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MSO relabellings are rational

Lax314295.RationalOfRelabelling · concepts/Lax314295/RationalOfRelabelling.lean · lax-314295

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    Natural Language Statement

    Theorem

    Every function definable by an mso relabelling is rational (Theorem C.4.4 of Transducers, Bloem–Engelfriet, the implication from relabelling to rational). By Claim C.4.6 the strings annotated at every position with the formula true there form a regular language, so an automaton with output guesses the annotation, checks it, and outputs the output map of the guessed formulas.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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    In the paper

    Lean source view on GitHub

    1import Lax132576.RationalFunctions
    2import Lax314295.MSORelabellings
    3
    4/-!
    5---
    6title: MSO relabellings are rational
    7type: theorem
    8---
    9Every function definable by an mso relabelling is rational (Theorem C.4.4 of
    10*Transducers*, Bloem–Engelfriet, the implication from relabelling to rational).
    11By Claim C.4.6 the strings annotated at every position with the formula true
    12there form a regular language, so an automaton with output guesses the
    13annotation, checks it, and outputs the output map of the guessed formulas.
    14
    15# Formalization notes
    16
    17Both alphabets are assumed finite, as in the book.
    18-/
    19
    20namespace Lax314295.RationalOfRelabelling
    21
    22open Lax132576.RationalFunctions Lax314295.MSORelabellings
    23
    24/-- A function definable by an mso relabelling is rational. -/
    25axiom isRationalFun_of_isMSORelabelling {A B : Type} [Finite A] [Finite B]
    26 {f : List A → List B} (hf : IsMSORelabelling f) : IsRationalFun f
    27
    28end Lax314295.RationalOfRelabelling
    29
    Show Proof
    Formalization notes

    Both alphabets are assumed finite, as in the book.

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