A hierarchy theorem for regular languages over free bisemigroups

In this article a question left open in [2] is answered. In particular, we show that it is essential that in the definition of parenthesizing automata an arbitrary number of parentheses can be used. Moreover, we prove that the classes Regm of languages accepted by a parenthesizing automaton with at most m pairs of parentheses form a strict hierarchy. In fact, this hierarchy is proper for all alphabets

Weighted Logics for Nested Words and Algebraic Formal Power Series

Nested words, a model for recursive programs proposed by Alur and Madhusudan, have recently gained much interest. In this paper we introduce quantitative extensions and study nested word series which assign to nested words elements of a semiring. We show that regular nested word series coincide with series definable in weighted logics as introduced by Droste and Gastin. For this we establish a connection between nested words and the free bisemigroup. Applying our result, we obtain characterizations of algebraic formal power series in terms of weighted logics. This generalizes results of Lautemann, Schwentick and Therien on context-free languages

Definable transductions and weighted logics for texts

AbstractA text is a word together with an additional linear order on it. We study quantitative models for texts, i.e. text series which assign to texts elements of a semiring. We introduce an algebraic notion of recognizability following Reutenauer and Bozapalidis as well as weighted automata for texts combining an automaton model of Lodaya and Weil with a model of Ésik and Németh. After that we show that both formalisms describe the text series definable in a certain fragment of weighted logics as introduced by Droste and Gastin. In order to do so, we study certain definable transductions and show that they are compatible with weighted logics