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Computability and polynomial-time equivalence of Turing machines and word RAMs

lax-51·formalized by Szymon Toruńczyk @szymtor·Codex 5.6·created 2026-08-09·GitHub @6fe5a3d·Lean v4.30.0 epoch · mathlib c5ea00351c28

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    Abstract

    Turing machines and word random access machines compute exactly the same total functions from finite lists of natural numbers to finite lists of natural numbers. The word RAM is the archive's existing canonical model. Because that machine has a finite word length, plain RAM computability includes a computable input-dependent threshold above which one uniform program must return the exact answer. The effective threshold rules out the strictly weaker notion of convergence without a computable modulus.

    The submission also states the polynomial-time refinement. Both machines measure an input by the length of one canonical binary encoding. On the RAM side, one polynomial bounds the sufficient word length and another bounds the number of instructions; this is exactly the restriction needed for a Turing simulation of unit-cost word operations to retain polynomial time.

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    thm✓proven claimdefdefinition

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    assumptions conclusionProven claimThis submissionProof — click to open

    Lean sources for these proofs: proofs/ on GitHub

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    Cite this

    @misc{lax-51,
  author = {Szymon Toruńczyk and Codex 5.6},
  title = {Computability and polynomial-time equivalence of Turing machines and word RAMs},
  year = {2026},
  howpublished = {Lax Archive, lax-51},
  url = {https://laxarchive.org/lax-51/},
  note = {draft},
}

    References

    1. Alfred V. Aho, John E. Hopcroft and Jeffrey D. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley, 1974.

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