Luc Boasson

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.

Partial words and a theorem of Fine and Wilf

A partial word is a word that is a partial mapping into an alphabet. We prove a variant of Fine and Wilfs theorem for partial words, and give extensions of some general combinatorial properties of words.

Adherences of languages

Abstract This paper studies context-free sets of finite and infinite words. In particular, it gives a natural way of associating to a language a set of infinite words. It then becomes possible to begin a study of families of sets of infinite words rather similar to the classical studies of families of languages.

Sur diverses familles de langages fermées par transduction rationnelle

SummaryThe main theorem gives a sufficient condition for an AFL to be the closure under union of the set of images under rational transductions of any of its sets of generators. All the AFLs known to have this property satisfy the given condition. As an application we give a short proof of the fact that every generator of the AFL of algebraic (context-free) languages is a faithful generator, i.e. can be mapped onto every algebraic language by a faithful (& free) rational transduction.

Two iteration theorems for some families of languages

The full Abstract Families of Languages, abbreviated full AFLs, were introduced by S. Ginsburg and S. Greibach in [I 1]. It is well known that the family of contextfree languages is a full AFL [9]. It is also a rational cone (according to Eilenbergs terminology) [5]. Let Dn* (respectively, D~*) be the Dyck language (respectively, semi-Dyck language) on 2n letters, ai , 5i , i = 1,..., n, i.e., the class of 1 in the congruence generated by (7i5 i = 5i (7 i ~ 1 (respectively, aiSi = 1) [9-14]. The ChomskySchtitzenberger theorem [9] implies that, for any n >~ 2, D~* (respectively, D~*) is a full generator [10-11] of the AFL as well as of the rational cone of the context-free languages. It seems, then, natural to look at the rational cones and the AFLs generated by the Dyck languages DI* and the semi-Dyck language DI* on two letters. We proved elsewhere [2] that the rational cone W generated by D~* can be characterized by the structure of the pushdown automata recognizing the languages in cg. The main restriction we impose on them is that they should use a single pushdown symbol. Since we can consider the pushdown store as a counter, we call such an automaton a one-counter automaton and we call one-counter languages the elements of ~7. It is rather important to notice that this family W is not the family ~ studied by Greibach in [13]. However, o~ and c# are closely related: o~ is the full AFL generated by Di* [13]. Consequently, from a result of [11] restated in [3], o~is the closer of under union, product, and star operation. In this paper, we prove two pumping lemmas (Theorems 3 and 4) which yield corollaries such as:

Context-free languages

The first section of this chapter contains the definitions of context-free or algebraic languages by means of context-free grammars and of systems of algebraic equations. In the second section, we recall without proof several constructions and closure properties of context-free languages. This section contains also the iteration lemmas for context-free languages. The third section gives a description of the various families of Dyck languages. They have two definitions, as classes of certain congruences, and as languages generated by some context-free grammars. The section ends with a proof of the Chomsky-Schutzenberger Theorem. Two other languages, the Lukasiewicz language and the language of completely parenthesized arithmetic expressions, are studied in the last section.

Balanced grammars and their languages

Balanced grammars are a generalization of parenthesis grammars in two directions. First, several kind of parentheses are allowed. Next, the set of right-hand sides of productions may be an infinite regular language. XML-grammars are a special kind of balanced grammars. This paper studies balanced grammars and their languages. It is shown that there exists a unique minimal balanced grammar equivalent to a given one. Next, balanced languages are characterized through a property of their syntactic congruence. Finally, we show how this characterization is related to previous work of McNaughton and Knuth on parenthesis languages.

NTS languages are deterministic and congruential

Abstract A context-free grammar is said to be NTS if the set of sentential forms it generates is unchanged when the rules are used both ways. We prove here that such grammars generate deterministic languages which are finite unions of congruence classes. Moreover, we show that this family of languages is closed under reversal and intersection with regular sets. A forthcoming paper will prove that, for this class, the equivalence problem is decidable.

Quelques problèmes ouverts en théorie des langages algébriques

— We present hère some open questions about the context-free languages.

Groups and NTS languages

Abstract The context-free groups are known to be exactly the finitely generated virtually free groups [19, 11]. We give here a new combinatorial property which characterizes these groups: they are “locally primary.” A corollary of this property is that the cylinder generated by the group languages is included in the family of NTS languages. In particular, every context-free group language is NTS.