Lax Lean Archive

Logarithmic space for inductive counting

Lax362205.CountMachine · concepts/Lax362205/CountMachine.lean · lax-362205

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    Natural Language Statement

    Lemma

    For a machine using logarithmic space, a nondeterministic machine implements its configuration counting predicate in logarithmic space. Vertices and bounded paths are generated sequentially. Every branch halts.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    In the paper

    • page 3 of this submission's paper

    Lean source view on GitHub

    1import Lax362205.ConfigurationCounting
    2import Lax434930.SpaceBounds
    3
    4/-!
    5---
    6title: Logarithmic space for inductive counting
    7type: lemma
    8---
    9For a machine using logarithmic space, a nondeterministic machine implements
    10its configuration counting predicate in logarithmic space. Vertices and
    11bounded paths are generated sequentially. Every branch halts.
    12-/
    13
    14namespace Lax362205.CountMachine
    15
    16open Lax434930.PolynomialTime Lax434930.SpaceMachines Lax434930.SpaceBounds
    17open ConfigurationCounting
    18
    19axiom implement (M : Machine) (c : ℕ) (hc : 0 < c)
    20 (hM : ∀ w, M.UsesSpace w (c * logSpace w.length)) :
    21 ∃ (d : ℕ) (D : Machine), 0 < d ∧
    22 (∀ w, D.HaltsOn w ∧ (D.Accepts w ↔ RejectsByCounting M w (c * logSpace w.length))) ∧
    23 ∀ w, D.UsesSpace w (d * logSpace w.length)
    24
    25end Lax362205.CountMachine
    26
    Show Proof
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