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The class DP

Lax564036.Difference · concepts/Lax564036/Difference.lean · lax-564036

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    Natural Language Statement

    Definition

    A decision problem PP is DP-definable when there are a Σ1\Sigma_1-definable problem SS and a Π1\Pi_1-definable problem TT over the same vocabulary such that, on nonempty finite structures, PP holds exactly when both SS and TT hold: the conjunction of an NP condition and a coNP condition, or equivalently the difference of two NP problems. DP is the class of the DP-definable problems, the class of Papadimitriou and Yannakakis, with the cofinal hardness of the NP core.

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    1import Mathlib.ModelTheory.Semantics
    2import Lax904597.Problems
    3import Lax904597.SecondOrder
    4import Lax904597.Classes
    5
    6/-!
    7---
    8title: The class DP
    9type: definition
    10---
    11A decision problem PP is DP-definable when there are a
    12Σ1\Sigma_1-definable problem SS and a Π1\Pi_1-definable problem TT over
    13the same vocabulary such that, on nonempty finite structures, PP holds
    14exactly when both SS and TT hold: the conjunction of an NP condition and a
    15coNP condition, or equivalently the difference of two NP problems. DP is the
    16class of the DP-definable problems, the class of Papadimitriou and
    17Yannakakis, with the cofinal hardness of the NP core.
    18-/
    19
    20namespace Lax564036.Difference
    21
    22open Lax904597.Problems Lax904597.SecondOrder Lax904597.Classes
    23
    24open FirstOrder
    25
    26open Language Structure
    27
    28/-- A decision problem is **DP-definable** if, on nonempty finite structures,
    29it is the conjunction of a `Σ₁`-definable and a `Π₁`-definable problem: an NP
    30condition and a coNP one, imposed together. Equivalently, it is the difference
    31`S \ Tᶜ` of two NP problems. -/
    32def DPDefinable {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L) : Prop :=
    33 ∃ S T : DecisionProblem L, SigmaSODefinable 1 S ∧ PiSODefinable 1 T ∧
    34 ∀ (A : Type) [L.Structure A] [Finite A] [Nonempty A], P A ↔ (S A ∧ T A)
    35
    36/-- **DP**: the class of the DP-definable problems, with cofinal hardness. -/
    37def DP : ComplexityClass :=
    38 ComplexityClass.ofMem fun P => DPDefinable P
    39
    40end Lax564036.Difference
    41
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    From Mathlib

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