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Do the following. ( a ) Provide the instructions of the URM program P that computes the function f ( n ) = n 2

Do the following.
(a) Provide the instructions of the URM program P that computes the function f(n)= n2. Use the online URM simulator to verify that your URM is correct. (10 pts)
(b) Briefly describe the idea behind your program and the purpose of each register. (10 pts)
2. Prove that the \pi encoding function described in lecture is a one-to-one correspondence between N\times N andN. Hint: foreachz in N,showthatthereisauniquepair(x,y) in N\times N for which \pi (x, y)= z.(15 pts)
3. Do the following.
(a) ProvidetheG odelnumberefortheprogramP =J(1,2,7),S(2),S(2),S(2),S(3),J(1,1,1),T(3,1).
Write e as a sum of powers of two minus one. (10 pts)
(b) Apply the decoding functions to e in order to verify that your answer to part a is correct.
Show all work. (10 pts)
(c) Describe \phi e(x), We, and Ee. Justify your answers. (10 pts)
4. Prove that there is a total computable function h(n) for which, for all n >=0,\phi h(n)(x)= xn. Hint: you may assume that xn is a computable function of two variables. (15 pts)

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