Weighted Logics for Nested Words and Algebraic Formal Power Series

TL;DR

This paper extends nested word models with quantitative features, connecting weighted logics and algebraic formal power series, and generalizes context-free language characterizations.

Contribution

It introduces weighted logics for nested words, linking them to algebraic formal power series and providing new characterizations of context-free languages.

Findings

Regular nested word series match those definable by weighted logics.

Established a connection between nested words and the free bisemigroup.

Generalized results to characterize algebraic formal power series.

Abstract

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.