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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)

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)

1) Introduction We examine the problem of coloring part of a k-colorable graph, while not knowing the rest of it. To illustrate some of our concepts, we describe a somewhat whimsical scenario, which we call the Mapmaker's Dilemma. Picture a 12th-century mapmaker who is given a map of Europe and the countries adjacent to Europe, and is told to 4-color the… (More)

Traditional work in inductive inference has been to model a learner receiving data about a function <italic>f</italic> and trying to learn the function. The data is usually just the values <italic>f</italic>(0), <italic>f</italic>(1),…. The scenario is modeled so that the learner is also allowed to ask questions about the data (e.g.,… (More)