By Jean Berstel

This booklet provides a idea of formal languages with major emphasis on rational transductions and their use for the type of context-free lan guages. The Ievel of presentation corresponds to that of starting graduate or complicated undergraduate paintings. necessities for this ebook are coated through a "standard" first-semester coursein formallanguages and automata concept: e.g. an information of Chapters 1-3 of Ginsburg [1966], or Chapters 3-4 of Hopcroft and Ullman [1971], or bankruptcy 2 of Salomaa [1973], or Chap ters 2 and four of Becker and Walter [1977] could suffice. The publication is self-contained within the experience that whole proofs are given for all theorems said, apart from a few easy effects explicitly summarized at the start of the textual content. bankruptcy IV and Chapters V-VIII are self sufficient from one another. the subject material is split into initial and 6 major chapters. The preliminary chapters include a normal survey of the "classical" conception of normal and context-free languages with an in depth description of a number of distinctive languages. bankruptcy III offers with the overall idea of rational transductions, handled in an algebraic type alongside the traces of Eilenberg, and on the way to be used systematically in next chapters. bankruptcy N is worried with the real specific case of rational services, and offers an entire remedy of the newest advancements, together with subsequential transductions, unambiguous trans ducers and selection difficulties.

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**Example text**

E ße8(a)}. 4 cannot be strengthened to assert the existence, for each factorization f = hgh' g' h" of f E L, of a factorization f = aubvc satisfying (i) and suchthat both u is a segrnent of g and v is a segment of g'. 3 Dy~k Languages The Dyck sets are among the most frequently cited context-free languages. In view of the Chomsky-Schützenberger Theorem proved below, they arealso the most "typical" context-free languages. In Chapter VII, we shall see another formulation of this fact: The Dyck languages are, up to four exceptions, generators of the rational cone of context-free languages.

54 III Rational Transductions Consequently AnleRec(l). Define now A={a}. Then AeRec(M), and A+A={O}, A+={O,a}, A*={O,s,a}. 4. The following theorem gives a description of the recognizable subsets of the product of two monoids. Eilenberg [1974] attributes it to Mezei. 5 (Mezei) Let M 1 , M 2 be monoids and M=M1 XM~. Then Be Rec(M) iff B is a finite union of sets of the form A 1 x A 2 , with A 1 E Rec(M1) and A 2 E Rec(M2). Proof. The condition is sufficient. Let indeed '7T;: M canonical projections.

15). 14). Thus r = 1 and w E D". 13} and w E D~. 12), u = b;ikvJJ;ikv2 for some vt>v 2 ED~. 14), v 1 EKi, v2 EKk, and arguing by induction, v 1 ED~nKi, v 2 ED~nKk. Thus wED~. c) D~nK, cM;, (i = 1, ... ,N). Let w E D~ n K,. 10). Otherwise, w = a,ikuä;ik for some indices j, k, and u E D~*. 12), u = b;ikv 1b;ikv 2 for some vt> v 2 E D~*. Moreover, v 1 E ~ and v 2 E Kk. Thus v 1 ED~*nKicD~nKi, and similarly v 2 ED~nKk by part b) of the proof. 9). Thus we proved i = 1, ... 11) follows. 1 Show that for any I c: {1, ...