Applied Automata Theory by Julius T. Tou (Eds.)

By Julius T. Tou (Eds.)

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The only trouble is that it is not insightful; that is to say, when one actually goes through this process with a given a and 0, the mechanical procedure that is used is usually not such as to provide any insight into the nature of these regular expressions. For example, a and ß may be unequal but closely related to each other. But the procedure would terminate with a no answer, and that would be the end of it; one would not discover any relationship. Even in cases where a yes answer results, after the computation is over there is usually a feeling that one lacks an understanding of why the two expressions are equal.

8a-c. For this reason, we can say that the set of graph structures is richer than the set of regular-expression structures. This fact contrasts with the fact that the class of events represented by graphs is exactly the same as the class of events represented by regular expressions. For it is well known that for every graph (including that of Fig. 4) there exists a regular expression representing the same event (although it has, in general, a vastly different structure). In fact, there is an algorithm to make this conversion (see Eggan [ 7 ] ) .

2. To begin our discussion of graphs, let us take another look at regular expressions, so as to see them as devices for generating words. For example, from [0(00)*1 W 11]*, one generates a sequence by beginning either with a 0 or a 1. If one selects the 0, then one has the option of wTiting 00 any number of times and then writing 1; but if one elects to begin with a 1, then one must write 1 immediately after. After that, one again has the option of writing 0 or 1, etc. Of course, by virtue of the meaning of the star, one could have settled for the null word at the very outset.

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