Bayesian semantics for the semantic web




НазваниеBayesian semantics for the semantic web
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Quiddity*Suite has been applied to many complex, real-world problems (e.g., Alghamdi et al., 2004; Fung et al., 2004; e.g., Alghamdi et al., 2005; Costa et al., 2005).  Its powerful representation and reasoning capabilities have provided solutions to problems that could not be solved with previously existing technology.  Development of Quiddity*Suite is ongoing, and new capabilities are being added on a continuing basis. As of the final phase of writing this dissertation, a new version is being released that incorporates significant advances in the use of Prolog rules to establish constraints in the slot instantiation process.

Those advances allow Quiddity models to replicate the context nodes of an MFrag, adding a major capability that as far as our knowledge goes is not implemented in any similar system. The time frame of this work prevent us to perform a more detailed analysis to evaluate the impact of those advances, and to verify whether full compatibility with MEBN logic has been achieved. Yet, the fact that we were able to built three logically equivalent versions of the Star Trek model using Quiddity*Suite, MEBN logic, and PR-OWL is a clear indication that in its current stage, Quiddity*Suite can be used as a reasoner in MEBN-based probabilistic ontologies.

The set of rules we have just described, combined with the extended version of MEBN logic we presented in the previous section, represent our solution for the two major issues preventing the development of a probabilistic ontology language. In the next chapter, we show how we have built upon what we implemented in “clearing the path” for probabilistic ontologies to develop PR-OWL, a probabilistic extension to the OWL Web ontology language.

  1. PR-OWL

As a means to realize the use of Bayesian theory for representing and reasoning under uncertainty in the Semantic Web, this Chapter proposes a standard knowledge representation formalism to express uncertain phenomena, performing plausible reasoning, and learning from data in the context of the Semantic Web. This formalism will provide a framework for executing those tasks in an interoperable way, so probabilistic ontologies that were built for different purposes, using diverse tools, and by knowledge engineers that were not mutually aware of each other’s work, would have a common underlying architecture guaranteeing the exchange of information in a useful and meaningful way.

A framework intended to provide means for building probabilistic ontologies for the Semantic Web must be compatible with the technologies being used in that environment. Thus, since OWL is the recognized ontology language of choice for the Semantic Web, it is also our base language for building the framework for probabilistic ontologies. That is, PR-OWL is an extension of OWL that enables the specification of probabilistic ontologies.

The OWL Web ontology language is a W3C Recommendation, which means it is the product of an exhaustive, consensus-based process in which many highly qualified participants from different countries composed various working groups and generated the technical reports which collectively comprise the final Recommendation (c.f. Jacobs, 2003). Extending a W3C Recommendation requires a similar process and implies a level of commitment from the W3C that makes it clearly outside the scope of a PhD Dissertation. Still, as explained in the previous chapters, the W3C’s vision for the Semantic Web can only be achieved with a sound and principled treatment of inconclusive, ambiguous, incomplete, unreliable, and dissonant data, all quite abundant in the current World Wide Web environment.

We saw in Chapter 3 that Bayesian probability theory provides a means for representing, reasoning and learning from all the above cited varieties of uncertain data, and is thus a natural candidate for providing the much-needed probabilistic framework for the Semantic Web. The development of a strategy for building that framework can be embraced as a doctoral research effort, and that is precisely the intention of the present work and the main focus of this chapter. Furthermore, the present work is envisioned as a basis for incorporating uncertainty in a future version of OWL.

The Chapter is divided into two main sections. The initial section establishes an implementation strategy for PR-OWL, which includes further considerations on probabilistic ontologies, the reasons for choosing MEBN logic as the underlying semantics of PR-OWL, and the intended scope of its definitions. The second part of the Chapter presents PR-OWL itself, and covers the major characteristics of the language.

1.20The Overall Implementation Strategy

Before devising a way of implementing a framework for building interoperable probabilistic ontologies, it is important to emphasize that a probabilistic ontology is not a probabilistic model (e.g. a model built using applications such as Netica, Hugin, or Quiddity*Suite) the same way that an ontology is not a database application.

