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SelMajSAT is PP-complete and SelEqSAT is C₌P-complete

Lax175070.SelectedSatComplete · concepts/Lax175070/SelectedSatComplete.lean · lax-175070

proven

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    Natural Language Statement

    Theorem

    SelMajSAT is PP-complete and SelEqSAT is C=_=P-complete. Membership compares the two counts of #SelSAT, both in #P. Hardness pairs the two kernels of a problem of the class into one Tseitin formula, with the selected variable choosing the kernel, so that the two counts of the formula are those of the problem.

    Concept map
    23 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 selEqSat_CeqP_complete proven

    2 selMajSat_PP_complete proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax485149.Problems
    3import Lax485149.Complement
    4import Lax904597.Classes
    5import Lax904597.Sat
    6import Lax904597.Interpretations
    7import Lax904597.Machines
    8import Lax564036.Hierarchy
    9import Lax366625.CountingProblems
    10import Lax366625.CountingClasses
    11import Lax366625.WitnessCounting
    12import Lax366625.CountingSat
    13import Lax366625.CountingRuns
    14import Lax175070.CountDefinability
    15import Lax175070.SelectedSat
    16
    17/-!
    18---
    19title: SelMajSAT is PP-complete and SelEqSAT is C₌P-complete
    20type: theorem
    21---
    22SelMajSAT is PP-complete and SelEqSAT is C=_=P-complete. Membership
    23compares the two counts of #SelSAT, both in #P. Hardness pairs the two
    24kernels of a problem of the class into one Tseitin formula, with the
    25selected variable choosing the kernel, so that the two counts of the formula
    26are those of the problem.
    27-/
    28
    29namespace Lax175070.SelectedSatComplete
    30
    31open FirstOrder FirstOrder.Language
    32open Lax904597.Problems Lax904597.Classes Lax904597.Sat Lax904597.Interpretations
    33 Lax904597.Machines Lax564036.Hierarchy
    34open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
    35 Lax366625.CountingSat
    36open Lax366625.CountingRuns Lax175070.CountDefinability Lax175070.SelectedSat Lax485149.Problems
    37open Lax485149.Complement
    38
    39/-- SelMajSAT is PP-complete. -/
    40axiom selMajSat_PP_complete :
    41 PP.Complete SelMajSAT
    42
    43/-- SelEqSAT is C₌P-complete. -/
    44axiom selEqSat_CeqP_complete :
    45 CeqP.Complete SelEqSAT
    46
    47end Lax175070.SelectedSatComplete
    48
    Show ProofShow Proof
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