The past few years have witnessed an increasingly mature body of research on the Semantic Web, with new standards being developed and more complex use cases being proposed and explored. As complexity increases in SW applications, so does the need for principled means to cope with uncertainty inherent to real world SW applications. Not surprisingly, several approaches addressing uncertainty representation and reasoning on the Semantic Web have emerged [3, 4, 6, 7, 10, 11, 13, 14]. For example, PR-OWL [3] provides OWL constructs for representing Multi-Entity Bayesian Network (MEBN) [8] theories. This paper reviews some shortcomings of PR-OWL 1 [2] and describes how they will be addressed in PR-OWL 2. A method is presented for mapping back and forth from triples into random variables (RV). The method applies to triples representing both predicates and functions. A complex example is given for mapping an n-ary relation using the proposed schematic.
Appreciation is growing within the Semantic Web community of the need to
represent and reason with uncertainty. In recognition of this need, 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 URW3-XG concluded that standardized representations were needed to
express uncertainty in Web-based information [
Compatibility with OWL was a major design goal for PR-OWL [
These problems have been noted in the literature [
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 justi es the need for a formal mapping between random variables de ned in PR-OWL and concepts de ned in OWL, and proposes an approach to such a mapping. We rst present a solution that is su cient for binary relations. Next, we present a more robust solution that allows the user to de ne PR-OWL random variables with arbitrarily many arguments, while maintaining a 2-way mapping to OWL concepts. Finally, we present a schematic for the mapping back and forth from triples into random variables. 2
PR-OWL was proposed as an extension to the OWL language based on MEBN, which can express a probability distribution on interpretations of any rst-order theory. In PR-OWL, a probabilistic ontology (PO) 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-OWL syntax, that link is expressed via the object property hasMFrag (which is the inverse of object property isMFragIn). Individuals of class MFrag are comprised of nodes. Each individual of class Node is a random variable (RV) and thus has a mutually exclusive, 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 Node in PR-OWL) have unconditional or conditional probability distributions, which are represented by class ProbabilityDistribution and linked to their respective nodes via the object property hasProbDist.
As a running example, we consider 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.
Now, imagine we want to de ne some uncertain relations about this domain.
For example, if an enterprise wins a procurement for millions of dollars, but the
responsible person for this enterprise makes less than 10 thousand dollars a year,
the responsible person may be a front. That is, we can identify potential fronts
by examining the value of the procurement and the income of the responsible
person. Figure 1 shows this probabilistic relation de ned using PR-OWL in an
open-source tool for probabilistic reasoning, UnBBayes [
Listing 1.1. De nition of WinnerOf RV in PR-OWL 1 1 <owl:Thing r d f : a b o u t="#WinnerOf RV"> 2 <r d f : t y p e r d f : r e s o u r c e="#Domain Res "/> 3 <h a s P o s s i b l e V a l u e s r d f : r e s o u r c e="#E n t e r p r i s e "/> 4 <i s R e s i d e n t N o d e I n r d f : r e s o u r c e="#ProcurementInfo MFrag "/> 5 <hasArgument r d f : r e s o u r c e="#WinnerOf 1 "/> 6 </ owl:Thing> 7 8 <owl:Thing r d f : a b o u t="#WinnerOf 1 "> 9 <r d f : t y p e r d f : r e s o u r c e="#S i m p l e A r g R e l a t i o n s h i p "/> 10 <hasArgNumber r d f : d a t a t y p e="&xsd ; i n t ">1</hasArgNumber> 11 <hasArgTerm r d f : r e s o u r c e= 12 "#ProcurementInfo MFrag . procurement "/> 13 <isArgumentOf r d f : r e s o u r c e="#WinnerOf RV"/> 14 </ owl:Thing> 15 16 <owl:Thing r d f : a b o u t="#ProcurementInfo MFrag . procurement "> 17 <r d f : t y p e r d f : r e s o u r c e="#OVariable "/> 18 <i s O V a r i a b l e I n r d f : r e s o u r c e="#ProcurementInfo MFrag "/> 19 <isSubsBy r d f : r e s o u r c e="#Procurement "/> 20 <isArgTermIn r d f : r e s o u r c e="#WinnerOf 1 "/> 21 </ owl:Thing>
We would like to be able to tie this fragment of probabilistic knowledge with domain knowledge already represented in an OWL ontology. That is, we might have a database containing instances of persons and enterprises, linked to an OWL ontology de ning their semantics (e.g., that persons can be responsible for enterprises). 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, for PR-OWL to make use of this knowledge, there must be a way to link PR-OWL random variables (RVs) with concepts de ned in OWL. The current version of PR-OWL has no standard way to establish such links.
