From truth to computability I

The recently initiated approach called computability logic is a formal theory of interactive computation. It understands computational problems as games played by a machine against the environment, and uses logical formalism to describe valid principles of computability, with formulas representing computational problems and logical operators standing for operations on computational problems. The concept of computability that lies under this approach is a nontrivial generalization of Church-Turing computability from simple, two-step (question/answer, input/output) problems to problems of arbitrary degrees of interactivity. Restricting this concept to predicates, which are understood as computational problems of zero degree of interactivity, yields exactly classical truth. This makes computability logic a generalization and refinement of classical logic. The foundational paper “Introduction to computability logic” [Annals of Pure and Applied Logic 123 (2003), pp. 1-99] was focused on semantics rather than syntax, and certain axiomatizability assertions in it were only stated as conjectures. The present contribution contains a verification of one of such conjectures: 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. CL3 is a conservative extention of classical predicate calculus in the language which, along with all of the (appropriately generalized) logical operators of classical logic, contains propositional connectives and quantifiers representing the so called choice operations. The atoms of this language are interpreted as elementary problems, i.e. predicates in the standard sense. Among the potential application areas for CL3 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. It is meant for a wide audience and does not assume that the reader has specialized knowledge in any particular subarea of logic or computer science. MSC: primary: 03B47; secondary: 03F50; 03B70; 68Q10; 68T27; 68T30; 91A05