# Description Logics Go Second-Order -- Extending EL with Universally Quantified Concepts

arXiv: https://arxiv.org/abs/2308.08252 · HTML: https://mlsanity.com/paper/2308.08252/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 30 references that are in the map.
- First posted on arXiv 16 Aug 2023.

## Builds on (most cited first, 2 of 2)

- [Generating Ontologies from Templates: A Rule-Based Approach for Capturing Regularity](https://mlsanity.com/paper/1809.10436/) (2018)
- [Generic Ontology Design Patterns at Work](https://mlsanity.com/paper/1906.08724/) (2019)

## Abstract (arXiv)

The study of Description Logics have been historically mostly focused on features that can be translated to decidable fragments of first-order logic. In this paper, we leave this restriction behind and look for useful and decidable extensions outside first-order logic. We introduce universally quantified concepts, which take the form of variables that can be replaced with arbitrary concepts, and define two semantics of this extension. A schema semantics allows replacements of concept variables only by concepts from a particular language, giving us axiom schemata similar to modal logics. A second-order semantics allows replacement of concept variables with arbitrary subsets of the domain, which is similar to quantified predicates in second-order logic. To study the proposed semantics, we focus on the extension of the description logic EL. We show that for a useful fragment of the extension, the conclusions entailed by the different semantics coincide, allowing us to use classical EL reasoning algorithms even for the second-order semantics. For a slightly smaller, but still useful, fragment, we were also able to show polynomial decidability of the extension. This fragment, in particular, can express a generalized form of role chain axioms, positive self restrictions, and some forms of (local) role-value-maps from KL-ONE, without requiring any additional constructors.