automata-3_fix - Automata Chapter3...

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Automata Chapter 3  Regular Languages and Regular  Grammars
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3.1 Regular Expressions Definition 3.1 Let   be a given alphabet. Then 1. ∅ , λ , and a ∈∑  are all regular expressions.  These are called primitive regular expressions. 1. If r 1  and r 2  are regular expressions, so are r 1 + r 2 r 1 r 2 , r 1 *, and (r 1 ). 2. A string is a regular expression iff it can be  derived from the primitive regular expressions  by a finite number of applications of the rules in  (2).
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3.1 Regular Expressions Definition 3.2 The Language L( r ) denoted by any  regular expression  r  is defined: 1. ∅  is a r.e. denoting the empty set. 2. λ  is a r.e. denoting { λ }. 1. For every a ∈∑ , a is a r.e. denoting {a}.  If  r 1  and  r 2  are regular expressions, then 4. L( r 1 r 2 ) = L( r 1 ) L( r 2 ), 5. L( r 1     r 2 ) = L( r 1 )L( r 2 ),  6. L(( r 1 )) = L( r 1 ), 7. L( r 1   *) = (L( r 1 ))*.
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3.1 Regular Expressions Ex. 3.1           (a+b    c)* (c +  ) Ex. 3.2        L(a    (a+b)) = L(a*) L(a+b)                           = (L(a))*(L(a) L(b))                           = { ,a,aa,aaa, …}{a,b} λ                           = {a,aa,aaa, …. ,b, ab, aab, …} 
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3.1 Regular Expressions Ex. 3.3           r = (a+b)* (a + bb) Ex. 3.4        r = (aa)   * (bb)*b                           
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3.1 Regular Expressions Ex. 3.5       L(r) = { w in {0,1}*| w has at least one pair of                         consecutive zeros} Ex. 3.6       L(r) = { w in {0,1}*| w has no pair of                         consecutive zeros}
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3.2 Connection Between Regular  Expressions and Regular Language Theorem 3.1 Let r be a r.e. Then  5  some NFA that  accepts L( r ).   (L( r ) is a regular language.) Proof:  q 0 q 1 q 0 q 1 λ q 0 q 1 a NFA accepts ,          { λ },                   {a}
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Expressions and Regular Language Assume we have M 1  and M 2  for L( r 1 ) and L( r 2 ). i)
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This note was uploaded on 11/03/2009 for the course CS automata taught by Professor Prof.jung during the Fall '09 term at 홍익대학교.

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automata-3_fix - Automata Chapter3...

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