Lax Lean Archive

DP is closed under first-order reductions

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

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Theorem

    Membership in DP travels backward along first-order reductions and along ordered first-order reductions: if a problem reduces to a problem of the class, it is in the class. Membership reads a problem on its finite instances only.

    The two halves of a DP definition are not individually invariant in the order, so an ordered reduction is not pulled back half by half: the order is quantified existentially on the NP half and universally on the coNP half.

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

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 DP_mem_congr_finite proven

    2 DP_mem_of_foReduction proven

    3 DP_mem_of_orderedReduction proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Interpretations
    3import Lax904597.Relativized
    4import Lax904597.SecondOrder
    5import Lax904597.Classes
    6import Lax904597.Sat
    7import Lax904597.Machines
    8import Lax485149.Problems
    9import Lax485149.Complement
    10import Lax535992.ClassPTIME
    11import Lax564036.Hierarchy
    12import Lax564036.Difference
    13import Lax564036.Tautology
    14import Lax564036.ThreeDnfTautology
    15import Lax564036.SatUnsat
    16import Lax564036.QuantifiedBooleanFormulas
    17import Lax564036.AlternatingMachines
    18
    19/-!
    20---
    21title: DP is closed under first-order reductions
    22type: theorem
    23---
    24Membership in DP travels backward along first-order reductions and along
    25ordered first-order reductions: if a problem reduces to a problem of the
    26class, it is in the class. Membership reads a problem on its finite
    27instances only.
    28
    29The two halves of a DP definition are not individually invariant in the
    30order, so an ordered reduction is not pulled back half by half: the order is
    31quantified existentially on the NP half and universally on the coNP half.
    32-/
    33
    34namespace Lax564036.DPClosure
    35
    36open FirstOrder FirstOrder.Language
    37open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    38open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    39open Lax485149.Problems Lax485149.Complement Lax535992.ClassPTIME
    40open Lax564036.Hierarchy Lax564036.Difference Lax564036.Tautology Lax564036.ThreeDnfTautology
    41open Lax564036.SatUnsat Lax564036.QuantifiedBooleanFormulas Lax564036.AlternatingMachines
    42
    43/-- Membership travels backward along first-order reductions. -/
    44axiom DP_mem_of_foReduction : ∀ {L L' : Language.{0, 0}} [L.IsRelational] [L'.IsRelational]
    45 {P : DecisionProblem L} {Q : DecisionProblem L'}, FOReduction P Q → DP.Mem Q → DP.Mem P
    46
    47/-- Membership travels backward along ordered first-order reductions. -/
    48axiom DP_mem_of_orderedReduction : ∀ {L L' : Language.{0, 0}} [L.IsRelational] [L'.IsRelational]
    49 {P : DecisionProblem L} {Q : DecisionProblem L'}, OrderedFOReduction P Q → DP.Mem Q → DP.Mem P
    50
    51/-- Membership only depends on the finite instances of a problem. -/
    52axiom DP_mem_congr_finite : ∀ {L : Language.{0, 0}} [L.IsRelational] {P Q : DecisionProblem L},
    53 (∀ (A : Type) [L.Structure A] [Finite A], P A ↔ Q A) → (DP.Mem P ↔ DP.Mem Q)
    54
    55end Lax564036.DPClosure
    56
    Show ProofShow ProofShow Proof
    Builds on
    Used by

    none

    From Mathlib

    none

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…