Lax Lean Archive

Polynomial many-one reductions and NP-completeness

Lax429075.Reductions · concepts/Lax429075/Reductions.lean · lax-429075

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    Natural Language Statement

    Definition

    A polynomial many-one reduction is one polynomial time computable function on binary words that preserves membership. A language is NP-complete if it belongs to NP and every language in NP reduces to it.

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    • page 3 of this submission's paper

    Lean source view on GitHub

    1import Lax434930.NondeterministicPolynomialTime
    2
    3/-!
    4---
    5title: Polynomial many-one reductions and NP-completeness
    6type: definition
    7---
    8A polynomial many-one reduction is one polynomial time computable function
    9on binary words that preserves membership. A language is NP-complete if it
    10belongs to NP and every language in NP reduces to it.
    11-/
    12
    13namespace Lax429075.Reductions
    14
    15open Lax434930.PolynomialTime Lax434930.NondeterministicPolynomialTime
    16
    17def ManyOne (A B : Language) : Prop :=
    18 ∃ f : Word → Word, Nonempty (Turing.TM2ComputableInPolyTime id id f) ∧
    19 ∀ x, x ∈ A ↔ f x ∈ B
    20
    21def NPComplete (B : Language) : Prop := B ∈ NP ∧ ∀ A : Language, A ∈ NP → ManyOne A B
    22
    23end Lax429075.Reductions
    24
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