The differences in the in-depth underlying concepts and technologies supporting ontologies and database schemas are not easily distinguishable, as the real differentiation between the two resides in their respective intended purposes. Ontologies represent domains in a way that should facilitate interoperability with other representations of that domain (i.e. other ontologies build by different people with different views and interests) or of domains that are not directly related but share some concepts. When a database solution for a given domain is conceived, its primary focus is not in representing all concepts of a domain in a way that makes it interoperable with current or future views of that domain, but in defining the concepts of that domain which would allow to coherently store the information the database stakeholders (and their customers) want to store and to retrieve that information in a way that best fits their requirements.

In a similar view, when a probabilistic model is built to solve (say) a radar data fusion problem, the main interest driving its creators is not in making sure that their definitions about radar domain concepts are interoperable with other definitions that might exist on those same concepts. In contrast, interoperability would definitely be a primary focus when building a probabilistic ontology for the domain of radar data fusion. Ontology engineers would attempt to express one view of that domain in a way that others (with possibly different views) may use/understand and thus build applications (databases, decision systems, etc) that are compatible with anything built under that view.

Furthermore, it is not necessary for an ontology to be an actually running database, yet a database application can be built on top of an ontology. Likewise, a probabilistic ontology does not necessarily need to be an actually running probabilistic model, yet a running probabilistic model (i.e. an executable application built using a probabilistic package) can be built on top of a probabilistic ontology if that fits the objectives of the application at hand. A subtle difference here is that anything built on top of an ontology can be built on top of a probabilistic ontology, but the converse is not always true, since the latter is an extension of the former that adds the above mentioned features of a probabilistic framework.

To comply with interoperability requirements and at the same time be useful enough for allowing a probabilistic model to be built on top of its definitions, a probabilistic ontology has to be based on a very flexible framework. Thus, the initial issue to be addressed is the definition of an underlying model for PR-OWL, one that allows representing uncertain data using OWL’s RDF based syntax. Clearly, it is desirable that the semantics of such model should have at least the same representational power of the semantics supporting OWL, so everything that can be represented in OWL could also be expressed in PR-OWL.

1.20.1Why MEBN as the semantic basis for PR-OWL?

In general, people faced with the complex challenge of representing uncertainty in languages like OWL tend to start their attempts by writing probabilities (i.e. priors and CPTs) as annotations (e.g. marked-up text describing some details related to a specific object or property). This is a palliative solution that addresses only part of the information that needs to be represented, since it fails to convey the structural intricacies that are present in even the simplest probabilistic models, such as conditional dependence (or independence) implied by connecting arcs (or lack of), double counting of influence on multiply connected graphs, and others.

Indeed, many researchers have pointed out the importance of structural information in probabilistic models (e.g. Shafer, 1986; Schum, 1994; Kadane & Schum, 1996). For instance, Schum (1994, page 271) shows that some questions about evidence can be answered entirely in structural terms.

In short, annotating the numerical probabilities of a probabilistic model in an ontology is just not enough, as too much information is lost to the lack of a good representational scheme that captures the structural nuances of the model. As noted in Chapter 2, one way of representing structural information of a probabilistic model is by extending OWL to represent Bayesian networks (e.g. Ding & Peng, 2004). However, even though such approach does capture some of the structural information of a probabilistic model, the limited expressiveness of Bayesian networks make it difficult to represent complex systems, as we could see from the Starship example in Chapter 3.

Probabilistic Relational Models provide a leap in representation power when compared with BNs, but as we could see in Chapter 4, PRMs alone cannot represent all that is needed for declarative representations that cover complex situations with tightly defined contexts (i.e. situations in which probability distributions are defined within very specific constrains). Therefore, to represent one of those specific situations in an ontology using PRMs as the underlying logic, either the instances of that ontology would also have to be declared (e.g. expressing that two starships are not the same individual by referring to two actual instances of starships25) or some combined approach would have to be used for constraining the context where the definitions apply (e.g. the use of Prolog rules in Quiddity*Suite).

The need to declare all instances in advance makes the first solution unsuitable for most use cases for the Semantic Web, where the ontologies generally have only T-Box information (or occasionally a few built-in A-Box definitions) and the A-Box is left for each specific situation/application based on that ontology. Thus, the second solution seems to be a more appropriate way of employing PRMs to build probabilistic ontologies. One successful example of that approach is the use of Prolog rules in Quiddity*Suite as a means to enforce constraints under which instances of a random variable are created.

