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Lax765601.MealyContinuity

Mealy machines are continuous

concepts/Lax765601/MealyContinuity.lean · lax-765601

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    In the paper

    Theorem

    Every function computed by a Mealy machine is continuous: the inverse image of a regular language under it is regular (Theorem A.1.4 of Transducers). The book derives this from closure under composition through the letter-to-letter lifting of a language, which labels every position of a string by whether the prefix ending there belongs to the language; a direct product construction works just as well, and is the one formalised.

    Lean source view on GitHub

    1import Lax765601.Continuity
    2import Lax765601.MealyMachine
    3
    4/-!
    5---
    6title: Mealy machines are continuous
    7type: theorem
    8---
    9Every function computed by a Mealy machine is continuous: the inverse image of a
    10regular language under it is regular (Theorem A.1.4 of *Transducers*). The book
    11derives this from closure under composition through the letter-to-letter lifting
    12of a language, which labels every position of a string by whether the prefix
    13ending there belongs to the language; a direct product construction works just
    14as well, and is the one formalised.
    15
    16# Formalization notes
    17
    18Finiteness of the two alphabets is assumed, as everywhere in the book; the
    19product of the machine with a deterministic automaton for the output language
    20recognises the inverse image.
    21-/
    22
    23namespace Lax765601.MealyContinuity
    24
    25open Lax765601.Continuity Lax765601.MealyMachine
    26
    27/-- A function computed by a Mealy machine is continuous. -/
    28axiom continuous_of_isMealy {A B : Type} [Finite A] [Finite B] {f : List A → List B}
    29 (hf : IsMealy f) : Continuous f
    30
    31end Lax765601.MealyContinuity
    32
    Show Proof

    Formalization notes

    Finiteness of the two alphabets is assumed, as everywhere in the book; the product of the machine with a deterministic automaton for the output language recognises the inverse image.

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