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QSAT is PSPACE-complete

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

proven

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    Natural Language Statement

    Theorem

    QSAT is PSPACE-complete under first-order reductions, a theorem of Stockmeyer and Meyer, and hence also coPSPACE-complete. It is SO(TC) definable, by a deterministic walk that computes the value of the formula. Hardness goes through SUCCINCT-REACH and the recursive doubling of Savitch's theorem: a path of length 2m2^m between two states is a quantified formula with mm alternations asking for a midpoint and for both halves.

    Concept map
    22 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 qsat_coPSPACE_complete proven

    2 qsat_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: QSAT is PSPACE-complete
    24type: theorem
    25---
    26QSAT is PSPACE-complete under first-order reductions, a theorem of
    27Stockmeyer and Meyer, and hence also coPSPACE-complete. It is SO(TC)
    28definable, by a deterministic walk that computes the value of the formula.
    29Hardness goes through SUCCINCT-REACH and the recursive doubling of
    30Savitch's theorem: a path of length 2m2^m between two states is a
    31quantified formula with mm alternations asking for a midpoint and for both
    32halves.
    33-/
    34
    35namespace Lax134656.QsatPSPACEComplete
    36
    37open FirstOrder FirstOrder.Language
    38open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    39open Lax904597.Classes Lax904597.Machines
    40open Lax485149.Problems Lax485149.Complement
    41open Lax535992.InflationaryFixedPoint Lax535992.DeterministicMachines Lax535992.ClassPTIME
    42open Lax564036.Hierarchy
    43open Lax134656.SecondOrderTransitiveClosure Lax134656.OrderFreeTransitiveClosure
    44open Lax134656.PartialFixedPoint
    45open Lax134656.Qsat Lax134656.SuccinctReach Lax134656.SpaceBoundedMachines Lax134656.ClassPSPACE
    46
    47/-- The problem is PSPACE-complete. -/
    48axiom qsat_PSPACE_complete : PSPACE.Complete QSAT
    49
    50/-- QSAT is coPSPACE-complete. -/
    51axiom qsat_coPSPACE_complete : coPSPACE.Complete QSAT
    52
    53end Lax134656.QsatPSPACEComplete
    54
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