Establishing such constraints is a vital asset for building probabilistic ontologies, since it allows one to express very detailed situations in which a given probability distribution holds. MEBN logic has a built-in form of representing such constraints (i.e. its context nodes), which makes it a flexible and simple technology that is also logically coherent (i.e. it can express a fully coherent joint probability distribution over instances that satisfy the constraints). These intrinsic features of MEBN logic makes it very suitable for being the basis of a probabilistic framework for the Semantic Web.

Approaches with limited expressiveness, such as BNs, are less suitable because the Semantic Web demands a certain level of flexibility those approaches cannot deliver. More expressive representational schemes such as PRMs, implementations of MEBN logic, and probabilistic logic programs theoretically have the basic conditions for supporting such a framework. In any case, there will always be a trade-off between flexibility and expressiveness when using a probabilistic logic to support a language meant for the Semantic Web. We found that MEBN logic provides a particularly attractive trade-off that made our work easier when extending the OWL Semantic Web language.

Laskey (2005, pages 22-27) shows that MEBN logic can express a joint probability distribution over models of any consistent finitely axiomatizable theory in classical first order logic. Thus, even the most specific situations can be represented in MEBN, provided they can represented in FOL. Furthermore, since MEBN is a first order Bayesian logic, using it as the underlying semantics of PR-OWL not only guarantees a formal mathematical background for a probabilistic extension to the OWL language (PR-OWL), but also ensures that the advantages of Bayesian Inference (e.g. natural “Occam’s Razor”, support for learning from data, etc.) will be available for using with any PR-OWL probabilistic ontology.

Therefore, we opted to use MEBN logic as the underlying semantics of OWL for its optimal combination of expressiveness and flexibility. Our next step is to lay out an overall plan for implementing it in a way that does not render current OWL ontologies incompatible with the extended language.

1.20.2Implementation Approach

OWL has intrinsic mechanisms to enable the development of extensions. The most basic means of extending OWL is to specify a vocabulary using a syntax that complies with its format. As an example, the Dublin Core metadata initiative is devoted to developing specialized metadata vocabularies and to promote the widespread adoption of interoperable metadata standards (Hillmann, 2001). Any OWL ontology can use the Dublin Core vocabulary to define additional semantics about its contents just by adding its namespace in the file header and encoding qualified Dublin Core Metadata in the RDF / XML format described in Kokkelink & Schwänzl (2001).

Yet, this basic level of extensibility is not enough to guarantee a coherent, widely used standard for more complex activities that demand a greater level of commitment from users of the standard. As an example, OWL-S (Martin et al., 2004) and the Web Service Modeling Ontology – WSMO (Polleres et al., 2005) have been acknowledged as Member Submissions, both proposing solutions for Web service ontologies. Their respective sponsoring organizations submitted the draft specifications with the hop that these can form the basis of a future standard and thus form a framework that would allow a much higher degree of automation, functionality and interoperability among the various types of services. Those specifications build upon and extend the foundation laid by OWL and other web standards; PR-OWL as described below intends to make a similar contribution.

Some extensions do change the semantics and abstract syntax of OWL. As an example, the Semantic Web Rule Language (SWRL) is a W3C Member Submission that proposes to extend OWL abstract syntax so it includes support for rules based on RuleML (Boley & Tabet, 2004) and provides a model-theoretic semantics defining the meaning of the rules written in the extended syntax (Horrocks et al., 2004). In addition, there is another W3C Member Submission that proposes extending SWRL so it would allow OWL ontologies containing the extended abstract syntax and semantics defined in SWRL to handle unary/binary first-order logic (Patel-Schneider, 2005).

The extensions listed in the above paragraph do add new elements to the abstract syntax and semantics of OWL, which means they augment the expressiveness of the language by enabling it to express concepts that are not possible to convey with standard OWL. On the other side, in order to make use of those extensions, it is necessary to develop new tools supporting the extended syntax and implied semantics of each extension.

