# Union and Intersection of all Justifications

arXiv: https://arxiv.org/abs/2109.11216 · HTML: https://mlsanity.com/paper/2109.11216/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 46 references that are in the map.
- First posted on arXiv 23 Sep 2021.

## Builds on (most cited first, 4 of 4)

- [Axiom Pinpointing](https://mlsanity.com/paper/2003.08298/) (2020)
- [Efficient MUS Enumeration of Horn Formulae with Applications to Axiom Pinpointing](https://mlsanity.com/paper/1505.04365/) (2015)
- [Consequence-based Reasoning for Description Logics with Disjunction, Inverse Roles, Number Restrictions, and Nominals](https://mlsanity.com/paper/1805.01396/) (2018)
- [Consequence-Based Axiom Pinpointing](https://mlsanity.com/paper/1805.10250/) (2018)

## Built on by (most cited first, 2 of 2)

- [Can You Tell the Difference? Contrastive Explanations for ABox Entailments](https://mlsanity.com/paper/2511.11281/) (2025)
- [How Hard is it to Decide if a Fact is Relevant to a Query?](https://mlsanity.com/paper/2604.22422/) (2026)

## Abstract (arXiv)

We present new algorithm for computing the union and intersection of all justifications for a given ontological consequence without first computing the set of all justifications. Through an empirical evaluation, we show that our approach works well in practice for expressive DLs. In particular, the union of all justifications can be computed much faster than with existing justification-enumeration approaches. We further discuss how to use these results to repair ontologies efficiently.