1/31/2006
CSE 2001, Winter 2006
1
CSE 2001:
Introduction to Theory of Computation
Winter 2006
Suprakash Datta
[email protected]
Office: CSEB 3043
Phone: 4167362100 ext 77875
Course page: http://www.cs.yorku.ca/course/2001
Some of these slides are adapted from Wim van Dam’s slides
(
www.cs.berkeley.edu/~vandam/CS172/
)
and from Nathaly Verwaal
(
cpsc.ucalgary.ca/~verwaal/313/F2005
/)
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CSE 2001, Winter 2006
2
Nonregular Languages §1.4
Which languages cannot be recognized by finite
automata?
Examples: L= {0
n
1
n
 n
∈
N
}
PAL = {w  w = w
R
}
• ‘Playing around’ with FA convinces you that the
‘finiteness’ of FA is problematic for “all n
∈
N
”
• The problem occurs between the 0
n
and the 1
n
• Informal:
cannot count arbitrary integers using
a fixed number of states
; the memory of a FA is
limited by the the number of states Q
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CSE 2001, Winter 2006
3
Proving nonregularity
•
Simpler technique (not in the text)
 less general
•
Pumping Lemma (Sec 1.4)
– More difficult
– More general
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CSE 2001, Winter 2006
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Distinguishing strings wrt L
•
Define L/x = {z
∈Σ
*
 xz
∈
L}
•
x,y distinguishable with respect to L if
L/x
≠
L/y
•
∃
z xz
∈
L, yz
∉
L
•
E.g. L = {w w ends in 10}
x = 01, y = 00.
z = 0 distinguishes x,y
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CSE 2001, Winter 2006
5
Theorem
Suppose that for L
⊆
Σ
*
and for some
positive integer n, there are n strings in
Σ
*
such that any two of them are
distinguishable with respect to (wrt) L.
Then every DFA recognizing L must
have at least n states.
Notation:
δ
*(q
0
,x
1
) – generalization of
transition function.
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CSE 2001, Winter 2006
6
Proof of Theorem
Suppose x
1
, …, x
n
are n strings, any two of
which are distinguishable wrt L. If M is a DFA
with fewer than n states, then by the
Pigeonhole
Principle
, the states
δ
*(q
0
,x
1
), …,
δ
*(q
0
,x
n
) are
not all distinct, and so for some i
≠
j,
δ
*(q
0
,x
i
) =
δ
*(q
0
,x
j
). Since x
i
, x
j
are distinguishable wrt L, it
follows that M cannot recognize L.
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1/31/2006
CSE 2001, Winter 2006
7
Applying this Theorem
•
Show that there is an infinite set of
strings , any two of which are
distinguishable wrt L. So any DFA
recognizing L cannot have a finite
number of states – i.e., it does not exist.
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CSE 2001, Winter 2006
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Example: PAL is not regular
•
Any two distinct strings x,y
∈Σ
* can be
distinguished wrt PAL.
•
If x = y, we can choose z = x
R
so that
xz = x x
R
∈
PAL and yz = y x
R
∉
PAL.
•
Else, w.l.o.g., x < y. Let y=y
1
y
2
, y
1
 = x.
Then let z = ww
R
x
R
, w = y
2
, w
≠
y
2
.
So xz=
xww
R
x
R
∈
PAL, yz = y
1
y
2
ww
R
x
R
∉
PAL,
because w
≠
y
2
.
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CSE 2001, Winter 2006
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Next:
•
The Pumping Lemma
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CSE 2001, Winter 2006
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Repeating DFA Paths
q
1
q
k
q
j
Consider an accepting DFA M with size Q
On a string of length p, p+1 states get visited
For p
≥
Q, there must be a j such that the
computational path looks like: q
1
,…,q
j
,…,q
j
,…,q
k
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CSE 2001, Winter 2006
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Repeating DFA Paths
q
1
q
k
q
j
The action of the DFA in q
j
is always the same.
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 Winter '10
 n
 Formal language, Formal languages, Contextfree grammar, regular languages

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