Differential Tree Automata

2024-07-11

AI Summary

This paper addresses the identification of tree structures underlying differential-algebraic (DA) power series generated by differential equations and the equivalence problem for differential tree automata (DTAs). We introduce, for the first time, the DTA model—a weighted tree automaton whose transitions are labeled with rational functions—establishing a precise one-to-one correspondence with DA power series. By combining symbolic differentiation and tree enumeration, we characterize the combinatorial structure of DA series and derive a general differential recurrence satisfied by their coefficient sequences. We extend Reutenauer’s matrix representation to differential recursive sequences. Finally, we present an effective algorithm for deciding DTA equivalence. Our results unify differential algebra, weighted tree automata, and generating function theory, providing the first structural characterization and computational framework for DA power series based on tree automata.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Machine Learning: Matrix & Tensor Methods

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation

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 $oldsymbol{y}' = F(oldsymbol{y})$, where $F$ is a vector of rational functions that is defined at $oldsymbol{y}(0)$. 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.

Problem

• Differential Equations
• RDA Power Series
• Tree Automata Equivalence

Innovation

• Differential Tree Automata
• Rational Dynamic Algebra
• Equivalence Checking