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Differential Tree Automata

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

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LIPIcs.LICS.2026.70.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Tree languages
  • Theory of computation → Quantitative automata
  • Mathematics of computing → Generating functions
Keywords
  • Combinatorial species
  • Differentially algebraic series
  • Weighted automata

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Abstract

In this paper we introduce the notion of a differential tree automaton. Differential tree automata generalise weighted tree automata (over a field) by allowing the transition weights to be rational functions of the tree size. Whereas the class of generating functions of weighted tree automata coincides with the class of algebraic power series, our main result is that the class of generating functions of differential tree automata coincides with the class of differentially algebraic power series. As a corollary, we obtain a decision procedure for determining equivalence of differential tree automata. In the course of proving our main result we identify a class of recurrences that characterises the sequence of coefficients of a differentially algebraic power series, generalising Reutenauer’s matrix representation of polynomially recursive sequences. We further identify a natural syntactic subset of differential tree automata whose generating functions are given by rational dynamical systems, that is, as components of the solution of a system of differential equations y' = F(y), where F is a vector of rational functions that is defined at y(0). We further show that this class of power series can be characterised in terms of the classical notion of weighted tree automata by using a labelled generating function on trees.

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Rida Ait El Manssour, Vincent Cheval, Mahsa Shirmohammadi, and James Worrell. Differential Tree Automata. In 41st Annual Symposium on Logic in Computer Science (LICS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 380, pp. 70:1-70:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.LICS.2026.70

Author Details

Rida Ait El Manssour
  • University of Oxford, UK
Vincent Cheval
  • University of Oxford, UK
Mahsa Shirmohammadi
  • CNRS & IRIF, Paris, France
James Worrell
  • University of Oxford, UK

Funding

  • Ait El Manssour, Rida: Supported by the EPSRC Fellowship (EP/X033813/1).
  • Cheval, Vincent: Supported by the ERC Synergy Grant VePaSS (agreement No. 101224640).
  • Shirmohammadi, Mahsa: Supported by the ERC Synergy Grant VePaSS (agreement No. 101224640) and the grant VeSyAM (ANR-22-CE48-0005).
  • Worrell, James: Supported by the EPSRC Fellowship (EP/X033813/1).

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