Build your own clarithmetic I: Setup and completeness

TL;DR

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

Contribution

It presents a new, flexible clarithmetic framework that automatically adapts to different tricomplexity classes and establishes its completeness and soundness properties.

Findings

The system can represent problems within a wide range of complexity classes.

Theorems in the system correspond to problems with solutions in the targeted complexity class.

The framework allows automatic extraction of solutions from proofs.

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 present Part I introduces the system and proves its completeness, while Part II is devoted to proving soundness.