This problem concerns the Arithmetic Hierarchy, which is at the top of the World-of-Computability-and Complexity diagram. We say that a set of natural numbers, S, is an element of , iff there is a PTIME predicate p, such that here Qk is V if k is even andif k is odd. Similarly, s is an element of iff. for some PTIME predicate . Here Q. is V if k is odd andif k is even. Define the Arithmetic Hierarchy to be Uk- Classify the following sets by writing a formula that places them as low as you can in the arithmetic hierarchy. You do not have to prove that they cannot be placed in a lower class, For example, TOTAL = {n | Mn halts on all inputs} is 2 because it can be written as TOTAL {n|Va3z(COMP(n.r, L(z), R(z)))). = [A PTIME predicate, , is just a PTIME computable decider - for each possible input it answers "true" or false" in PTIME. COMP(n, r,c, y) is the PTIME predicate meaning that c is a complete halting com- putation of TM Mn on input r and its output is y. Recall that we defined Cantor's pairing function: P : N x NN, given by P(i, j)-writi+1) t i and it's inverses, L, R where L(P(x,y)) 2, R(P(x,y)) = y and P(L(z), R(z)) = z. The pairing function makes it clear that in the above definition there is no change if we have multiple quantifiers of the same kind, ie. dy has the same power as From now on, let's let N = {0, i, . . . (d) FINITE-M = {n I w. is finite) This problem concerns the Arithmetic Hierarchy, which is at the top of the World-of-Computability-and Complexity diagram. We say that a set of natural numbers, S, is an element of , iff there is a PTIME predicate p, such that here Qk is V if k is even andif k is odd. Similarly, s is an element of iff. for some PTIME predicate . Here Q. is V if k is odd andif k is even. Define the Arithmetic Hierarchy to be Uk- Classify the following sets by writing a formula that places them as low as you can in the arithmetic hierarchy. You do not have to prove that they cannot be placed in a lower class, For example, TOTAL = {n | Mn halts on all inputs} is 2 because it can be written as TOTAL {n|Va3z(COMP(n.r, L(z), R(z)))). = [A PTIME predicate, , is just a PTIME computable decider - for each possible input it answers "true" or false" in PTIME. COMP(n, r,c, y) is the PTIME predicate meaning that c is a complete halting com- putation of TM Mn on input r and its output is y. Recall that we defined Cantor's pairing function: P : N x NN, given by P(i, j)-writi+1) t i and it's inverses, L, R where L(P(x,y)) 2, R(P(x,y)) = y and P(L(z), R(z)) = z. The pairing function makes it clear that in the above definition there is no change if we have multiple quantifiers of the same kind, ie. dy has the same power as From now on, let's let N = {0, i, . . . (d) FINITE-M = {n I w. is finite)
