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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)

- William Gasarch
- 2004

e will onsider lter formultion y ghor et lF PQF e model dtse s @IA n nE it string x = x 1 x 2 · · · x n D together with @PA omputtionl gent tht n do omputtions sed on oth x nd queries mde to itF elie wnts to otin x i suh tht the dtse does not lern iF etully elie wnts more thn tht! she wnts the dtse… (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)

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)

A two-dimensional grid is a set G n,m = [n] × [m]. A grid G n,m is c-colorable if there is a function χ n,m : G n,m → [c] such that there are no rectangles with all four corners the same color. We address the following question: for which values of n and m is G n,m c-colorable? This problem can be viewed as a bipartite Ramsey problem and is related to the… (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)