Magic Sets vs. SLD-Resolution

It is by now folklore that the bottom-up evaluation of a program after the “magic set” transformation is “as efficient as” top-down evaluation. There are a number of formalizations of this in the literature. However, the naive formalization is false: As shown by Ross, SLD-resolution can be much more efficient than bottom-up evaluation with magic sets on tail-recursive programs. We show that this happens only for tail-recursive programs, and that the only problem of magic sets is the materialization of “lemmas”. So magic sets are always “as goal-directed as” SLD-resolution. These results are not surprising, but we believe that the variants given here are especially useful for teaching purposes. We also give rather simple proofs. Furthermore, we demonstrate that SLD-resolution can be directly simulated by bottom-up evaluable programs if all recursions are tailrecursive. This is based on the meta-interpreter approach of Bry and seems to be very promising.