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AC⁰ is closed under complement

Lax895169.ACZeroComplement · concepts/Lax895169/ACZeroComplement.lean · lax-895169

proven

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    Natural Language Statement

    Theorem

    If a decision problem PP is AC⁰ definable, then so is its complement PcP^c: it suffices to negate the defining sentence.

    Concept map
    22 concepts
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Classes
    3import Lax485149.Complement
    4import Lax485149.FirstOrderDefinability
    5import Lax485149.DeterministicTransitiveClosure
    6import Lax485149.ClassNL
    7import Lax485149.ClassL
    8import Lax535992.LeastFixedPoint
    9import Lax535992.InflationaryFixedPoint
    10import Lax535992.ClassPTIME
    11import Lax895169.BitPredicate
    12import Lax895169.ArithmeticLogic
    13import Lax895169.BitLogic
    14import Lax895169.LogTimeMachines
    15
    16/-!
    17---
    18title: AC⁰ is closed under complement
    19type: theorem
    20---
    21If a decision problem PP is AC⁰ definable, then so is its complement
    22PcP^c: it suffices to negate the defining sentence.
    23-/
    24
    25namespace Lax895169.ACZeroComplement
    26
    27open FirstOrder FirstOrder.Language
    28open Lax904597.Problems Lax904597.Classes
    29open Lax485149.Complement Lax485149.FirstOrderDefinability Lax485149.DeterministicTransitiveClosure
    30open Lax485149.ClassNL Lax485149.ClassL
    31open Lax535992.LeastFixedPoint Lax535992.InflationaryFixedPoint Lax535992.ClassPTIME
    32open Lax895169.BitPredicate Lax895169.ArithmeticLogic Lax895169.BitLogic Lax895169.LogTimeMachines
    33
    34/-- The complement of an AC⁰ definable problem is AC⁰ definable. -/
    35axiom ac0Definable_compl : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    36 AC0Definable P → AC0Definable (DecisionProblem.compl P)
    37
    38end Lax895169.ACZeroComplement
    39
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