PR-OWL is an extension that enables OWL ontologies to represent complex Bayesian probabilistic models in a way that is flexible enough to be used by diverse Bayesian probabilistic tools (e.g. Netica, Hugin, Quiddity*Suite, JavaBayes, etc.) based on different probabilistic technologies (e.g. PRMs, BNs, etc.).

That level of flexibility can only be achieved using the underlying semantics of first-order Bayesian logic, which is not a part of the standard OWL semantics and abstract syntax. Therefore, it seems clear that PR-OWL can only be realized via extending the semantics and abstract syntax of OWL the same way as the above examples of SWRL, RuleML and SWRL-FOL.

Indeed, an ideal full implementation of a probabilistic ontology would follow the steps defined by the W3C (Jacobs – ed., 2003) until it becomes an official standard. As demonstrated in Chapter 3, all the information needed to process probabilistic queries in a MEBN models is contained in the model’s generative MTheory and the findings related to the query of interest. Also, we have shown that one of the advantages of MEBN logic is the ability to express very specific situations via context nodes, which are declarative statements with FOL expressiveness.

Therefore, and that constitutes one of the major contributions of the present work, it is possible to define an upper ontology for probabilistic systems that can be used as a framework for developing probabilistic ontologies (as defined in the beginning of this Chapter) that are expressive enough to represent even the most complex probabilistic models.

Defining such a framework is the initial step towards a full PR-OWL specification, and a basic requirement for the development of probabilistic ontologies. With that in mind, the implementation strategy that guided our actions in the present research effort consisted of the following steps:

  1. Define the formal foundation (based on Bayesian first-order logic) needed to specify general probabilistic ontologies.

  2. Present an operational concept to provide a general guidance on the development of plug-ins and/or applications that make it easier for the average user to write probabilistic ontologies.

  3. As a step towards standardization by the W3C, establish a future vision for the PR-OWL specification, and a plan for realizing that vision.

Steps “a” and “b” are covered in the remainder of this Chapter, while the last step is addressed in Chapter 6.

1.21An Upper Ontology for Probabilistic Systems

Our initial step towards a Bayesian framework for the Semantic Web is to create an upper ontology to guide the development of probabilistic ontologies. DaConta et al. define an upper ontology as a set of integrated ontologies that characterizes a set of basic commonsense knowledge notions (2003, page 230). In this preliminary work on PR-OWL as an upper ontology, these basic commonsense notions are related to representing uncertainty in a principled way using OWL syntax. If PR-OWL were to become a W3C Recommendation, this collection of notions would be formally incorporated into the OWL language as a set of constructs that can be employed to build probabilistic ontologies.

The PR-OWL upper ontology for probabilistic systems is presented in Appendix B. It consists of a set of classes, subclasses and properties that collectively form a framework for building probabilistic ontologies. The first step toward building a probabilistic ontology in compliance with our Definition 3 (pages 101/102) is to import into any OWL editor an OWL file containing the PR-OWL classes, subclasses, and properties. In fact, this is exactly what we did when we built the Star Trek probabilistic ontology. We used the Protégé import feature to download the PR-OWL upper ontology from a website we had previously set up.

After importing the PR-OWL definitions, the next step in ontology design is to construct domain-specific concepts, using the PR-OWL definitions to represent uncertainty about their attributes and relationships. As an example, the concepts of the Star Trek probabilistic ontology were either subclasses or instances of the imported PR-OWL upper ontology. Using this procedure, an ontology engineer is not only able to build a coherent generative MTheory and other probabilistic ontology elements, but also make it compatible with other ontologies that use PR-OWL concepts.

Because we designed PR-OWL with the objective of eventually turning it into a W3C submission, we wanted it to be as general purpose as possible. That is, we attempted to avoid unnecessary restrictions that would initially make the job easier for the designer of a specific application, but would limit its flexibility for a broader set of applications. Imposing such limitations would render this preliminary work less suitable as the starting point for a W3C Recommendation process. Thus, even though we did establish a fixed set of classes, subclasses and instances for the upper ontology, which was necessary to enforce consistency with MEBN logic standards, we intentionally avoided unnecessary restrictions on how a modeler would develop her/his own probabilistic ontology. It is clear to us that such approach is valid for the scope of this work, but the natural tendency for the process towards a W3C Recommendation is to impose extra restrictions that would achieve an optimal trade-off between flexibility and enforcing the rules of the underlying logic. In other words, an upper ontology is enough as a starting point to represent uncertainty in a principled way using PR-OWL, but it cannot prevent unintentional misuse of its elements that would lead to inconsistencies in the resulting probabilistic ontology.

