Description logic
Description logic (DL) is the dominant knowledge representation formalism, used by both OBO and OWL (Semantic Web).
Nomenclature
DL has a self-describing naming system [Baader et al, 2007, Appendix: DL Terminology]. Unfortunately, it is not entirely consistent across the literature.
My attempt at a summary is below. See also Evgeny Zolin’s encyclopedic page on complexity of reasoning in description logics .
- \(\mathcal{A}\mathcal{L}\) [Attributive Concept Language, the most basic DL]
- Atomic concept (\(A\)), universal concept (\(\top\)), bottom concept (\(\bot\))
- Atomic negation (\(\neg A\))
- Concept intersection (\(C \cap D\))
- Value restriction (\(\forall R.C\))
- Limited existential quantification (\(\exists R.\top\))
- Concept hierarchy (\(C \subseteq D\), \(C \equiv D\))
- \(\mathcal{A}\mathcal{L}\mathcal{C}\) (equivalently \(\mathcal{A}\mathcal{L}\mathcal{U}\mathcal{E}\))
- \(\mathcal{C}\): Concept negation (\(\neg C\))
- \(\mathcal{U}\): Concept union (\(C \cup D\))
- \(\mathcal{E}\): “Full” existential quantification (\(\exists R.C\))
- \(\mathcal{H}\): Role hierarchies (\(R \subseteq S\), \(R \equiv S\))
- \(\mathcal{I}\): Inverse roles (\(R^{-}\))
- \(\mathcal{N}\): (Unqualified) number restriction (\(\geq n~R\), \(\leq n~R\), \(= n~R\))
- \(\mathcal{Q}\): Qualified number restriction (\(\geq n~R.C\), \(\leq n~R.C\), \(= n~R.C\))
- \(\mathcal{O}\): Nominals [class literals] (\(\{x_1,x_2,\ldots,x_n\}\))
- \(\mathcal{F}\): Depending on author, EITHER
- Functional roles (\(\leq 1~R\)), a restricted form of \(\mathcal{Q}\), OR
- Agreement and disagreement [Baader et al, 2007, Table A.1]
- \(\mathcal{R}\): Depending on author, EITHER
- Role intersection (\(R \cap S\)), OR
- Regular role inclusion axioms [regular RIAs] (\(R_1 \circ R_2 \circ \cdots \circ R_n \subseteq S\))
- Regular means acyclic, which ensures decidability
- May include other features: disjoint roles, reflexive, irreflexive, etc.
- \((\mathcal{D})\): Concrete domains, e.g., natural numbers with \(<,\leq,=,>,\geq\)
Languages
Literature
Books
- Baader et al, 2007: The Description Logic Handbook, 2nd ed.
- Most comprehensive reference on DL
- Covers OWL 1 (Ch. 14) but not OWL 2
- Robinson & Bauer, 2011: Introduction to Bio-ontologies
- DL from a bioinformatics perspective
- Hitzler et al, 2010: Foundations of Semantic Web Technologies
- Coverage of RDFS and OWL 2 with more mathematical formality than usual
Surveys and tutorials
- Rudolf, 2011: Foundations of description logics (doi, pdf)
- Krotzsch, Simancik, Horrocks, 2012: A description logic primer (arxiv)
DL in Semantic Web
- Horrocks, Kutz, Sattler, 2005: The irresistible SRIQ (pdf)
- Horrocks, Kutz, Sattler, 2006: The even more irresistible SROIQ (pdf)
- Knorr, Alferes, Hitzler, 2011: Local closed world reasoning with description logics under the well-founded semantics (doi, pdf)
- Sengupta, Krisnadhi, Hitzler: 2011: Local closed world semantics: Grounded circumscription for OWL (doi, pdf)
Critical perspectives on DL