Download Transductions and Context-Free Languages by Jean Berstel PDF

By Jean Berstel

This e-book provides a idea of formal languages with major emphasis on rational transductions and their use for the class of context-free lan guages. The Ievel of presentation corresponds to that of starting graduate or complex undergraduate paintings. must haves for this e-book are lined via 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 booklet is self-contained within the experience that whole proofs are given for all theorems acknowledged, with the exception of a few simple effects explicitly summarized first and foremost of the textual content. bankruptcy IV and Chapters V-VIII are self reliant from one another. the subject material is split into initial and 6 major chapters. The preliminary chapters comprise a common survey of the "classical" concept of normal and context-free languages with an in depth description of numerous exact languages. bankruptcy III bargains with the overall concept 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 $64000 targeted case of rational services, and offers a whole therapy of the most recent advancements, together with subsequential transductions, unambiguous trans ducers and selection difficulties.

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Extra info for Transductions and Context-Free Languages

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7 The language Df is context-free. X;gx;g. 3) iel First, we introduce the notation Thus YI = xn if I= 0, and YI = Zn if I= {1, ... , n}. Proof. The grammar GI is strict. 3). Assurne w = zw' zw" with w'""" w"""" 1 (mod oi) and z E YI. Then w""" zz""" 1 (mod oi) and w E Df. This shows the inclusion iel l~k~n Conversely, Iet w E Df, w f= 1. 4. Since w f= 1. there is a Ietter z E Y, such that w e"'- zz. 6. w factorizes in with d 0 , d 1 , d 2 E Df. If d 0 = 1, then w E zDfzDf. If d 0 f= 1, then ldol

For any subset A of G, Iet (A) denote the subgroup generated by A, and Iet A- 1 ={x- 1 1 xeA}. Then (A)=(A uA- 1)*. This shows that a finitely generated subgroup of G is rational. In order to prove the converse we first consider the following situation. x2 Ti· · · X.. 6) with Xt. + 1e G, Tt. , T,. c G, and define Y; = X1X2 · · · X; S;=y;T;yi1 AI= Yn+1 u s1 u 0 0 0 i = 1, ... , n + 1 i=1, ... 7) u s .. 8) 0 Then we claim: (A)=(A'). 6), Yn+l> y;;-l 1 E (A). +l)y;;-ll· Thus S; c (A), whence A' c (A) and (A') c (A).

Next, we prove (iii)~ (iv). 4). Thus A = {(a(yh'), ß(yh')): h' E K'} and the morphisms aoy, ßoy are alphabetic. Assurne now X n Y = fl), and define 1r: (XU Y)*-+ X* x Y* by 1rh = (7Txh, 7Tyh}. Obviously 7T is a surjective morphism. 2, a regular language K c (XU Y)* such that 1r(K) = A. This proves (i)~(v). 2. Finally we prove (i)~(iii). Assurne A E Rat(X* x Y*). 1f Xn Y = fl), then (iii) follows from (v). Otherwise, Iet X'* x Y'* be a copy of X* x Y*with X' n Y' = fl), Iet wx :X*-+ X'*, wy : Y*-+ Y'* be the copy isomorphisms and set A'={(wxf,wyg}:(f, g)EA}.

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