Jean Berstel

Context-free languages and pushdown automata

This chapter is devoted to context-free languages. Context-free languages and grammars were designed initially to formalize grammatical properties of natural languages [9]. They subsequently appeared to be well adapted to the formal description of the syntax of programming languages. This led to a considerable development of the theory.

The origins of combinatorics on words

We investigate the historical roots of the field of combinatorics on words. They comprise applications and interpretations in algebra, geometry and combinatorial enumeration. These considerations gave rise to early results such as those of Axel Thue at the beginning of the 20th century. Other early results were obtained as a by-product of investigations on various combinatorial objects. For example, paths in graphs are encoded by words in a natural way, and conversely, the Cayley graph of a group or a semigroup encodes words by paths. We give in this text an account of this two-sided interaction.

Formal properties of XML grammars and languages

Abstract. We consider XML documents described by a document type definition (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.Résumé. Nous considérons des documents XML décrits par une définition de type de document (DTD). Une grammaire XML est une grammaire formelle qui retient les aspects syntaxiques dune DTD. Nous étudions les propriétés de cette famille de grammaires. Nous montrons quun langage XML a essentiellement une seule grammaire XML. Nous donnons deux caractérisations des langages engendrés par les grammaires XML, la première est ensembliste, la deuxième est par une propriété de saturation. Nous examinons des problèmes de décision et nous prouvons que certaines propriétés qui sont indécidables pour les langages context-free généraux deviennent décidables pour les langages XML. Nous caractérisons également les grammaires XML qui engendrent des langages rationnels.

A scalable formal method for design and automatic checking of user interfaces

The paper addresses the formal specification, design and implementation of the behavioral component of graphical user interfaces. Dialogs are specified by means of modular, communicating grammars called VEG (Visual Event Grammars), which extend traditional BNF grammars to make the modeling of dialogs more convenient.nA VEG specification is independent of the actual layout of the GUI, but it can be easily integrated with various layout design toolkits. The specification may be verified with the model checker Spin, in order to test consistency and correctness, to detect deadlocks and unreachable states, and also to generate test cases for validation purposes. Efficient code is automatically generated by the VEG toolkit, based on compiler technology.nRealistic applications have been specified, verified and implemented, like a Notepad-style editor, a graph construction library and a large real application to medical software. The complete VEG toolkit is going to be available soon as free software.

On the complexity of hopcroft’s state minimization algorithm

Hopcroft’s algorithm for minimizing a deterministic automaton has complexity O(n log n). We show that this complexity bound is tight. More precisely, we provide a family of automata of size n = 2k on which the algorithm runs in time k2k. These automata have a very simple structure and are built over a one-letter alphabet. Their sets of final states are defined by de Bruijn words.

Growth of repetition-free words: a review

This survey reviews recent results on repetitions in words, with emphasis on the estimations for the number of repetition-free words.

Mixed languages

Let T = A ∪ B ∪ C be an alphabet that is partitioned into three subalphabets. The mixing product of a word g over A ∪ B and of a word d over A ∪ C is the set of words w over T such that its projection onto A ∪ B gives g and its projection onto A ∪ C gives d.Let R be a regular language over T such that xbcy is in R if and only if xcby is in R for any two letters b in B and c in C. In other words, R is commutative over B and C. Is this property structural in the sense that R can then be obtained as a mixing product of a regular language over A ∪ B and of a regular language over A ∪ C?This question has a rather easy answer, but there are many cases where the answer is negative. A more interesting question is whether R can be represented as a finite union of mixed products of regular languages. For the moment, we do not have an answer to this question. However, we prove that it is decidable whether, for a given k, the language R is a union of at most k mixed products of regular languages.

Crochemore factorization of sturmian and other infinite words

The Crochemore factorization was introduced by Crochemore for the design of a linear time algorithm to detect squares in a word. We give here the explicit description of the Crochemore factorization for some classes of infinite words, namely characteristic Sturmian words, (generalized) Thue-Morse words, and the period doubling sequence.

An Exercise on Fibonacci Representations

We give a partial answer to a question of Carlitz asking for a closed formula for the number of distinct representations of an integer in the Fibonacci base.

Operations preserving regular languages

Given a strictly increasing sequence s of non-negative integers, filtering a word a0a1.... an by s consists in deleting the letters ai such that i is not in the set {s0,s1,...}. By a natural generalization, denote by L[s], where L is a language, the set of all words of L filtered by s. The filtering problem is to characterize the filters s such that, for every regular language L, L[s] is regular. In this paper, the filtering problem is solved, and a unified approach is provided to solve similar questions, including the removal problem considered by Seiferas and McNaughton. Our approach relies on a detailed study of various residual notions, notably residually ultimately periodic sequences and residually rational transductions.