Standard semantic technologies propose powerful means for knowledge representation as well as enhanced reasoning capabilities to modern applications. However, the question of dealing with uncertainty, which is ubiquitous and inherent to real world domain, is still considered as a major deficiency. We need to adapt those technologies to the context of uncertain representation of the world. Here, this issue is examined through the evidential theory, in order to model and reason about uncertainty in the assertional knowledge of the ontology. The evidential theory, also known as the DempsterShafer theory, is an extension of probabilities and proposes to assign masses on specific sets of hypotheses. Further on, thanks to the semantics (hierarchical structure, constraint axioms and properties defined in the ontology) associated to hypotheses, a consistent frame of this theory is automatically created to apply the classical combinations of information and decision process offered by this mathematical theory.
Uncertainty is an important characteristic of data and information handled by
realworld applications. The term "uncertainty" refers to a variety of forms of imperfect
knowledge, such as incompleteness, vagueness, randomness, inconsistency and
ambiguity. In this approach, we consider only the epistemic uncertainty, due to lack
of knowledge (incompleteness) and the inconsistency, due to conflicting testimonies
or reports. This paper presents a proposal on a possible way to tackle the issue of
representing and reasoning on this type of uncertainty in semantic applications, by
using the Dempster–Shafer theory [
The following section of this paper introduces the basic definitions and notations of the Dempster–Shafer theory. Section 3 presents our ontology modeling of the representation of uncertainty, using evidential theory. In the fourth section, we address how to reason with the evidential theory while benefiting from the semantics included in the domain ontology. Section 5 proposes to position our approach by comparing it with already existing works in the domain of uncertainty and the Semantic Web. 2
The Dempster–Shafer theory [
Let Ω be the universal set, also called the discernment frame. It is the set of all the N states (hypothesis) under consideration: Ω = {H1 , H 2 ,..H N }.
The universal set is supposed to be exhaustive and all hypotheses are exclusives. Exhaustivity refers to the closed-world principle. From this universal set, we can define a set, noted 2Ω. It is called the power set and is the set of all possible sub-sets of Ω, including the empty set. It is defined as follows:
2Ω = {A A ⊆ Ω} = {Ø,{H1},...,{H N },{H1, H 2 },...,Ω}. 2.2
A source, who believes that one or more states in the power set of Ω might be true,
can assign belief mass to these states. Formally, a mass function is defined by:
m : 2Ω → [
(2) A∈2Ω This quantity differs from a probability since the total mass can be given either to singleton hypothesis Hn or to composite ones.
The main other belief measures are belief and plausibility. Belief bel(A) for a set A is defined as the sum of all the masses of the subsets of the set of interest: bel( A) = ∑ m(B) B B⊆ A ∀A ⊆ Ω .
It is the degree of evidence that directly supports the given hypothesis A at least in part, forming a lower bound. The plausibility pl(A) is the sum of all the masses of the sets B that intersect the set of interest A: pl( A) =
∑ m(B) B B∩ A≠Ø ∀A ⊆ Ω . pl(A) can be interpreted as the part of belief which could be potentially allocated to A, taking into account the elements that do not contradict this hypothesis. It is seen as an upper bound.
Modeling by masses through the evidential theory would be useless without an adequate combination enabling the fusion of a set of information sources. This is especially the role of the Dempster’s rule of combination. Namely, it combines two independent sets of mass assignments (i.e. from difference sources). The combination (called the joint mass) is calculated from the two sets of masses m1 and m2 in the following manner: ∑ m1(B)m2 (C) B∩C=A (m1 ⊕ m2 )( A) = 1 - K12 0
A ≠ Ø A = Ø . where K12 = ∑m1(B)m2 (C) . (6)
B∩C=Ø
K is a measure of the amount of conflict between the two mass sets. K is ranging from
0 to 1. Dempster’s rule corresponds to the normalized conjunctive operator. Other
combination rules exist, such as the disjunctive combination and other operators that
reassign the amount of conflict differently [
The first step of our approach is to model and represent the uncertainty through ontologies. Modeling is proposed through a specific ontology that needs to be imported in the initial domain ontology. This initial domain ontology is the ontology we want to instantiate in an uncertain way. The imported ontology is called DSOntology. It is described in the following. (3) (4) (5)
This ontology is a formal representation of the theory of Dempster-Shafer, as it
proposes a shared understanding of the main concepts: mass, belief, plausibility,
source, etc. It is non-domain specific, since one can use it in every area of knowledge.
