cs3102: Theory of Computation
Class 5:
NonRegular Languages
Spring 2010
University of Virginia
David Evans
Menu
•
PS1, Problem 8
•
Nonregular languages
PS1 General Comments
•
Proofs are for
making convincing arguments
, not
for obfuscation.
–
e.g., If you assumed pizzas can only be cut through
their center, it is obvious each cut makes 2 new
pieces, and the number of pieces is 2
n
.
Adding an
inductive proof only adds unnecessary confusion!
•
Pledges are to remind you to be honorable
–
I assume you are all honorable whether you write a
pledge or not
–
Writing a rote pledge (not what the PS collaboration
policy says) doesn’t work
Problem 8
DFA that recognizes:
{
w

w
∈
[
a
,
b
]*
and
w
does not contain two
consecutive
a
s }
noa
onea
b
a
b
twoa
a
a, b
How many strings of length
n
in this language?
noa
onea
b
a
b
twoa
a
a, b
n
End in noa state
End in onea state
Total of length n
0
1
0
1
1
1
1
2
2
1+1 = 2
1
3
3
2+1 = 3
2
5
4
3+2 = 5
3
8
n
> 2
E
0
(
n
1)+
E
1
(
n
1)
E
0
(
n
1)
2E
0
(
n
1) +
E
1
(
n
1)
Fibonacci Strings!
E
1
(
n
) =
E
0
(
n
1)
E
0
(
n
) =
E
0
(
n
1)+
E
0
(
n
2)
T(n)
=
2E
0
(
n
1) +
E
0
(
n
2)
=
2
(
E
0
(
n
2) +
E
0
(
n
3)) +
E
0
(
n
2)
= 3
E
0
(
n
2) + 2
E
0
(
n
3)
T(n 1)
=
2E
0
(
n
2) +
E
0
(
n
3)
+ T(n 2)
=
2E
0
(
n
3) +
E
0
(
n
4)
=
2E
0
(
n
2) + 3
E
0
(
n
3)
+ E
0
(
n
4)
=
2E
0
(
n
2) + 2
E
0
(
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 Spring '08
 Humphreys,G
 Formal languages, Regular expression, Regular language, regular languages

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