Before analysing this problem , please have a look at my answer for the problem : https://gateoverflow.in/46562/cmi2012-b-02a .
This problem is a slight modification to the problem that I mentioned above. But the idea of construction of FA completely differs for this problem.
Analysing the input string:
Consider the input: $01011010$
$(a0 a1 a2 a3)$ $=$ $0011$
$(b0 b1 b2 b3)$ $=$ $1100$
val$(b0 b1 b2 b3)$ $=$ $4$ . val$(a0 a1 a2 a3)$
So this input is one of the strings belonging to the language.
For any string belonging to this language, the least significant $2$ bits of $($$b0$ $b1$ $b2$ $b3$....$bn$$-$$1$$)$ should be zeros because val$($$b0$ $b1$ $b2$ $b3$$)$ $=$ $4$ . val($a0$ $a1$ $a2$ $a3$$)$. [Any bit sequence which is a multiple of 4 will end with 00]
The $2$ most significant bit positions of $($$a$0 $a1$ $a2$ $a3$.......$an-1$$)$ should be zeros so only there exists a possibility for $($$b0$ $b1$ $b2$ ....$bn-1$$)$ to be $4$ times the value of $($$a0$ $a1$ $a2$ $a3$.......$an$$-$$1$$)$.
These are the $2$ basic requirements for the input to be a part of the language.
Idea of construction of Finite Automaton :
Consider the input 01011010
$(a0 a1 a2 a3)$ $=$ $0011$
$(b0 b1 b2 b3)$ $=$ $1100$
We can clearly see from the above example that
$a0=a1=0$
$b0=a2$
$b1=a3.$
$b2=b3=0$
For any input in the language,
$a0=a1=0$
$b0=a2$
$b1=a3$
$b2=a4$
......
$bn-3 =an-1$
$bn-2=bn-1=0$
So, this input pattern should have to recognized by the Finite automaton.
The corresponding Finite automaton is:
Regular Expression : $00 + (000+010(11)^*01).(00+10(11)^*01)^*.0$