GATE | GATE CS 2011 | Question 65

2) Consider the language L1,L2,L3 as given below.
L1={0^{p}1^{q} | p,q \in N}
L2={0^{p}1^{q} | p,q \in N and p=q}
L3={0^{p}1^{q}0^{r} | p,q,r \in N and p=q=r}

Which of the following statements is NOT TRUE?
(A) Push Down Automata (PDA) can be used to recognize L1 and L2
(B) L1 is a regular language
(C) All the three languages are context free
(D) Turing machine can be used to recognize all the three languages


Answer: (C)

Explanation: L1 is regular. Its DFA is given as

       gate2011A35

L2 is not regular, can be proved using pumping lemma (refer to Ullman). But L2 is CFL.

        S → AB
        A → 0A|ε
        B → 1B|ε

L3 is not CFL, can be proved using pumping lemma (refer to Ullman). But L3 is Recursive.

gate2011A35b

Every regular language is also a CFL. So PDA can be used to recognized L1 and L2.
As a CFL and Regular language is algo a Recursive language. Hence, Turing machine can be used to recognise
L1, L2 and L3.
L2 is not regular, can be proved using pumping lemma (refer to Ullman). But L2 is CFL.

        S → AB
        A → 0A|ε
        B → 1B|ε

L3 is not CFL, can be proved using pumping lemma (refer to Ullman). But L3 is Recursive.

Every regular language is also a CFL. So PDA can be used to recognised L1 and L2.
As a CFL and Regular language is algo a Recursive language. Hence, turing machine can be used to recognise
L1, L2 and L3.

Source: http://clweb.csa.iisc.ernet.in/rahulsharma/gate2011key.html

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