250+ TOP MCQs on Reversal-Homomorphism and Inverse Homomorphism and Answers

Automata Theory Multiple Choice Questions on “Reversal-Homomorphism and Inverse Homomorphism”.

1. If L is a language, the reversal of the language can be represented as:
a) L’
b) L^c
c) L^r
d) more than one option is correct

Answer: c
Clarification: L^r is defined as the reversal of a language. L^r is a set of strings whose reversal is in L.
Example: L={0, 01, 100}
Lr={0, 10, 001}

2. If L is a regular language, ____ is also regular.
a) L^r
b) L’
c) L*
d) All of the mentioned

Answer: d
Clarification: L^r, L’, L* i.e. reversal, complementation and kleene all are the closure properties of regular language.

3. If E=F+G;
E^r=?
a) F^r+G^r
b) (F+G)^r
c) Both (a) and (b)
d) None of the mentioned

Answer: a
Clarification: If E is a symbol a, e, or f, then Er=E. Other inductive properties include union of reversals, concatenation and Kleene.

4. If E= FG, E^r=?
a) F^rG^r
b) G^rF^r
c) Both (a) and (b)
d) None of the mentioned

Answer: b
Clarification: If E= FG, E^r=G^rF^r . Example: (01*)R=(1*)R(0)R

5. Simplify the following identity:
E=01*+10*
E^R=?
a) (1*0+0*1)
b) (01*10*)^R
c) (0*1+10*)
d) All of the mentioned

Answer: a
Clarification: 01*+10*
ER=(01*)^R+(10*)^R=>(1*)^R0^R+(0*)^R1^R=>1*0+0*1

6. Which of the following obey the closure properties of Regular language?
a) Homomorphism
b) Inverse Homomorphism
c) Reversal
d) All of the mentioned

Answer: d
Clarification: Homomorphism on an aphabet is a function that gives a string for each symbol in that alphabet. Example: h(0)=ab, etc.

7. Let h(L) be a language of regular expression abe*+e(ab)*. Simplify the h(L)
a) (ab)*+eab*
b) abe*+ea*b*
c) (ab)*
d) None of the mentioned

Answer:
abe*+e(ab)*(Using the identities e=e*, eE=Ee=E)
=ab+(ab)*=> ab will contain inside (ab)*, thus =>(ab)*.

8. Let h(0)=ab; h(1)=e
Let L={abab,baba}
h-1(L)=_______
a) the language of two one’s and any number of zeroes
b) the language of two zeroes and any number of one’s
c) the language of two zeroes and two one’s
d) none of the mentioned

Answer: b
Clarification: h-1(L) is the language with two 0’s and any number of 1’s=>(1*01*01*).

9. While proving Inverse Homomorphism, which of the following steps are needed?
a) Start with a DFA Ain L
b) Construct a DFA B for h-1(L)
c) The set of states, initial and final states should be same.
d) All of the mentioned

Answer: d
Clarification: While constructing DFA B, we need to take care of the following:
a) The same set of states
b) The same start state
c) The same final state
d) Input alphabet = the symbols to which homomorphism h applies.

10. 8. Let h(0)=ab; h(1)=e
Let L={abab,baba}
h-1(L)= the language of two zeroes and any number of one’s.
The given example belongs to which of the following?
a) Homomorphism
b) Inverse Homomorphism
c) Both (a) and (b)
d) None of the mentioned

Answer: b
Clarification: Let h be a homomorphism and L a language whose alphabet is the output language of h.
h-1(L) = {w | h(w) is in L}.