Compatibility Formalization Between PR-OWL and OWL

Abstract

As stated in [5], a major design goal for PR-OWL was to attain compatibility with OWL. However, this goal has been only partially achieved as yet, primarily due to several key issues not fully addressed in the original work. This paper describes several important issues of compatibility between PR-OWL and OWL, and suggests approaches to deal with them. To illustrate the issues and how they can be addressed, we use procurement fraud as an example application domain [2]. First, we describe the lack of mapping between PR-OWL random variables (RVs) and the concepts de ned in OWL, and then show how this mapping can be done. Second, we describe PR-OWL's lack of compatibility with existing types already present in OWL, and then show how every type de ned in PR-OWL can be directly mapped to concepts already present in OWL.

Keywords

• OWL
• PR-OWL
• MEBN
• probabilistic ontology
• semantic web
• compatibility

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The Semantic Web (SW) is predicated upon radical notions of information sharing, which include [ 1 ]: (i) the Anyone can say Anything about Any topic (AAA) requirement; (ii) the open world assumption, i.e. there may exist more information of which we are not aware, and (iii) nonunique naming, meaning that di erent people can assign di erent names to the same concept. The Semantic Web (SW) di ers from the document web in that it is intended to provide not only information sharing, but also knowledge synergy. We call such an environment characterized a Radical Information Sharing (RIS) environment. While the SW promises great power and exibility, RIS environments present fundamental challenges, and can lead to chaos, disagreement and con ict.

The challenge facing SW architects is therefore to avoid the natural chaos to which RIS environments are prone, and move to a state characterized by information sharing, cooperation and collaboration. According to [ 1 ], one solution to this challenge lies in modeling. Modeling is the process of organizing information for community use. Modeling supports information sharing in four ways: (1) It provides a framework for human communication; (2) it provides a means for explaining conclusions; (3) it provides a basis for formalization and automation of reasoning; and (4) it provides a structure for managing varying viewpoints.

There is an immense variety of modeling approaches. Di erent approaches and processes are supported by di erent modeling languages. One of special interest to this research is the Web Ontology Language (OWL) [ 18, 9 ]. OWL was developed with the aim of enabling achievement of the full SW potential. According to [ 18 ] OWL is intended for use when the information contained in documents needs to be processed by applications, as opposed to situations in which the content need only be presented to humans. OWL can be used to explicitly and formally represent the meaning of terms in vocabularies and the relationships between those terms. This representation of terms and their interrelationships is called an ontology.

One of the rst de nitions of ontology in the context of the Semantic Web was given by Thomas Gruber [ 10 ].

An ontology is an explicit speci cation of a conceptualization. The term is borrowed from philosophy, where an Ontology is a systematic account of Existence. For Arti cial Intelligence (AI) systems, what \exists" is that which can be represented. A conceptualization is an abstract, simpli ed view of the world that we wish to represent for some purpose. Every knowledge base, knowledge-based system, or knowledge-level agent is committed to some conceptualization, explicitly or implicitly.

In the past few years, as the Semantic Web community has developed standards and more complex use cases, the need for principled approaches for representing and reasoning under uncertainty has received increasing appreciation. As a consequence, the World Wide Web Consortium (W3C) created the Uncertainty Reasoning for the World Wide Web Incubator Group (URW3-XG) in 2007 to identify requirements for reasoning with and representing uncertain information in the World Wide Web. The work of the URW3-XG provided an important beginning for characterizing the range of uncertainty that a ects reasoning on the scale of the World Wide Web, and the issues to be considered in designing a standard representation of that uncertainty. However, the work to date likely falls short of what would be needed to charter an e ort to develop that representation. A candidate representation for uncertainty reasoning in the semantic web is Probabilistic OWL (PR-OWL) [ 5 ], an OWL upper ontology for representing probabilistic ontologies based on Multi-Entity Bayesian Networks (MEBN) [ 15 ].

As stated in [ 5 ], a major design goal for PR-OWL was to attain compatibility with OWL. However, this goal has been only partially achieved as yet, due to several key issues not fully addressed in the original work. First, there is no mapping in PR-OWL to properties of OWL. Second, although PR-OWL has the concept of meta-entities, which allows the de nition of complex types, it lacks compatibility with existing types already present in OWL.

These problems have been noted in the literature [20]:

PR-OWL does not provide a proper integration of the formalism of

MEBN and the logical basis of OWL on the meta level. More speci cally, as the connection between a statement in PR-OWL and a statement in OWL is not formalized, it is unclear how to perform the integration of ontologies that contain statements of both formalisms.

This paper is structured as follows. Section 2 brie y describes PR-OWL and its underlying logic, MEBN. Section 3 presents PR-OWL's lack of mapping to OWL and our suggested solution to the problem. Section 4 presents the lack of compatibility between types in OWL and PR-OWL and describes how they could be integrated. Finally, Section 5 presents the conclusion of the proposed extension to PR-OWL language. 2

PR-OWL and MEBN Logic

Ontologies are becoming increasingly popular as a means to ensure formal semantic support for knowledge sharing [ 3, 4, 7, 8, 13, 22 ]. Representing and reasoning with uncertainty is becoming recognized as an essential capability in many domains. The nave approach of simply annotating ontologies with numerical probabilities is inadequate, because it cannot capture complex relational probabilistic dependencies. More expressive representation formalisms are needed [ 16 ].