Listing 1.1 presents how the RV WinnerOf RV from Figure 1 is de ned in PR-OWL today. This RV is de ned as follows: { It is a domain resident node (line 2) { Its possible values (range) are instances of Enterprise (line 3) { Its home MFrag is ProcurementInfo MFrag (line 4) { It has one argument (domain) WinnerOf 1 (line 5) { WinnerOf 1 is the rst argument (line 10) { WinnerOf 1 is related to the variable ProcurementInfo MFrag.procurement (lines 11-12) { ProcurementInfo MFrag.procurement is an ordinary variable (line 17) { ProcurementInfo MFrag.procurement is de ned in the ProcurementInfo
MFrag (line 18) { ProcurementInfo MFrag.procurement can only be replaced by instances of Procurement (line 19)
Listing 1.2 is a suggested de nition of the object property winnerOf in OWL. This property is de ned as follows: { It is an object property (line 1) { It is a functional property (line 2) { Its domain is the instances of Procurement (line 3) { Its range is the instances of Enterprise (line 4)
Listing 1.2. De nition of winnerOf object property in OWL 1 <o w l : O b j e c t P r o p e r t y r d f : a b o u t="#winnerOf "> 2 <r d f : t y p e r d f : r e s o u r c e="&owl ; F u n c t i o n a l P r o p e r t y "/> 3 <r d f s : d o m a i n r d f : r e s o u r c e="#Procurement "/> 4 <r d f s : r a n g e r d f : r e s o u r c e="#E n t e r p r i s e "/> 5 </ o w l : O b j e c t P r o p e r t y>
Comparing the two de nitions winnerOf and WinnerOf RV, we can see that they are consistent, since their domain/arguments and range/possible values are the same, Procurement and Enterprise, respectively. However, there is no property that explicitly relates these two concepts, and there is no implicit way of guring out that they should be related besides the fact that their names are similar (winnerOf and WinnerOf RV). Therefore, we would not have access to the semantics of the term winnerOf de ned in our ontology when de ning its probabilistic relations using the new and unrelated term WinnerOf RV de ned in our probabilistic ontology.
This simple example demonstrates the need to de ne a reference from every probabilistic de nition involving a concept to its OWL de nition. In other words, full compatibility with OWL requires modi cations to PR-OWL that guarantee the preservation of OWL's semantics.
A simple solution to this mapping problem is presented in Listing 1.3. By adding the property defineUncertaintyOf which states that a random variable de nes the uncertainty relations of a speci c property, we could state that WinnerOf RV de nes the uncertainty of the object property winnerOf (line 3). In order to make this de nition consistent we would need to add some axioms to our language stating that the possible values of the RV must be the same as the range de ned in the property for which this RV de nes the uncertainty. Similar axioms would be needed for its domain.
