Consider the following statements regarding Turing machine:
S1 = Let L = { |M is a Turing machine that accepts a string of length 210 over {a, b}.} is decidable.
S2 = Given a TM M and a string s, M accept s is decidable.
S3 = Turing recognizable languages are not closed under union but closed under complementation.
Which of the following statement/s is/are false?
(A) S1, S3
(B) S1, S2
(C) S3 only
(D) S1 only
Answer: (C)
Explanation: According to Rice\’s theorem, any non-trivial property of L() is undecidable. Input string of length 210 is present in L() is non-trivial property.
Undecidable. Otherwise, we could be able to decide the problem ‘does M accepts ε ?’. The latter is undecidable by the Rice Theorem since it corresponds to a non-trivial property of r.e. sets ε ∈ L(M).
Turing recognizable languages are closed under union and complementation.
So, option (C) is correct.
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