Decidability and undecidability results for boundedness of linear recursive queries

If it is possible to eliminate recursion from a Datalog program P, then P is said to be bounded. It was shown by Gaifman et al that the problem of deciding whether a given Datalog program is bounded is undecidable, even for linear programs that has one 4-ary intensional predicate. We sharpen that result by showing that the problem of deciding whether a given Datalog program is bounded is undecidable, even for linear programs that has one binary intensional predicate. We then consider linear programs with a single recursive rule. We show that if the intensional predicate is binary, then the boundedness problem for such program is decidable, in fact, it is NP-complete.