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2110711
THEORY OF COMPUTATION
Athasit Surarerks 06
NUMBER OF STATES
IMPORTANT
2
The number of distinct states the finite state
machine needs in order to recognize a language is
related to the number of distinct strings that must
be distinguished from each other.
EXAMPLE
3
(0+1)*(000+111)(0+1)*
11
011011
00011
1
1
q0
0
1
q4
0
0
q1
0
011010
00110
q5
1
0,1
1
0
q2
0
q3
DISTINGUISHABLE
4
Definition
Let L be a language in *. Two strings x and y
in * are distinguishable with respect to L if
there is a string z in * so that exactly one of the
strings xz and yz is in L. The string z is said to
distinguish x and y with respect to L.
DISTINGUISHABLE
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Definition
Let L be a language in *. Two strings x and y
in * are distinguishable with respect to L if L/x ≠
L/y where
L/x = { z  * | xz  L }
L/y = { z  * | yz  L }.
EXAMPLE
6
Let  = { 0, 1 }.
Let L be the language associated with
(0+1)*10.
L.
Two strings x = 01101 and y = 010 in *.
Since there is a string z = 0 in * such that
xz = 011010 is in L but yz = 0100 is not in L,
x and y are distinguishable with respect to
We may say that x and y are
indistinguishable with respect to L if there is no
such string z.
The strings 0 and 100 are indistinguishable
LEMMA
7
Suppose that L  *, and M = (Q,,q0,A,). If x
and y are two strings in * for which *(q0,x) =
*(q0,y) then x and y are indistinguishable with
respective to L.
Note: *(q0,x)= q j means that there is a path
from q0 to q j with respect to x, *(q0,x) =
((…((q0,x1),x2),…),xj) = q j
where x = x1x2…xj.
THEOREM
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Suppose that L  *, and for some positive
integer n, there are n strings in *, any two of
which are distinguishable with respect to L. Then
there can be no finite state machine recognizing L
with fewer than n states.
THEOREM
9
PALINDROME language over the alphabet
{0,1} cannot be accepted by any finite automaton.
Proof: Any two strings in {0,1}* are
distinguishable with respect to PALINDROME
language.
EXERCISE
10
How many pair-wise distinguished strings
are there with respect to the following
languages:
•
•
Language of all bit strings, not containing any
double characters.
Language of all bit strings, ending with at least 3
zeroes.