GATE CS 2015 Set 2 — Question 20
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Algorithms → Complexity Theory → Church-Turing Thesis
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Question
Consider the following statements.I. The complement of every Turing decidable language is Turing decidable
II. There exists some language which is in NP but is not Turing decidable
III. If is a language in NP, is Turing decidableWhich of the above statements is/are true?
II. There exists some language which is in NP but is not Turing decidable
III. If is a language in NP, is Turing decidableWhich of the above statements is/are true?
Correct answer
(D) Only I and III
Solution
Let's evaluate each statement:
- Statement I: Turing decidable languages (also known as Recursive languages) are closed under complementation. If a language is decidable, there exists a Turing machine that halts on all inputs. We can construct a machine for by simply swapping the accept and reject states of the machine for . Thus, Statement I is true.
- Statement II: The class NP is a subset of the class of decidable languages (). Therefore, every language in NP is decidable. There is no language in NP that is undecidable. Thus, Statement II is false.
- Statement III: As established above, . Therefore, if , then is Turing decidable. Thus, Statement III is true.
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