I recently had the pleasure to present at the OntoSpot meeting at EBI to help my colleagues gain an intuitive understanding of ontology semantics and reasoning. In this talk I assume that you have a very basic understanding of what an ontology is, but I assume no previous knowledge wrt logic. I provide a number of examples and graphics to explain logic and description logic (DL) concepts.

Here I provide both the slides of this presentation and the link to the recording. If you have any questions or suggestions, please let me know in the comments. I have already had the very helpful suggestion for adding a reference of DL symbols, which I will do shortly.

Errata:

In the section on speaking about propositional logic, I accidentally said predicate logic instead of propositional logic.

At the end while answering questions, I said RFD rather than RDF.

This video will also be made available at the OBO Academy.

In creating their first OWL ontology, there are at least two aspects of EquivalentTo and SubClassOf that perplex users. The first is when to use EquivalentTo and when to use SubClassOf. The second problem is best illustrated by the following example:

ObjectProperty: a_to_b
Class: A1
EquivalentTo: (a_to_b some B)
Class: A2
SubClassOf: (a_to_b some B)
Class: B
Individual: b1
Types:
B
Individual: x
Facts:
a_to_b b1

When running a reasoner on this example, the individual x is inferred to be of type A1. What perplex users sometimes is that x is not inferred to be of type A2 as well. This is shown in the next figure.

The difference between EquivalentTo and SubClassOf

The first thing to be aware of wrt equivalentTo is that

Class: C
EquivalentTo: D

is an abbreviation for

Class: C
SubClassOf: D
Class: D
SubClassOf: C

The semantics of SubClassOf is subset. Thus, the above states that the set C is a subset of the set D and the set D is a subset of the set C. Which means that the sets C and D are exactly the same set. We say they are equivalent.

Note that if I know that the classes C1 and C2 are both subclasses of class C, there is nothing more I can say about how class C1 relates to class C2. This is a bit like knowing that bicycles and trucks are both vehicles – I can say nothing more about how bicycles relate to trucks beyond knowing that they are both vehicles.

Back to our initial example

Understanding the semantics of EquivalentTo we can see that indeed the individual x is an instance of A1. Understanding the semantics of SubClassOf helps us to understand why x is not inferred to be of type A2. We know that A2 is a subclass of a_to_b some B and that x is an instance of a_to_b some B, but there is nothing that can force the reasoner to infer that x is necessarily an instance of the class A2. This is illustrated in the next figure.

When to use EquivalentTo versus SubClassOf

EquivalentTo is used for definitions. That is when you want to state the necessary and sufficient conditions for a concept.

SubClassOf is used when you want to define a hierarchy from the most general to the most specific. I.e., it is typically what you see in taxonomies or in object oriented programming languages where one can define class hierarchies. In fact there is a strong relation between OWL 2 ontologies and object orientation which I explore here in more detail.

Conclusion

In this post I explained the difference between EquivalentTo versus SubClassOf and how they are used, as well as some inferences thatmay be confusing to new users. You can find the example ontology on GitHub.