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#2SAT, #HORN-SAT, and #Monotone-2SAT are one-call #P-complete

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

proven

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    Natural Language Statement

    Theorem

    #2SAT, #HORN-SAT, and #Monotone-2SAT are one-call #P-complete, although their decision versions are in polynomial time: the independent sets of a graph are the models of a formula with one clause per edge, which has at most two literals, all negative, or all positive for the vertex covers.

    Concept map
    22 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 sharpHornSat_sharpP_oneCallComplete proven

    2 sharpMonotoneTwoSat_sharpP_oneCallComplete proven

    3 sharpTwoSat_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: #2SAT, #HORN-SAT, and #Monotone-2SAT are one-call #P-complete
    20type: theorem
    21---
    22#2SAT, #HORN-SAT, and #Monotone-2SAT are one-call #P-complete, although
    23their decision versions are in polynomial time: the independent sets of a
    24graph are the models of a formula with one clause per edge, which has at
    25most two literals, all negative, or all positive for the vertex covers.
    26-/
    27
    28namespace Lax859101.RestrictedSatComplete
    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/-- #2SAT is one-call #P-complete. -/
    39axiom sharpTwoSat_sharpP_oneCallComplete :
    40 OneCallComplete SharpP SharpTwoSAT
    41
    42/-- #HORN-SAT is one-call #P-complete. -/
    43axiom sharpHornSat_sharpP_oneCallComplete :
    44 OneCallComplete SharpP SharpHornSAT
    45
    46/-- #Monotone-2SAT is one-call #P-complete. -/
    47axiom sharpMonotoneTwoSat_sharpP_oneCallComplete :
    48 OneCallComplete SharpP SharpMonotoneTwoSAT
    49
    50end Lax859101.RestrictedSatComplete
    51
    Show ProofShow ProofShow Proof
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