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# Category Archives: The mathematics of object orientation

## The Correspondence between DLs/OWL and OO

The analogy between DLs and object-orientation can be observed when it is considered that the basic task in constructing an ontology is classification. Explicit subsumption relationships between concepts can be defined in the TBox. In object-orientation this can be achieved by definition of an inheritance hierarchy between classes. Classification is further solidified as the basis of DLs in that the core reasoning capabilities they provide are subsumption and instance checking. Subsumption computes a subsumption hierarchy, which essentially categorizes concepts into superconcept/subconcept relationships. Instance checking verifies whether a given individual is an instance of a specific concept [1].

In object-orientation the domain of interest is described in terms of classes that have properties, which are defined via attributes and/or associations. Classes in essence have a set-theoretic semantics, i.e. a class represents a set of objects in the domain of interest which shares attributes. Objects that are classified by a class are called instances of the class. The analogy with DLs is that classes, attributes/associations and instances (or sometimes called objects) correspond respectively with concepts, roles and individuals in DLs, which in OWL corresponds respectively to classes, properties and individuals.

This correspondence between object orientation, DLs and OWL 2 is summarized in the table below.

Object orientation DLs OWL 2
Class Concept Class
Attribute/association Role Property
Object Individual Individual

### Bibliography

[1] F. Baader, D. Calvanese, D. L. McGuinness, D. Nardi and P. F. Patel-Schneider, The Description Logic Handbook: Theory, Implementation and Applications, Cambridge University Press, 2007.

## Theoretical Basis of Object-oriented Analysis

Classification is the core activity of object-oriented analysis. Classification is the means via which people order knowledge according to the similarities they recognize between different objects they observe in the world. The specific classification approach that is applied when doing object-oriented analysis is called classical categorization. Classification, in specific classical categorization, does not pertain to object-orientation alone, but rather, it reflects how people think in general about the world [1] [2] [3]. Olive explains the need and use of classification as follows [2]:

Classification provides cognitive economy because it allows us to structure knowledge about objects into two levels: concept and instance. At the concept level, we find the properties (both defining and nondefining) common to all instances of the concept. At the instance level, we find only the concept of which the object is an instance, and the particular properties of that instance. In the absence of classification, we would have to associate every instance with all of its properties. Classification reduces the amount of information we have to remember, communicate, and process; the extent to which it is reduced depends on the number of properties of the concept.

It is precisely this cognitive economy, provided by a complete and consistent object-oriented conceptual schema, that is the essence of enabling efficient communication between stakeholders of software development projects. When software development projects decide to forgo the creation of conceptual schemas, it is at the cost of efficient communication.

Notice that classification is inherently set theoretic. Sets have a characteristic function that essentially determines from a possible universe of elements, which elements belong to the set. With classification commonalities between instances are recognized which cause us to classify these instances as belonging to the same concept. In this way concepts and sets are equivalent.

#### Bibliography

[1] G. Booch, R. A. Maksimchuk, M. W. Engel, B. J. Young, J. Conallen and K. A. Houston, Object-oriented analysis and design with applications, Addison-Wesley Professional, 2007.

[2] A. Olive, Conceptual modeling of information systems, Springer, 2007.

[3] G. Lakoff, Women, fire and dangerous things: what categories reveal about the mind, Chicago: University of Chicago Press, 1990.

## What are Description Logics?

Description logics (DLs) are syntactic variants of first-order logic that are specifically designed for the conceptual representation of an application domain in terms of concepts and relationships between concepts [1].

Expressions in DLs are constructed from atomic concepts (unary predicates), atomic roles (binary predicates) and individuals (constants). Complex expressions can be built inductively from these atomic elements using concept constructors. Formally a concept represents a set of individuals and a role a binary relation between individuals [2].

Formally every DL ontology consists of a set of axioms that are based on finite sets of concepts, roles and individuals. Axioms in a DL ontology are divided into the TBox, the RBox and the ABox. A TBox is used to define concepts and relationships between concepts (that is the terminology or taxonomy) and an ABox is used to assert knowledge regarding the domain of interest (i.e. that an individual is a member of a concept). Depending on the expressivity of the DL used, an ontology may include an RBox. An RBox is used to define relations between roles as well as properties of roles [2].

A feature of DLs is that they have decidable reasoning procedures for standard reasoning tasks.  This means these reasoning procedures will give an answer, unlike undecidable reasoning procedures which may not terminate and thus may not give an answer.  A fundamental goal of DL research is to preserve decidability to the point that decidability is considered to be a precondition for claiming that a formalism is a DL. Standard DL reasoning algorithms are sound and complete and, even though the worst-case computational complexity of these algorithms is ExpTime and worse, in practical applications they are well-behaved [3].

Standard reasoning procedures for DLs are the following [2].

• Satisfiability checking checks that every axiom in an ontology can be instantiated. Axioms that cannot be instantiated indicates that modelling errors exist within the ontology.
• Consistency checking checks whether there are axioms that contradict each other, which again is indicative of modelling errors.
• Subsumption checking checks whether an axiom subsumes another axiom, which is used for classifying axioms into a parent-child taxonomy.

Various DLs exist with different levels of expressivity and computational complexity. The most widely supported DL is SROIQ(D) which forms the mathematical basis of the W3C OWL 2 standard [4]. In OWL concepts are referred to as classes, roles are referred to as properties and individuals are still referred to as individuals.

In subsequent posts I will provide an intuitive understanding of OWL 2 and explain some of its uses. If you are using OWL or other semantic technologies, I will love to hear from you. Please leave a comment and feel free to explain the novel ways in which you use semantic technologies.

### Bibliography

[1] D. Berardi, D. Calvanese and G. De Giacomo, “Reasoning on UML class diagrams,” Artificial Intelligence, vol. 168, no. 1-2, p. 70–118, 2005.

[2] F. Baader, D. Calvanese, D. L. McGuinness, D. Nardi and P. F. Patel-Schneider, The Description Logic Handbook: Theory, Implementation and Applications, Cambridge University Press, 2007.

[3] F. Baader, “What’s new in Description Logics,” Informatik-Spektrum, vol. 34, no. 5, p. 434–442, 2011.

[4] W3C, “OWL 2 Web Ontology Language – Document Overview (Second Edition),” W3C, 11 December 2012. [Online]. Available: https://www.w3.org/TR/owl2-overview/. [Accessed 9 September 2017].