The Complexity and Expressive Power of Limit Datalog

Motivated by applications in declarative data analysis, in this article, we study Datalog(Z)-an extension of Datalog with stratified negation and arithmetic functions over integers. This language is known to be undccidable, so we present the fragment of limit Datalog(Z) programs, which is powerful enough to naturally capture many important data analysis tasks. In limit Datalog(Z), all intensional predicates with a numeric argument are limit predicates that keep maximal or minimal bounds on numeric values. We show that reasoning in limit Datalog(Z )is decidable if a linearity condition restricting the use of multiplication is satisfied. In particular, limit-linear Datalog(Z )is complete for Delta(EXP)(2) and captures Delta(P)(2) over ordered datasets in the sense of descriptive complexity. We also provide a comprehensive study of several fragments of limit-linear Datalog(Z). We show that semi-positive limit-linear programs (i.e., programs where negation is allowed only in front of extensional atoms) capture coNP over ordered datasets; furthermore, reasoning becomes coN EXP-complete in combined and coN P-complete in data complexity, where the lower bounds hold already for negation-free programs. In order to satisfy the requirements of data-intensive applications, we also propose an additional stability requirement, which causes the complexity of reasoning to drop to EXP in combined and to P in data complexity, thus obtaining the same bounds as for usual Datalog. Finally, we compare our formalisms with the languages underpinning existing Datalog-based approaches for data analysis and show that core fragments of these languages can be encoded as limit programs; this allows us to transfer decidability and complexity upper bounds from limit programs to other formalisms. Therefore, our article provides a unified logical framework for declarative data analysis which can be used as a basis for understanding the impact on expressive power and computational complexity of the key constructs available in existing languages.