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MSO transductions define regular functions

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

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    Natural Language Statement

    Theorem

    Every function defined by a string-to-string mso transduction is regular (Theorem C.4.8 of Transducers, Engelfriet–Hoogeboom, the implication from transduction to regular). The transduction is normalised to one universe and one letter formula per copy and one order formula per pair of copies (Lemma C.4.9); the questions the formulas ask are precomputed by a letter-to-letter rational function (Lemma C.4.10); and a two-way transducer walks through the output order, moving to the successor of the current element by running the automaton of the precomputed language on the infix between them.

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    In the paper

    Lean source view on GitHub

    1import Lax916827.RegularFunctions
    2import Lax314295.MSOTransductions
    3
    4/-!
    5---
    6title: MSO transductions define regular functions
    7type: theorem
    8---
    9Every function defined by a string-to-string mso transduction is regular
    10(Theorem C.4.8 of *Transducers*, Engelfriet–Hoogeboom, the implication from
    11transduction to regular). The transduction is normalised to one universe and
    12one letter formula per copy and one order formula per pair of copies (Lemma
    13C.4.9); the questions the formulas ask are precomputed by a letter-to-letter
    14rational function (Lemma C.4.10); and a two-way transducer walks through the
    15output order, moving to the successor of the current element by running the
    16automaton of the precomputed language on the infix between them.
    17
    18# Formalization notes
    19
    20Both alphabets are assumed finite, as in the book.
    21-/
    22
    23namespace Lax314295.RegularOfMSOTransduction
    24
    25open Lax916827.RegularFunctions Lax314295.MSOTransductions
    26
    27/-- A function defined by an mso transduction is regular. -/
    28axiom isRegularFun_of_isMSOTransduction {A B : Type} [Finite A] [Finite B]
    29 {f : List A → List B} (hf : IsMSOTransduction f) : IsRegularFun f
    30
    31end Lax314295.RegularOfMSOTransduction
    32
    Show Proof
    Formalization notes

    Both alphabets are assumed finite, as in the book.

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