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.

The chase is a ubiquitous algorithm in database theory. However, for existential rules (aka tuple-generating dependencies), its termination is not guaranteed, and even undecidable in general. The problem of termination becomes particularly difficult for the restricted (or standard) chase, for which the order of rule application matters. Thus, decidability of restricted chase termination is still open for many well-behaved classes such as linear or guarded multi-headed rules. We make a step forward by showing that all-instances restricted chase termination is decidable in the linear multi-headed case.

The chase is a fundamental algorithm with ubiquitous uses in database theory. Given a database and a set of existential rules (aka tuple-generating dependencies), it iteratively extends the database to ensure that the rules are satisfied in a most general way. This process may not terminate, and a major problem is to decide whether it does. This problem has been studied for a large number of chase variants, which differ by the conditions under which a rule is applied to extend the database. Surprisingly, the complexity of the universal termination of the restricted (aka standard) chase is not fully understood. We close this gap by placing universal restricted chase termination in the analytical hierarchy. This higher hardness is due to the fairness condition, and we propose an alternative condition to reduce the hardness of universal termination.

We present Nemo, a toolkit for rule-based reasoning and data processing that emphasises robustness and ease of use. Nemo’s core is a scalable and efficient main-memory reasoner that supports an expressive extension of Datalog with support for datatypes, existential rules, aggregates, and (stratified) negation. Built around this core is a versatile system of libraries and applications for interfacing with several data formats and programming languages, use as a progressive web application, and IDE integration. In this system description, we present this toolkit and discuss relevant application areas in rule-based knowledge representation, knowledge graph processing, and reasoner prototyping. Our evaluation on a range of tasks from these areas demonstrates Nemo’s robust performance in comparison to state-of-the-art rule engines.

The disjunctive skolem chase is a sound, complete, and potentially non-terminating procedure for solving boolean conjunctive query entailment over knowledge bases of disjunctive existential rules. We develop novel acyclicity and cyclicity notions for this procedure; that is, we develop sufficient conditions to determine chase termination and non-termination. Our empirical evaluation shows that our novel notions are significantly more general than existing criteria.

The disjunctive restricted chase is a sound and complete procedure for solving boolean conjunctive query entailment over knowledge bases of disjunctive existential rules. Alas, this procedure does not always terminate and checking if it does is undecidable. However, we can use acyclicity notions (sufficient conditions that imply termination) to effectively apply the chase in many real-world cases. To know if these conditions are as general as possible, we can use cyclicity notions (sufficient conditions that imply non-termination). In this paper, we discuss some issues with previously existing cyclicity notions, propose some novel notions for non-termination by dismantling the original idea, and empirically verify the generality of the new criteria.