Lax Lean Archive

While this submission is a draft, it cannot be used by other submissions.

FP by binary digits and by quantitative terms

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

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Theorem

    A counting problem is in FP if and only if it is digit-definable: its binary digits are relations of a least fixed point. And the value of a closed QFO term, with no fixed point at all, defines a problem in FP.

    Concept map
    28 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 digitDefinable_iff_mem_FP proven

    2 fpDefinable_of_qfo 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 by binary digits and by quantitative terms
    26type: theorem
    27---
    28A counting problem is in FP if and only if it is digit-definable: its binary
    29digits are relations of a least fixed point. And the value of a closed QFO
    30term, with no fixed point at all, defines a problem in FP.
    31-/
    32
    33namespace Lax366625.FPByDigits
    34
    35open FirstOrder FirstOrder.Language
    36open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    37open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    38open Lax535992.ClassPTIME
    39open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
    40open Lax366625.SecondOrderCounting
    41open Lax366625.QuantitativeLogic Lax366625.CountingSat Lax366625.MachineNumbers
    42open Lax366625.CountingRuns
    43open Lax366625.NumberedCircuits Lax366625.HornNumbers
    44
    45/-- FP is digit definability. -/
    46axiom digitDefinable_iff_mem_FP : ∀ {L : Language.{0, 0}} [L.IsRelational] (C : CountingProblem L),
    47 DigitDefinable C ↔ FP.Mem C
    48
    49/-- The value of a closed quantitative first-order term is in FP. -/
    50axiom fpDefinable_of_qfo :
    51 ∀ {L : Language.{0, 0}} [L.IsRelational] {C : CountingProblem L} (t : QTerm
    52 (L.sum Language.order) Empty),
    53 (∀ (A : Type) [L.Structure A] [LinearOrder A] [Finite A] [Nonempty A], C A = t.value A) →
    54 FP.Mem C
    55
    56end Lax366625.FPByDigits
    57
    Show ProofShow Proof
    Builds on
    Used by

    none

    From Mathlib

    none

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…