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#SAT is parsimoniously #P-complete

Lax366625.SharpSatComplete · concepts/Lax366625/SharpSatComplete.lean · lax-366625

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    Natural Language Statement

    Theorem

    #SAT is parsimoniously #P-complete, a theorem of Valiant: it is in #P, and every problem of #P reduces to it by an ordered parsimonious reduction. The reduction is that of the Cook–Levin theorem, made parsimonious: the assignments of a block satisfying a first-order kernel are in bijection with the models of the formula it produces, the auxiliary variables of the Tseitin encoding being determined.

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

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Interpretations
    3import Lax904597.Relativized
    4import Lax904597.SecondOrder
    5import Lax904597.Classes
    6import Lax904597.Sat
    7import Lax904597.Machines
    8import Lax535992.HornSat
    9import Lax535992.CircuitValue
    10import Lax535992.DeterministicMachines
    11import Lax535992.ClassPTIME
    12import Lax366625.CountingProblems
    13import Lax366625.CountingClasses
    14import Lax366625.WitnessCounting
    15import Lax366625.SecondOrderCounting
    16import Lax366625.QuantitativeLogic
    17import Lax366625.CountingSat
    18import Lax366625.MachineNumbers
    19import Lax366625.CountingRuns
    20import Lax366625.NumberedCircuits
    21import Lax366625.HornNumbers
    22
    23/-!
    24---
    25title: #SAT is parsimoniously #P-complete
    26type: theorem
    27---
    28#SAT is parsimoniously #P-complete, a theorem of Valiant: it is in #P, and
    29every problem of #P reduces to it by an ordered parsimonious reduction. The
    30reduction is that of the Cook–Levin theorem, made parsimonious: the
    31assignments of a block satisfying a first-order kernel are in bijection with
    32the models of the formula it produces, the auxiliary variables of the
    33Tseitin encoding being determined.
    34-/
    35
    36namespace Lax366625.SharpSatComplete
    37
    38open FirstOrder FirstOrder.Language
    39open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    40open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    41open Lax535992.ClassPTIME
    42open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
    43open Lax366625.SecondOrderCounting
    44open Lax366625.QuantitativeLogic Lax366625.CountingSat Lax366625.MachineNumbers
    45open Lax366625.CountingRuns
    46open Lax366625.NumberedCircuits Lax366625.HornNumbers
    47
    48/-- #SAT is parsimoniously #P-complete. -/
    49axiom sharpSat_sharpP_parsimoniousComplete : SharpP.ParsimoniousComplete SharpSAT
    50
    51end Lax366625.SharpSatComplete
    52
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