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

Abstract

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.

Algorithm for the k-Position Tree Automaton Construction

“…The two functions First and Follow are sufficient to construct the k-position tree automaton from a regular expression E. Definition 1. [12] Let E be a regular expression , f and g be symbols in Σ and f j and g i be positions in…”

“…It has been shown in [12] that the k-position tree automaton of E accepts E , hence the following theorem: Theorem 1. [12] Let E be a regular expression, then…”

“…In the following sections, we will show how we can efficiently compute the function Follow(E, f j , k). This algorithm can be used in different constructions such us the equation automaton [7], k-C-continuation automaton [10,12] and Follow Automaton [12].…”

Bottom Up Quotients and Residuals for Tree Languages

“…Some of these methods have already been extended to tree expression: the position tree automaton was introduced in [8], and the top-down partial derivative automaton [7] (see [11,12] for an other version of the position tree automaton and its morphic links with other methods), producing non-deterministic and linear-sized tree automaton. As far as top-down deterministic tree automata are concerned, there exist regular languages that can not be recognized; Therefore, the notion of (top-down) derivative cannot be well-defined but it is not the case for bottom-up tree automata.…”

Construction of rational expression from tree automata using a generalization of Arden's Lemma

“…Next, Laugerotte et al [9] generalize position automata to trees. Finally, the morphic links between these constructions have been defined in [12].…”