375 research outputs found

    Defining Logical Systems via Algebraic Constraints on Proofs

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    We comprehensively present a program of decomposition of proof systems for non-classical logics into proof systems for other logics, especially classical logic, using an algebra of constraints. That is, one recovers a proof system for a target logic by enriching a proof system for another, typically simpler, logic with an algebra of constraints that act as correctness conditions on the latter to capture the former; for example, one may use Boolean algebra to give constraints in a sequent calculus for classical propositional logic to produce a sequent calculus for intuitionistic propositional logic. The idea behind such forms of reduction is to obtain a tool for uniform and modular treatment of proof theory and provide a bridge between semantics logics and their proof theory. The article discusses the theoretical background of the project and provides several illustrations of its work in the field of intuitionistic and modal logics. The results include the following: a uniform treatment of modular and cut-free proof systems for a large class of propositional logics; a general criterion for a novel approach to soundness and completeness of a logic with respect to a model-theoretic semantics; and a case study deriving a model-theoretic semantics from a proof-theoretic specification of a logic.Comment: submitte

    The Complexity of Reasoning about Spatial Congruence

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    In the recent literature of Artificial Intelligence, an intensive research effort has been spent, for various algebras of qualitative relations used in the representation of temporal and spatial knowledge, on the problem of classifying the computational complexity of reasoning problems for subsets of algebras. The main purpose of these researches is to describe a restricted set of maximal tractable subalgebras, ideally in an exhaustive fashion with respect to the hosting algebras. In this paper we introduce a novel algebra for reasoning about Spatial Congruence, show that the satisfiability problem in the spatial algebra MC-4 is NP-complete, and present a complete classification of tractability in the algebra, based on the individuation of three maximal tractable subclasses, one containing the basic relations. The three algebras are formed by 14, 10 and 9 relations out of 16 which form the full algebra

    Efficient Constraints on Possible Worlds for Reasoning About Necessity

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    Modal logics offer natural, declarative representations for describing both the modular structure of logical specifications and the attitudes and behaviors of agents. The results of this paper further the goal of building practical, efficient reasoning systems using modal logics. The key problem in modal deduction is reasoning about the world in a model (or scope in a proof) at which an inference rule is applied - a potentially hard problem. This paper investigates the use of partial-order mechanisms to maintain constraints on the application of modal rules in proof search in restricted languages. The main result is a simple, incremental polynomial-time algorithm to correctly order rules in proof trees for combinations of K, K4, T and S4 necessity operators governed by a variety of interactions, assuming an encoding of negation using a scoped constant ⊥. This contrasts with previous equational unification methods, which have exponential performance in general because they simply guess among possible intercalations of modal operators. The new, fast algorithm is appropriate for use in a wide variety of applications of modal logic, from planning to logic programming

    Efficient Constraints on Possible Worlds for Reasoning about Necessity

    Get PDF
    Modal logics offer natural, declarative representations for describing both the modular structure of logical specifications and the attitudes and behaviors of agents. The results of this paper further the goal of building practical, efficient reasoning systems using modal logics. The key problem in modal deduction is reasoning about the world in a model (or scope in a proof) at which an inference rule is applied—a potentially hard problem. This paper investigates the use of partial-order mechanisms to maintain constraints on the application of modal rules in proof search in restricted languages. The main result is a simple, incremental polynomial-time algorithm to correctly order rules in proof trees for combinations of K, K4, T and S4 necessity operators governed by a variety of interactions, assuming an encoding of negation using a scoped constant ┴. This contrasts with previous equational unification methods, which have exponential performance in general because they simply guess among possible intercalations of modal operators. The new, fast algorithm is appropriate for use in a wide variety of applications of modal logic, from planning to logic programming

    MetTeL: A Generic Tableau Prover.

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