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Lax314295.BuchiTheorem

Büchi's theorem: regular languages are the MSO-definable ones

concepts/Lax314295/BuchiTheorem.lean · lax-314295

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    In the paper

    Theorem

    A language L⊆A∗L \subseteq A^* is regular if and only if it is definable in monadic second-order logic (Theorem C.4.1 of Transducers, Büchi, Elgot and Trakhtenbrot). The two implications are the separate statements MSODefinableOfRegularMSODefinableOfRegular and RegularOfMSODefinableRegularOfMSODefinable; this statement is their conjunction.

    Lean source view on GitHub

    1import Lax314295.MSOLogic
    2
    3/-!
    4---
    5title: Büchi's theorem: regular languages are the MSO-definable ones
    6type: theorem
    7---
    8A language L⊆A∗L \subseteq A^* is regular if and only if it is definable in
    9monadic second-order logic (Theorem C.4.1 of *Transducers*, Büchi, Elgot and
    10Trakhtenbrot). The two implications are the separate statements
    11`MSODefinableOfRegular` and `RegularOfMSODefinable`; this statement is their
    12conjunction.
    13
    14# Formalization notes
    15
    16The alphabet is assumed finite.
    17-/
    18
    19namespace Lax314295.BuchiTheorem
    20
    21open Lax314295.MSOLogic
    22
    23/-- A language is regular if and only if it is mso-definable. -/
    24axiom isRegular_iff_msoDefinable {A : Type} [Finite A] (L : Language A) :
    25 L.IsRegular ↔ MSODefinable L
    26
    27end Lax314295.BuchiTheorem
    28
    Show Proof

    Formalization notes

    The alphabet is assumed finite.

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