Abstract
A graph G = ( V, E) is recursive if every node of G has a finite number of neighbors, and both V and E are recursive (i.e., decidable). We examine the complexity of identifying the number of connected components of an infinite recursive graph, and other related problems, both when an upper bound to that value is given a priori or not. The problems that we deal with are unsolvable, but are recursive in some level of the arithmetic hierarchy. Our measure of the complexity of these problems is precise in two ways: the Turing degree of the oracle, and the number of queries to that oracle. Although they are in several different levels of the arithmetic hierarchy, all problems addressed have the same upper and lower bounds for the number of queries as the binary search problem, both in the bounded and in the unbounded case.
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