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Invariance and characterization of machine acceptance in bounded space

Lax134656.SpaceBoundedMachineInvariance · concepts/Lax134656/SpaceBoundedMachineInvariance.lean · lax-134656

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    Natural Language Statement

    Lemma

    Being a well-formed machine instance accepting in bounded space, and being moreover deterministic, are invariant under isomorphism of instances; an instance is a yes-instance of machine acceptance in bounded space, respectively of its deterministic variant, exactly when it has the corresponding property.

    Concept map
    22 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 dtmAcceptSpace_iff proven

    2 dtmAcceptsSpace_iso proven

    3 ntmAcceptSpace_iff proven

    4 ntmAcceptsSpace_iso proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Interpretations
    3import Lax904597.Relativized
    4import Lax904597.SecondOrder
    5import Lax904597.Classes
    6import Lax904597.Machines
    7import Lax485149.Problems
    8import Lax485149.Complement
    9import Lax535992.InflationaryFixedPoint
    10import Lax535992.DeterministicMachines
    11import Lax535992.ClassPTIME
    12import Lax564036.Hierarchy
    13import Lax134656.SecondOrderTransitiveClosure
    14import Lax134656.OrderFreeTransitiveClosure
    15import Lax134656.PartialFixedPoint
    16import Lax134656.Qsat
    17import Lax134656.SuccinctReach
    18import Lax134656.SpaceBoundedMachines
    19import Lax134656.ClassPSPACE
    20
    21/-!
    22---
    23title: Invariance and characterization of machine acceptance in bounded space
    24type: lemma
    25---
    26Being a well-formed machine instance accepting in bounded space, and being
    27moreover deterministic, are invariant under isomorphism of instances; an
    28instance is a yes-instance of machine acceptance in bounded space,
    29respectively of its deterministic variant, exactly when it has the
    30corresponding property.
    31-/
    32
    33namespace Lax134656.SpaceBoundedMachineInvariance
    34
    35open FirstOrder FirstOrder.Language
    36open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    37open Lax904597.Classes Lax904597.Machines
    38open Lax485149.Problems Lax485149.Complement
    39open Lax535992.InflationaryFixedPoint Lax535992.DeterministicMachines Lax535992.ClassPTIME
    40open Lax564036.Hierarchy
    41open Lax134656.SecondOrderTransitiveClosure Lax134656.OrderFreeTransitiveClosure
    42open Lax134656.PartialFixedPoint
    43open Lax134656.Qsat Lax134656.SuccinctReach Lax134656.SpaceBoundedMachines Lax134656.ClassPSPACE
    44
    45/-- Being well-formed and accepting in bounded space is isomorphism-invariant. -/
    46axiom ntmAcceptsSpace_iso : ∀ {A B : Type} [turing.Structure A] [turing.Structure B],
    47 (A ≃[turing] B) → (NTMAcceptsSpace A ↔ NTMAcceptsSpace B)
    48
    49/-- Being well-formed, deterministic and accepting in bounded space is
    50isomorphism-invariant. -/
    51axiom dtmAcceptsSpace_iso : ∀ {A B : Type} [turing.Structure A] [turing.Structure B],
    52 (A ≃[turing] B) → (DTMAcceptsSpace A ↔ DTMAcceptsSpace B)
    53
    54/-- The yes-instances of acceptance in bounded space. -/
    55axiom ntmAcceptSpace_iff : ∀ (A : Type) [turing.Structure A], NTMAcceptSpace A ↔ NTMAcceptsSpace A
    56
    57/-- The yes-instances of deterministic acceptance in bounded space. -/
    58axiom dtmAcceptSpace_iff : ∀ (A : Type) [turing.Structure A], DTMAcceptSpace A ↔ DTMAcceptsSpace A
    59
    60end Lax134656.SpaceBoundedMachineInvariance
    61
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