Computer Studies of Turing Machine Problems
Journal of the ACM (JACM)
A Theory of Program Size Formally Identical to Information Theory
Journal of the ACM (JACM)
A new kind of science
CADE-18 Proceedings of the 18th International Conference on Automated Deduction
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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.