In order to illustrate our conceptual approach, consider the question of whether to represent an MFrag template such as the Zone MFrag from our Star Trek generative MTheory (see Figure 10, page 70) as a class or an instance. If we choose the first option, we would create it as a subclass of the imported PR-OWL class Domain MFrag (see Appendix B, page 230). That newly created subclass will thus inherit all the properties from the PR-OWL Domain MFrag class that enforce the structural and logical constraints of a MEBN Fragment (e.g. it must have at least one resident node, it might have context and input nodes, etc.). The instances of that subclass would then be copies of the Zone MFrag template that have all of its inherited elements. This approach seems appropriate when the ontology being built is supposed to represent the many copies of Zone MFrags created by SSBN construction procedures started to answer a given query.

If, instead, we opt for the second approach and represent the Zone MFrag template as a direct instance of the PR-OWL class Domain MFrag, then such instance would still carry all the properties of a Domain MFrag (e.g. it must have at least one resident node, etc.) that enforce the structural and logical constraints of MEBN logic. In this case, unless we want to use a second order representation (i.e. use instances of instances), the ontology itself could not contain instances of the Zone MFrag template. We could add instances of the random variables that appear in the Zone MFrag (e.g., ZoneMD(!Z0)) to the ontology, but there would be no instance of the MFrag template explicitly represented in the ontology. The PR-OWL instance we created to represent the Zone MFrag template would contain all the information that an application external to the ontology (i.e. a decision support system) needs to build copies of Zone MFrags when building SSBNs to answer a query.

It is important to keep in mind that no matter what approach an ontology designer uses in the light of his/her objectives, the structural and logical constraints of MEBN logic will be inherited. Since the other elements of the “probabilistic part” of the ontology will also be either instances or subclasses of the imported PR-OWL upper ontology, then all will inherit the structural and logical constraints that collectively enforce the compliance with MEBN rules, thus guaranteeing that such an ontology would be a coherent, logically consistent MEBN Theory.

Although we did not establish any constraints on this specific issue, we considered the pros and cons of modeling our concepts as subclasses or instances of PR-OWL classes in the design of our Star Trek probabilistic ontology. Our experience leads us to conclude that the objectives and characteristics of the probabilistic ontology being built will dictate how to make this choice. In general, ontologies that are expected to represent many instances of a given concept (e.g. copies of Zone MFrag in the illustration above) should characterize that concept as a subclass of PR-OWL. Conversely, if a given concept is not going to have its instances represented in the ontology (e.g. only the Zone MFrag template is of interest) then the concept itself might be characterized as an instance of a PR-OWL class. The advantage of doing so is to avoid unnecessary duplications (e.g. many copies of a Zone MFrag template that would not be used by the client applications of the Star Trek ontology). In the Star Trek probabilistic ontology, most of the concepts directly related to the generative MTheory were modeled as instances, whereas Object entities such as Starships and other concepts for which we expect to have its instances populating the ontology were modeled as classes.

Our choice took into account that representing uncertainty within an ontology is not the same thing as building a probabilistic system. In our Star Trek case study, the generative MTheory is used in conjunction with information about domain entities (e.g. instances of starships) to build SSBNs to answer queries about those entities. In this case, the Enterprise’s decision support system would carry out the process of building situation-specific models (i.e. instantiating and combining MFrags) to answer the relevant queries, evaluate the perceived situation, and update the system’s knowledge accordingly. The generative MTheory can be seen as the part of the system that holds the domain knowledge used in this process. In other words, the process of building, working and storing the instantiated MFrags in this case is not part of the Star Trek probabilistic ontology.

