Differential Tree Automata
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866916586136272896 |
|---|---|
| author | Manssour, Rida Ait El Cheval, Vincent Shirmohammadi, Mahsa Worrell, James |
| author_facet | Manssour, Rida Ait El Cheval, Vincent Shirmohammadi, Mahsa Worrell, James |
| contents | A rationally dynamically algebraic (RDA) power series is one that arises as (a component of) the solution of a system of differential equations of the form $\boldsymbol{y}' = F(\boldsymbol{y})$, where $F$ is a vector of rational functions that is defined at $\boldsymbol{y}(0)$. RDA power series subsume algebraic power series and are a proper subclass of differentially algebraic power series (those that satisfy a univariate polynomial-differential equation). We give a combinatorial characterisation of RDA power series in terms of exponential generating functions of regular languages of labelled trees. Motivated by this connection, we define the notion of a differential tree automaton. Differential tree automata generalise weighted tree automata by allowing the transition weights to be rational functions of the tree size. Our main result is that the ordinary generating functions of the formal tree series recognised by differential tree automata are exactly the differentially algebraic power series. The proof of this result establishes a general form of recurrence satisfied by the sequence of coefficients of a differentially algebraic power series, generalising Reutenauer's matrix representation of polynomially recursive sequences. As a corollary we obtain a procedure for determining equality of differential tree automata. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08218 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Differential Tree Automata Manssour, Rida Ait El Cheval, Vincent Shirmohammadi, Mahsa Worrell, James Formal Languages and Automata Theory Discrete Mathematics A rationally dynamically algebraic (RDA) power series is one that arises as (a component of) the solution of a system of differential equations of the form $\boldsymbol{y}' = F(\boldsymbol{y})$, where $F$ is a vector of rational functions that is defined at $\boldsymbol{y}(0)$. RDA power series subsume algebraic power series and are a proper subclass of differentially algebraic power series (those that satisfy a univariate polynomial-differential equation). We give a combinatorial characterisation of RDA power series in terms of exponential generating functions of regular languages of labelled trees. Motivated by this connection, we define the notion of a differential tree automaton. Differential tree automata generalise weighted tree automata by allowing the transition weights to be rational functions of the tree size. Our main result is that the ordinary generating functions of the formal tree series recognised by differential tree automata are exactly the differentially algebraic power series. The proof of this result establishes a general form of recurrence satisfied by the sequence of coefficients of a differentially algebraic power series, generalising Reutenauer's matrix representation of polynomially recursive sequences. As a corollary we obtain a procedure for determining equality of differential tree automata. |
| title | Differential Tree Automata |
| topic | Formal Languages and Automata Theory Discrete Mathematics |
| url | https://arxiv.org/abs/2407.08218 |