The complexity of ODDnA

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:and, for m > 2,where #nA. (x1….. xn) = A(x1) + A(xn)(We identify with , 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 ODDnA and MODmnA:• ODDnA can be decided with n parallel queries to A, but not with n − 1.• ODDnA can be decided with ⌈log(n + 1)⌉ sequential queries to A but not with ⌈log(n + 1)⌉ − 1.• MODmnA can be decided with ⌈n/m⌉ + ⌊n/m⌋ parallel queries to A but not with ⌈n/m⌉ + ⌊n/m⌋ − 1.• MODmnA can be decided with ⌈log(⌈n/m⌉ + ⌊n/m⌋ + 1)⌉ sequential queries to A but not with ⌈log(⌈n/m⌉ + ⌊n/m⌋ + 1)⌉ − 1.The lower bounds above hold for nonrecursive recursively enumerable sets A as well. (Interestingly, the lower bounds for recursively enumerable 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 ODDnA cannot be decided with n − 1 parallel queries to A. Since every truth-table degree also contains a set B such that ODDnB 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 recursively enumerable, 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.

Frequency computations and the cardinality theorem

In 1960 G. F. Rose [R] made the following definition: A function f: ω → ω is (m, n)-computable, where 1 ≤ m ≤ n, iff there exists a recursive function R: ωn → ωn such that, for all n-tuples (x1,…, xn) of distinct natural numbers,J. Myhill (see [McN, p. 393]) asked if f had to be recursive if m was close to n; B. A. Trakhtenbrot [T] responded by showing in 1963 that f is recursive whenever 2m > n. This result is optimal, because, for example, the characteristic function of any semirecursive set is (1,2)-computable. Trakhtenbrot's work was extended by E. B. Kinber [Ki1], using similar techniques. In 1986 R. Beigel [B] made a powerful conjecture, much more general than the above results. Partial verification, falling short of a full proof, appeared in [O]. Using new techniques, M. Kummer has recently established the conjecture, which will henceforth be referred to as the cardinality theorem (CT). It is the goal of this paper to show the connections between these various theorems, to review the methods used by Trakhtenbrot, and to use them to prove a special case of CT strong enough to imply Kinber's theorem (see §3). We thus have a hierarchy of results, with CT at the top. We will also include a discussion of Kummer's methods, but not a proof of CT.

Weakly semirecursive sets

We introduce the notion of “semi-r.e.” for subsets of ω, a generalization of “semirecursive” and of “r.e.”, and the notion of “weakly semirecursive”, a generalization of “semi-r.e.”. We show that A is weakly semirecursive iff, for any n numbers x1, …,xn, knowing how many of these numbers belong to A is equivalent to knowing which of these numbers belong to A. It is shown that there exist weakly semirecursive sets that are neither semi-r.e. nor co-semi-r.e. On the other hand, we exhibit nonzero Turing degrees in which every weakly semirecursive set is semirecursive. We characterize the notion “A is weakly semirecursive and recursive in K” in terms of recursive approximations to A. We also show that if a finite Boolean combination of r.e. sets is semirecursive then it must be r.e. or co-r.e. Several open questions are raised.