Abstract. We study congruences on words in order to characterize the class of visibly pushdown languages (Vpl), a subclass of context-free languages. For any language L, we define a natural congruence on words that resembles the syntactic congruence for regular languages, such that this congruence is of finite index if, and only if, L is a Vpl. We then study the problem of finding canonical minimal deterministic automata for Vpls. Though Vpls in general do not have unique minimal automata, we consider a subclass of VPAs called k-module single-entry VPAs that correspond to programs with recursive procedures without input parameters, and show that the class of well-matched Vpls do indeed have unique minimal k-module single-entry automata. We also give a polynomial time algorithm that minimizes such k-module single-entry VPAs. 1 Introduction The class of visibly pushdown languages (Vpl), introduced in [1], is a subclassof context-free languages accepted by pushdown automata in which the input letter determines the type of operation permitted on the stack. Visibly push-down languages are closed under all boolean operations, and problems such as inclusion, that are undecidable for context-free languages, are decidable for Vpl. Vpls are relevant to several applications that use context-free languages suchas the model-checking of software programs using their pushdown models [1-3]. Recent work has shown applications in other contexts: in modeling semanticsof effects in processing XML streams [4], in game semantics for programming languages [5], and in identifying larger classes of pushdown specifications thatadmit decidable problems for infinite games on pushdown graphs [6].

Contents 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.1 Grammars : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.2 Examples : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2. Systems of equations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.1 Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.2 Resolution : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 2.3 Linear systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 2.4 Parikh's theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

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. 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. 1

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XML documents are described by a document type de nition (DTD) . An XML-grammar is a formal grammar that captures the syntactic features of a DTD. We investigate properties of this family of grammars. We show that every XML-language basically has a unique XML-grammar. We give two characterizations of languages generated by XML-grammars, one is set-theoretic, the other is by a kind of saturation property. We investigate decidability problems and prove that some properties that are undecidable for general context-free languages become decidable for XML-languages. We also characterize those XML-grammars that generate regular XML-languages.

Abstract. We present an approach for statically reasoning about the behavior of Web applications that are developed using Java Servlets and JSP. Specifically, we attack the problems of guaranteeing that all output is well-formed and valid XML and ensuring consistency of XHTML form fields and session state. Our approach builds on a collection of program analysis techniques developed earlier in the JWIG and Xact projects, combined with work on balanced context-free grammars. Together, this provides the necessary foundation concerning reasoning about output streams and application control flow. 1

. XML documents are described by a document type denition (DTD). An XML-grammar is a formal grammar that captures the syntactic features of a DTD. We investigate properties of this family of grammars. We show that an XML-language basically has a unique XML-grammar. We give two characterizations of languages generated by XML-grammars, one is set-theoretic, the other is by a kind of saturation property. We investigate decidability problems and prove that some properties that are undecidable for general context-free languages become decidable for XML-languages. 1 Introduction XML (eXtensible Markup Language) is a format recommended by W3C in order to structure a document. The syntactic part of the language describes the relative position of pairs of corresponding tags. This description is by means of a document type denition (DTD). In addition to its syntactic part, each tag may also have attributes. If the attributes in the tags are ignored, a DTD appears to be a special kind...

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. Finding computationally valuable representations of models of predicate logic formulas is an important issue in the field of automated theorem proving, e.g. for automated model building or semantic resolution. In this article we treat the problem of representing single models independently of building them and discuss the power of different mechanisms for this purpose. We start with investigating context-free languages for representing single Herbrand models. We show their computational feasibility and prove their expressive power to be exactly the finite models. We show an equivalence with "ground atoms and ground equations" concluding equal expressive power. Finally we indicate how various other well known techniques could be used for representing essentially infinite models (i.e. models of not finitely controllable formulas), thus motivating our interest in relating model properties with syntactical properties of corresponding Herbrand models and in investigating connections betwe...

XML graphs have shown to be a simple and effective formalism for representing sets of XML documents in program analysis. It has evolved through a six year period with variants tailored for a range of applications. We present a unified definition, outline the key properties including validation of XML graphs against different XML schema languages, and provide a software package that enables others to make use of these ideas. We also survey the use of XML graphs for program analysis with four very different languages: Xact (XML in Java), Java Servlets (Web application programming), XSugar (transformations between XML and non-XML data), and XSLT (stylesheets for transforming XML documents). 1.