Queries and Algorithms Computable by Polynomial Time Existential Reeective Machines Extended Abstract for Pods'97 Submission

We consider two kinds of reeective relational machines: the usual ones, which use rst order queries, and existential reeective machines, which use only rst order existential queries. We compare these two computation models. We build on already existing results for standard relational machines, obtained by Abiteboul, Papadimitriou and Vianu APV], so we prove only results for existential machines. First we show that for both standard and existential reeective machines the set of polynomial time computable Boolean queries consists precisely of all PSPACE computable queries. The we go farther and compare which classes of algorithms both kinds of machines represent. It appears that, unless PSPACE = PTIME NP ; there are PSPACE queries which cannot be computed by polynomial time existential reeective machines which use only polynomial amount of relational memory, while it is always possible for standard reeective machines. We conclude that existential reeective machines, although equivalent in expressive power to unrestricted machines, provide substantially worse algorithms than the latter. On the positive side, we demonstrate that the above quality diierence does not aaect computing of PTIME queries, what can be done by existential reeective machines in PTIME; even if we take into account query evaluation costs, as well. Therefore within PTIME we use ner distinctions of the quality of algorithms. We demonstrate that any existential reeective machine computing the transitive closure query must use at least (1 ? o(1)) log n variables, while for reeective relational machines using full rst order queries already 3 variables suuce.