# Foundations of Declarative Data Analysis Using Limit Datalog Programs

## TL;DR

This paper introduces and analyzes fragments of an extended Datalog language with arithmetic, providing complexity results and demonstrating its applicability to declarative data analysis tasks.

## Contribution

It defines the limit Datalog_{Z} fragments, analyzes their computational complexity, and shows their usefulness for declarative data analysis applications.

## Key findings

- Fact entailment is coNExpTime-complete in combined complexity.
- Adding stability reduces complexity to ExpTime and PTime.
- Stable Datalog_{Z} can express many data analysis tasks.

## Abstract

Motivated by applications in declarative data analysis, we study $\mathit{Datalog}_{\mathbb{Z}}$---an extension of positive Datalog with arithmetic functions over integers. This language is known to be undecidable, so we propose two fragments. In $\mathit{limit}~\mathit{Datalog}_{\mathbb{Z}}$ predicates are axiomatised to keep minimal/maximal numeric values, allowing us to show that fact entailment is coNExpTime-complete in combined, and coNP-complete in data complexity. Moreover, an additional $\mathit{stability}$ requirement causes the complexity to drop to ExpTime and PTime, respectively. Finally, we show that stable $\mathit{Datalog}_{\mathbb{Z}}$ can express many useful data analysis tasks, and so our results provide a sound foundation for the development of advanced information systems.