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FO(LFP) is closed under complement

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

proven

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    Natural Language Statement

    Theorem

    If a decision problem PP is FO(LFP) definable, then so is its complement PcP^c: the output sentence of a definition may negate fixed-point atoms, so it suffices to negate it.

    Concept map
    24 concepts
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    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: FO(LFP) is closed under complement
    27type: theorem
    28---
    29If a decision problem PP is FO(LFP) definable, then so is its complement
    30PcP^c: the output sentence of a definition may negate fixed-point atoms, so
    31it suffices to negate it.
    32-/
    33
    34namespace Lax535992.LeastFixedPointComplement
    35
    36open FirstOrder FirstOrder.Language
    37open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    38open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    39open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms
    40open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    41open Lax485149.ClassNL Lax485149.ClassL
    42open Lax535992.HornFragment Lax535992.LeastFixedPoint Lax535992.InflationaryFixedPoint
    43open Lax535992.HornSat Lax535992.CircuitValue Lax535992.Game Lax535992.DeterministicMachines
    44open Lax535992.ClassPTIME
    45
    46/-- The complement of an FO(LFP) definable problem is FO(LFP) definable. -/
    47axiom lfpDefinable_compl : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    48 LFPDefinable P → LFPDefinable (DecisionProblem.compl P)
    49
    50end Lax535992.LeastFixedPointComplement
    51
    Show Proof
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