Recursive Rules with Aggregation: A Simple Unified Semantics
This work addresses a foundational problem in logic programming and knowledge representation, providing a simple and unified approach for researchers and practitioners dealing with complex reasoning tasks.
The paper tackles the challenge of defining a clear semantics for recursive logical rules with aggregation operations like count and sum, which are essential for practical applications but have lacked consensus. It introduces a unified semantics that supports different underlying assumptions and uses the usual meaning of aggregations, proving properties and demonstrating efficient inference that matches desired results on all studied examples.
Complex reasoning problems are most clearly and easily specified using logical rules, but require recursive rules with aggregation such as count and sum for practical applications. Unfortunately, the meaning of such rules has been a significant challenge, leading to many disagreeing semantics. This paper describes a unified semantics for recursive rules with aggregation, extending the unified founded semantics and constraint semantics for recursive rules with negation. The key idea is to support simple expression of the different assumptions underlying different semantics, and orthogonally interpret aggregation operations using their simple usual meaning. We present a formal definition of the semantics, prove important properties of the semantics, and compare with prior semantics. In particular, we present an efficient inference over aggregation that gives precise answers to all examples we have studied from the literature. We also apply our semantics to a wide range of challenging examples, and show that our semantics is simple and matches the desired results in all cases. Finally, we describe experiments on the most challenging examples, exhibiting unexpectedly superior performance over well-known systems when they can compute correct answers.