A R S D I G I T A
V N I V E R S I T Y
Month 8: Theory of Computation
Problem Set 2 Solutions  Mike Allen and Dimitri Kountourogiannis
1. Minimizing DFAs
2. Regular or Not?
a. NOT. (1.17b) {www  w is {a,b}*}
Assume that the language is regular. Let p be the pumping length, and choose s to be the
string 0
p
10
p
10
p
1. Now we try to break it up into s=xyz. Since xy <= p and y>0, y can
only contain 0's. When we pump the string even just once we get xy
2
z = 0
p+y
10
p
10
p
1, and
this is not of the form www, since y > 0. This contradicts the pumping lemma, so the
language is not regular.
b. NOT. (1.23c) {0
m
1
n
 m is not equal to n}
We know that
{0
n
1
n
 n >= 0}
= {0
m
1
n
 m,n >= 0}^{0*1*}
c
,
where we are using ^ to denote intersection and
c
to denote complement. The proof is by
contradiction. If {0
m
1
n
 m is not equal to n} really were regular then {0
n
1
n
 n >= 0}
would also be regular because 0*1* is regular and because of the closure properties of
regular sets. Therefore it can't be regular
There is a direct way to prove it as well: If p is the pumping length and we take the
string s = 0
p
1
p+p!
, then no matter what the decomposition s = xyz is the string xy
1+p!/y
z
will equal 0
p+p!
1
p+p!
which is not in the language.
c. NOT. (1.23a) {0
m
1
n
0
m
 m,n >= 0}
Assume that the language is regular. Let p be the pumping length, and choose s to be the
string 0
p
10
p
. Now we try to break it up into s=xyz. Since xy <= p, y can only have zeros
in it. Now xy
j
z = 0
p+ (j1) y
10
p
, and since y>0 the number of 0's on the left and right
sides of xy
j
z will not be the same for any j>1 so xy
j
z will not be in the language,
a. Convert to a DFA
b. Convert to a minimal DFA
c. Conver to a regular expression:
0 + (00 + 1 + 11)00*
d. Convert to a regular grammar using the NFA:
A > 0  0B  1C  1D
B > 0D
C > 1D
D > 0  0D
Using the DFA:
S > 0A  1B
A > 0C  e
B > 0D  1C
C > 0D
Using the regular expression:
S > 0  110A  000A  10A
A > 0  0A  e
页码，
1/4(W)
w
2/19/2011
http://aduni.org/courses/theory/courseware/psets/Problem_Set_02_Solutions.html
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 Spring '11
 Friesen
 Formal languages, Regular expression, Regular language

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