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Differential Tree Automata

Rida Ait El Manssour, Vincent Cheval, Mahsa Shirmohammadi, James Worrell

Abstract

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.

Differential Tree Automata

Abstract

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 , where is a vector of rational functions that is defined at . 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.
Paper Structure (23 sections, 50 theorems, 215 equations, 2 figures)

This paper contains 23 sections, 50 theorems, 215 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Combinatorial Classes, Automata, and Power Series
  3. Tree Automata and their Generating Functions
  4. Main Results
  5. Related Work
  6. Overview
  7. Tree Automata
  8. Weighted Tree Automata
  9. Differential Tree Automata
  10. Multi-holonomic sequences
  11. Proof of Theorem \ref{['thm:MAIN1']}
  12. Proof of Theorem \ref{['thm:MAIN2']}
  13. From Multi-Holonomic Sequences to Differentially Algebraic Power Series
  14. From Differentially Algebraic Power Series to Multi-Holonomic Sequences
  15. Equivalence of differential tree automata and their generating function
  16. ...and 8 more sections

Key Result

Theorem 1

The following are equivalent for a power series $g(x) = \sum_{n=0}^\infty a_n x^n$ in $\mathbb{K}[\![x]\!]$:

Figures (2)

  • Figure 1: Examples of Combinatorial encodings
  • Figure 2: Example of Species with RDA exponential generating series. Unspecified initial values indicate that all power series solutions are Rationally Dynamically Algebraic, no matter the initial values.

Theorems & Definitions (98)

  • Theorem 1
  • Theorem 2
  • Example 1: Partitions
  • Proposition 1
  • Example 2: Labelled trees
  • Example 3
  • Example 4
  • Theorem 3
  • Proposition 2
  • proof : Proof sketch
  • ...and 88 more