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Lax314295.FOOfAperiodic

Aperiodic automata recognise first-order definable languages

concepts/Lax314295/FOOfAperiodic.lean · lax-314295

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    In the paper

    Theorem

    Every language recognised by an aperiodic deterministic automaton is definable in first-order logic (Theorem C.4.11 of Transducers, the implication from aperiodic to definable). The automaton, read as a Mealy machine, is a composition of flip-flops by Theorem A.2.8, and the state of a flip-flop at a position is determined by the last resetting letter before it, which first-order logic can express; composing the formulas along the decomposition gives a first-order description of the run.

    Lean source view on GitHub

    1import Lax765601.StateTransformations
    2import Lax314295.MSOLogic
    3
    4/-!
    5---
    6title: Aperiodic automata recognise first-order definable languages
    7type: theorem
    8---
    9Every language recognised by an aperiodic deterministic automaton is
    10definable in first-order logic (Theorem C.4.11 of *Transducers*, the
    11implication from aperiodic to definable). The automaton, read as a Mealy
    12machine, is a composition of flip-flops by Theorem A.2.8, and the state of a
    13flip-flop at a position is determined by the last resetting letter before it,
    14which first-order logic can express; composing the formulas along the
    15decomposition gives a first-order description of the run.
    16
    17# Formalization notes
    18
    19Aperiodicity is `TransAperiodic` on the transition function of a mathlib
    20`DFA` with a finite state set. The alphabet is assumed finite.
    21-/
    22
    23namespace Lax314295.FOOfAperiodic
    24
    25open Lax765601.StateTransformations Lax314295.MSOLogic
    26
    27/-- The language of an aperiodic dfa is first-order definable. -/
    28axiom foDefinable_of_aperiodic_dfa {A σ : Type} [Finite A] [Finite σ] (M : DFA A σ)
    29 (hM : TransAperiodic M.step) : FODefinable M.accepts
    30
    31end Lax314295.FOOfAperiodic
    32
    Show Proof

    Formalization notes

    Aperiodicity is TransAperiodicTransAperiodic on the transition function of a mathlib DFADFA with a finite state set. The alphabet is assumed finite.

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