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AC⁰ definability reads finite instances only

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

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    Natural Language Statement

    Theorem

    AC⁰ definability only depends on the finite instances of a problem: two problems with the same finite yes-instances are both AC⁰ definable or both not.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    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 definability reads finite instances only
    19type: theorem
    20---
    21AC⁰ definability only depends on the finite instances of a problem: two
    22problems with the same finite yes-instances are both AC⁰ definable or both
    23not.
    24-/
    25
    26namespace Lax895169.ACZeroFinite
    27
    28open FirstOrder FirstOrder.Language
    29open Lax904597.Problems Lax904597.Classes
    30open Lax485149.Complement Lax485149.FirstOrderDefinability Lax485149.DeterministicTransitiveClosure
    31open Lax485149.ClassNL Lax485149.ClassL
    32open Lax535992.LeastFixedPoint Lax535992.InflationaryFixedPoint Lax535992.ClassPTIME
    33open Lax895169.BitPredicate Lax895169.ArithmeticLogic Lax895169.BitLogic Lax895169.LogTimeMachines
    34
    35/-- AC⁰ definability only depends on the finite instances of a problem. -/
    36axiom ac0Definable_congr_finite :
    37 ∀ {L : Language.{0, 0}} [L.IsRelational] {P Q : DecisionProblem L},
    38 (∀ (A : Type) [L.Structure A] [Finite A], P A ↔ Q A) → (AC0Definable P ↔ AC0Definable Q)
    39
    40end Lax895169.ACZeroFinite
    41
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