Which of the following problems is decidable?

- I. Given a TM M and a string y, does M ever write the symbol # on its tape on input y?
- II. Given a context free grammar G over {a, b}, does G generate all the strings of the language {a, b}
^{*}of length ≤ 381? - III. Given a TM M, are there infinitely many TMs M′accepting the same recursive enumerable set A = L(M)?
- IV. Given a TM M and a string y, does M accept y?

**(A)** I and II

**(B)** II and III

**(C)** I, II and III

**(D)** II and IV

**Answer:** **(B)** **Explanation:**

- I. Undecidable, since the halting problem is reducible to it. Indeed, given M (without # among the tape alphabet) build M′ as follows: M′simulate M but each time M wants to halt M′first prints # and then halts. It is clear,that M on y if and only if M’ writes ]on input y. Therefore, if we could decide (a) we would be able to decide the halting problem.
- II. Decidable: each individual problem x ∈ L(G ) is decidable, and we have to check whether each string of length ≤ 381 (a given finite = set of them) is in L(G).
- III. Decidable. This is a trivial property, since for each TM M there are infinitely many TMs M′ accepting the same r.e. set A = L(M).
- IV. Undecidable. Otherwise we could be able to decide the problem ‘do es 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).

So, option (B) is correct.

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