Lax Lean Archive

Invariance and characterization of alternating machine acceptance

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

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    Natural Language Statement

    Lemma

    Being a well-formed alternating machine instance with well-formed blocks that accepts is invariant under isomorphism of instances, and an instance is a yes-instance of alternating machine acceptance exactly when it has this property.

    Concept map
    23 concepts
    100%
    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 atmAccept_iff proven

    2 atmAccepts_iso 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: Invariance and characterization of alternating machine acceptance
    22type: lemma
    23---
    24Being a well-formed alternating machine instance with well-formed blocks
    25that accepts is invariant under isomorphism of instances, and an instance is
    26a yes-instance of alternating machine acceptance exactly when it has this
    27property.
    28-/
    29
    30namespace Lax564036.AlternatingMachineInvariance
    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/-- Being well-formed and accepting is isomorphism-invariant. -/
    40axiom atmAccepts_iso : ∀ (k : ℕ) (start : Bool) {A B : Type} [(turingAlt k).Structure A]
    41 [(turingAlt k).Structure B], (A ≃[turingAlt k] B) →
    42 (ATMAccepts k start A ↔ ATMAccepts k start B)
    43
    44/-- The yes-instances of alternating machine acceptance are exactly the
    45well-formed accepting instances. -/
    46axiom atmAccept_iff : ∀ (k : ℕ) (start : Bool) (A : Type) [(turingAlt k).Structure A],
    47 ATMAccept k start A ↔ ATMAccepts k start A
    48
    49end Lax564036.AlternatingMachineInvariance
    50
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