Finite Representation of In nite Query Answers

We deene here a formal notion of nite representation of innnite query answers in logic programs. We apply this notion to Datalog nS (Datalog with n successors): an extension of Datalog capable of representing innnite phenomena like ow of time or plan construction. Predicates in Datalog nS can have arbitrary unary and limited n-ary function symbols in one xed position. This class of logic programs is known to be decidable. However, least Herbrand models of Datalog nS programs may be innnite and consequently queries may have innnite answers. We present a method to nitely represent innnite least Herbrand models of Datalog nS programs as relational speciications. A relational speciication consists of a nite set of facts and a nitely speciied congruence relation. A relational speciication has the following desirable properties. First, it is explicit in the sense that once it is computed, the original Datalog nS program (and its underlying computational engine) can be forgotten. Given a query to be evaluated, it is easy to obtain from the relational speciication nitely many answer substitutions that represent innnitely many answer substitutions to the query. The method involved is a combination of a simple, uniication-less, computational mechanism (graph traversal, congruence closure, or term rewriting) and standard relational query evaluation methods. Second, a relational speciication is eeectively computable and its computation is no harder, in the sense of the complexity class, than answering yes-no queries. Our method is applicable to every range-restricted Datalog nS program. We also show that for some very simple non-Datalog nS logic programs, nite representations of query answers do not exist.