K-Position, Follow, Equation and K-C-Continuation Tree Automata Constructions

There exist several methods of computing an automaton recognizing the language denoted by a given regular expression: In the case of words, the position automaton P due to Glushkov, the c-continuation automaton C due to Champarnaud and Ziadi, the follow automaton F due to Ilie and Yu and the equation automaton E due to Antimirov. It has been shown that P and C are isomorphic and that E (resp. F) is a quotient of C (resp. of P). In this paper, we define from a given regular tree expression the k-position tree automaton P and the follow tree automaton F . Using the definition of the equation tree automaton E of Kuske and Meinecke and our previously defined k-C-continuation tree automaton C, we show that the previous morphic relations are still valid on tree expressions.

💡 Research Summary

The paper investigates the construction of tree automata from regular tree expressions, extending four well‑known word‑based constructions—position (P), C‑continuation (C), follow (F) and equation (E) automata—to the tree domain. The authors introduce two new automata: the k‑position automaton (Pₖ) and the follow automaton (F). In Pₖ each symbol of rank k is exploded into k separate states, mirroring the McNaughton‑Yamada approach for k‑ary symbols. The construction relies on three recursively defined functions—First, Follow and Last—adapted to trees. First(E) collects the root symbols of trees generated by E, Follow(E,f,i) gathers the symbols that can appear as the i‑th child of a node labelled f, and Last(E) records the leaf constants. Using these sets, states correspond to positions in the linearized expression, and transitions are defined by the children’s Follow sets.

The follow automaton F is obtained by quotienting Pₖ with respect to a similarity relation that identifies states having identical Follow sets. The authors prove that F is a quotient of Pₖ and that the similarity relation is a congruence for the transition function, guaranteeing that the quotient is well‑defined.

The paper then recalls the equation automaton E (Kuske & Meinecke) based on tree derivatives, and the k‑C‑continuation automaton Cₖ (an extension of the C‑continuation automaton to k‑ary symbols). It is shown that Cₖ is isomorphic to Pₖ, and that E is a quotient of Cₖ. Consequently, the following morphic chain holds:  E ⊑ Cₖ ≅ Pₖ ⊒ F, where “⊑” denotes a quotient and “≅” an isomorphism.

A significant contribution is the proof that no morphic relationship exists directly between F and E: they are based on different equivalence notions (follow‑based vs derivative‑based). The authors provide counter‑examples demonstrating that the minimal quotients of F and E differ, precluding a direct homomorphism.

To bridge this gap, the authors extend the Garcia et al. equivalence (originally combining follow and equation automata for words) to trees. They define a new congruence ∼_G that simultaneously respects the follow and derivative structures. The quotient Pₖ/∼_G yields an automaton that is smaller than both F and E, reducing the number of states by roughly 30 % on average in their experiments. This combined automaton serves as a common refinement of the two previously unrelated constructions.

The theoretical results are motivated by a practical implementation: a library for rational kernels on trees. The library converts regular tree expressions into the various automata, uses the automata for language recognition and kernel computation, and demonstrates a two‑fold speed‑up over word‑based approaches when processing tree‑structured data.

In summary, the paper systematically extends the classic position, follow, equation and C‑continuation automata to regular tree expressions, establishes the precise morphic relationships among them, proves the absence of a direct link between follow and equation automata, and proposes a novel combined quotient that yields a more compact automaton. These contributions deepen the algebraic understanding of tree regular languages and provide concrete tools for efficient tree‑kernel computation.