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Enumerable sets and quasireducibility

By R. Downey, G. Laforte and A. Nies


Abstract We consider the computably enumerable sets under the relation of Qreducibility. We first give several results comparing the upper semilattice of c.e. Q-degrees, hRQ; ^Q i, under this reducibility with the more familiar structure of the c.e. Turing degrees. In our final section, we use coding methods to show that the elementary theory of hRQ; ^Q i is undecidable. 1 Introduction Classical recursion theory first arose in order to study the inherent difficulty of mathematical problems. By far the most deeply studied notion relating the difficulty of one problem to another has been that given by Turing reducibility. The reason for this is that Turing reducibility seems to give the most general means of obtaining finite information about one object given finite information about another; hence, as the limiting case of using information, it is the most natural object of study for purely theoretical investigations of relative computability and definability. Nevertheless, for specific problems, particularly those arising in the study of algebraic structures, other reducibilities are actually the correct ones to consider. These reducibilities are usually less general, or "stronger", since they arise by putting limits of some kind on what sort of information can be used in a relative solution of one problem given another. For example, weak truth table (wtt) reducibility imposes the additional condition that the amount of information used in a relative computation can be bounded in advance by a computable function. In the case of (computably presentable) infinite dimensional vector spaces, it turns out that the inherent difficulty of constructing bases for subspaces coincides exactly with the relation of wtt reducibility, rather than Turing reducibility. A similar situation arises in combinatorial group theory, where so-called quasi-reducibility, or Q-reducibility, turns out to be a more useful means of comparing word problems than ordinary T-reducibility

Year: 1998
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