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Lax132576.RationalOfWeightedPrecomposition

Functions that weighted automata can be pre-composed with are rational

concepts/Lax132576/RationalOfWeightedPrecomposition.lean · lax-132576

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

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    In the paper

    Theorem

    If pre-composition with a function f:A∗→B∗f : A^* \to B^* preserves computability by weighted automata over every semiring, then ff is rational: the implication ⇐ of Theorem B.3.6 of Transducers, its content. The book applies the hypothesis to the weighted automaton over the semiring of regular languages that maps a string to the singleton language of itself — the rational relations are the weighted automata over that semiring — and reads a rational relation computing the graph of ff off the resulting automaton.

    Lean source view on GitHub

    1import Lax132576.RationalFunctions
    2import Lax132576.WeightedAutomata
    3
    4/-!
    5---
    6title: Functions that weighted automata can be pre-composed with are rational
    7type: theorem
    8---
    9If pre-composition with a function f:A∗→B∗f : A^* \to B^* preserves computability by
    10weighted automata over every semiring, then ff is rational: the implication
    11⇐ of Theorem B.3.6 of *Transducers*, its content. The book applies the
    12hypothesis to the weighted automaton over the semiring of regular languages
    13that maps a string to the singleton language of itself — the rational relations
    14are the weighted automata over that semiring — and reads a rational relation
    15computing the graph of ff off the resulting automaton.
    16
    17# Formalization notes
    18
    19The hypothesis quantifies over all semirings `S : Type` in the same universe as
    20the alphabets, which is where the semiring of regular languages over `B` lives.
    21Both alphabets are assumed finite.
    22-/
    23
    24namespace Lax132576.RationalOfWeightedPrecomposition
    25
    26open Lax132576.RationalFunctions Lax132576.WeightedAutomata
    27
    28/-- A function with which every weighted automaton can be pre-composed is
    29rational. -/
    30axiom isRationalFun_of_weighted_precomp {A B : Type} [Finite A] [Finite B]
    31 (f : List A → List B)
    32 (h : ∀ (S : Type) (_ : Semiring S) (h : List B → S), IsWeighted h → IsWeighted (h ∘ f)) :
    33 IsRationalFun f
    34
    35end Lax132576.RationalOfWeightedPrecomposition
    36
    Show Proof

    Formalization notes

    The hypothesis quantifies over all semirings S:TypeS : Type in the same universe as the alphabets, which is where the semiring of regular languages over BB lives. Both alphabets are assumed finite.

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