Please answer the following questions with full formal proofs (when asked to prove something):

Phineas Phoole has decided that Turing machines need ringtones, so he has invented a new model of computation, the Phooling machine. A Phooling machine is exactly like the Turing machines presented in our textbook, except that every transition, in addition to entering a new state, writing to the tape, and (possibly) moving the tape head, also has the option of playing the ringtone (i.e., transitions do all the things you're used to, plus they either play the ringtone or don't).

A. Our book defines a Turing machine transition function as a mapping from pairs of state and tape symbol to triples of state, tape symbol, and direction. Extend this definition to work for Phooling machine transition functions.

B. Define the ringing problem to be to determine whether a given Phooling machine run on a given input ever plays its ringtone. More formally, define R to be the set { : M plays its ringtone when run on w }. Use diagonalization to show that R is undecidable.

C. Use reduction to show that R as defined in part B is undecidable.