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Π_k is co-Σ_k

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

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    Natural Language Statement

    Theorem

    For every kk, a decision problem is in Πkp\Pi_k^p if and only if its complement is in Σkp\Sigma_k^p. At level 0 this is the definition of coPTIME; above, negating a second-order sentence exchanges the two kinds of quantifier blocks. In particular the complement of a problem is in coNP exactly when the problem is in NP.

    Concept map
    23 concepts
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    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 compl_mem_coNP_iff proven

    2 mem_piP_iff 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: Π_k is co-Σ_k
    22type: theorem
    23---
    24For every kk, a decision problem is in Πkp\Pi_k^p if and only if its
    25complement is in Σkp\Sigma_k^p. At level 0 this is the definition of
    26coPTIME; above, negating a second-order sentence exchanges the two kinds of
    27quantifier blocks. In particular the complement of a problem is in coNP
    28exactly when the problem is in NP.
    29-/
    30
    31namespace Lax564036.HierarchyDuality
    32
    33open FirstOrder FirstOrder.Language
    34open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    35open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    36open Lax485149.Problems Lax485149.Complement Lax535992.ClassPTIME
    37open Lax564036.Hierarchy Lax564036.Difference Lax564036.Tautology Lax564036.ThreeDnfTautology
    38open Lax564036.SatUnsat Lax564036.QuantifiedBooleanFormulas Lax564036.AlternatingMachines
    39
    40/-- `Πₖᵖ` is the class of complements of `Σₖᵖ` problems. -/
    41axiom mem_piP_iff : ∀ (k : ℕ) {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    42 (PiP k).Mem P ↔ (SigmaP k).Mem (DecisionProblem.compl P)
    43
    44/-- The complement of a problem is in coNP exactly when the problem is in NP. -/
    45axiom compl_mem_coNP_iff : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    46 coNP.Mem (DecisionProblem.compl P) ↔ NP.Mem P
    47
    48end Lax564036.HierarchyDuality
    49
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