Context-Free Series on Trees

Abstract

We investigate context-free (CF) series on trees with coefficients on a semiring; they are obtained as components of the least solutions of systems of equations having polynomials on their right-hand sides. The relationship between CF series on trees and CF tree-grammars and recursive program schemes is also examined. Polypodes, a new algebraic structure, are introduced in order to study in common series on trees and words and applications are given.

Formal Tree Series

“…Moreover, we prove a Kleene Theorem due to Bozapalidis [11]. Guessarian [32] introduced the notion of a (top-down) pushdown tree automaton and showed that these pushdown tree automata recognize exactly the class of context-free tree languages.…”

“…We do not consider generalizations of the IO-substitution. Bozapalidis [11], Engelfriet, Fülöp, Vogler [19] and Fülöp, Vogler [23] consider these generalizations to formal tree series.…”

“…These algebraic tree systems are a generalization of the context-free tree grammars (see Rounds [51] and Gécseg, Steinby [25]). They are a particular instance of the second-order systems of Bozapalidis [11]. The presentation follows Kuich [43].…”

“…Our algebraic tree systems are second-order systems in the sense of Bozapalidis [11] and are a generalization of the context-free tree grammars. (See Rounds [51], and Engelfriet, Schmidt [20], especially Theorem 3.4.…”

“…Formal tree series were introduced by Berstel, Reutenauer [5], and then extensively studied by Bozapalidis [7,8,9,10,11], Bozapalidis, Rahonis [12], Kuich [38,39,40,41,43], Engelfriet, Fülöp, Vogler [19] and Fülöp, Vogler [23].…”