Ans 1)
It’s the set consisting of 0s and 1 where the number of ones is equal to the number of
blocks of zeroes. Ones are never consecutive. All strings end with 1 and begin with 0.
There can be odd as well as even blocks of zeroes. The relationship between the (2n1)th
and the (2n)th block of zeros is that the latter has twice the number of zeros as the prior
block. The relationship between the (2n)th and the (2n+1)th block is that the latter has
one more zero than the prior numbered block. The shortest string is 01.
A few strings
{01, 0101, 010
2
10
3
10
6
1, 010
2
10
3
10
6
10
7
10
14
1,
...}
The last string 
010
2
10
3
10
6
10
7
10
14
1 is the string with 6 blocks.
Ans 2)
a)
h(0120)
=
aabbaa
b)
h(21120)
=
baababbaa
c)
h(L)
=
a(ab)*ba
d)
h(L)
=
a+abba
e)
Inv_h(ababa)
=
{022,110,102}
f)
Inv_h(L)
=
L(02*) U L(1*02*)
Ans 3)
Consider any language L over alphabets {a,b}
Define homomorphism h1 as below
h1(a) =
a
h1(a’) =
a
h1(b) =
b
consider the regular language L1
defined by the regular expression (a+b)*a’
Define homomorphism h2 as below
h2(a) =
a
h2(a’) =
є
h2(b) =
b
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 Fall '06
 HOPCROFT
 Formal language, Regular expression, Regular language, final states

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