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Unformatted text preview: Chapter 6 Properties of Regular Languages 2 Regular Sets and Languages Claim(1) . The family of languages accepted by FSAs consists of precisely the regular sets over a given alphabet. Every regular set is accepted by some NFA , a) : b) : c) a : Defn. 2.3.1 (on regular sets ): Let be an alphabet. The regular sets over are defined recursively as follows: i) Basis: , { }, and { a }, a , are regular sets over ii) Recursion: Let X and Y be regular sets over . The sets X Y , XY , and X * are regular sets over . iii) Closure: Every regular set is obtained from(i) by a finite number of application of (ii) q q f q q f a q q f q or 3 Regular Sets and Languages Let M 1 and M 2 be two FSAs, and let S m 1 , F m 1 , S m 2 , and F m 1 , be the new start and accepting states of M 1 and M 2 , respectively: a) L ( M 1 ) L( M 2 ): a string can be processed by M 1 and M 2 in parallel b) L ( M 1 ) L( M 2 ): a string is processed by the composite machine sequentially c) L ( M 1 )* S m 2 F m 2 M 1 M 2 S F S m 1 F m 1 M 1 M 2 S m 1 S m 2 F m 1 F m 2 M 1 S m 1 F m 1 F S 4 3.3 Regular Grammars A grammar is defined as a quadruple ( V , , P , S ), where V is a finite set of nonterminals is a finite set of terminals S V , is a start symbol , and P is a finite set of rewrite/production rules of the form , where ( V )*, ( V )* Defn 3.3.1 . A regular grammar is a grammar in which each rule has one of the following forms, where A , B V and a (i) A a (ii) A aB (iii) A A language is regular if it can be generated by a regular grammar Regular grammars generate precisely the languages defined by regular expressions Regular expressions are used to abbreviate the descriptions of regular sets 5 3.3 Regular Grammars Example . Given G = ( V , , P , S ), where P = { S xX X yY Y xX Y }...
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 Winter '12
 DennisNg

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