Complexity and Expressive Power of Disjunction and Negation in Limit Datalog

Limit Datalog is a fragment of Datalog(Z)-the extension of Datalog with arithmetic functions over the integers-which has been proposed as a declarative language suitable for capturing data analysis tasks. In limit Datalog programs, all intensional predicates with a numeric argument are limit predicates that keep maximal (or minimal) bounds on numeric values. Furthermore, to ensure decidability of reasoning, limit Datalog imposes a linearity condition restricting the use of multiplication in rules. In this paper, we study the complexity and expressive power of limit Datalog programs extended with disjunction in the heads of rules and non-monotonic negation under the stable model semantics. We show that allowing for unrestricted use of negation leads to undecidability of reasoning. Decidability can be restored by stratifying the use of negation over predicates carrying numeric values. We show that the resulting language is Pi(EXP)(2)-complete in combined complexity and that it captures Pi(P)(2) over ordered structures in the sense of descriptive complexity. We also provide a study of several fragments of this language: we show that the complexity and expressive power of the full language are already reached for disjunction-free programs; furthermore, we show that semi-positive disjunctive programs are coNEXP-complete and that they capture coNP.