basic Constructive Modal Logic
 In Proceedings of Methods for Modalities
, 2001
"... this paper. This calculus satises cutelimination, as for instance shown (in a more complicated form) in [Wij90]. This calculus is dierent from what is usually taken as the basic constructive system K, as we do not assume the distribution of possibility (3) over disjunctions neither in its binary f ..."
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Cited by 10 (2 self)
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this paper. This calculus satises cutelimination, as for instance shown (in a more complicated form) in [Wij90]. This calculus is dierent from what is usually taken as the basic constructive system K, as we do not assume the distribution of possibility (3) over disjunctions neither in its binary form 3(A _ B) ! (3A _ 3B) nor in its nullary form 3? ! ? The sequent calculus above corresponds to an axiomatic formulation given by axioms for intuitionistic logic, plus axioms: 2(A ! B) ! (2A ! 2B) 2(A ! B) ! (3A ! 3B) 2A3B ! 3(A B) together with rules for Modus Ponens and Necessitation: ` A ! B ` A ` B MP ` A ` 2A Nec Wijesekera proved a Craig interpolation theorem, one of the usual consequences of syntactic cutelimination and produced Kripke, algebraic and topological semantics for a calculus very similar to the one above. The only dierence is that he does assume 3? ! ?. From our \wish list" for logical systems only a natural deduction formulation and a categorical semantics are missing. These we proceed to discuss
A Uniform Tableau Method for Intuitionistic Modal Logics I
 STUDIA LOGICA
, 1993
"... We present tableau systems and sequent calculi for the intuitionistic analogues IK, ID, IT, IKB, IKDB, IB, IK4, IKD4, IS4, IKB4, IK5, IKD5, IK45, IKD45 and IS5 of the normal classical modal logics. We provide soundness and completeness theorems with respect to the models of intuitionistic logic e ..."
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We present tableau systems and sequent calculi for the intuitionistic analogues IK, ID, IT, IKB, IKDB, IB, IK4, IKD4, IS4, IKB4, IK5, IKD5, IK45, IKD45 and IS5 of the normal classical modal logics. We provide soundness and completeness theorems with respect to the models of intuitionistic logic enriched by a modal accessibility relation, as proposed by G. Fischer
Paiva: Constructive CK for Contexts
 Propositional Lax Logic. Information and Computation
, 1997
"... Abstract. This note describes possible world semantics for a constructive modal logic CK. The system CK is weaker than other constructive modal logics K as it does not satisfy distribution of possibility over disjunctions, neither binary (3(A ∨ B) → 3A ∨ 3B) nor nullary (3 ⊥ → ⊥). We are intereste ..."
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Abstract. This note describes possible world semantics for a constructive modal logic CK. The system CK is weaker than other constructive modal logics K as it does not satisfy distribution of possibility over disjunctions, neither binary (3(A ∨ B) → 3A ∨ 3B) nor nullary (3 ⊥ → ⊥). We are interested in this version of constructive K for its application to contexts in AI [dP03]. However, our previous work on CK described only a categorical semantics [BdPR01] for the system, while most logicians interested in contexts prefer their semantics possible worlds style. This note fills the gap by providing the possible worlds model theory for the constructive modal system CK, showing soundness and completeness of the proposed semantics, as well as the finite model property and (hence) decidability of the system. Wijesekera [Wij90] investigated possible worlds semantics of a system similar to CK, without the binary distribution, but satisfying the nullary one. The semantics presented here for CK is new and considerably simpler than the one of Wijesekera. 1
Markov’s principle for propositional type theory
 Computer Science Logic, Proceedings of the 10 th Annual Conference of the EACSL
, 2001
"... Abstract. In this paper we show how to extend a constructive type theory with a principle that captures the spirit of Markov’s principle from constructive recursive mathematics. Markov’s principle is especially useful for proving termination of specific computations. Allowing a limited form of class ..."
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Abstract. In this paper we show how to extend a constructive type theory with a principle that captures the spirit of Markov’s principle from constructive recursive mathematics. Markov’s principle is especially useful for proving termination of specific computations. Allowing a limited form of classical reasoning we get more powerful resulting system which remains constructive and valid in the standard constructive semantics of a type theory. We also show that this principle can be formulated and used in a propositional fragment of a type theory.
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 ..."
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Cited by 7 (3 self)
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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 ..."
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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
Exponential Speedup in UL Subsumption Checking Relative to General TBoxes for the Constructive Semantics
"... Abstract. The complexity of the subsumption problem in description logics can vary widely with the choice of the syntactic fragment and the semantic interpretation. In this paper we show that the constructive semantics of concept descriptions, which includes the classical descriptive semantics as a ..."
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Abstract. The complexity of the subsumption problem in description logics can vary widely with the choice of the syntactic fragment and the semantic interpretation. In this paper we show that the constructive semantics of concept descriptions, which includes the classical descriptive semantics as a special case, offers exponential speedup in the existentialdisjunctive fragment UL of ALC. 1