Marrying Words and Trees

Abstract

We discuss the model of nested words for representation of data with both a linear ordering and a hierarchically nested matching of items. Examples of data with such dual linear-hierarchical structure include annotated linguistic data, executions of structured programs, and HTML/XML documents. Nested words generalize both words and ordered trees, and allow both word and tree operations. We define nested word automata—finite-state acceptors for nested words, and show that the resulting class of regular languages of nested words has all the appealing theoretical properties that the classical regular word languages enjoy such as determinization, closure under a variety of operations, decidability of emptiness as well as equivalence, and characterization using monadic second order logic. The linear encodings of nested words gives the class of visibly pushdown languages of words, and this class lies between balanced languages and deterministic context-free languages. We argue that for algorithmic verification of structured programs, instead of viewing the program as a context-free language over words, one should view it as a regular language of nested words (or equivalently, as a visibly pushdown language), and this would allow model checking of many properties (such as stack inspection, pre-post conditions) that are not expressible in existing specification logics. We also study the relationship between ordered trees and nested words, and the corresponding automata: while the analysis complexity of nested word automata is the same as that of classical tree automata, they combine both bottom-up and top-down traversals, and enjoy expressiveness and succinctness benefits over tree automata. There is a rapidly growing literature related to nested words, and we will briefly survey results on languages infinite nested words, nested trees, temporal logics over nested words, and new decidability results based on visibility.