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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Q. Consider the language L1,L2,L3 as given below.

L1={0^p1^q | p,q ∈ N}
L2={0^p1^q| p,q ∈ N and p=q}
L3={0^p1^q0^r | p,q,r ∈N and p=q=r}
Which of the following statements is NOT TRUE?

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Q. Definition of a language L with alphabet {a} is given as following. L= { a^(nk) | k > 0, and n is a positive integer constant}. What is the minimum number of states needed in a DFA to recognize L?

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Q. Which of the following pairs have DIFFERENT expressive power?

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Q. Which of the following problems are decidable?

1) Does a given program ever produce an output?
2) If L is a context-free language, then is L’ (complement of L) also context-free?
3) If L is a regular language, then is L’ also regular?
4) If L is a recursive language, then, is L’ also recursive?

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