Even though we understand the above option might be desirable in some applications, we preferred to adopt a different approach that avoids duplications by restricting the user defined classes only to the elements we expect to be instantiated in the ontology itself (as distinct from an application that uses the ontology). In short, we opted to represent the generative MTheory concepts as instances of PR-OWL built-in classes, while representing the object entities, random variables (i.e. resident nodes), and its distribuitions as user defined classes. As an example, Starship would be a user-defined class (subclass of PR-OWL ObjectEntity class) whose instances would be something such as !ST0, !ST1, etc., whereas the Zone MFrag is modeled as an instance of PR-OWL built-in Domain MFrag. This approach is consistent with the fact that a generative MTheory contains all the domain-specific information that is needed in conjunction with information on the object entities for the targeted application of our ontology to conduct its reasoning processes (e.g. the Enterprise’s decision support system). In the end, we believed our choice to be preferable in most cases in which an ontology is needed, because it results in a more concise ontology that still can be used for applications as the basis for conducting their respective reasoning process.

A generative MTheory can express domain-specific ontologies that capture statistical regularities in a particular domain of application, and MTheories with findings can augment statistical information with particular facts germane to a given reasoning problem (Laskey, 2005). From our definition, it is possible to realize that nothing prevents a probabilistic ontology from being “partially probabilistic”. That is, a knowledge engineer can choose the concepts that he/she is interested to be in the “probabilistic part” of the ontology, while writing the other concepts in standard OWL.

In this specific case, the “probabilistic part” refers to the concepts written using PR-OWL definitions and that collectively form an MTheory. There is no need for all the concepts in a probabilistic ontology to be probabilistic, but at least some have to form a valid MTheory. Of course, only the concepts being part of the MTheory will be subject to the advantages of the probabilistic ontology over a deterministic one.

The subtlety here is that legacy OWL ontologies can be upgraded to probabilistic ontologies only with respect to the concepts for which the modeler wants to have uncertainty represented in a principled manner, make plausible inferences from that uncertain evidence, or to learn its parameters from incoming data using Bayesian learning.

The ability to perform probabilistic reasoning with incomplete or uncertain information conveyed through an ontology is a major advantage of PR-OWL. However, it should be noted that in some cases solving a probabilistic query might be intractable or even undecidable. In fact, providing the means to ensure decidability was the reason why the W3C defined three different version of the OWL language. While OWL Full is more expressive, it enables an ontology to represent knowledge that can lead to undecidable queries. OWL-DL imposes some restrictions to OWL in order to eliminate these cases. Similarly, restrictions of PR-OWL could be developed that limit expressivity to avoid undecidable queries or guarantee tractability. This initial step is focused on the most expressive version of PR-OWL.

In this section, the “probabilistic part” of PR-OWL ontologies will be covered, and the main objective is to show how to represent any generative MTheory (with no regard to its level of complexity) and also Finding MFrags using PR-OWL concepts. An overview of the general concepts involved in the definition of an MTheory in PR-OWL is depicted in Figure 27. In this diagram, the ovals represent general classes, while the major relationship between those classes are symbolized by arrows. A probabilistic ontology has to have at least one individual of class MTheory, which is basically a label linking a group of MFrags that collectively form a valid MTheory. In actual PR-OLW syntax, that link is expressed via the object property hasMFrag (which is the inverse of object property isMFragIn).



  1. Overview of a PR-OWL MTheory Concepts

Individuals of class MFrag are comprised of nodes, which can be resident, input, or context nodes (not shown in the picture). Each individual of class Node is a random variable and thus has a mutually comprehensive, collectively exhaustive set of possible states. In PR-OWL, the object property hasPossibleValues links each node with its possible states, which are individuals of class Entity. Finally, random variables (represented by the class Nodes in PR-OWL) have unconditional or conditional probability distributions, which are represented by class Probability Distribution and linked to its respective nodes via the object property hasProbDist.

The scheme in Figure 27 is intended to present just a general view and thus fails to show many of the intricacies of an actual PR-OWL representation of an MTheory. Figure 28 shows an expanded version conveying the main elements in Figure 27, its subclasses, the secondary elements that are needed for representing an MTheory and the reified relationships that were necessary for expressing the complex structure of a Bayesian probabilistic model using OWL syntax.

