Construction of tree automata from regular expressions

Abstract

Since recognizable tree languages are closed under the rational operations, every regular tree expression denotes a recognizable tree language. We provide an alternative proof to this fact that results in smaller tree automata. To this aim, we transfer Antimirov's partial derivatives from regular word expressions to regular tree expressions. For an analysis of the size of the resulting automaton as well as for algorithmic improvements, we also transfer the methods of Champarnaud and Ziadi from words to trees.

An Efficient Algorithm for the Equation Tree Automaton via the k-C-Continuations

“…Recall that the k-Pseudo-Continuation identification can be achieved in O(| E | 2 ) [4,8] using Paige and Tarjan's sorting algorithm [12]. In what follows we show that this step amounts to the minimization of the acyclic deterministic word automaton B T E = (Q B , Σ B , {ν T }, {ν E }, δ B ) defined with ν T / ∈ Nodes(E) and…”

“…In [8], Kuske and Meinecke extend the notion of word partial derivatives [1] to tree partial derivatives in order to compute from a regular expression E a tree automaton recognizing E . Due to the notion of ranked alphabet, partial derivatives are no longer sets of expressions, but sets of tuples of expressions.…”

“…In [8], the computation of the k-C-Continuations requires a preprocessing step which is the identification of subexpression of E in O(| E | 2 ) time and space complexity. We propose an algorithm for the computation of the set of states with an O(| E |) time and space complexity.…”

“…They also present how to compute them extending from words to trees [8] the k-C-Continuation algorithm by Champarnaud and Ziadi [3]. They obtain an algorithm with O(R · | E | · | E |) space and time complexity where R is the maximal rank of a symbol occurring in the finite ranked alphabet Σ and | E | is the size of the regular expression.…”

“…This constitutes an improvement in comparison with Kuske and Meinecke algorithm [8]. The paper is organized as follows: Section 2 outlines finite tree automata over ranked trees, regular tree expressions, and linearized regular tree expressions which allows the set of positions to be defined.…”