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APSPACE = EXPTIME

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

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    Natural Language Statement

    Theorem

    Acceptance by alternating Turing machines in bounded space is EXPTIME-complete: APSPACE = EXPTIME. Membership expands the instance to its configurations, on which acceptance is the game problem of PTIME; hardness runs a second-order alternating game, which defines every problem of EXPTIME, on an alternating machine whose tape holds two block assignments.

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

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    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: APSPACE = EXPTIME
    19type: theorem
    20---
    21Acceptance by alternating Turing machines in bounded space is
    22EXPTIME-complete: APSPACE = EXPTIME. Membership expands the instance to its
    23configurations, on which acceptance is the game problem of PTIME; hardness
    24runs a second-order alternating game, which defines every problem of
    25EXPTIME, on an alternating machine whose tape holds two block assignments.
    26-/
    27
    28namespace Lax480241.APSPACEIsEXPTIME
    29
    30open FirstOrder FirstOrder.Language
    31open Lax904597.Problems Lax904597.Classes Lax904597.Machines Lax485149.Problems
    32 Lax485149.Complement Lax485149.ClassNL
    33open Lax535992.ClassPTIME Lax564036.Hierarchy Lax564036.AlternatingMachines Lax134656.ClassPSPACE
    34open Lax480241.Expansions Lax480241.SecondOrderFixedPoints Lax480241.AlternatingSpace
    35 Lax480241.ExponentialClasses
    36
    37/-- Alternating acceptance in bounded space is EXPTIME-complete. -/
    38axiom atmAcceptSpace_EXPTIME_complete :
    39 EXPTIME.Complete ATMAcceptSpace
    40
    41end Lax480241.APSPACEIsEXPTIME
    42
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