Reification of relationships in PR-OWL is necessary because of the fact that properties in OWL are binary relations (i.e. link two individuals or an individual and a value), while many of the relations in a probabilistic model include more than one individual (i.e. N-ary relations). The use of reification for representing N-ary relations on the Semantic Web is covered by a working draft from the W3C’s Semantic Web Best Practices Working Group (Noy & Rector, 2004).

Although the scheme in Figure 28 shows all the elements that are needed for representing a complete MTheory, it is clear that any attempt at a complete description would render the diagram cluttered and incomprehensible. Therefore, a complete account of the classes, properties and the code of PR-OWL are given in Appendix B



  1. Elements of a PR-OWL Probabilistic Ontology

The material provided in the appendix defines an upper ontology for probabilistic systems, and it can be used to represent any system that can be represented using the extended version of MEBN logic presented in Chapter 4. In order to show the applicability of the presented framework, the next Subsections explain how it can be used to build a probabilistic ontology.

In order to demonstrate the applicability of PR-OWL in diverse levels of complexity, initially a generic explanation is given for each major aspect of the modeling process, then an illustrative example based on the Starship case study is provided as a means to facilitate the understanding over the most important steps. In both cases, the examples were built using the open source software Protégé26, an ontology editor developed by the by Stanford Medical Informatics at the Stanford University School of Medicine (Noy et al., 2000; Noy et al., 2001), and its OWL plugin (Knublauch et al., 2004).

At the present experimental stage, writing probabilistic ontologies in PR-OWL is a process that requires importing the upper ontology provided in Section B.4 in the appendices. Figure 29 shows the header of the Starship probabilistic ontology developed as a case study for this research. There, it is possible to see the owl:imports feature being used for downloading the PR-OWL upper ontology utilized as the base block for building the Starship probabilistic ontology.

Even though the above example was written in Protégé, any ontology tool capable of editing OWL ontologies, such as SWOOP27 (Kalyanpur et al., 2004) or webODE28 (Arpírez et al., 2001), can be used for editing a PR-OWL ontology.



  1. Header of the Starship Probabilistic Ontology

1.21.1Creating an MFrag

Figure 30 shows the initial Protégé screen after importing the PR-OWL ontologies and defining the classes of object entities that will be part of the ontology. In Protégé, concepts of imported ontologies appear with a light colored dot icon and the namespace abbreviation at the left side of the concept’s name, as it can be seen in the Asserted Hierarchy window on the left side of the picture.

The darker icons (Starship, Zone, Sensor Report, and TimeStep) correspond to the classes created as a first step to build the Starship probabilistic ontology. PR-OWL object entities correspond to frames in frame systems and to objects in object-oriented systems. The simple model used in this research contains only four object entities; so four classes were created under the PR-OWL ObjectEntity Class (i.e. Starship, Zone, SensorReport, and TimeStep). These are the user-defined classes that convey the equivalent of what a standard ontology would represent about a domain, so its individuals are the concepts and entities that would populate a non-probabilistic description of that domain. In our Starship ontology, the domain instances will be individual zones, sensor reports, starships, and time steps, all represented as individuals of the domain classes created by the user.



  1. Initial Starship Screen with Object Properties Defined

The other PR-OWL classes shown in the picture are directly fulfilled by individuals representing the elements of a generative MTheory. The user does not create new classes here, but individuals that convey the information necessary for creating elements of an SSBN. In other words, these individuals express the probabilistic aspects of the domain MTheory, and can be seen as templates that a probabilistic reasoner uses for building an SSBN to answer a query. Examples of those aspects are the characteristics of domain instances (e.g. the possible nature of a zone, class of a starship, etc), its possible states, its probability distributions, etc.

When Quiddity*Suite (or another probabilistic reasoner chosen by the user) receives a query on (say) the status of zones !Z0, !Z1 and !Z8 (all individuals of the user-defined domain class Zone) it will build an SSBN based on the individuals of PR-OWL classes representing the generative MTheory and the evidence available in form of findings. In this case, the reasoner will certainly build three copies of RV ZoneNature(z) based on the information contained in the individual Z_ZoneNature of the PR-OWL class Domain_res.

