The origins of this monograph lie in my Ph.D. dissertation of 1987 at the University of Pennsylvania, which was concerned with proof procedures for the Horn clause subset of logic. The rise of logic programming has made this an important area of study. All Prologs are based on a variant of resolution, and inherit various properties related to this proof method. This monograph studies the paradigm of logic programming in the context of graph-based proof procedures which are unrelated to resolution. The monograph is not a general introduction to logic programming, although it is self-contained with respect to the mathematics used. It should appeal to the computer scientist or mathematician interested in the general area we now call computational logic. A large part of the monograph is devoted to detailed proofs that the methods we present are sound and complete, which in the context of the logic programming, means that the operational and denotational semantics agree.