Complexity of Arithmetic in Warded Datalog+-

TL;DR

This paper introduces a new, expressive Warded Datalog+- language extended with arithmetic, proving its P-completeness and providing an efficient reasoning algorithm, thus enabling advanced reasoning in AI systems like Knowledge Graphs.

Contribution

It defines a novel arithmetic-extended Warded Datalog+- language, proves its computational complexity, and offers an efficient reasoning algorithm, filling a key gap in logic-based data analytics.

Findings

The new language is P-complete.

An efficient reasoning algorithm is developed.

Descriptive complexity results are established.

Abstract

Warded Datalog+- extends the logic-based language Datalog with existential quantifiers in rule heads. Existential rules are needed for advanced reasoning tasks, e.g., ontological reasoning. The theoretical efficiency guarantees of Warded Datalog+- do not cover extensions crucial for data analytics, such as arithmetic. Moreover, despite the significance of arithmetic for common data analytic scenarios, no decidable fragment of any Datalog+- language extended with arithmetic has been identified. We close this gap by defining a new language that extends Warded Datalog+- with arithmetic and prove its P-completeness. Furthermore, we present an efficient reasoning algorithm for our newly defined language and prove descriptive complexity results for a recently introduced Datalog fragment with integer arithmetic, thereby closing an open question. We lay the theoretical foundation for highly expressive Datalog+- languages that combine the power of advanced recursive rules and arithmetic while guaranteeing efficient reasoning algorithms for applications in modern AI systems, such as Knowledge Graphs.