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Lax132576.RationalViaWeighted

Rational functions characterised by weighted automata

concepts/Lax132576/RationalViaWeighted.lean · lax-132576

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    In the paper

    Theorem

    A string-to-string function is rational if and only if weighted automata, over every semiring, are closed under pre-composition with it (Theorem B.3.6 of Transducers). The implication ⇒ is Lemma B.3.5 (WeightedPrecompositionWeightedPrecomposition) and the implication ⇐ is RationalOfWeightedPrecompositionRationalOfWeightedPrecomposition; this statement is their conjunction.

    Lean source view on GitHub

    1import Lax132576.RationalFunctions
    2import Lax132576.WeightedAutomata
    3
    4/-!
    5---
    6title: Rational functions characterised by weighted automata
    7type: theorem
    8---
    9A string-to-string function is rational if and only if weighted automata, over
    10every semiring, are closed under pre-composition with it (Theorem B.3.6 of
    11*Transducers*). The implication ⇒ is Lemma B.3.5 (`WeightedPrecomposition`) and
    12the implication ⇐ is `RationalOfWeightedPrecomposition`; this statement is
    13their conjunction.
    14
    15# Formalization notes
    16
    17Both alphabets are assumed finite; the semirings range over `Type`.
    18-/
    19
    20namespace Lax132576.RationalViaWeighted
    21
    22open Lax132576.RationalFunctions Lax132576.WeightedAutomata
    23
    24/-- A function is rational if and only if every weighted automaton can be
    25pre-composed with it. -/
    26axiom isRationalFun_iff_weighted_precomp {A B : Type} [Finite A] [Finite B]
    27 (f : List A → List B) :
    28 IsRationalFun f ↔
    29 ∀ (S : Type) (_ : Semiring S) (h : List B → S), IsWeighted h → IsWeighted (h ∘ f)
    30
    31end Lax132576.RationalViaWeighted
    32
    Show Proof

    Formalization notes

    Both alphabets are assumed finite; the semirings range over TypeType.

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