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Unformatted text preview: ECS 120 Lesson 4 Closure Properties of Regular Languages, Pt. 1 Oliver Kreylos Friday, April 6th, 2001 1 Operations on Languages We defined a (formal) language L over an alphabet as a set of words: L * . Today we introduce operations on languages, and how the class of regular languages behaves under such operations. Since languages are sets of words, the first operations we look at are the set operations complement, intersection and union. Let A * be a language. Then we define the complement A of A as A := w * w / A , or, equivalently, A = * \ A . Let A, B * be languages over the same alphabet. Then we define the intersection A B of A and B as A B := w * w A w B , the usual set intersection; and we define the union A B of A and B as A B := w * w A w B , the usual set union. The next two operations are specific to sets of words: Concatenation and Kleene Star. Before we introduce them, we have to formally define the concatenation of two words. Let x 1 , x 2 * be two words over the same alphabet. Then we define the concatenation x 1 x 2 of x 1 and x 2 recursively as follows: 1. If x 1 , i. e., x 1 = , then we define x 1 x 2 := x 2 . 2. If x 1 * \ { } , i. e., x 1 is not the empty word, we can split x 1 into a character a and a word x 1 * : x 1 = ax 1 . Then we define x 1 x 2 = ( ax 1 ) x 2 := a ( x 1 x 2 ). 1 Less formally, two words are concatenated by writing their characters as a sin gle word: If x 1 = a 1 a 2 . . . a n and x 2 = b 1 b 2 . . . b m , then x 1 x 2 = a 1 a 2 . . . a n b 1 b 2 . . . b n ....
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This note was uploaded on 05/20/2010 for the course ECS 120 taught by Professor Filkov during the Spring '07 term at UC Davis.
 Spring '07
 Filkov

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