====== Reading Rice's Theorem ======

==== Establish the hardness of the following ====

{{##
  * $math[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.]
##}}
  * $math[f(M) = \left\{\begin{array}{ll} 1 & M \text{ does not terminate for every input } \\ 0 & \text{otherwise} \end{array}\right.]

==== Reading Rice's Theorem ====

$prop[Rice]
Let $math[\mathcal{C} \subseteq RE]. Given a Turing Machine $math[M], we ask:// "The problem accepted by $math[M] is in $math[\mathcal{C}]?". //Answering this question is not in $math[R] (not decidable).
$end

  * 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.