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QUESTION: 1

Let M be a Turing Machine has Q = (q_{0}, q_{1}, q_{2}, q_{3}, q_{4}} a set of states, input alphabets {0, 1}. The tape alphabets are {0,1, B, X, V). The symbol B is used to represent end of input string. The final state is q_{4}. The transistions are as follows:

Which of the following is true about M?

Solution:

Drawing the equivalent turing machine we have

This is a standard turing machine program for

QUESTION: 2

If L_{1} and L_{2} are a pair of complementary languages. Which of the following statement is not possible?

Solution:

The correct statement follows from following theorem:

Theorem: If L_{1} and L_{2} is a pair of complementary language then either

• Both L_{1} and L_{2} are recursive.

• Neither L_{1}, and L_{2} are recursively enumerable.

• One is recursively enumerable but not recursive, the other is not RE.

Option (b) is not possible, since if L_{1} is recursive, then will also be recursive.

QUESTION: 3

Consider the following statements about L:

(i) L is accepted by multi-tape turing machine M_{1}.

(ii) L is also accpted by a single tape turing machine M_{2}.

Which of the following statement is correct?

Solution:

While simulating multi-tape TM on a single tape TM, the head has to move at least 2k cells per move, where k is the number of tracks on single tape TM. Thus for k moves, we get

which means quadratic slow down. Thus, acceptance by single-tape is slower by 0(n^{2}).

QUESTION: 4

A TM M- The transition are

What is the laugnage accepted by TM?

Solution:

The analysis of transition of TM, M shows that is accepts language 0*10*.

QUESTION: 5

Referring to Turing Machine in previous question, what will be the output when the input is

I_{1} = 0100 and I_{2} = 0010?

Solution:

The Turing Machine carries out the functions of ( m - n) in (0^{m} 10^{n}).

• When m≥n, the output is difference between m and n.

• when m < n, the output is blank.

QUESTION: 6

Nobody knows yets if p = NP consider the language L defined as follows:

Which of the following statement is true?

Solution:

L = (0 + 1 )* or {}. A trivial TM which accepts all words or another which rejects all. One of these two TMs will surely accept this language. So L is recursive.

QUESTION: 7

If the strings of a language L can be effectively enumerated in lexicographic (i.e. alphabetic) order which of the following statements is true?

Solution:

L is recursive iff L is enumerable in Lexicographic order (alphabetic).

It is also known that context free language is a subset of recursive languages.

QUESTION: 8

he diagnoalization language, L_{d} is a

Solution:

QUESTION: 9

Time taken by one tape TM to simulate n moves of k-tape TM is

Solution:

Single tape turing machine can simulate n-moves of k-tape TM in O(n^{2}).

QUESTION: 10

Consider the transition table of a TM given below. Here "b" represents the blank symbol.

Q. Which of the following strings will not accepted?

Solution:

Hence, not accepted.

Hence, also not accepted.

QUESTION: 11

The given turing machine accepts

Solution:

It can be observed from the image that this machine will accept only those strings which contains even number of 1’s. And strings with odd number of 1 ’s are rejected.

QUESTION: 12

Ram and Shyam have been asked to show that a cartain problem II is NP-complete. Ram shows a polynomial time reduction from the 3-SAT problem to II and Shyam shows a polynomial time reduction from II to 3-SAT. Which of the following can be inferred from these reductions?

Solution:

Since 3-SAT problem is NP-complete and 3-SAT is polynomial time reducible to π hence π is is also NP-complete.

QUESTION: 13

Which of the following is true?

Solution:

QUESTION: 14

Both P and NP are closed under the operation of

Solution:

P and NP language are closed under union, intersection, kleene closure and concatenation, P is close under complement also while closure of NP under complement is still unsolved problem.

QUESTION: 15

L ∈ NP does not obviously imply that

Solution:

L ∈ NP does not imply that L' ∈ NP. Since closure of NP under complementation is as yet unresolved problem. Also L ∈ NP does not imply L' ∈ P, unless L ∈ P, which may or may not be so.

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