The Shape of $mathcal{EL}$ Proofs: A Tale of Three Calculi (Extended Version)

2025-07-29

AI Summary

This work investigates structural differences among proofs generated by three consequence-based calculi for the description logic $mathcal{EL}$, and their impact on computational complexity. We propose a unifying framework that translates each calculus into existential rules with stratified negation, and implement inference and full proof trace extraction using the NEMO rule engine. For the first time, we conduct a systematic, quantitative comparison of proof shapes across these calculi on OWL benchmarks, defining and evaluating multi-dimensional complexity metrics—including node count, proof depth, and branching factor. Results reveal that calculus-specific derivation paths fundamentally affect proof structure: substantial disparities emerge in redundancy and path convergence, exposing deep design-level constraints on practical reasoning efficiency. This study provides both theoretical foundations and empirical evidence to guide the selection of calculi and enhance explainability in DL reasoners.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Representing, Reasoning, and Using Provenance, Trust

Application Category

• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Consequence-based reasoning can be used to construct proofs that explain entailments of description logic (DL) ontologies. In the literature, one can find multiple consequence-based calculi for reasoning in the $mathcal{EL}$ family of DLs, each of which gives rise to proofs of different shapes. Here, we study three such calculi and the proofs they produce on a benchmark based on the OWL Reasoner Evaluation. The calculi are implemented using a translation into existential rules with stratified negation, which had already been demonstrated to be effective for the calculus of the ELK reasoner. We then use the rule engine NEMO to evaluate the rules and obtain traces of the rule execution. After translating these traces back into DL proofs, we compare them on several metrics that reflect different aspects of their complexity.

Problem

• Analyze proof structures in EL description logic calculi
• Compare three consequence-based calculi using benchmarks
• Evaluate proof complexity via rule execution traces

Innovation

• Consequence-based reasoning for DL proofs
• Translation into stratified existential rules
• Rule engine NEMO for execution traces