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#P = ΣQSO(FO)

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

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    Natural Language Statement

    Theorem

    A counting problem is in #P if and only if it is ΣQSO(FO)-definable: the witness counts of existential second-order sentences are exactly the values of the terms of ΣQSO(FO), as Arenas, Muñoz, and Riveros showed. A witness count is the term that sums the indicator of the kernel over the assignments of the block; conversely every construction of the logic is a closure property of witness counts.

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

    Each proof establishes this claim relative to its assumptions.

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    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: #P = ΣQSO(FO)
    26type: theorem
    27---
    28A counting problem is in #P if and only if it is ΣQSO(FO)-definable: the
    29witness counts of existential second-order sentences are exactly the values
    30of the terms of ΣQSO(FO), as Arenas, Muñoz, and Riveros showed. A witness
    31count is the term that sums the indicator of the kernel over the assignments
    32of the block; conversely every construction of the logic is a closure
    33property of witness counts.
    34-/
    35
    36namespace Lax366625.SharpPAsQuantitativeLogic
    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/-- #P is ΣQSO(FO) definability. -/
    49axiom mem_sharpP_iff_sqDefinable : ∀ {L : Language.{0, 0}} [L.IsRelational] (C : CountingProblem L),
    50 SharpP.Mem C ↔ SQDefinable C
    51
    52end Lax366625.SharpPAsQuantitativeLogic
    53
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