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Consider the function F A k (x 1 ; : : :; x k) = A(x 1) A(x k). We show that if F A k can be computed with less than k queries to some set X then A 2 P=poly. A generalization of this result has applications to bounded query classes, circuits, and enumerability. In particular we obtain the following. (1) Assuming p 3 6 = p 3 the hierarchy of functions… (More)

Let C(x) and K(x) denote plain and prefix Kolmogorov complexity, respectively, and let R C and R K denote the sets of strings that are " random " according to these measures; both R K and R C are undecidable. Earlier work has shown that every set in NEXP is in NP relative to both R K and R C , and that every set in BPP is polynomial-time truth-table… (More)

- William Gasarch, Carl H Smith
- 1997

A certain two-person infinite game (between " Robin Hood " and the " Sheriff ") has been studied in the context of set theory. In certain cases, it is known that for any deterministic strategy of Robin Hood's, if the Sheriff knows Robin Hood's strategy, he can adapt a winning counter-strategy. We show that in these cases, Robin Hood wins with " probability… (More)

This paper investigates the computational complexity of approximating several NP-optimization problems using the number of queries to an NP oracle as a complexity measure. The results show a trade-off between the closeness of the approximation and the number of queries required. For an approximation factor k(n), log log k(n) n queries to an NP oracle can be… (More)

We classify functions in recursive graph theory in terms of how many queries to K (or ∅ ′′ or ∅ ′′′) are required to compute them. We show that (1) binary search is optimal (in terms of the number of queries to K) for finding the chromatic number of a recursive graph and that no set of Turing degree less than 0 ′ will suffice, (2) determining if a recursive… (More)