Listing 1.3. De nition of WinnerOf RV with mapping information to its OWL concept 1 <owl:Thing r d f : a b o u t="#WinnerOf RV"> 2 <r d f : t y p e r d f : r e s o u r c e="#Domain Res "/> 3 <d e f i n e U n c e r t a i n t y O f r d f : r e s o u r c e="#winnerOf "/> 4 <h a s P o s s i b l e V a l u e s r d f : r e s o u r c e="#E n t e r p r i s e "/> 5 <i s R e s i d e n t N o d e I n r d f : r e s o u r c e="#ProcurementInfo MFrag "/> 6 <hasArgument r d f : r e s o u r c e="#WinnerOf 1 "/> 7 </ owl:Thing> 8 9 <owl:Thing r d f : a b o u t="#WinnerOf 1 "> 10 <r d f : t y p e r d f : r e s o u r c e="#S i m p l e A r g R e l a t i o n s h i p "/> 11 <hasArgNumber r d f : d a t a t y p e="&xsd ; i n t ">1</hasArgNumber> 12 <hasArgTerm r d f : r e s o u r c e= 13 "#ProcurementInfo MFrag . procurement "/> 14 <isArgumentOf r d f : r e s o u r c e="#WinnerOf RV"/> 15 </ owl:Thing> 16 17 <owl:Thing r d f : a b o u t="#ProcurementInfo MFrag . procurement "> 18 <r d f : t y p e r d f : r e s o u r c e="#OVariable "/> 19 <i s O V a r i a b l e I n r d f : r e s o u r c e="#ProcurementInfo MFrag "/> 20 <isSubsBy r d f : r e s o u r c e="#Procurement "/> 21 <isArgTermIn r d f : r e s o u r c e="#WinnerOf 1 "/> 22 </ owl:Thing> 3
In Section 2 we presented a simple solution to map OWL concepts to random variables de ned in PR-OWL. In this section we will show that the presented solution is not enough to cover the full expressiveness of PR-OWL. In particular, this solution cannot represent uncertainty for n-ary functions and relations.
Imagine extending our example to a situation in which a group of enterprises can win a procurement. Moreover, there will be a price associated with each enterprise on the contract. Therefore, instead of comparing the value of the procurement as a whole to try to identify the owner of the enterprise as a front (as shown on Figure 1), we need to consider only the part of that total associated to that speci c enterprise, as shown in Figure 2.
Note that we now have a ternary relation which associates an enterprise, a contract, and the amount awarded by the contract to the enterprise. As a functional relation, this is represented by the two-argument function priceOf(contract,enterprise).
Suppose that we want to represent that enterprise1 was hired for $10,000.00 and enterprise2 for $500,000.00 both in contract1. The problem is that OWL supports only binary relations. As shown in Listing 1.4, if we tried to represent this situation using binary relations with the class Contract, we would be unable to distinguish whether enterprise1 has price of $10,000.00, price1, or $500,000.00, price2.
Listing 1.5. De ning n-ary relations in OWL
As shown in Figure 3, one way to overcome this problem is to create a blank
node which has three functions mapping to each of the 3 arguments of our ternary
relation. Notice that these 3 binary relations (contractOf, enterpriseOf, and
priceOf) have to be functions, otherwise we would have the same problem we
had in Listing 1.4. Listing 1.5 presents this representation in OWL (for more
details on how to de ne n-ary relations in OWL, see [
When we try to apply the simple solution given in Section 2, we realize that it is not suitable for RVs with more than one argument. This is due to the fact that we assume the following: 1. The range from the property associated to the defineUncertaintyOf has to be the same type as the value of the RV's hasPossibleValues property; and 2. The domain from the property associated to the defineUncertaintyOf has to be the same type as the only RV's argument (hasArgument ! hasArgTerm ! isSubsBy).
So, what happens with the other arguments of the RV? What do they map to? Notice also that there is no argument in priceOf(contract,enterprise) that relates to the domain of the OWL property priceOf. In other words, there is no argument that \points" to the blank node we de ned in Figure 3 and Listing 1.5. Besides, having only the property defineUncertaintyOf relating to the OWL property priceOf tells us nothing about what contract and enterprise are and where they come from. As a matter of fact, we need to have a reference to all the binary properties that we use to represent the n-ary relation we want. Therefore, in this case, we also need to have a mapping to both contractOf and enterpriseOf.
