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Lax132576.RationalUnambiguousBimachine

Eilenberg's theorem: rational functions, unambiguous automata and bimachines

concepts/Lax132576/RationalUnambiguousBimachine.lean · lax-132576

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    In the paper

    Theorem

    For a string-to-string function the following are equivalent (Theorem B.2.3 of Transducers, Eilenberg): (1) it is a rational relation which happens to be functional; (2) it is computed by an unambiguous nondeterministic automaton with output; (3) it is computed by a bimachine. The book proves (3) ⇒ (1) ⇒ (2) ⇒ (3); here the three implications with content are separate statements — RationalOfBimachineRationalOfBimachine, UnambiguousOfRationalUnambiguousOfRational and BimachineOfRationalBimachineOfRational — and this statement is their conjunction, the implication (2) ⇒ (1) being immediate since an unambiguous automaton is an automaton.

    Lean source view on GitHub

    1import Mathlib.Data.List.TFAE
    2import Lax132576.RationalFunctions
    3import Lax132576.Bimachines
    4
    5/-!
    6---
    7title: Eilenberg's theorem: rational functions, unambiguous automata and bimachines
    8type: theorem
    9---
    10For a string-to-string function the following are equivalent (Theorem B.2.3 of
    11*Transducers*, Eilenberg): (1) it is a rational relation which happens to be
    12functional; (2) it is computed by an unambiguous nondeterministic automaton
    13with output; (3) it is computed by a bimachine. The book proves
    14(3) ⇒ (1) ⇒ (2) ⇒ (3); here the three implications with content are separate
    15statements — `RationalOfBimachine`, `UnambiguousOfRational` and
    16`BimachineOfRational` — and this statement is their conjunction, the
    17implication (2) ⇒ (1) being immediate since an unambiguous automaton is an
    18automaton.
    19
    20# Formalization notes
    21
    22`List.TFAE` is mathlib's "the following are equivalent" for a list of
    23propositions. Both alphabets are assumed finite.
    24-/
    25
    26namespace Lax132576.RationalUnambiguousBimachine
    27
    28open Lax132576.RationalRelations Lax132576.RationalFunctions Lax132576.Bimachines
    29
    30/-- Eilenberg's theorem: a function is rational, computed by an unambiguous
    31automaton with output, or computed by a bimachine, equivalently. -/
    32axiom tfae_rational_unambiguous_bimachine {A B : Type} [Finite A] [Finite B]
    33 (f : List A → List B) :
    34 [IsRationalFun f, IsUnambiguousRel (fun w v => v = f w), IsBimachine f].TFAE
    35
    36end Lax132576.RationalUnambiguousBimachine
    37
    Show Proof

    Formalization notes

    List.TFAEList.TFAE is mathlib's "the following are equivalent" for a list of propositions. Both alphabets are assumed finite.

    Builds on

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    From Mathlib

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