5,604 research outputs found

    Propositional computability logic I

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    In the same sense as classical logic is a formal theory of truth, the recently initiated approach called computability logic is a formal theory of computability. It understands (interactive) computational problems as games played by a machine against the environment, their computability as existence of a machine that always wins the game, logical operators as operations on computational problems, and validity of a logical formula as being a scheme of "always computable" problems. The present contribution gives a detailed exposition of a soundness and completeness proof for an axiomatization of one of the most basic fragments of computability logic. The logical vocabulary of this fragment contains operators for the so called parallel and choice operations, and its atoms represent elementary problems, i.e. predicates in the standard sense. This article is self-contained as it explains all relevant concepts. While not technically necessary, however, familiarity with the foundational paper "Introduction to computability logic" [Annals of Pure and Applied Logic 123 (2003), pp.1-99] would greatly help the reader in understanding the philosophy, underlying motivations, potential and utility of computability logic, -- the context that determines the value of the present results. Online introduction to the subject is available at http://www.cis.upenn.edu/~giorgi/cl.html and http://www.csc.villanova.edu/~japaridz/CL/gsoll.html .Comment: To appear in ACM Transactions on Computational Logi

    From truth to computability I

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    The recently initiated approach called computability logic is a formal theory of interactive computation. See a comprehensive online source on the subject at http://www.cis.upenn.edu/~giorgi/cl.html . The present paper contains a soundness and completeness proof for the deductive system CL3 which axiomatizes the most basic first-order fragment of computability logic called the finite-depth, elementary-base fragment. Among the potential application areas for this result are the theory of interactive computation, constructive applied theories, knowledgebase systems, systems for resource-bound planning and action. This paper is self-contained as it reintroduces all relevant definitions as well as main motivations.Comment: To appear in Theoretical Computer Scienc

    The intuitionistic fragment of computability logic at the propositional level

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    This paper presents a soundness and completeness proof for propositional intuitionistic calculus with respect to the semantics of computability logic. The latter interprets formulas as interactive computational problems, formalized as games between a machine and its environment. Intuitionistic implication is understood as algorithmic reduction in the weakest possible -- and hence most natural -- sense, disjunction and conjunction as deterministic-choice combinations of problems (disjunction = machine's choice, conjunction = environment's choice), and "absurd" as a computational problem of universal strength. See http://www.cis.upenn.edu/~giorgi/cl.html for a comprehensive online source on computability logic

    Introduction to Cirquent Calculus and Abstract Resource Semantics

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    This paper introduces a refinement of the sequent calculus approach called cirquent calculus. While in Gentzen-style proof trees sibling (or cousin, etc.) sequents are disjoint sequences of formulas, in cirquent calculus they are permitted to share elements. Explicitly allowing or disallowing shared resources and thus taking to a more subtle level the resource-awareness intuitions underlying substructural logics, cirquent calculus offers much greater flexibility and power than sequent calculus does. A need for substantially new deductive tools came with the birth of computability logic (see http://www.cis.upenn.edu/~giorgi/cl.html) - the semantically constructed formal theory of computational resources, which has stubbornly resisted any axiomatization attempts within the framework of traditional syntactic approaches. Cirquent calculus breaks the ice. Removing contraction from the full collection of its rules yields a sound and complete system for the basic fragment CL5 of computability logic. Doing the same in sequent calculus, on the other hand, throws out the baby with the bath water, resulting in the strictly weaker affine logic. An implied claim of computability logic is that it is CL5 rather than affine logic that adequately materializes the resource philosophy traditionally associated with the latter. To strengthen this claim, the paper further introduces an abstract resource semantics and shows the soundness and completeness of CL5 with respect to it.Comment: To appear in Journal of Logic and Computatio
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