# Automata Theory Questions and Answers – Reversal-Homomorphism and Inverse Homomorphism

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This set of Automata Theory Multiple Choice Questions & Answers (MCQs) focuses on “Reversal-Homomorphism and Inverse Homomorphism”.

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

Explanation: Lr is defined as the reversal of a language. Lr 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) Lr
b) L’
c) L*
d) All of the mentioned

Explanation: Lr, L’, L* i.e. reversal, complementation and kleene all are the closure properties of regular language.

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

Explanation: If E is a symbol a, e, or f, then Er=E. Other inductive properties include union of reversals, concatenation and Kleene.
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4. If E= FG, Er=?
a) FrGr
b) GrFr
c) FrGr and GrFr
d) None of the mentioned

Explanation: If E= FG, Er=GrFr . Example: (01*)R=(1*)R(0)R

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

Explanation: 01*+10*
ER=(01*)R+(10*)R=>(1*)R0R+(0*)R1R=>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

Explanation: 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

Explanation: 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

Explanation: 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?
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

Explanation: 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) Homomorphism and Inverse Homomorphism
d) None of the mentioned

Explanation: 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}.

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