in Theory of Computation retagged by
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20 votes

Which of the following statements is false?

  1. Every NFA can be converted to an equivalent DFA

  2. Every non-deterministic Turing machine can be converted to an equivalent deterministic Turing machine

  3. Every regular language is also a context-free language

  4. Every subset of a recursively enumerable set is recursive

in Theory of Computation retagged by
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4 Answers

26 votes
 
Best answer

There exists a set of languages which is RE but not REC ( i.e. Recursively Enumerable but not Recursive), this set is a subset of RE but is Not Recursive.

Option D tells us that every subset of RE is REC this is false.
Hence, option D is chosen.

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2 Comments

is B correct?

every non-deterministic  turning machine is equal to deterministic one.?
0
yes, but it is "equivalent" not "equal". Equal requires them to be exactly same- which is not. But they are equivalent in the sense that both accept the same class of languages - recursively enumerable set.
14
8 votes
A language is recursively enumerable if there exists a Turing machine that accepts every string of the language, and does not accept strings that are not in the language. Strings that are not in the language may be rejected or may cause the Turing machine to go into an infinite loop.
A recursive language can't go into an infinite loop, it has to clearly reject the string, but a recursively enumerable language can go into an infinite loop.
So, every recursive language is also recursively enumerable. Thus, the statement ‘Every subset of a recursively enumerable set is recursive’ is false.
 
Thus, option (D) is the answer.
2 votes
(D) is correct answer
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2 Comments

Yes. $\Sigma^*$ is recursively enumerable and any non recursively language must be a subset of this.
8
Do recursively enumerable language and recursive language are synonyms or different?
0
1 vote
Every NFA can be converted into DFA (as there exist a standard procedure to convert NFA into DFA). Also, every non-deterministic Turing machine can be converted to an equivalent deterministic Turing machine. Every regular language is also a CFL , since if a language can be recognized by Finite automata, then it must also be recognize by PDA (as PDA is more powerful than FA). But every subset of recursively enumerable need not be recursive.
Answer:

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