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#NAE-SAT and #Set Splitting are one-call #P-complete

Lax859101.NaeSatComplete · concepts/Lax859101/NaeSatComplete.lean · lax-859101

proven

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    Natural Language Statement

    Theorem

    #NAE-SAT is one-call #P-complete, by a one-call reduction from #SAT, and #Set Splitting is, by a one-call reduction from #NAE-SAT.

    Concept map
    22 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 sharpNaeSat_sharpP_oneCallComplete proven

    2 sharpSetSplitting_sharpP_oneCallComplete proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Sat
    3import Mathlib.ModelTheory.Graph
    4import Lax799700.SetFamily
    5import Lax366625.CountingProblems
    6import Lax366625.CountingClasses
    7import Lax366625.WitnessCounting
    8import Lax366625.CountingSat
    9import Lax859101.OneCallReductions
    10import Lax859101.SubtractiveReductions
    11import Lax859101.CountingDnf
    12import Lax859101.CountingNaeSat
    13import Lax859101.CountingRestrictedSat
    14import Lax859101.CountingAllSets
    15import Lax859101.CountingBipartite
    16
    17/-!
    18---
    19title: #NAE-SAT and #Set Splitting are one-call #P-complete
    20type: theorem
    21---
    22#NAE-SAT is one-call #P-complete, by a one-call reduction from #SAT, and
    23#Set Splitting is, by a one-call reduction from #NAE-SAT.
    24-/
    25
    26namespace Lax859101.NaeSatComplete
    27
    28open FirstOrder FirstOrder.Language FirstOrder.Language.Structure
    29open Lax904597.Problems Lax904597.Sat Lax799700.SetFamily
    30open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
    31 Lax366625.CountingSat
    32open Lax859101.OneCallReductions Lax859101.SubtractiveReductions Lax859101.CountingDnf
    33 Lax859101.CountingNaeSat Lax859101.CountingRestrictedSat Lax859101.CountingAllSets
    34 Lax859101.CountingBipartite
    35
    36/-- #NAE-SAT is one-call #P-complete. -/
    37axiom sharpNaeSat_sharpP_oneCallComplete :
    38 OneCallComplete SharpP SharpNAESAT
    39
    40/-- #Set Splitting is one-call #P-complete. -/
    41axiom sharpSetSplitting_sharpP_oneCallComplete :
    42 OneCallComplete SharpP SharpSetSplitting
    43
    44end Lax859101.NaeSatComplete
    45
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