Lax Lean Archive

Draft — mutable and not usable as a dependency; its citation marks the draft state.

Lax314295.FOIffAperiodic

First-order definable languages are exactly the aperiodic ones

concepts/Lax314295/FOIffAperiodic.lean · lax-314295

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.

    Concept map

    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

    Evidence

    Each proof establishes this claim relative to its assumptions.

    In the paper

    Theorem

    A language is definable in first-order logic if and only if it is recognised by an aperiodic deterministic automaton (Theorem C.4.11 of Transducers, Schützenberger, McNaughton and Papert). The two implications are the separate statements AperiodicOfFOAperiodicOfFO and FOOfAperiodicFOOfAperiodic; this statement is their conjunction.

    Lean source view on GitHub

    1import Lax765601.StateTransformations
    2import Lax314295.MSOLogic
    3
    4/-!
    5---
    6title: First-order definable languages are exactly the aperiodic ones
    7type: theorem
    8---
    9A language is definable in first-order logic if and only if it is recognised
    10by an aperiodic deterministic automaton (Theorem C.4.11 of *Transducers*,
    11Schützenberger, McNaughton and Papert). The two implications are the separate
    12statements `AperiodicOfFO` and `FOOfAperiodic`; this statement is their
    13conjunction.
    14
    15# Formalization notes
    16
    17The alphabet is assumed finite.
    18-/
    19
    20namespace Lax314295.FOIffAperiodic
    21
    22open Lax765601.StateTransformations Lax314295.MSOLogic
    23
    24/-- A language is first-order definable if and only if some aperiodic dfa
    25recognises it. -/
    26axiom foDefinable_iff_aperiodic_dfa {A : Type} [Finite A] (L : Language A) :
    27 FODefinable L ↔
    28 ∃ (σ : Type) (_ : Finite σ) (M : DFA A σ), TransAperiodic M.step ∧ M.accepts = L
    29
    30end Lax314295.FOIffAperiodic
    31
    Show Proof

    Formalization notes

    The alphabet is assumed finite.

    Builds on

    Used by

    none

    From Mathlib

    none

    Community review

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above; your ORCID profile must share a public name.

    0 comments

    Loading discussion…