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Lax314295.RegularOfMSODefinable

MSO-definable languages are regular

concepts/Lax314295/RegularOfMSODefinable.lean · lax-314295

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    In the paper

    Theorem

    Every language definable in monadic second-order logic is regular (Theorem C.4.1 of Transducers, Büchi–Elgot–Trakhtenbrot, the implication from definable to regular). It follows from Lemma C.4.2 on formulas with free variables, by induction on the formula: the annotated strings satisfying a formula form a regular language, Boolean connectives are Boolean operations on languages, and a quantifier is a projection of the annotation.

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    1import Lax314295.MSOLogic
    2
    3/-!
    4---
    5title: MSO-definable languages are regular
    6type: theorem
    7---
    8Every language definable in monadic second-order logic is regular (Theorem
    9C.4.1 of *Transducers*, Büchi–Elgot–Trakhtenbrot, the implication from
    10definable to regular). It follows from Lemma C.4.2 on formulas with free
    11variables, by induction on the formula: the annotated strings satisfying a
    12formula form a regular language, Boolean connectives are Boolean operations on
    13languages, and a quantifier is a projection of the annotation.
    14
    15# Formalization notes
    16
    17The alphabet is assumed finite, as in the book.
    18-/
    19
    20namespace Lax314295.RegularOfMSODefinable
    21
    22open Lax314295.MSOLogic
    23
    24/-- An mso-definable language is regular. -/
    25axiom isRegular_of_msoDefinable {A : Type} [Finite A] {L : Language A} (hL : MSODefinable L) :
    26 L.IsRegular
    27
    28end Lax314295.RegularOfMSODefinable
    29
    Show Proof

    Formalization notes

    The alphabet is assumed finite, as in the book.

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