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Exact correctness of deterministic computations

Lax489179.DeterministicSemantics · concepts/Lax489179/DeterministicSemantics.lean · lax-489179

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    Natural Language Statement

    Lemma

    For a deterministic program the bounded-error specification is equivalent to exact correctness on the all-zero random tape. Thus the common probability-based definition gives ordinary deterministic decision and computation when randomness is forbidden.

    Concept map
    5 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 turing_exact proven

    2 word_exact proven

    Lean source view on GitHub

    1import Lax489179.TuringMachine
    2import Lax489179.WordPrograms
    3
    4/-!
    5---
    6title: Exact correctness of deterministic computations
    7type: lemma
    8---
    9For a deterministic program the bounded-error specification is
    10equivalent to exact correctness on the all-zero random tape. Thus
    11the common probability-based definition gives ordinary deterministic
    12decision and computation when randomness is forbidden.
    13-/
    14
    15namespace Lax489179.DeterministicSemantics
    16
    17axiom turing_exact (M : TuringMachine.Machine) (h : TuringMachine.Deterministic M)
    18 (input : List Bool) (answer : Prop) (t : ℕ) :
    19 TuringMachine.DecidesWithin M input answer t ↔
    20 ∃ b, (TuringMachine.run M t (fun _ => false) (TuringMachine.init M input)).answer =
    21 some b ∧ (b = true ↔ answer)
    22
    23axiom word_exact (w : ℕ) (p : WordPrograms.Program) (h : WordPrograms.Deterministic p)
    24 (input : List ℕ) (correct : List ℕ → Prop) (t : ℕ) :
    25 WordPrograms.ComputesWithin w p input correct t ↔
    26 ∃ y, WordPrograms.output (WordPrograms.run w p t (fun _ => false)
    27 (WordPrograms.init input)) = some y ∧ correct y
    28
    29end Lax489179.DeterministicSemantics
    30
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