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#3-Colorability is one-call #P-complete

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

proven

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    Natural Language Statement

    Theorem

    #3-Colorability is one-call #P-complete, by a one-call reduction from #SAT: the graph built from a formula has 6⋅8g6 \cdot 8^g proper 3-colorings for each model of the formula, gg being its number of gadgets, and the post-processing term divides the oracle answer by that number.

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

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    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: #3-Colorability is one-call #P-complete
    20type: theorem
    21---
    22#3-Colorability is one-call #P-complete, by a one-call reduction from #SAT:
    23the graph built from a formula has 6⋅8g6 \cdot 8^g proper 3-colorings for each
    24model of the formula, gg being its number of gadgets, and the
    25post-processing term divides the oracle answer by that number.
    26-/
    27
    28namespace Lax859101.ColoringComplete
    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/-- #3-Colorability is one-call #P-complete. -/
    39axiom sharpThreeCol_sharpP_oneCallComplete :
    40 OneCallComplete SharpP SharpThreeCol
    41
    42end Lax859101.ColoringComplete
    43
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