Build your own clarithmetic II: Soundness

TL;DR

This paper introduces a parameterized clarithmetic system based on computability logic, capable of representing and solving interactive computational problems within various tricomplexity constraints, and proves its soundness.

Contribution

It presents a new parameterized system CLA11(P1,P2,P3,P4) that automatically adapts to different complexity classes and establishes its soundness for these classes.

Findings

The system is sound for a wide range of tricomplexity classes.

Automatic extraction of solutions from proofs is possible.

The system can be tuned for extensional and intensional completeness.

Abstract

Clarithmetics are number theories based on computability logic (see http://www.csc.villanova.edu/~japaridz/CL/ ). Formulas of these theories represent interactive computational problems, and their "truth" is understood as existence of an algorithmic solution. Various complexity constraints on such solutions induce various versions of clarithmetic. The present paper introduces a parameterized/schematic version CLA11(P1,P2,P3,P4). By tuning the three parameters P1,P2,P3 in an essentially mechanical manner, one automatically obtains sound and complete theories with respect to a wide range of target tricomplexity classes, i.e. combinations of time (set by P3), space (set by P2) and so called amplitude (set by P1) complexities. Sound in the sense that every theorem T of the system represents an interactive number-theoretic computational problem with a solution from the given tricomplexity class and, furthermore, such a solution can be automatically extracted from a proof of T. And complete in the sense that every interactive number-theoretic problem with a solution from the given tricomplexity class is represented by some theorem of the system. Furthermore, through tuning the 4th parameter P4, at the cost of sacrificing recursive axiomatizability but not simplicity or elegance, the above extensional completeness can be strengthened to intensional completeness, according to which every formula representing a problem with a solution from the given tricomplexity class is a theorem of the system. This article is published in two parts. The previous Part I has introduced the system and proved its completeness, while the present Part II is devoted to proving soundness.