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Introduction to ontology semantics and reasoning
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.
- 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.
A Common Misconception regarding OWL Properties
A misconception w.r.t. OWL properties that I come across from time-to-time is that people mistakenly think that properties express relations between classes rather than relations between individuals. In this post
- I explain how this misconception fails,
- I explain what is the meaning of OWL properties, and
- I explain how you can model relations between classes if that is really what you want to do.
Throughout this post I will refer to object properties only, even though what I say applies to data properties as well, except of course that object properties relate objects to objects whereas data properties relate objects to data type values.
As an example ontology I will use the following:
Class: Employer Class: Employee ObjectProperty: employs Domain: Employer Range: Employee
How thinking that Properties express Relations between Classes fails
Thinking that properties express relations between classes fails in two different ways:
- The first way in which the expectations of users fail under this misconception is that their assumption is that domain and range axioms behave as constraints. That is if we extend our
Employerontology with something nonsensical like
Class: Cheese Class: Fridge Individual: companyFridge Types: Fridge Individual: blueCheese Types: Cheese Facts: employs companyFridge
the reasoner will indicate that our ontology is inconsistent because the individual
blueCheeseis not of type
Employerand neither is the individual
companyFridgean instance of
Employee. However, this is not the case. This ontology is in fact consistent because domain and range axioms do not behave as constraints.
- The other way in which the assumption of users is shattered is that they tend to think that domain and range axioms mean that in our
Employerontology it means that instances of the
Employerclass must necessarily be linked via the
employsproperty to an instance (or instances?) of
Employee. Thus, the expectation is that if we have an instance
Employerthat is not linked to an instance of
employsproperty, the reasoner should give an inconsistency. Again, this is wrong. This ontology will still be consistent.
The Real Meaning of OWL Object Properties
The OWL specification is very explicit about the meaning of object properties. It states:
Object properties connect pairs of individuals.
So what is the meaning of domain and range axioms then? Domain and range axioms are not constraints to be checked, but rather they are axioms from which the reasoner can make inferences. What domain and range axioms state is that whenever two instances are linked via the
employs property, for example, it means that the first instance is of type
Employer and the second instance is of type
Employee. Thus, in our cheese and fridge example, no matter how silly it is, the reasoner will infer that
blueCheese is an instance of
companyFridge is an instance of
Finally, the underlying mathematical formalization of object properties themselves does not enable linking of pairs of classes. An OWL object
r is represented as a role r in description logics. Stating that the role r has the domain C and the range D, where C and D are concepts, is achieved through the following axioms:
whereis the set representing the application domain. An object property
r between instances
b is expressed as a role assertion
in description logics. Note that this assertion contains no information regarding description logic concepts (i.e. classes in OWL).
If you really, really want to have a Link between Classes
So if you really, really want to express a relation between classes, how can this be done? For our
Employer ontology we can state the following:
Class: Employer SubClassOf: employs some Employee
This states that the
Employer class is a subclass of the class consisting of the instances that are linked to at least 1 instance of the
Employee class. If we now have an instance of
Employer that is not linked via
employs to an instance of
Employee, the reasoner will find that our ontology is inconsistent. To state that
acme is an employer without employees, we state it as follows:
Individual: acme Types: Employer Facts: employs max 0 Employee
In this post I explained that object properties link individuals (not classes!) and that thinking otherwise can lead to various errors when designing an ontology.