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Rational functions

Lax132576.RationalFunctions · concepts/Lax132576/RationalFunctions.lean · lax-132576

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    Natural Language Statement

    Definition

    A rational function (Definition B.2.1 of Transducers) is the special case of a rational relation in which each input string is related to exactly one output string. The definition is semantic: the underlying automaton is nondeterministic and merely happens to have a unique output for every input, possibly through several runs; Theorem B.2.3 gives the deterministic model, the bimachine, that computes exactly these functions. Rational functions inherit closure under composition and continuity from rational relations, and every Mealy machine is a rational function.

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    In the paper

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    1import Lax132576.RationalRelations
    2
    3/-!
    4---
    5title: Rational functions
    6type: definition
    7---
    8A *rational function* (Definition B.2.1 of *Transducers*) is the special case
    9of a rational relation in which each input string is related to exactly one
    10output string. The definition is semantic: the underlying automaton is
    11nondeterministic and merely happens to have a unique output for every input,
    12possibly through several runs; Theorem B.2.3 gives the deterministic model, the
    13bimachine, that computes exactly these functions. Rational functions inherit
    14closure under composition and continuity from rational relations, and every
    15Mealy machine is a rational function.
    16
    17# Formalization notes
    18
    19A function is rational if its graph `fun w v => v = f w` is a rational
    20relation. Since `f` is a total function, this is exactly "a rational relation
    21in which each input has exactly one output"; partial rational functions do not
    22occur in the book's statements.
    23-/
    24
    25namespace Lax132576.RationalFunctions
    26
    27open Lax132576.RationalRelations
    28
    29/-- A string-to-string function is rational if its graph is a rational relation.
    30-/
    31def IsRationalFun {A B : Type} (f : List A → List B) : Prop :=
    32 IsRationalRel (fun w v => v = f w)
    33
    34end Lax132576.RationalFunctions
    35
    Formalization notes

    A function is rational if its graph funwv=>v=fwfun w v => v = f w is a rational relation. Since ff is a total function, this is exactly "a rational relation in which each input has exactly one output"; partial rational functions do not occur in the book's statements.

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