Probabilistic Ontologies [ 5, 6 ] have been proposed as a more expressive formalism for representing knowledge in domains characterized by uncertainty. The PR-OWL probabilistic ontology language [ 5, 6 ] has its logical basis in MultiEntity Bayesian Networks (MEBN), an extension of Bayesian networks (BNs) to achieve rst-order expressive power [ 14, 15 ]. MEBN represents knowledge as a collection of MEBN Fragments (MFrags), which are organized into MEBN Theories (MTheories). Figure 1 presents the main concepts needed to de ne an MTheory in PR-OWL. In the diagram, the ellipses represent the general classes, while the arcs represent the main relationships among the classes.

An MFrag contains random variables (RVs) and a fragment graph representing dependencies among these RVs. An MFrag is a template for a fragment of a Bayesian network. It is instantiated by binding its arguments to domain entity identi ers to create instances of its RVs. There are three kinds of RV: context, resident and input. Context RVs represent conditions that must be satis ed for the distributions represented in the MFrag to apply. Input nodes represent RVs that may in uence the distributions de ned in the MFrag, but whose distributions are de ned in other MFrags. Distributions for resident RV instances are de ned in the MFrag. Distributions for resident RVs are de ned by specifying local distributions conditioned on the values of the instances of their parents in the fragment graph.

A set of MFrags represents a joint distribution over instances of its random variables. MEBN provides a compact way to represent repeated structure in a BN. An advantage of MEBN is that there is no xed limit on the number of RV instances, and the random variable instances can be dynamically instantiated as needed.

An MTheory is a set of MFrags that satis es conditions of consistency ensuring the existence of a unique joint probability distribution over its random variable instances.

To apply an MTheory to reason about particular scenarios, one needs to provide the system with speci c information about the individual entity instances involved in the scenario. Throughout the remainder of the paper, we use procurement fraud as an example application domain [ 2 ]. On receipt of information about a particular procurement scenario, Bayesian inference can be used both to answer speci c questions of interest (e.g., how likely is it that a particular procurement is being directed to a speci c enterprise?) and to re ne the MTheory (e.g., each new investigation provides additional statistical data about relevant indicators for a given category of fraud). Bayesian inference is used to perform both problem speci c inference and learning in a sound, logically coherent manner (for more details see [ 15, 17 ]). 3

Mapping PR-OWL Random

Concepts

Variables to OWL

Suppose we have an OWL ontology for the public procurement domain. The ontology de nes concepts such as procurement, winner of a procurement, members of a committee responsible for a procurement, etc. Figure 2 shows a light-weight ontology for this domain represented in Uni ed Modeling Language (UML) [ 21 ].

Now, imagine we want to de ne some uncertain relations about this domain, e.g. it is common to identify a front for an enterprise by looking at his/her income and the value of a procurement the enterprise he/she represents won, meaning, if the enterprise won a procurement of millions of dollars, but the responsible person for this enterprise makes less than 10 thousand dollars a year, it is likely that this person is a front. Figure 3 shows this probabilistic relation de ned using PR-OWL in an open-source tool for probabilistic reasoning, UnBBayes.

As expected, we would need to ensure some conditions were met in order to make assertions about this probabilistic relationship. One of these conditions is that the person we are trying to determine as a possible front has to be responsible for the enterprise we are analyzing.

It is natural to think that the data we have about this domain would be associated with the ontological markups de ned in OWL. In other words, our database would have instances of persons and enterprises, and these instances would be linked to their semantic meaning de ned in the OWL ontology.

Accessing this information should be trivial once the de nitions in the ontology were made available and permission was granted to retrieve data from the database. However, this can only be achieved by developing a link between PR-OWL random variables (RVs) and the concepts de ned in OWL. In its current state, though, the relations de ned in PR-OWL are not formally linked to the relevant concepts in the OWL procurement ontology. That is, the relation IsResponsibleFor should be linked to the OWL concepts representing persons and enterprises.

From this simple example, it is clear that every probabilistic de nition involving a concept must keep a reference to its semantic de nition. In other words, full compatibility with OWL requires modi cations to PR-OWL that guarantee the preservation of OWL's semantics.

Figure 4 shows a suggested approach to map concepts in OWL to random variables in PR-OWL. In this case, the relation is represented by a class (see [ 11 ] for details on how to de ne n ary relations in OWL) named HasContract which represents a 4-ary relation that relates a contract that has a public agency as a contractor, has its origin in a procurement, and has an enterprise contracted. This relation is mapped as a predicate because in this example it is possible to have more than one contracted enterprise for the same contract.

The idea is to keep a reference to the main relation (OWL concept) when creating its probabilistic de nition (PR-OWL random variable). In this case, the random variable HasContract is a function that de nes the probabilistic characteristics of the concept HasContract, in this case a class that has a role of a relation as explained above. As it can be seen on Figure 4 the range of the random variable is a boolean.

