Differential Tree Automata

TL;DR

This paper introduces differential tree automata, a new computational model that characterizes differentially algebraic power series through combinatorial and automata-theoretic methods.

Contribution

It defines differential tree automata and proves their recognized series correspond exactly to differentially algebraic power series, linking automata theory with differential algebra.

Findings

Differential tree automata recognize exactly the differentially algebraic power series.

A recurrence relation for coefficients of differentially algebraic series is established.

A procedure for deciding equality of differential tree automata is developed.

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.