Consider the following statements( ) : S1 : There exists no algorithm for…

Answer: Both statements are correct.

• Statement 1: Language equivalence for Turing machines is undecidable. Sketch: If we had an algorithm that decides whether two Turing machines accept the same language, then for any machine M we could decide whether L(M) is empty by comparing M with a machine that rejects every input. Because emptiness of a Turing-recognizable language is undecidable, language equivalence must be undecidable as well.

• Statement 2: Determining whether a Turing machine halts on any input (i.e., whether there exists at least one input on which it halts) is undecidable. Sketch: Reduce the standard halting problem to this question. Given an instance (M,w) of the halting problem, construct M' that on any input ignores that input, simulates M on w, and halts exactly if M halts on w. Then M' halts on some input if and only if M halts on w. If we could decide whether a machine halts on some input, we could decide the halting problem, contradiction. Hence the problem is undecidable.