The Complexity of ODDn

For a fixed set A, the number of queries to A needed in order to decide a set S is a measure of S’s complexity. We consider the complexity of certain sets defined in terms of A: ODDn = {(x1, . . . , xn) : #n (x1, . . . , xn) is odd} and, for m ≥ 2, MODmn = {(x1, . . . , xn) : #n (x1, . . . , xn) 6≡ 0 (mod m)}, where #n (x1, . . . , xn) = A(x1) + · · ·+A(xn). (We identify A(x) with χA(x), where χA is the characteristic function of A.) If A is a nonrecursive semirecursive set or if A is a jump, we give tight bounds on the number of queries needed in order to decide ODDn and MODm A n : • ODDn can be decided with n parallel queries to A, but not with n− 1. • ODDn can be decided with dlog(n+ 1)e sequential queries to A but not with dlog(n+ 1)e − 1. • MODmn can be decided with dn/me+ bn/mc parallel queries to A but not with dn/me+ bn/mc − 1. • MODmn can be decided with dlog(dn/me+ bn/mc+ 1)e sequential queries to A but not with dlog(dn/me+ bn/mc+ 1)e − 1. The lower bounds above hold for nonrecursive r.e. setsA as well. (Interestingly, the lower bounds for r.e. sets follow by a general result from the lower bounds for semirecursive sets.) In particular, every nonzero truth-table degree contains a set A such that ODDn cannot be decided with n− 1 parallel queries to A. Since every truth-table degree also contains a set B such that ODDn can be decided with one query to B, a set’s query complexity depends more on its structure than on its degree. For a fixed set A, Q(n,A) = {S : S can be decided with n sequential queries to A}, Q‖(n,A) = {S : S can be decided with n parallel queries to A}. We show that if A is semirecursive or r.e., but is not recursive, then these classes form non-collapsing hierarchies: • Q(0, A) ⊂ Q(1, A) ⊂ Q(2, A) ⊂ · · · • Q‖(0, A) ⊂ Q‖(1, A) ⊂ Q‖(2, A) ⊂ · · · The same is true if A is a jump.