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SAT-UNSAT is DP-complete

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

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    Natural Language Statement

    Theorem

    SAT-UNSAT is DP-complete under first-order reductions, a theorem of Papadimitriou and Yannakakis. It is in DP, each side projecting onto a plain CNF instance; and every DP-definable problem reduces to it, by running the Cook–Levin reduction of its NP half and of the complement of its coNP half side by side into one paired instance.

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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 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: SAT-UNSAT is DP-complete
    22type: theorem
    23---
    24SAT-UNSAT is DP-complete under first-order reductions, a theorem of
    25Papadimitriou and Yannakakis. It is in DP, each side projecting onto a
    26plain CNF instance; and every DP-definable problem reduces to it, by running
    27the Cook–Levin reduction of its NP half and of the complement of its coNP
    28half side by side into one paired instance.
    29-/
    30
    31namespace Lax564036.SatUnsatDPComplete
    32
    33open FirstOrder FirstOrder.Language
    34open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    35open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    36open Lax485149.Problems Lax485149.Complement Lax535992.ClassPTIME
    37open Lax564036.Hierarchy Lax564036.Difference Lax564036.Tautology Lax564036.ThreeDnfTautology
    38open Lax564036.SatUnsat Lax564036.QuantifiedBooleanFormulas Lax564036.AlternatingMachines
    39
    40/-- SAT-UNSAT is DP-complete. -/
    41axiom satUnsat_DP_complete : DP.Complete SATUNSAT
    42
    43end Lax564036.SatUnsatDPComplete
    44
    Show Proof
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