Abstract. We propose 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 executions of structured programs, annotated linguistic data, 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 enjoys: deterministic nested word automata are as expressive as their nondeterministic counterparts; the class is closed under union, intersection, complementation, concatenation, Kleene-*, prefixes, and language homomorphisms; membership, emptiness, language inclusion, and language equivalence are all decidable; and definability in monadic second order logic corresponds exactly to finite-state recognizability. We also consider regular languages of infinite nested words and show that the closure properties, MSO-characterization, and decidability of decision problems carry over. The linear encodings of nested words give 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, 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. This research was partially supported by ARO URI award DAAD19-01-0473 and NSF awards CCR-0306382 and CPA-0541149. This article unifies and extends results that have appeared in Alur and Madhusudan [2004, 2006] North Goodwin Avenue, Urbana, IL 61801, e-mail: madhu@cs.uiuc.edu Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax +1 (212) 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.
The model of nested words was proposed in REF for representation of data with both a linear ordering and a hierarchically nested matching of items.
Adding nesting structure to words
JACM
Computer Science