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Alternating acceptance in bounded space, characterized

Lax480241.AlternatingSpaceValue · concepts/Lax480241/AlternatingSpaceValue.lean · lax-480241

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    Natural Language Statement

    Lemma

    The conditions of acceptance by alternating machines in bounded space are invariant under isomorphism, so the problem holds of an instance exactly when the instance is well formed, its states are split in two, and it accepts in bounded space.

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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 Lax904597.Machines
    4import Lax485149.Problems
    5import Lax485149.Complement
    6import Lax485149.ClassNL
    7import Lax535992.ClassPTIME
    8import Lax564036.Hierarchy
    9import Lax564036.AlternatingMachines
    10import Lax134656.ClassPSPACE
    11import Lax480241.Expansions
    12import Lax480241.SecondOrderFixedPoints
    13import Lax480241.AlternatingSpace
    14import Lax480241.ExponentialClasses
    15
    16/-!
    17---
    18title: Alternating acceptance in bounded space, characterized
    19type: lemma
    20---
    21The conditions of acceptance by alternating machines in bounded space are
    22invariant under isomorphism, so the problem holds of an instance exactly
    23when the instance is well formed, its states are split in two, and it
    24accepts in bounded space.
    25-/
    26
    27namespace Lax480241.AlternatingSpaceValue
    28
    29open FirstOrder FirstOrder.Language
    30open Lax904597.Problems Lax904597.Classes Lax904597.Machines Lax485149.Problems
    31 Lax485149.Complement Lax485149.ClassNL
    32open Lax535992.ClassPTIME Lax564036.Hierarchy Lax564036.AlternatingMachines Lax134656.ClassPSPACE
    33open Lax480241.Expansions Lax480241.SecondOrderFixedPoints Lax480241.AlternatingSpace
    34 Lax480241.ExponentialClasses
    35
    36/-- Alternating acceptance in bounded space holds exactly when its conditions do. -/
    37axiom atmAcceptSpace_iff :
    38 ∀ (A : Type) [(turingAlt 2).Structure A],
    39 ATMAcceptSpace A ↔ TMData.WellFormed (atmData 2 A).toTMData ∧
    40 ATMData.BlocksSplit (atmData 2 A) ∧ ATMData.AltAcceptsSpace (atmData 2 A) true
    41
    42end Lax480241.AlternatingSpaceValue
    43
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