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Lax51.RamToTuringGenericTime

Generic-time simulation of word RAMs by Turing machines

concepts/Lax51/RamToTuringGenericTime.lean · lax-51

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    Theorem

    An arbitrary finite execution of a fixed word-RAM program, using a word length polynomially bounded in the encoded input size, can be simulated by one finite multi-tape Turing machine. The Turing running time has fixed polynomial overhead in the encoded input size plus the number of executed RAM instructions.

    No regularity assumption is made on the RAM running time. A polynomial word length bound is retained because a Turing machine must explicitly process the bits of every RAM word.

    Lean source view on GitHub

    1import Lax51.RamPolytime
    2import Mathlib.Computability.TuringMachine.Computable
    3
    4/-!
    5---
    6title: Generic-time simulation of word RAMs by Turing machines
    7type: theorem
    8---
    9An arbitrary finite execution of a fixed word-RAM program, using a word
    10length polynomially bounded in the encoded input size, can be simulated by
    11one finite multi-tape Turing machine. The Turing running time has fixed
    12polynomial overhead in the encoded input size plus the number of executed
    13RAM instructions.
    14
    15No regularity assumption is made on the RAM running time. A polynomial word
    16length bound is retained because a Turing machine must explicitly process
    17the bits of every RAM word.
    18-/
    19
    20namespace Lax51.RamToTuringGenericTime
    21
    22open Lax13.Ram
    23open Lax51.BinaryWordEncoding Lax51.RamPolytime
    24open Polynomial Turing
    25
    26/-- At a polynomially bounded selected word length, every `t`-step execution
    27of a fixed word-RAM program is simulated by one finite Turing machine in a
    28fixed polynomial of the input bit-size plus `t`. -/
    29axiom ramInTime_to_turingInPolynomialOverhead
    30 (p : Program) (wordBound : Polynomial ℕ) :
    31 ∃ (H : TM2ComputableAux Symbol Symbol)
    32 (widthOverhead timeOverhead : Polynomial ℕ),
    33 (∀ n, wordBound.eval n ≤ widthOverhead.eval n) ∧
    34 ∀ (x output : List ℕ) (t : ℕ),
    35 FitsInWords (wordBound.eval (bitSize x))
    36 ((x.length :: x) ++ output) →
    37 RunsTo (widthOverhead.eval (bitSize x)) p
    38 (x.length :: x) output t →
    39 Nonempty (TM2OutputsInTime H.tm
    40 (List.map H.inputAlphabet.invFun (encode x))
    41 (some (List.map H.outputAlphabet.invFun (encode output)))
    42 (timeOverhead.eval (bitSize x + t)))
    43
    44end Lax51.RamToTuringGenericTime
    45
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