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Lax916827.TwoWayComposition

Two-way transducers are closed under composition

concepts/Lax916827/TwoWayComposition.lean · lax-916827

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    In the paper

    Theorem

    Functions computed by two-way transducers are closed under composition (Theorem C.2.5 of Transducers, Chytil and Jákl). The composed transducer computes the reachable configuration graph of the first transducer by a rational function (Lemma C.2.3), which two-way transducers can be pre-composed with (Corollary C.2.7), and then simulates the second transducer on the represented path, walking backwards along the path — which is possible because the run of a deterministic transducer never revisits a configuration — when the second transducer moves left.

    Lean source view on GitHub

    1import Lax916827.TwoWayTransducers
    2
    3/-!
    4---
    5title: Two-way transducers are closed under composition
    6type: theorem
    7---
    8Functions computed by two-way transducers are closed under composition
    9(Theorem C.2.5 of *Transducers*, Chytil and Jákl). The composed transducer
    10computes the reachable configuration graph of the first transducer by a
    11rational function (Lemma C.2.3), which two-way transducers can be pre-composed
    12with (Corollary C.2.7), and then simulates the second transducer on the
    13represented path, walking backwards along the path — which is possible because
    14the run of a deterministic transducer never revisits a configuration — when the
    15second transducer moves left.
    16
    17# Formalization notes
    18
    19All three alphabets are assumed finite, as in the book.
    20-/
    21
    22namespace Lax916827.TwoWayComposition
    23
    24open Lax916827.TwoWayTransducers
    25
    26/-- The composition of two functions computed by two-way transducers is computed by
    27a two-way transducer. -/
    28axiom isTwoWay_comp {A B C : Type} [Finite A] [Finite B] [Finite C]
    29 {f : List A → List B} {g : List B → List C} (hf : IsTwoWay f) (hg : IsTwoWay g) :
    30 IsTwoWay (g ∘ f)
    31
    32end Lax916827.TwoWayComposition
    33
    Show Proof

    Formalization notes

    All three alphabets are assumed finite, as in the book.

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