Let F IN = {M | L(M) is finite}, and recall HP = {M#w | M halts on w}.

(a) Prove HP m F IN, where HP is the complement of the halting problem. That is, show there exists a computable function f such that M#w HP iff f(M#w) F IN.

(b) Prove HP m F IN. That is, show there exists a computable function f such that M#w HP iff f(M#w) F IN.

(c) Is F IN decidable? Is it recognizable? Justify your answer.

Let FIN = {M L(M) is finite), and recall HP = {M #w | M halts on w). (a) Prove HP