From truth to computability II

TL;DR

This paper extends the soundness and completeness results of computability logic from system CL3 to the more expressive system CL4, which includes elementary and general atoms, and explores decidability aspects.

Contribution

It proves soundness and completeness for CL4, an extension of CL3 with more expressive atoms and logical operators, advancing the axiomatization of computability logic.

Findings

CL4 extends CL3 with elementary and general atoms.

Removing blind quantifiers from CL4 makes it decidable.

CL4 remains first-order despite increased expressiveness.

Abstract

Computability logic is a formal theory of computational tasks and resources. Formulas in it represent interactive computational problems, and "truth" is understood as algorithmic solvability. Interactive computational problems, in turn, are defined as a certain sort games between a machine and its environment, with logical operators standing for operations on such games. Within the ambitious program of finding axiomatizations for incrementally rich fragments of this semantically introduced logic, the earlier article "From truth to computability I" proved soundness and completeness for system CL3, whose language has the so called parallel connectives (including negation), choice connectives, choice quantifiers, and blind quantifiers. The present paper extends that result to the significantly more expressive system CL4 with the same collection of logical operators. What makes CL4 expressive is the presence of two sorts of atoms in its language: elementary atoms, representing elementary computational problems (i.e. predicates, i.e. problems of zero degree of interactivity), and general atoms, representing arbitrary computational problems. CL4 conservatively extends CL3, with the latter being nothing but the general-atom-free fragment of the former. Removing the blind (classical) group of quantifiers from the language of CL4 is shown to yield a decidable logic despite the fact that the latter is still first-order. A comprehensive online source on computability logic can be found at http://www.cis.upenn.edu/~giorgi/cl.html