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TAUT is coNP-complete

Lax564036.TautCoNPComplete · concepts/Lax564036/TautCoNPComplete.lean · lax-564036

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    Natural Language Statement

    Theorem

    TAUT, the tautology problem for DNF formulas, is coNP-complete under first-order reductions: swapping the signs of the literals turns a DNF tautology into an unsatisfiable CNF formula and back, so the result follows from the Cook–Levin theorem by complementation.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    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 Lax535992.ClassPTIME
    11import Lax564036.Hierarchy
    12import Lax564036.Difference
    13import Lax564036.Tautology
    14import Lax564036.ThreeDnfTautology
    15import Lax564036.SatUnsat
    16import Lax564036.QuantifiedBooleanFormulas
    17import Lax564036.AlternatingMachines
    18
    19/-!
    20---
    21title: TAUT is coNP-complete
    22type: theorem
    23---
    24TAUT, the tautology problem for DNF formulas, is coNP-complete under
    25first-order reductions: swapping the signs of the literals turns a DNF
    26tautology into an unsatisfiable CNF formula and back, so the result follows
    27from the Cook–Levin theorem by complementation.
    28-/
    29
    30namespace Lax564036.TautCoNPComplete
    31
    32open FirstOrder FirstOrder.Language
    33open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    34open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    35open Lax485149.Problems Lax485149.Complement Lax535992.ClassPTIME
    36open Lax564036.Hierarchy Lax564036.Difference Lax564036.Tautology Lax564036.ThreeDnfTautology
    37open Lax564036.SatUnsat Lax564036.QuantifiedBooleanFormulas Lax564036.AlternatingMachines
    38
    39/-- TAUT is coNP-complete. -/
    40axiom taut_coNP_complete : coNP.Complete TAUT
    41
    42end Lax564036.TautCoNPComplete
    43
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