Undecidable Boundedness Problem For Datalog Program

A given Datalog program is bounded if its depth of recursion is independent of the input database. Deciding boundedness is a basic task for the analysis of database logic programs. The undecidability of Datalog boundedness was first demonstrated by Gaifman et al. We introduce new techniques for proving the undecidability of various kinds of boundedness, which allow us to considerably strengthen the results of Gaifman et al. In particular: (1) We use a new generic reduction technique to show that program boundedness is undecidable for arity 2 predicates, even with linear rules. (2) We use the mortality problem of Turing machines to show that uniform boundedness is undecidable for arity 3 predicates and for arity 1 predicates when $\ne$ is also allowed. (3) We use a single rule polymorphically to show that program (resp., predicate) boundedness is undecidable for two linear rules (resp., one rule and a projection), one initialization rule and small arity predicates.