On the system CL12 of computability logic

Abstract

Computability logic (see http://www.csc.villanova.edu/~japaridz/CL/) is a long-term project for redeveloping logic on the basis of a constructive game semantics, with games seen as abstract models of interactive computational problems. Among the fragments of this logic successfully axiomatized so far is CL12 --- a conservative extension of classical first-order logic, whose language augments that of classical logic with the so called choice sorts of quantifiers and connectives. This system has already found fruitful applications as a logical basis for constructive and complexity-oriented versions of Peano arithmetic, such as arithmetics for polynomial time computability, polynomial space computability, and beyond. The present paper introduces a third, indispensable complexity measure for interactive computations termed amplitude complexity, and establishes the adequacy of CL12 with respect to A-amplitude, S-space and T-time computability under certain minimal conditions on the triples (A,S,T) of function classes. This result very substantially broadens the potential application areas of CL12. The paper is self-contained, and targets readers with no prior familiarity with the subject.