🤖 AI Summary
This work addresses the theoretical complexity and lack of a unified, reproducible formal foundation for the chase algorithm in existential rule reasoning by developing a formal framework within the Lean theorem prover. The framework clarifies subtle discrepancies in existing chase definitions, unifies various termination criteria—including the MFA class—and, for the first time in a formal system, supports existential rules with constants while characterizing the chase result as the core model in the absence of “homomorphic matches.” Leveraging Lean’s dependent type theory and libraries such as Mathlib and CSlab, the project formally verifies key properties of the chase-generated universal model, completes an initial proof of the core model theorem, and delivers a reusable, machine-checkable formal library for future research on existential rules.
📝 Abstract
The chase is a sound, complete, but possibly non-terminating algorithm for reasoning with existential rules (aka. tuple-generating dependencies), a highly expressive knowledge representation language. Although the procedure appears simple, research on theoretical properties and optimization for practical implementations has grown to a point where verifying correctness and reproducing proofs becomes challenging and intuition can sometimes be misleading. Lean is a purely functional programming language and interactive theorem prover whose community actively develops formal libraries for mathematics (Mathlib) and computer science (CSLib). In this work, we present our own endeavor of crafting a Lean framework around existential rules and the chase. We discuss design decisions concerning the nuances of chase definitions commonly found in the literature and show how these translate into Lean. To illustrate the framework's capabilities using known results, we show that the result of a chase is a universal model and outline the formalization for proving that without so-called "alternative matches" it is even a core. Beyond existing literature, we unify sufficient chase termination conditions in the likeness of Model-Faithful Acyclicity (MFA) into a common framework while also adding support for constants in rules.