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Lax132576.StringHomomorphisms

String homomorphisms

concepts/Lax132576/StringHomomorphisms.lean · lax-132576

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    Concept map

    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

    Definition

    A string homomorphism h:A∗→B∗h : A^* \to B^* is a function that applies a fixed map A→B∗A \to B^* to every letter of the input and concatenates the results; a letter may be erased or replaced by a longer string. It is a homomorphism of the free monoids, h(uv)=h(u)h(v)h(uv) = h(u)h(v). Homomorphisms are among the prime rational functions (Theorem B.2.6), and the complement of the graph of a homomorphism is a rational relation (Claim B.1.7), which is the observation behind the undecidability of equivalence of rational relations.

    Lean source view on GitHub

    1import Mathlib.Data.List.Basic
    2
    3/-!
    4---
    5title: String homomorphisms
    6type: definition
    7---
    8A *string homomorphism* h:A∗→B∗h : A^* \to B^* is a function that applies a fixed map
    9A→B∗A \to B^* to every letter of the input and concatenates the results; a letter
    10may be erased or replaced by a longer string. It is a homomorphism of the free
    11monoids, h(uv)=h(u)h(v)h(uv) = h(u)h(v). Homomorphisms are among the prime rational
    12functions (Theorem B.2.6), and the complement of the graph of a homomorphism is
    13a rational relation (Claim B.1.7), which is the observation behind the
    14undecidability of equivalence of rational relations.
    15
    16# Formalization notes
    17
    18`homOf φ` is the homomorphism determined by `φ : A → List B`.
    19-/
    20
    21namespace Lax132576.StringHomomorphisms
    22
    23/-- The string homomorphism that applies `φ` to every letter and concatenates the
    24results. -/
    25def homOf {A B : Type} (φ : A → List B) : List A → List B := fun w => (w.map φ).flatten
    26
    27end Lax132576.StringHomomorphisms
    28

    Formalization notes

    homOfφhomOf φ is the homomorphism determined by φ:A→ListBφ : A → List B.

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    Used by

    From Mathlib

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