{
  "id": "one-tape-time-in-two-fifths-power-space",
  "title": "One-tape time in two-fifths-power space",
  "description": "Machines with one writable tape: blocks, visits, controllers and the local replay of a list of controllers. Finite fields of characteristic two, polynomial extensions of Boolean functions, and an evaluator that computes words defined by rules with three shared registers. The theorem: T steps of a machine with one writable tape are simulated in about T^(2/5) bits of storage, with a complete proof. Several writable tapes in about sqrt(T) bits; a lower bound for time from the space hierarchy theorem, which is a named hypothesis. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "one-tape-machines-blocks-and-visits",
      "title": "One-tape machines, blocks and visits",
      "reader": "one-tape-machines-blocks-and-visits.html",
      "source": "src/one-tape-machines-blocks-and-visits.md",
      "source_sha256": "be6b524069dc1fdcc95fa4d65d9da9c6f24ac220231cec1168b7a8bbb58c37a3",
      "status": "author_self_checked"
    },
    {
      "id": "polynomials-for-boolean-functions",
      "title": "Polynomials for Boolean functions",
      "reader": "polynomials-for-boolean-functions.html",
      "source": "src/polynomials-for-boolean-functions.md",
      "source_sha256": "0f6da4f6212a891ef23151f89c3a9d39ea02c397f8ec9edcc8e81742ed3beef8",
      "status": "author_self_checked"
    },
    {
      "id": "an-evaluator-with-one-pool-of-registers",
      "title": "An evaluator with one pool of registers",
      "reader": "an-evaluator-with-one-pool-of-registers.html",
      "source": "src/an-evaluator-with-one-pool-of-registers.md",
      "source_sha256": "86af91349ef902b96bdb7d60d108537e300f3accff060e2ebf3b0073fd8a3b3e",
      "status": "author_self_checked"
    },
    {
      "id": "histories-checkpoints-and-the-theorem",
      "title": "Histories, checkpoints and the theorem",
      "reader": "histories-checkpoints-and-the-theorem.html",
      "source": "src/histories-checkpoints-and-the-theorem.md",
      "source_sha256": "8559645269bb3a52254db6616e422bf2be24a5ac86ff83387965ab3ab14c7453",
      "status": "author_self_checked"
    },
    {
      "id": "several-writable-tapes-and-lower-bounds-for-time",
      "title": "Several writable tapes, and lower bounds for time",
      "reader": "several-writable-tapes-and-lower-bounds-for-time.html",
      "source": "src/several-writable-tapes-and-lower-bounds-for-time.md",
      "source_sha256": "f4b434b6f086403e9787ca3423e41413a9bba1fe2270f8512045bb469e733943",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Mathematical Logic, Set Theory, and Computability (C80): Turing machines",
    "polynomials over a field; finite fields are constructed in the second lesson",
    "the lessons prove every other fact they use, except the named hypothesis (SH) of the fifth lesson, which is used only for Corollary 3.1 there"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}