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PTIME = coPTIME

Lax535992.PTIMEEqCoPTIME · concepts/Lax535992/PTIMEEqCoPTIME.lean · lax-535992

proven

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    Natural Language Statement

    Theorem

    The classes PTIME and coPTIME are equal: a problem is SO-Horn definable if and only if its complement is. The Horn fragment has no negation of its own; the closure comes from FO(LFP), where complementing is negating the output sentence, through the equivalence of the two logics.

    Concept map
    24 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 PTIME_eq_coPTIME proven

    2 sigmaSOHornDefinable_compl proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Interpretations
    3import Lax904597.Relativized
    4import Lax904597.SecondOrder
    5import Lax904597.Classes
    6import Lax904597.Sat
    7import Lax904597.Machines
    8import Lax485149.Problems
    9import Lax485149.Complement
    10import Lax485149.SecondOrderAtoms
    11import Lax485149.TransitiveClosure
    12import Lax485149.DeterministicTransitiveClosure
    13import Lax485149.ClassNL
    14import Lax485149.ClassL
    15import Lax535992.HornFragment
    16import Lax535992.LeastFixedPoint
    17import Lax535992.InflationaryFixedPoint
    18import Lax535992.HornSat
    19import Lax535992.CircuitValue
    20import Lax535992.Game
    21import Lax535992.DeterministicMachines
    22import Lax535992.ClassPTIME
    23
    24/-!
    25---
    26title: PTIME = coPTIME
    27type: theorem
    28---
    29The classes PTIME and coPTIME are equal: a problem is SO-Horn definable if
    30and only if its complement is. The Horn fragment has no negation of its
    31own; the closure comes from FO(LFP), where complementing is negating the
    32output sentence, through the equivalence of the two logics.
    33-/
    34
    35namespace Lax535992.PTIMEEqCoPTIME
    36
    37open FirstOrder FirstOrder.Language
    38open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    39open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    40open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms
    41open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    42open Lax485149.ClassNL Lax485149.ClassL
    43open Lax535992.HornFragment Lax535992.LeastFixedPoint Lax535992.InflationaryFixedPoint
    44open Lax535992.HornSat Lax535992.CircuitValue Lax535992.Game Lax535992.DeterministicMachines
    45open Lax535992.ClassPTIME
    46
    47/-- The complement of an SO-Horn definable problem is SO-Horn definable. -/
    48axiom sigmaSOHornDefinable_compl : ∀ {L : Language.{0, 0}} [L.IsRelational]
    49 {P : DecisionProblem L},
    50 SigmaSOHornDefinable P → SigmaSOHornDefinable (DecisionProblem.compl P)
    51
    52/-- PTIME is closed under complement. -/
    53axiom PTIME_eq_coPTIME : PTIME = coPTIME
    54
    55end Lax535992.PTIMEEqCoPTIME
    56
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