1. Converting Reg. Exps. To -NFAs: An Example
a•b+c*  a  b     
•  L ¢   c   a b c c
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2. What is Rijk ?
In any DFA,
xRijk  *(qi,x)=qj with the following properties:
A similar relation holds also for –NFAs.
1) Whole x is consumed on the path going out of qi and entering qj ,and
2) The suffix of any state on the path (except for qi and qj) is at most k.
•  L ¢  x qi qj
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3. Converting DFAs To Reg. Exps: Example 1 (Based on the construction of the proof of the theorem)
R110={}, R120={a,b}, R130={c},
R210= , R220={}, R230=,
R310= , R320=, R330={d,},
R111=R110R110(R110)*R110 ={}{}{}*{}={},
R121=R120R110(R110)*R120={a,b},
R131=R130R110(R110)*R130={c},
R211=, R221={}, R231=R311=R321= , R331={d,},
R112={} , R122={a,b}, R132={c},
R212=, R222={}, R232=, R312=R322=, R332={d,},
R113={}, R123={a,b},
R133=R132R132(R332)*R332={c}{c}{d,}*{d,}={c}{d}*,
L(M)=R113 R133={}{c}{d}*.
=||+cd*||
•  L ¢  q1 q2
a,b c d q3
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4. Converting -NFAs To Reg. Exps: Example 2
Based on the meaning of Rijk
R114=R113=R112=R111=R110={}, R124={a},
R130=R131={c}, R132={a,c}, R144=R143={,ab}, R444=R443={}, R434=R433=, R221={}, R231={e}, R321=,
R332=R331=R330R310(R110)*R130={,d},
R133=R132R132(R332)*R332
={a,c}{a,c}{,d}*{,d}={a,c}{d}*,
R134=R133R143(R443)*R433
={a,c}{d}*{,ab}{}*={a,c}{d}*.
L(M)=R134 R144={a,c}{d}*{,ab}
=||(a+c)d*++ab||.
•  L ¢   q1 q2 q4 a b  c d
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5. Regular Sets
L is called a regular set if L=|||| for some reg. exp. ;  is said to denote L.
If L and L’ are regular sets, then so are LL’, LL’, and L*.
Why?
THEOREM (S.C.Kleene) The class of regular sets, which we denote by R, is the smallest class of languages that contains all finite languages and closed under , •, and *.
THEOREM R=L(DFA)=L(NFA)= L(-NFA)=L(Reg. Exp.).
•  L ¢ 
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6. Examples & Exercises =  ||||=|||| Examples Excercises
def
=  ||||=||||
Examples
1. a*+a*=a*, 2. a*a*=a*, 3. (a*)*=a*,
4. (a+b)*=(b+a)*, 5. (ab)*=a(ba)*b+,
5. (ab*+a*b+)(a+)+a*b*
=(ab*a+a*ba+ab*+a*b)+a*b*
=ab*a+a*ba+a*b*a*b*(a+)
||ab*a+a*ba+a*b*|| is a proper subset of ||a*b*(a+)|| (Consider aabba).
6. ||a(b+c)+ab||=||a(b+c)||||ab||= ||a||•||b+c||||a||•||b||={a}(||b||||c||) {a}{b}={a}{b,c}{ab}={ab,ac}.
Excercises
1. (a+b)*(b+a)*=(a+b)*? 2. (a+b)*(a+b)+=(a+b)*?
3. (ab+ba)*=(ab)*+(ba)*? 4. (a*+b*)*=(a*b*)*?
5. ||**+aa*||=?, 6. ||0*10*10*||=?,
7. Give reg. exps. for (a) the set of words over {0,1} containing 000 as a subword, (b) the set of odd nonnegative integers over {0,1,2,…,9}, (c) the set of words over {a,b} containing even number of a’s and b’s, (d) the set of integers which are multiples of 3, (e) the set of positive integers over {0,1,…,9} that are dividable by 10.
•    L ¢ 
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