Unbounded search and recursive graph problems

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, when no bound is given a priori. The Turing degree of the oracle required is already established in the literature. In this paper we analyze the number of queries required. We show that this problem is related to unbounded search in two ways:

• 1. If f is a non-decreasing recursive function, and ∑_i≥02^−f(i) ≤ 1 is effectively computable, then the number of components of a recursive graph G _e , nC(G _e ), can be found with f(nC(G _e )) queries to ø, and

• 2. If G is an infinite recursive graph and there is a set X such that nC(G) can be computed using f(nC(G)) queries to X, then f satisfies Kraft's inequality.

Part (2) above can be interpreted as a lower bound to the number of queries to ø, which is shown to be optimal, even if free queries to weaker oracles are allowed.

Our main result is a generalization of a key theorem of Beigel and Gasarch's, which allows us to conclude that part (2) also applies to a wide class of problems, including the problems of finding the number of finite components and finding the number of infinite components of an infinite recursive graph.