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PSPACE by Turing machines in bounded space

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

proven

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    Natural Language Statement

    Theorem

    Machine acceptance in bounded space and its deterministic variant are both PSPACE-complete under first-order reductions, and every problem of PSPACE reduces to the deterministic one: the class defined by SO(TC) is polynomial space on Turing machines, deterministic or not, which is PSPACE = NPSPACE in this setting. A configuration is an assignment of a block of three relation variables and a step a first-order condition on two consecutive assignments, which gives membership. Hardness is proved for the deterministic problem, by a machine that evaluates a quantified Boolean formula with one bit of recursion stack per variable.

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

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 dtmAcceptSpace_PSPACE_complete proven

    2 le_dtmAcceptSpace_of_mem_PSPACE proven

    3 ntmAcceptSpace_PSPACE_complete 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: PSPACE by Turing machines in bounded space
    24type: theorem
    25---
    26Machine acceptance in bounded space and its deterministic variant are both
    27PSPACE-complete under first-order reductions, and every problem of PSPACE
    28reduces to the deterministic one: the class defined by SO(TC) is polynomial
    29space on Turing machines, deterministic or not, which is PSPACE = NPSPACE
    30in this setting. A configuration is an assignment of a block of three
    31relation variables and a step a first-order condition on two consecutive
    32assignments, which gives membership. Hardness is proved for the
    33deterministic problem, by a machine that evaluates a quantified Boolean
    34formula with one bit of recursion stack per variable.
    35-/
    36
    37namespace Lax134656.SpaceMachinesPSPACEComplete
    38
    39open FirstOrder FirstOrder.Language
    40open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    41open Lax904597.Classes Lax904597.Machines
    42open Lax485149.Problems Lax485149.Complement
    43open Lax535992.InflationaryFixedPoint Lax535992.DeterministicMachines Lax535992.ClassPTIME
    44open Lax564036.Hierarchy
    45open Lax134656.SecondOrderTransitiveClosure Lax134656.OrderFreeTransitiveClosure
    46open Lax134656.PartialFixedPoint
    47open Lax134656.Qsat Lax134656.SuccinctReach Lax134656.SpaceBoundedMachines Lax134656.ClassPSPACE
    48
    49/-- The problem is PSPACE-complete. -/
    50axiom dtmAcceptSpace_PSPACE_complete : PSPACE.Complete DTMAcceptSpace
    51
    52/-- The problem is PSPACE-complete. -/
    53axiom ntmAcceptSpace_PSPACE_complete : PSPACE.Complete NTMAcceptSpace
    54
    55/-- Every problem of PSPACE reduces to deterministic acceptance in bounded space. -/
    56axiom le_dtmAcceptSpace_of_mem_PSPACE :
    57 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    58 PSPACE.Mem P → Nonempty (RelOrderedFOReduction P DTMAcceptSpace)
    59
    60end Lax134656.SpaceMachinesPSPACEComplete
    61
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