Transductions and Context-Free Languages by Jean Berstel

By Jean Berstel

This ebook offers a thought 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 by way of a "standard" first-semester coursein formallanguages and automata idea: 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] may suffice. The booklet is self-contained within the feel that whole proofs are given for all theorems said, with the exception of a few easy effects explicitly summarized firstly of the textual content. bankruptcy IV and Chapters V-VIII are self sustaining 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 numerous precise languages. bankruptcy III offers with the final conception of rational transductions, taken care of in an algebraic type alongside the traces of Eilenberg, and that allows you to be used systematically in next chapters. bankruptcy N is worried with the real specific case of rational features, and provides a whole therapy of the most recent advancements, together with subsequential transductions, unambiguous trans ducers and choice difficulties.

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10 (Fiiess [1971]) Rat(x<*>) is closed under intersection and Proof. lt suffices to show closure under complementation. Let K E Rat(x<*>). 9. Next p(Z*) is regular, thus p(Z*)\t(K) is regular. 2. 8. For this, we first establish a Iemma derived from Fliess [1971]. Consider an alphabet Y, and Iet Ac Y* be an arbitrary language. Define a function AA from Y* into the subsets of Y* as follows. , ... , a,. EA, x 1 , ••• , x,. -1 x,.. Thus AA(w) consists of all subwords of w obtained by deleting, in w, factors in A which are separated by letters.

1 can be used to draw a pictorial representation of a word w in L. This is given by the graph of the function w'>-+llw'll, where w' ranges over Clearly, Proposition the left factors of w. Thus, for w = aabaabbabbabaaabbbb, we obtain Fig. 3. -1 aabaabbabbaba Fig. e. X1 = b; Then D~* is defined by D~*= 1 UaD~*bD~*. Multiply this equation by b on the right. This gives D~*b = b U aD~*bD~*b. 1), and therefore D~*b = f.... 2 Let w EX*. Then w E D~* iff w satisfies: (i) llwii=O; (ii) llw'll;.. 0 for any left factor w' of w.

Thus i=1, ... 7). Conversely, Iet B = (Bh ... 7). Then i = 1, ... ij(u){Hq){J(v). 8) =Pi. 6). 8) {J(Pk) = {l(Pk \a) U {l{u){l( Qk){l(v); IJ(Qk) = {l(Pk \a) U {l(u){l(Pk){l(v) by definition. 6). 1 A context-free grarnmar G =(V, X, P) is called proper if each production g-+a E P verifies a t 1 U V. A vector A = (A,, ... , AN) of languages is called proper if 1 t A; for i = 1, ... , N. Prove that the system of equations associated to a proper grammar has a unique proper solution. 2 remains true if La is replaced by becomes false.

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