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PTIME at the bottom of the polynomial hierarchy

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

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    Natural Language Statement

    Theorem

    Polynomial time is the bottom of the hierarchy: PTIME is contained in coNP, and in Σkp\Sigma_k^p for every kk; with PTIME ⊆\subseteq NP this places it in NP ∩\cap coNP. The inclusion in coNP is the dual of the inclusion in NP, PTIME being closed under complement.

    Concept map
    23 concepts
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    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 piP_zero_eq proven

    2 PTIME_subset_coNP proven

    3 PTIME_subset_sigmaP 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: PTIME at the bottom of the polynomial hierarchy
    22type: theorem
    23---
    24Polynomial time is the bottom of the hierarchy: PTIME is contained in coNP,
    25and in Σkp\Sigma_k^p for every kk; with PTIME ⊆\subseteq NP this places
    26it in NP ∩\cap coNP. The inclusion in coNP is the dual of the inclusion
    27in NP, PTIME being closed under complement.
    28-/
    29
    30namespace Lax564036.PolynomialTimeInHierarchy
    31
    32open FirstOrder FirstOrder.Language
    33open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    34open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    35open Lax485149.Problems Lax485149.Complement Lax535992.ClassPTIME
    36open Lax564036.Hierarchy Lax564036.Difference Lax564036.Tautology Lax564036.ThreeDnfTautology
    37open Lax564036.SatUnsat Lax564036.QuantifiedBooleanFormulas Lax564036.AlternatingMachines
    38
    39/-- PTIME is contained in coNP. -/
    40axiom PTIME_subset_coNP : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    41 PTIME.Mem P → coNP.Mem P
    42
    43/-- PTIME is contained in every `Σ` level. -/
    44axiom PTIME_subset_sigmaP : ∀ (k : ℕ) {L : Language.{0, 0}} [L.IsRelational]
    45 (P : DecisionProblem L), PTIME.Mem P → (SigmaP k).Mem P
    46
    47/-- The two classes of level `0` coincide. -/
    48axiom piP_zero_eq : PiP 0 = SigmaP 0
    49
    50end Lax564036.PolynomialTimeInHierarchy
    51
    Show ProofShow ProofShow Proof
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