Category Archives: OWL

EquivalentTo versus SubClassOf

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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.

x inferred to be of type A1

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.

A2 and x wrt the set (a_to_b some B)

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.

Understanding OWL min vs max vs exactly Property Restrictions

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The open world assumption trips people up in many ways. In this post we will be looking at how the open world assumption affects the semantics of the property restrictions min, max and exactly.

The example we will use throughout this post is that of a product that may have no price, 1 price, exactly 1 price, or many prices. We will firstly assume a product must have at least 1 price, then that it can have a maximum of 1 price, and finally we will assume that it must have exactly 1 price. We therefore assume that we have a hasPrice data property that is defined as follows:

DataProperty: hasPrice
  Range: 
    xsd:decimal

The example ontology can be found on GitHub. To be able to distinguish the different product types having different rules regarding how many prices they have, we will create 3 different classes called ProductWith_Min_1_Price, ProductWith_Max_1_Price and ProductWith_Exactly_1_Price respectively.

Before we look at examples and the semantics of the min, max, exactly property restrictions, let us briefly recall what is meant by the open world assumption.

Open World Assumption versus Closed World Assumption

OWL has been designed with the explicit intention to be able to deal with incomplete information. Consequently, OWL intentionally does not make any assumptions with regards to information that is not known. In particular, no assumption is made about the truth or falsehood of facts that cannot be deduced from the ontology. This approach is known as the open world assumption. This approach is in contrast with the closed world assumption typically used in information systems. With the closed world assumption facts that cannot be deduced from a knowledge base (i.e. database) are implicitly understood as being false [1, 2, 3].

As an example, in a database when a product does not have a price, the general assumption is that the product does not have price. Moreover, in a database if a product has 1 price, the assumption is that the product has only 1 price and no other prices. This in stark contrast to OWL. If an OWL ontology defines a product for which no explicit price is given, the assumption is not that the product has no price. Rather, no assumption is made as to whether the product has a price, has many prices or whether it has no price. Futhermore, if a product has a price, the assumption is not that this is necessarily the only price for that product. Rather, it allows for the possibility that no other price may exist, or that many other prices may exist, which is merely not known. The only information that holds in an ontology is information that is either explicitly stated, or that can be derived form explicit information.

The min Property Restriction

To define a product that must have at least 1 price we define it as follows:

Class: ProductWith_Min_1_Price
  SubClassOf: 
    hasPrice min 1 xsd:decimal
  
Individual: productWithoutPrice
  Types:  
    ProductWith_Min_1_Price

If we now create an individual of type ProductWith_Min_1_Price, say productWithoutPrice, which has no price information, we will find that the reasoner will not give an inconsistency. The reason for this is that the reasoner has no information with regards to whether productWithoutPrice has any price information. Hence, it is possible
that productWithoutPrice has a price (or prices) that is merely unknown. To make explicit that productWithoutPrice has no price information, we can define it as follows:

Individual: productWithoutPrice
  Types:  
    ProductWith_Min_1_Price,
    hasPrice max 0 xsd:decimal

This revised definition of productWithoutPrice will now result in the reasoner detecting an inconsistency. However, note that ProductWith_Min_1_Price allows for products
that have more than 1 price. Hence, the following will not result in an inconsistency.

Individual: productWithManyPrices
  Types:  
    ProductWith_Min_1_Price
  Facts:  
    hasPrice 2.5,
    hasPrice 3.25

The max Property Restriction

To define a product that cannot have more than 1 price, we can define it as
follows:

Class: ProductWith_Max_1_Price
  SubClassOf: 
    hasPrice max 1 xsd:decimal

If we now define an individual productWithMoreThan1Price with more than 1 price (as shown in the example below), the reasoner will detect an inconsistency.

Individual: productWithMoreThan1Price
  Types: 
    ProductWith_Max_1_Price
  Facts:  
    hasPrice 2.5,
    hasPrice 3.25

Note that individuals of type ProductWith_Max_1_Price can also have no price information without resulting in the reasoner giving an inconsistency. I.e., if we define the individual productWithoutPrice as

Individual: productWithoutPrice
  Types:  
    ProductWith_Max_1_Price,
    hasPrice max 0 xsd:decimal

it will not give an inconsistency.

The exactly Property Restriction

Let us now define ProductWith_Exactly_1_Price with the individual productWithExactly1Price as follows:

Class: ProductWith_Exactly_1_Price
  SubClassOf: 
    hasPrice exactly 1 xsd:decimal    
    
Individual: productWithExactly1Price
  Types: 
    ProductWith_Exactly_1_Price  
  Facts:
    hasPrice 7.1

The exactly property is essentially syntactical shorthand for specifying both the min and max restrictions using the same cardinality. Thus, we could just as well have defined ProductWith_Exactly_1_Price as:

Class: ProductWith_Exactly_1_Price
  SubClassOf: 
     hasPrice min 1 xsd:decimal,
     hasPrice max 1 xsd:decimal

or, given the classes we have already defined in the ontology, we can define it as:

Class: ProductWith_Exactly_1_Price
  SubClassOf: 
    ProductWith_Max_1_Price,
    ProductWith_Min_1_Price

Prefer exactly

Given that the exactly property restriction is syntactical sugar, should we prefer using the combination of min and max directly as shown above? My answer to this is no. My motivation for this is that the semantics of exactly is only equivalent to the intersection of min and max if the cardinalities are the same and the data/object types are the same. As such specifying

Class: ProductWith_Exactly_1_Price
  SubClassOf: 
    hasPrice exactly 1 xsd:decimal

has less opportunities for mistakes than specifying 

Class: ProductWith_Exactly_1_Price
  SubClassOf: 
    hasPrice min 1 xsd:decimal,
    hasPrice max 1 xsd:decimal

Conclusion

In this post we looked at some of the ways in which the min, max and exactly property restrictions can trip people up due to the open world assumption. Please feel free to leave a comment if you have questions or suggestions about this post.

References

1. F. Baader and W. Nutt, Basic Description Logics, The Description Logic Handbook: Theory, Implementation and Applications (F. Baader, D. Calvanese, D. L. McGuinness, D. Nardi, and P. Patel-Schneider, eds.), Cambridge University Press, New York, USA, 2007, pp. 45–104.

2. M. Krötzsch, F. Simančı́k, and I. Horrocks, A Description Logic Primer, Computing Research Repository (CoRR) abs/1201.4089 (2012).

3. S. Rudolph, Foundations of Description Logics, Proceedings of the 7th International Conference on Reasoning Web: Semantic Technologies for the Web of Data (A. Polleres, C. d’Amato, M. Arenas, S. Handschuh, P. Kroner, S. Ossowski, and P. F. Patel-Schneider, eds.), Lecture Notes in Computer Science, vol. 6848, Springer, 2011, pp. 76–136.

A Common Misconception regarding OWL Properties

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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

  1. I explain how this misconception fails,
  2. I explain what is the meaning of OWL properties, and
  3. 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:

  1. 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 Employer ontology 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 blueCheese is not of type Employer and neither is the individual companyFridge an 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.

  2. 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 Employer ontology it means that instances of the Employer class must necessarily be linked via the employs property to an instance (or instances?) of Employee. Thus, the expectation is that if we have an instance Employer that is not linked to an instance of Employee via the employs property, 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 Employer and companyFridge is an instance of Employee.

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:

DomainRangeAxiom whereTopis the set representing the application domain. An object property r between instances a and b is expressed as a role assertion
RoleAssertion 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

Conclusion

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.