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#BIS and #PP2DNF are one-call #P-complete

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

proven

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    Natural Language Statement

    Theorem

    #BIS is one-call #P-complete, by a one-call reduction from counting all independent sets. #PP2DNF is, by a one-call reduction from #BIS: on a bipartite graph, the sets of vertices that are not independent are the models of its partitioned positive 2-DNF formula.

    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 sharpBIS_sharpP_oneCallComplete proven

    2 sharpPP2DNF_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: #BIS and #PP2DNF are one-call #P-complete
    20type: theorem
    21---
    22#BIS is one-call #P-complete, by a one-call reduction from counting all
    23independent sets. #PP2DNF is, by a one-call reduction from #BIS: on a
    24bipartite graph, the sets of vertices that are not independent are the
    25models of its partitioned positive 2-DNF formula.
    26-/
    27
    28namespace Lax859101.BipartiteComplete
    29
    30open FirstOrder FirstOrder.Language FirstOrder.Language.Structure
    31open Lax904597.Problems Lax904597.Sat Lax799700.SetFamily
    32open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
    33 Lax366625.CountingSat
    34open Lax859101.OneCallReductions Lax859101.SubtractiveReductions Lax859101.CountingDnf
    35 Lax859101.CountingNaeSat Lax859101.CountingRestrictedSat Lax859101.CountingAllSets
    36 Lax859101.CountingBipartite
    37
    38/-- #BIS is one-call #P-complete. -/
    39axiom sharpBIS_sharpP_oneCallComplete :
    40 OneCallComplete SharpP SharpBIS
    41
    42/-- #PP2DNF is one-call #P-complete. -/
    43axiom sharpPP2DNF_sharpP_oneCallComplete :
    44 OneCallComplete SharpP SharpPP2DNF
    45
    46end Lax859101.BipartiteComplete
    47
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