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FP-complete problems

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

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    Natural Language Statement

    Theorem

    The number written by a circuit, the number written by unit propagation on a Horn formula, and the number written by a deterministic Turing machine are parsimoniously FP-complete, and a counting problem is in FP if and only if it reduces to the machine problem by an ordered parsimonious reduction: FP is the class of the numbers written by deterministic polynomial-time machines.

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

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 circuitNumber_FP_parsimoniousComplete proven

    2 dtmNumber_FP_parsimoniousComplete proven

    3 hornNumber_FP_parsimoniousComplete proven

    4 mem_FP_iff_le_dtmNumber proven

    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: FP-complete problems
    26type: theorem
    27---
    28The number written by a circuit, the number written by unit propagation on
    29a Horn formula, and the number written by a deterministic Turing machine are
    30parsimoniously FP-complete, and a counting problem is in FP if and only if
    31it reduces to the machine problem by an ordered parsimonious reduction: FP
    32is the class of the numbers written by deterministic polynomial-time
    33machines.
    34-/
    35
    36namespace Lax366625.FPComplete
    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/-- The number written by a circuit is parsimoniously FP-complete. -/
    49axiom circuitNumber_FP_parsimoniousComplete : FP.ParsimoniousComplete CircuitNumber
    50
    51/-- The number written by unit propagation is parsimoniously FP-complete. -/
    52axiom hornNumber_FP_parsimoniousComplete : FP.ParsimoniousComplete HornNumber
    53
    54/-- The number written by a deterministic machine is parsimoniously FP-complete. -/
    55axiom dtmNumber_FP_parsimoniousComplete : FP.ParsimoniousComplete DTMNumber
    56
    57/-- FP is reducibility to the number written by a deterministic machine. -/
    58axiom mem_FP_iff_le_dtmNumber : ∀ {L : Language.{0, 0}} [L.IsRelational] (C : CountingProblem L),
    59 FP.Mem C ↔ Nonempty (OrderedParsimoniousReduction C DTMNumber)
    60
    61end Lax366625.FPComplete
    62
    Show ProofShow ProofShow ProofShow Proof
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