Taking a closer look, we realize that all three properties of interest (priceOf, contractOf, and enterpriseOf) have the same domain (the blank node) and their range, Money, Contract, and Enterprise, map directly to the possible values of our RV of interest, to the argument contract, and to the argument enterprise, respectively. Listing 1.6 shows a more complex and robust solution that covers this case and any other n-ary relation for which we might want to de ne uncertainty.
Listing 1.6 states that the RV priceOf RV de nes the probabilistic semantics of the property priceOf, which already has an OWL semantics (line 3). Lines 4 and 5 ensure that the domain and range from the OWL property match the RV domain (hasDomain) and range (hasPossibleValues), respectively. Lines 7 and 8 say that this RV has two arguments. Lines 13-15 de ne the rst argument as being the variable contract, and lines 30-32 de ne the second argument as the variable enterprise. Lines 22-24 specify that the contract variable is used as the object (objectIn) of the OWL property contracOf, thus it can only be substituted by (isSubsBy) the class that is the range of the contractOf property, which is Contract. In addition, the domain also has to be the same (hasDomain), which, in this case, is the blank node :id1. Listing 1.6. Robust solution for de ning n-ary RVs and mapping them to the OWL concepts that de ne their semantics 1 <owl:Thing r d f : a b o u t="#priceOf RV "> 2 <r d f : t y p e r d f : r e s o u r c e="#Domain Res"/> 3 <d e f i n e U n c e r t a i n t y O f r d f : r e s o u r c e="#p r i c e O f "/> 4 <hasDomain r d f : r e s o u r c e=" : i d 1 "/> 5 <h a s P o s s i b l e V a l u e s r d f : r e s o u r c e="#Money"/> 6 <i s R e s i d e n t N o d e I n r d f : r e s o u r c e="#FrontOfEnterprise MFrag "/> 7 <hasArgument r d f : r e s o u r c e="#p r i c e O f 1 "/> 8 <hasArgument r d f : r e s o u r c e="#p r i c e O f 2 "/> 9 </ owl:Thing> 10 11 <owl:Thing r d f : a b o u t="#p r i c e O f 1 "> 12 <r d f : t y p e r d f : r e s o u r c e="#S i m p l e A r g R e l a t i o n s h i p "/> 13 <hasArgNumber r d f : d a t a t y p e="&xsd ; i n t ">1</hasArgNumber> 14 <hasArgTerm r d f : r e s o u r c e= 15 "#FrontOfEnterprise MFrag . c o n t r a c t "/> 16 <isArgumentOf r d f : r e s o u r c e="#priceOf RV "/> 17 </ owl:Thing> 18 19 <owl:Thing r d f : a b o u t="#ProcurementInfo MFrag . c o n t r a c t "> 20 <r d f : t y p e r d f : r e s o u r c e="#OVariable "/> 21 <i s O V a r i a b l e I n r d f : r e s o u r c e="#FrontOfEnterprise MFrag "/> 22 <o b j e c t I n r d f : r e s o u r c e="#c o n t r a c t O f "/> 23 <hasDomain r d f : r e s o u r c e=" : i d 1 "/> 24 <isSubsBy r d f : r e s o u r c e="#Contract "/> 25 <isArgTermIn r d f : r e s o u r c e="#p r i c e O f 1 "/> 26 </ owl:Thing> 27 28 <owl:Thing r d f : a b o u t="#p r i c e O f 2 "> 29 <r d f : t y p e r d f : r e s o u r c e="#S i m p l e A r g R e l a t i o n s h i p "/> 30 <hasArgNumber r d f : d a t a t y p e="&xsd ; i n t ">2</hasArgNumber> 31 <hasArgTerm r d f : r e s o u r c e= 32 "#FrontOfEnterprise MFrag . e n t e r p r i s e "/> 33 <isArgumentOf r d f : r e s o u r c e="#priceOf RV "/> 34 </ owl:Thing> 35 36 <owl:Thing r d f : a b o u t="#ProcurementInfo MFrag . e n t e r p r i s e "> 37 <r d f : t y p e r d f : r e s o u r c e="#OVariable "/> 38 <i s O V a r i a b l e I n r d f : r e s o u r c e="#FrontOfEnterprise MFrag "/> 39 <o b j e c t I n r d f : r e s o u r c e="#e n t e r p r i s e O f "/> 40 <hasDomain r d f : r e s o u r c e=" : i d 1 "/> 41 <isSubsBy r d f : r e s o u r c e="#E n t e r p r i s e "/> 42 <isArgTermIn r d f : r e s o u r c e="#p r i c e O f 2 "/> 43 </ owl:Thing>
