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EXPTIME = coEXPTIME and EXPSPACE = coEXPSPACE

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

proven

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    Natural Language Statement

    Theorem

    EXPTIME and EXPSPACE are closed under complement: complementation commutes with reading a class one exponential up, and PTIME and PSPACE are closed under complement.

    Concept map
    25 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 mem_coEXPSPACE_iff proven

    2 mem_coEXPTIME_iff proven

    3 mem_EXPSPACE_compl_iff proven

    4 mem_EXPTIME_compl_iff proven

    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: EXPTIME = coEXPTIME and EXPSPACE = coEXPSPACE
    19type: theorem
    20---
    21EXPTIME and EXPSPACE are closed under complement: complementation commutes
    22with reading a class one exponential up, and PTIME and PSPACE are closed
    23under complement.
    24-/
    25
    26namespace Lax480241.ExponentialComplements
    27
    28open FirstOrder FirstOrder.Language
    29open Lax904597.Problems Lax904597.Classes Lax904597.Machines Lax485149.Problems
    30 Lax485149.Complement Lax485149.ClassNL
    31open Lax535992.ClassPTIME Lax564036.Hierarchy Lax564036.AlternatingMachines Lax134656.ClassPSPACE
    32open Lax480241.Expansions Lax480241.SecondOrderFixedPoints Lax480241.AlternatingSpace
    33 Lax480241.ExponentialClasses
    34
    35/-- EXPTIME is closed under complement. -/
    36axiom mem_EXPTIME_compl_iff :
    37 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    38 EXPTIME.Mem (DecisionProblem.compl P) ↔ EXPTIME.Mem P
    39
    40/-- EXPSPACE is closed under complement. -/
    41axiom mem_EXPSPACE_compl_iff :
    42 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    43 EXPSPACE.Mem (DecisionProblem.compl P) ↔ EXPSPACE.Mem P
    44
    45/-- coEXPTIME is EXPTIME. -/
    46axiom mem_coEXPTIME_iff :
    47 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    48 coEXPTIME.Mem P ↔ EXPTIME.Mem P
    49
    50/-- coEXPSPACE is EXPSPACE. -/
    51axiom mem_coEXPSPACE_iff :
    52 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    53 coEXPSPACE.Mem P ↔ EXPSPACE.Mem P
    54
    55end Lax480241.ExponentialComplements
    56
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