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 undecidable, 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 Δ
2
EXP
and captures Δ
2
P
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 coNEXP-complete in combined and coNP-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.