The intuitionistic fragment of computability logic at the propositional level

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

The logic of interactive Turing reduction

The paper gives a soundness and completeness proof for the implicative fragment of intuitionistic calculus with respect to the semantics of computability logic, which understands intuitionistic implication as interactive algorithmic reduction. This concept -- more precisely, the associated concept of reducibility -- is a generalization of Turing reducibility from the traditional, input/output sorts of problems to computational tasks of arbitrary degrees of interactivity. See http://www.cis.upenn.edu/~giorgi/cl.html for a comprehensive online source on computability logic

Propositional computability logic I

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

Intuitionistic computability logic

Computability logic (CL) is a systematic formal theory of computational tasks and resources, which, in a sense, can be seen as a semantics-based alternative to (the syntactically introduced) linear logic. With its expressive and flexible language, where formulas represent computational problems and "truth" is understood as algorithmic solvability, CL potentially offers a comprehensive logical basis for constructive applied theories and computing systems inherently requiring constructive and computationally meaningful underlying logics. Among the best known constructivistic logics is Heyting's intuitionistic calculus INT, whose language can be seen as a special fragment of that of CL. The constructivistic philosophy of INT, however, has never really found an intuitively convincing and mathematically strict semantical justification. CL has good claims to provide such a justification and hence a materialization of Kolmogorov's known thesis "INT = logic of problems". The present paper contains a soundness proof for INT with respect to the CL semantics. A comprehensive online source on CL is available at http://www.cis.upenn.edu/~giorgi/cl.htm

From truth to computability I

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