Download e-book for iPad: Automata, Languages, and Machines/Part A: v. A by Samuel Eilenberg

By Samuel Eilenberg

ISBN-10: 008087374X

ISBN-13: 9780080873749

ISBN-10: 0122340019

ISBN-13: 9780122340017

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Additional info for Automata, Languages, and Machines/Part A: v. A

Sample text

3) shows that it suffices to define 40. 4). 5) 'Ys z { q I there exists a path c : p q with p -+ E X and I c I = s}. 3). 6) I &' 1 = {s I Is n 'I' f 8) For any s E L'*, the set Is can be effectively calculated (essentially in I s 1 steps). T h u s we can decide whether Is n 'I' f (3 and thus decide whether s E I 1. 'The automaton a' is said to be nccessible if for every state q there exists a path c : i + q with i E I . , the states in Q IL'* are removed. By doing so we obtain an accessible automaton ~ = I F , T;l = 7' n p, such that I &' I &:I' I.

2), iys = isg, f @. 3p), i's E T ' . Consequently S E l d ' I I Ill. 2. 1 LY' 1 = I Lw" I. Let parts of ,u/ and JW' that the diagram I . 1. , independent of y), piPoper, and is n surjectiue function. i, T ) and d ' = (Q', i', T ' ) . 7) If i's E Q't, then is E Qt. Qlt. Indeed if is E Qt, then isv E T for some PI E P". T h u s sa E I ,3 I and therefore also sel E I 1. 'Thus i'sv E T' and therefore i's E Q". Since i's c iys c (is)p it follows that i's = (is)y. 6). Next assume i's E Q't. T h c n i'sv E T ' for some a E S" and thus sv E I Ld'I = I ,dI.

Let S be a finite alphabet and A a recognizable subset of S+. There exists then a finite alphabet Q, a very fine morphism f: sZ* + Yr", and a local subset B of Q* such that Bf = A I Local sets play a considerable role in the subsequent development. We shall therefore establish some of their properties here. II. 2. The intersection of two local subsets of 2" is a local set. Observe that the union of two local sets need not be local ; for instance, aJ" n P a , and a 2 P n P a , are local sets while their union is not if a1 f (T, , al, a2 E 2.

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Automata, Languages, and Machines/Part A: v. A by Samuel Eilenberg


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