Reading Rice's Theorem

• $ f(M_1,M_2) = \left\{\begin{array}{ll} 1 & M_1 \text{ takes as much time as } M_2 \text{ on every input } \\ 0 & \text{otherwise} \end{array}\right.$
• $ f(M) = \left\{\begin{array}{ll} 1 & M \text{ does not terminate for every input } \\ 0 & \text{otherwise} \end{array}\right.$

Proposition (Rice):

Let $ \mathcal{C} \subseteq RE$ . Given a Turing Machine $ M$ , we ask: “The problem accepted by $ M$ is in $ \mathcal{C}$ ?”. Answering this question is not in $ R$ (not decidable).

• Rice's theorem establishes undecidability of a problem. Which problem is that?
• What does this result entail?
• Examine the transformation from the lecture, and prove both directions by yourself.