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PSPACE = coPSPACE

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

proven

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    Natural Language Statement

    Theorem

    The classes PSPACE and coPSPACE are equal: a problem is SO(TC) definable if and only if its complement is. The complement of a walk is not a walk, so the closure is not syntactic. Every SO(TC) definable problem reduces to QSAT, and the walk that decides QSAT is deterministic, computing the value of the formula; reading its answer the other way round decides the complement.

    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 PSPACE_eq_coPSPACE proven

    2 sotcDefinable_compl 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 = coPSPACE
    24type: theorem
    25---
    26The classes PSPACE and coPSPACE are equal: a problem is SO(TC) definable if
    27and only if its complement is. The complement of a walk is not a walk, so
    28the closure is not syntactic. Every SO(TC) definable problem reduces to
    29QSAT, and the walk that decides QSAT is deterministic, computing the value
    30of the formula; reading its answer the other way round decides the
    31complement.
    32-/
    33
    34namespace Lax134656.PSPACEEqCoPSPACE
    35
    36open FirstOrder FirstOrder.Language
    37open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    38open Lax904597.Classes Lax904597.Machines
    39open Lax485149.Problems Lax485149.Complement
    40open Lax535992.InflationaryFixedPoint Lax535992.DeterministicMachines Lax535992.ClassPTIME
    41open Lax564036.Hierarchy
    42open Lax134656.SecondOrderTransitiveClosure Lax134656.OrderFreeTransitiveClosure
    43open Lax134656.PartialFixedPoint
    44open Lax134656.Qsat Lax134656.SuccinctReach Lax134656.SpaceBoundedMachines Lax134656.ClassPSPACE
    45
    46/-- The complement of an SO(TC) definable problem is SO(TC) definable. -/
    47axiom sotcDefinable_compl : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    48 SOTCDefinable P → SOTCDefinable (DecisionProblem.compl P)
    49
    50/-- PSPACE is closed under complement. -/
    51axiom PSPACE_eq_coPSPACE : PSPACE = coPSPACE
    52
    53end Lax134656.PSPACEEqCoPSPACE
    54
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