Computer Runtimes and the Length of Proofs
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.
Keywordshalting problem halting probability proof length automatic theorem proving Busy Beaver problem program-size complexity small Turing machines
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- 2.Baumgartner, P., Zhang, H.: On Using Ground Joinable Equations in Equational Theorem Proving. In: Proceedings of the 3rd International Workshop on First Order Theorem Proving (St Andrews, Scotland), Fachberichte Informatik 5/2000, pp. 33–43. Universität Koblenz-Landau (2000)Google Scholar
- 5.Chaitin, G.J.: Computing the Busy Beaver function. Information, Randomness & Incompleteness, 74–76 (1984)Google Scholar
- 7.Delahaye, J.-P., Zenil, H.: Numerical Evaluation of Algorithmic Complexity for Short Strings: A Glance Into the Innermost Structure of Randomness. Appl. Math. Comput. (in press, 2011)Google Scholar
- 8.Joosten, J., Soler-Toscano, F., Zenil, H.: Program-size Versus Time Complexity, Speed-up and Slowdown Phenomena in Small Turing Machines. International Journal of Unconventional Computing (2011)Google Scholar
- 13.Wolfram, S.: A New Kind of Science. Wolfram Media (2002)Google Scholar
- 15.Zenil, H.: Busy Beaver, from the Wolfram Demonstrations Project (2009), http://demonstrations.wolfram.com/BusyBeaver/