Computer Runtimes and the Length of Proofs

With an Algorithmic Probabilistic Application to Waiting Times in Automatic Theorem Proving
  • Hector Zenil
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7160)


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.


halting problem halting probability proof length automatic theorem proving Busy Beaver problem program-size complexity small Turing machines 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Hector Zenil
    • 1
    • 2
  1. 1.Dept. of Computer ScienceUniversity of SheffieldUK
  2. 2.Special ProjectsWolfram Research, Inc.USA

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