Q. Let G be the CFG, l be the number of left most derivations, r be the number of right most derivations and P be the number of parse trees. Assume l , r and P are computed for a particular string. For a given CFG ‘G’ and given string ‘w’, what is the relation between l , P , r ?

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Q. Which of the following statements is/are FALSE?

1. For every non-deterministic Turing machine,
there exists an equivalent deterministic Turing machine.
2. Turing recognizable languages are closed under union
and complementation.
3. Turing decidable languages are closed under intersection
and complementation.
4. Turing recognizable languages are closed under union
and intersection.

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Q. Let L1 be a recursive language. Let L2 and L3 be languages that are recursively enumerable but not recursive. Which of the following statements is not necessarily true?

(A) L2 – L1 is recursively enumerable.
(B) L1 – L3 is recursively enumerable
(C) L2 ∩ L1 is recursively enumerable
(D) L2 ∪ L1 is recursively enumerable.

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Q. If L and L' are recursively enumerable, then L is

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Q. Let L be a language and L' be its complement. Which one of the following is NOT a viable possibility?

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Q. Let A ≤m B denotes that language A is mapping reducible (also known as many-to-one reducible) to language B. Which one of the following is FALSE?

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Q. For S ∈ (0 + 1) * let d(s) denote the decimal value of s (e.g. d(101) = 5). Let L = {s ∈ (0 + 1)* d(s)mod5 = 2 and d(s)mod7 != 4}. Which one of the following statements is true?

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Q. Let L1 be a recursive language, and let L2 be a recursively enumerable but not a recursive language. Which one of the following is TRUE?

L1' --> Complement of L1
L2' --> Complement of L2

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Q. A single tape Turing Machine M has two states q0 and q1, of which q0 is the starting state. The tape alphabet of M is {0, 1, B} and its input alphabet is {0, 1}. The symbol B is the blank symbol used to indicate end of an input string. The transition function of M is described in the following table

 0 1 B
 q0  q1, 1, R q1, 1, R  Halt
 q1  q1, 1, R   q0, 1, L   q0, B, L  

The table is interpreted as illustrated below. The entry (q1, 1, R) in row q0 and column 1 signifies that if M is in state q0 and reads 1 on the current tape square, then it writes 1 on the same tape square, moves its tape head one position to the right and transitions to state q1. Which of the following statements is true about M ?

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Q. For any two languages L1 and L2 such that L1 is context free and L2 is recursively enumerable but not recursive, which of the following is/are necessarily true?

1. L1' (complement of L1) is recursive
2. L2' (complement of L2) is recursive
3. L1' is context-free
4. L1' ∪ L2 is recursively enumerable

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Q. Let X be a recursive language and Y be a recursively enumerable but not recursive language. Let W and Z be two languages such that Y' reduces to W, and Z reduces to X' (reduction means the standard many-one reduction).
Which one of the following statements is TRUE ?

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Q. Consider the following types of languages:

L1 Regular,
L2: Context-free,
L3: Recursive,
L4: Recursively enumerable.

Which of the following is/are TRUE?

I. L3' U L4 is recursively enumerable
II. L2 U L3 is recursive
III. L1* U L2 is context-free
IV. L1 U L2' is context-free

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Q. Which of the following are decidable?

I. Whether the intersection of two regular languages is infinite
II. Whether a given context-free language is regular
III. Whether two push-down automata accept the same language
IV. Whether a given grammar is context-free

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Q. Which of the following problems is undecidable?

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Q. Let < M > be the encoding of a Turing machine as a string over {0, 1}. Let L = { < M > |M is a Turing machine that accepts a string of length 2014 }. Then, L is

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Q. Which of the following problems is undecidable?

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Q. Consider three decision problems P1, P2 and P3. It is known that P1 is decidable and P2 is undecidable. Which one of the following is TRUE?

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Q. Consider the following statements:

1. The complement of every Turning decidable
language is Turning decidable
2. There exists some language which is in NP
but is not Turing decidable
3. If L is a language in NP, L is Turing decidable

Which of the above statements is/are True?

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Q. Let L={w ∈ (0 + 1)*|w has even number of 1s}, i.e. L is the set of all bit strings with even number of 1s. Which one of the regular expression below represents L?

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Q. Let P be a regular language and Q be context-free language such that Q ⊆ P. (For example, let P be the language represented by the regular expression p*q* and Q be {p^nq^n|n ∈ N}). Then which of the following is ALWAYS regular?

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