However, it is not enough to map a PR-OWL random variable to an OWL concept whose probabilistic characteristics are being de ned. It is also necessary to map the arguments of the random variable to their respective classes or data types in OWL. In this example we have that: the argument contract is mapped to the class Contract, which is the range of the property hasContract ; the argument contracted is mapped to the class Enterprise, which is the range of the property hasContracted ; the argument origin is mapped to the class Procurement, which is the range of the property hasOrigin; and the the argument contractor is mapped to the class PublicAgency, which is the range of the property hasContractor.

Finally, in First Order Logic (FOL) the range of any n-ary predicate is a boolean. However, due to the lack of n-ary relations in OWL, this predicate was modeled in PR-OWL as a class. Therefore, it has no de ned range. In fact, the only possible value of a class is an instantiation. So there is one last mapping to be done.

We need to map the existence of an instance to a random variable with value true. This is a one to one mapping. I.e., there is only one RV that describes the uncertainty of an OWL instance and there is only one OWL instance that describes the semantics of a RV. If there is no such instance, then the value of the RV is either false (if we are assuming a closed world) or unknown (if we are assuming an open world). The fact is that we need to have an extra parameter in our RV to state to which instance of the class HasContract it is related to.

So, once the random variable is instantiated as a node in a Bayesian network for a speci c situation, it is necessary to maintain the mapping we had in our PR-OWL random variable. Figure 5 shows a suggestion for how to perform this mapping. In this example, as we have an OWL assertion that HasContract1 is a predicate that states that the Contract1 contract had origin in Procurement1, has IRS as its contractor, and has contracted ITEnterprise, we have the PR-OWL counterpart stating that the node HasContract ITEnterprise Procurement1 IRS Contract1 has the state True with probability 100%. The actual mapping between the OWL instance and the node is kept because the node is in fact an instance of the random variable de ned in PR-OWL, which in turn is an OWL class. As the PR-OWL random variable has the mapping, so does its instance.

The mapping described in this section provides the basis for a formal de nition of consistency between a PR-OWL probabilistic ontology and an OWL ontology, in which rules in the OWL ontology correspond to probability one assertions in the PR-OWL ontology. A formal notion of consistency can lead to development of consistency checking algorithms. 4

Extending PR-OWL to Use OWL's Types

One of the main concerns when developing OWL [ 12 ] was to keep the same semantics of its predecessors, RDF and XML, which meant reusing all the concepts already de ned in those languages, including primitive types, such as string, boolean, decimal, etc. On the other hand, PR-OWL does not make use of the primitive types used in OWL. For instance, PR-OWL de nes Boolean as an individual of the class MetaEntity, as shown in Figure 6, but does not keep any relation to the boolean type used in OWL.

If we wanted to de ne a continuous random variable for the annual income of a person in PR-OWL, we would need to de ne the real numbers, even though they are already de ned in OWL. Moreover, concepts that use this primitive type in OWL would not be understood in PR-OWL, in other words, they lack compatibility as far as primitive types are concerned.

Figure 7 shows the di erent types of entities de ned in PR-OWL. A possible approach to keep OWL's semantics is to avoid de ning new types of entities and use what is already available in OWL. For instance, the class ObjectEntity can be substituted by the OWL class Thing, after all, according to [ 5 ] ObjectEntity aggregates the MEBN entities that are real world concepts of interest in a domain. They are akin to objects in Object-Oriented (OO) models and to frames in frame-based knowledge systems. In other words, they are nothing more than OWL classes.

According to [ 5 ] CategoricalRVState is used to represent a list of mutually exclusive, collectively exhaustive states, which in turn are possible states of random variables, represented by nodes in PR-OWL. Therefore, it can be replaced by DataOneOf if it needs to enumerate data types or ObjectOneOf if it needs to enumerate objects. These concepts allow the enumeration of literals and individuals, respectively (see [ 19 ] for more details).

BooleanRVState can be replaced by the boolean data type present in OWL. Finally, the MetaEntity class, which includes all the entities that convey speci c de nitions about entities (e.g. typelabels that name the possible types of entities), can be eliminated since all other entities were replaced by a concept already present in OWL. 5

Conclusion

We described the main issues with PR-OWL probabilistic ontology language with respect to its compatibility with the OWL ontology language and presented possible approaches to deal with these issues.

The rst issue described was the lack of mapping between PR-OWL random variables (RVs) and the concepts de ned in OWL. In its current state, though, the relations de ned in PR-OWL are not formally linked to concepts in OWL. We have shown through an example how this mapping can be done.

The second issue described was that PR-OWL does not make use of the primitive types used in OWL, as OWL did with respect to RDF and XML. For this reason, concepts already de ned in one language must be rede ned in the other. We have shown that every type de ned in PR-OWL can be directly mapped to concepts already present in OWL without any loss of generality.

This paper has provided qualitative descriptions and examples of how to deal with these compatibility issues. We are currently working on formalizing these qualitative descriptions and on modifying PR-OWL's syntax and semantics to incorporate the approaches presented here.

Acknowledgments. The authors would like to thank the Brazilian O ce of the Comptroller General (CGU) for their active support since 2008 and for providing the human resources necessary to conduct this research.

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