Do not throw out the baby: Clarithmetics as alternatives to weak arithmetics

TL;DR

This paper introduces clarithmetics, CoL-based axiomatic theories that extend Peano arithmetic with explicit computational resource control, offering advantages over traditional bounded arithmetic in extracting optimal algorithms.

Contribution

It presents new clarithmetic systems (CLA4-CLA7 and CLA11), demonstrating their soundness, completeness, and advantages over weak arithmetics in computational and logical expressiveness.

Findings

Clarithmetics support direct extraction of near-optimal algorithms from proofs.

They maintain full arithmetical strength while controlling computational resources.

They outperform bounded arithmetic in preserving useful arithmetical information.

Abstract

Computability logic (CoL) provides a semantic foundation in which formulas represent interactive computational problems and validity corresponds to uniform algorithmic solvability. Building on this foundation, clarithmetics -- CoL-based axiomatic number theories -- combine the full arithmetical strength of Peano arithmetic (PA) with explicit control over computational resources. In contrast to traditional bounded arithmetic and related complexity-oriented systems, they strengthen rather than weaken PA. This paper, after briefly surveying the relevant fragment of CoL, presents the systems CLA4-CLA7 and CLA11 of clarithmetic, and outlines their soundness and completeness with respect to natural classes of time, space, and so called amplitude complexities. We argue that, by weakening PA, traditional complexity-oriented systems of arithmetic throw out the baby with the bathwater, discarding large amounts of innocent and useful arithmetical information and losing intensional flexibility essential for natural specification and program extraction. Clarithmetics avoid this loss while supporting direct extraction of optimal or near-optimal algorithms from proofs and providing strong intensional completeness properties absent from bounded arithmetic and related systems. A central message of the paper is that the argued advantages of clarithmetics deserve either acknowledgment or serious refutation from the weak-arithmetics community. To date, neither has occurred.