Lax Lean Archive

Mealy machines are closed under composition

Lax765601.MealyComposition · concepts/Lax765601/MealyComposition.lean · lax-765601

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    Natural Language Statement

    Theorem

    If f:A∗→B∗f : A^* \to B^* and g:B∗→C∗g : B^* \to C^* are computed by Mealy machines, then so is their composition A∗→fB∗→gC∗A^* \xrightarrow{f} B^* \xrightarrow{g} C^* (Theorem A.1.3 of Transducers). The proof is a product construction: the composed machine runs both machines in lockstep, feeding each output letter of the first to the second, so its state space is the product of the two state spaces.

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    In the paper

    Lean source view on GitHub

    1import Lax765601.MealyMachine
    2
    3/-!
    4---
    5title: Mealy machines are closed under composition
    6type: theorem
    7---
    8If f:A∗→B∗f : A^* \to B^* and g:B∗→C∗g : B^* \to C^* are computed by Mealy machines, then
    9so is their composition A∗→fB∗→gC∗A^* \xrightarrow{f} B^* \xrightarrow{g} C^* (Theorem
    10A.1.3 of *Transducers*). The proof is a product construction: the composed
    11machine runs both machines in lockstep, feeding each output letter of the first
    12to the second, so its state space is the product of the two state spaces.
    13
    14# Formalization notes
    15
    16The book writes the composition as f⋅gf \cdot g, first ff then gg; in Lean it is
    17`g ∘ f`. The intermediate alphabet is required to be finite, as in the closure
    18under composition of `CompositionClosure`; nothing is assumed of the outer
    19alphabets.
    20-/
    21
    22namespace Lax765601.MealyComposition
    23
    24open Lax765601.MealyMachine
    25
    26/-- The composition of two functions computed by Mealy machines is computed by a
    27Mealy machine. -/
    28axiom isMealy_comp {A B C : Type} [Finite B] {f : List A → List B} {g : List B → List C}
    29 (hf : IsMealy f) (hg : IsMealy g) : IsMealy (g ∘ f)
    30
    31end Lax765601.MealyComposition
    32
    Show Proof
    Formalization notes

    The book writes the composition as f⋅gf \cdot g, first ff then gg; in Lean it is g∘fg ∘ f. The intermediate alphabet is required to be finite, as in the closure under composition of CompositionClosureCompositionClosure; nothing is assumed of the outer alphabets.

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