Even though the names chosen for the four object entity classes match their respective intended meaning, this is not a requirement. PR-OWL uses a UID as a means to enforce its unique naming assumption, and the name of each concept has no meaning for the logic under PR-OWL (MEBN logic). As an example, choosing a name such as “Umbrella” as the reference to the class including all sensor reports in the model would make no difference for the tasks performed by the reasoner, but would certainly confuse any human reader trying to understand the model. Therefore, as a means to facilitate human understanding and to improve interoperability with other systems (which probably have humans as API builders), an optional naming convention is proposed in Section B.3 in the appendices and was used in the Starship probabilistic ontology built for this research.

Figure 31 illustrates the PR-OWL representation of the Zone MFrag, which uses the above-cited naming convention. An MFrag can be seen as a hub connecting a collection of related random variables that together represent an atomic "piece of knowledge" about a domain.  The context nodes of the MFrag represent conditions under which the relationship holds. A coherent set of those “pieces” form a joint probability distribution over the included random variables, also known as an MTheory.



  1. Zone MFrag Represented in PR-OWL

A common method for handling cognitive tasks is the “divide and conquer” approach, which breaks a problem into smaller, simpler parts. Thus, building “pieces of knowledge” about a domain in a way that allows “gluing” them together to handle more complex issues within that domain is a natural technique for modeling probabilistic systems. Not surprisingly, a very usual way of starting a probabilistic ontology is by defining its generative MFrags or, in PR-OWL, the individuals of class Domain_MFrag.

PR-OWL includes all the necessary elements of MEBN logic that are necessary to represent an MFrag. Figure 31 shows MEBN’s representation of the Zone MFrag in comparison with its PR-OWL counterpart. Following the bullets within the figure, every individual of class Domain_MFrag is related to one or more MTheory via the object property isMFragOf [1]29.

The Zone MFrag is represented as an individual of class Domain_MFrag, having name Zone_MFrag. The MFrag has four resident nodes, three input nodes and six context nodes. Its PR-OWL represents those 13 nodes using 11 individuals of subclasses of Node, which are linked to the Zone_MFrag via the object property hasNode [2]. The mismatch between the number of MEBN nodes and their respective PR-OWL description is caused by the fact there is not a straightforward one-to-one correspondence between MEBN and PR-OWL constructs. Table 5 shows the details of how each node is portrayed in both representations.

As shown in the table, the “IsA” context nodes are not explicit represented in PR-OWL MFrags, since the notion of subtyping is already conveyed in the definition of the arguments of each resident node. In MEBN, the “IsA” context nodes are meant to define which type of entities can substitute the ordinary variables in an MFrag. In PR-OWL, this constraint is expressed by the object property isSubsBy, linking individuals of class OVariable to the individuals of class Entity that are allowed to substitute for them. As an example, Zone MFrag has four ordinary variables (st, t, tprev, and z) that are represented in PR-OWL as four individuals of class OVariable (Z_st, Z_t, Z_tprev, and Z_z). Thus, while in MEBN logic the context node IsA(Starship, st) is meant to restrict the ordinary variable st so that only entities of type Starship can substitute for it, in PR-OWL the equivalent construction is isSubsBy(Z_st, Starship_Label), meaning that only individuals that have property hasType equal to Starship_Label can substitute for Z_st.

  1. Zone_MFrag Nodes in MEBN and PR-OWL

MEBN MFrag

PR-OWL Representation

IsA(TimeStep, tprev)

Implicit in the type declaration

IsA(Zone, z)

IsA(Starship, st)

IsA(TimeStep, t)

tprev = Prev(t)

Z_TprevPrevT_context

Z_TprevPrevT_inner_prevT

z = StarshipZone(st)

Z_ZSZoneST_context

Z_ZSZoneST_inner_SZoneST

CloakMode(st)

Z_CloakMode_input

ZoneMD(z, tprev)

Z_ZoneMD_input

t = !T0

Z_TequalT0_inpu

ZoneMD(z, t)

Z_ZoneMD

ZoneNature(z)

Z_ZoneNature

ZoneFShips(z)

Z_ZoneFShips

ZoneEShips(z)

Z_ZoneEShips
1   ...   4   5   6   7   8   9   10   11   ...   20

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