Computer Runtimes and the Length of Proofs: On an Algorithmic Probabilistic Application to Waiting Times in Automatic Theorem Proving

TL;DR

This paper investigates the relationship between Turing machine runtimes and proof lengths in formal systems, suggesting a non-linear tradeoff that could optimize automatic theorem proving timeouts.

Contribution

It provides an experimental analysis linking machine runtimes and proof lengths, proposing a probabilistic approach to optimize waiting times in theorem proving.

Findings

Evidence of non-linear tradeoff between runtime and proof size

Statistics for small parameter choices in Turing machines and theorem provers

Potential for optimizing automatic theorem proving timeouts

Abstract

This paper is an experimental exploration of the relationship between the runtimes of Turing machines and the length of proofs in formal axiomatic systems. We compare the number of halting Turing machines of a given size to the number of provable theorems of first-order logic of a given size, and the runtime of the longest-running Turing machine of a given size to the proof length of the most-difficult-to-prove theorem of a given size. It is suggested that theorem provers are subject to the same non-linear tradeoff between time and size as computer programs are, affording the possibility of determining optimal timeouts and waiting times in automatic theorem proving. I provide the statistics for some small choices of parameters for both of these systems.