The same thing goes for the enterprise variable. In lines 39-41 we de ne that the enterprise variable is in fact used as the object (objectIn) of the OWL property enterpriseOf, thus it can only be substituted by (isSubsBy) the class that is the range of the enterpriseOf property, which is Enterprise. In addition, the domain also has to be the same (hasDomain), which, in this case, is the blank node :id1. 4
The key to building the bridge that connects the deterministic ontology de ned in OWL and its probabilistic extension de ned in PR-OWL is to understand how to translate one to the other. On the one hand, given a concept de ned in OWL, how should its uncertainty be de ned in PR-OWL in a way that maintains its semantics de ned in OWL? On the other hand, given a random variable de ned in PR-OWL, how should it be represented in OWL in a way that respects its uncertainty already de ned in PR-OWL? Examples of our proposed translation were given above. Here, a schematic is given in Figure 4 for the 2-way mapping between triples and random variables. Functions and predicates are considered as separate cases.
If a property (hasB or dOf) is de ned in OWL, then its domain and range are already represented (A and B; C and D, respectively). The rst thing to be done is to create the corresponding RV in PR-OWL (hasB RV and dOf RV, respectively) and link it to this OWL property through the property defineUncertaintyOf.
For binary relations, the domain of the property (A and C, respectively) will usually be the type (isSubsBy) of the variable ( MFrag.a and MFrag.c, respectively) used in the rst argument (hasB RV 1 and dOf RV 1, respectively) of the RV. For n-ary relations see Section 3.
If the property is non-functional (hasB), then it represents a predicate that may be true or false. Thus, instead of having the possible values of the RV in PROWL (hasB RV) being the range of the OWL property (B), it must be Boolean. So, its range (B) has to be mapped to the second argument (hasB RV 2) of the RV, the same way the domain (A) was mapped to the rst argument (hasB RV 1) of the RV. On the other hand, if the the property is functional (dOf), the possible values of its RV (dOf RV) must be the same as its range (B).
It is important to note that not only is the RV linked to the OWL property by the defineUncertaintyOf, but also to the variables by either subjectIn or objectIn, depending on what they refer to (domain or range of the OWL property, respectively). This feature is especially important when dealing with n-ary relations, where each variable will be associated with a di erent OWL property (see Section 3) for details).
Finally, if the RV is already de ned in PR-OWL with all its arguments and its possible values, the only thing that needs to be done is to create the corresponding OWL property, link the RV to it using the defineUncertaintyOf and make sure that the domain and range of the property matches the RV de nition, as explained previously.
The mapping described in this Section provides the basis for a formal definition 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. 5
Although the semantics was not formally de ned, this paper provided both the syntax and a more in depth description of one of the major changes in PR-OWL 2: a formal mapping between OWL concepts and PR-OWL random variables. First, the importance of a formal mapping was justi ed through an example. Second, a simple solution su cient for 2-way relations was presented. Next, a more complex and robust solution covering n-ary random variables was presented. Finally, a schematic was given for how to do the mapping back and forth between PR-OWL random variables and OWL triples (both predicates and functions).
As future work, this schematic will be formally de ned by explicitly de ning
its semantics. This will be a major contribution of PR-OWL 2. Moreover, a
formalization of an algorithm for performing the mapping from OWL concepts
to PR-OWL RVs, and vice-versa, will be proposed. In addition, PR-OWL 2 will
address other issues described in [