Consider the properties of recursively enumerable sets : (A) Finiteness (B)…

Answer: All three properties (finiteness, context-freedom, emptiness) are undecidable for recursively enumerable languages.

Key idea: Rice's theorem and simple reductions from the halting problem.

• Emptiness: Given a Turing machine M and input w, build a new machine M' that on any input simulates M on w and accepts if and only if that simulation halts. Then the language of M' is empty exactly when M does not halt on w. Because the halting problem is undecidable, emptiness for r.e. languages is undecidable.

• Finiteness: One can reduce the halting problem to finiteness: given M and w, construct a machine that, if M halts on w, accepts infinitely many distinct strings (for example, all strings of the form 0^n), and otherwise accepts none. Thus determining whether the language is finite is undecidable.

• Context-freeness: The property “is context-free” is a nontrivial language property of r.e. languages (some r.e. languages are context-free, some are not). By Rice's theorem, any nontrivial property of recursively enumerable languages is undecidable, so it is undecidable whether a given r.e. language is context-free.

Therefore, finiteness, emptiness, and context-freeness are all undecidable questions for recursively enumerable sets.