It has been coded in OWL2 language [
The main classes are Uncertain_concept and DS_concept. The DS_concept class links the hypothesis, with the source and the numerical amount of belief related to the hypothesis. The hypothesis consists either of a singleton or a union of hypotheses. Hypotheses are in fact instances of the domain ontology. Instances are either individuals of classes or instances of properties. The Uncertain_concept class links together all the DS_concept that are related to the same context. Indeed, the uncertainty is embodied by several candidate instances (with an assigned belief) and the uncertainty is concretely instantiated through one instance of Uncertain_concept. Uncertain_concept enables to retrieve the set of hypotheses under consideration, i.e. the power set 2Ω.
In order to represent uncertainty both on individuals and on asserted properties, DS_concept and Uncertain_concept have been specialized. They are specified in subclasses XX_class and XX_property (XX prefix representing both DS and Uncertain). Uncertain_concept is now an equivalent class to the union of Uncertain_property and Uncertain_class, while the latter two are disjoint. Respectively, this holds for DS-concept and its subclasses.
The hasDS_hypothesis object property relates an instance of DS_class to a set of candidate individuals. Concerning candidate properties, things have been done differently. Indeed, OWL properties are not first-class citizens, contrary to OWL classes; as such OWL properties cannot be related to each others: a property cannot be the subject or object of another property. To get around this, an object property hasUncertain_property has been introduced. The original subject of the candidate property is the subject of hasUncertain_property. The domain of hasUncertain_property is intuitively the class Uncertain_Property. Then, DS_Property instances are directly the subject of the candidate properties while their object remains unchanged.
An illustration of the use of the DS-Ontology is given in the next section.
As with the Dempster-Shafer theory, the modeling of ignorance is made possible. It
is realized through an instance of DS_concept linked to all hypothetical instances.
Ontologies evolve within the open world assumption. However, the original
evidential theory assumes a closed world and that is why the measure of the amount
of conflict exists. Therefore, we should for instance opt for an Open Extended World
extension of the Dempster-Shafer theory [
We represent numerical evidential belief through a specificUncertaintyDatatype which is a user-defined datatype defined in our DS-Ontology to restrict its value to an xsd:double ranging from 0 to 1.
In our model, Uncertain_concept and DS_concept are classes that let grouping
together collected pieces of information about an uncertain instance we want to
model and reason about. It can be viewed as a reification process, where an
addressable object is created as a proxy for non-addressable objects. Informally,
reification is often referred to as “making something a first-class citizen” within the
scope of a particular system. Reification is one of the most frequently used techniques
of conceptual analysis and knowledge representation. Even if RDF language enables
reification process [
Our applications aim mainly at observing real world situations through different perspectives (sources) and give an understandable and fused analysis of what is going on in this situation to the final decision maker. This simplified scenario involves here two distinct sources. One is a human while the other is an automatic sensor, such as radar. They both want to express that something is going into a specific direction; the “something” entity is the same object for both sources; however, they are not sure about how to identify this object. Indeed, the radar source can only distinguish a land vehicle from an aircraft; it assigns here a more important belief on the fact that it is an instance of a land vehicle. The second source is a human, who has a slight and far away view of the situation is assigning different beliefs to an instance of car which looks like red, or a fire truck or a more imprecisely one to a land vehicle. In most cases, we do not have to assess the belief assigned to hypotheses by ourselves, it is directly given by the sources according to their condition of use (e.g. meteorology, proximity, etc.) and we apply possibly a weakening coefficient according to the source reliability. The structural knowledge of this domain is modeled through an ontology (http://ontology-uri.owl), whose hierarchical structure is captured in figure 2.
In addition to the hierarchical structure of the knowledge, domain and range of properties are also defined, as well as additional information concerning a priori information about the world. For instance, in this domain ontology, it is mentioned that a fire truck individual is always associated to the property hasMainColor with the value red. According to the sources and to the domain ontology, the assertional knowledge of this ontology involves: - http://ontology-uri.owl#direction: an individual of class http://ontologyuri.owl#Direction - respectively #landVehicle for class #LandVehicle - respectively #aircraft for class #Aircraft - respectively #fireTruck for class #FireTruck - respectively #red for class #Color - respectively #car for class #Car which is linked to the individual #red through the #hasMainColor property.
The set of candidate instances are: {#landVehicle, #aircraft, #fireTruck, #car}. We refer here to IRI instances only with their local name, omitting the namespace. Regarding the Dempster-Shafer theory, the masses are assigned by the sources as: • mradar({#landVehicle}) = 0.6 ; mradar({#aircraft}) = 0.1 ; mradar({#landVehicle, #aircraft}) = 0.3 • mhuman({#car}) = 0.2 ; mhuman({#fireTruck}) = 0.4 ; mhuman({#landVehicle}) = 0.4 This domain ontology imports the DS-Ontology, in order to represent all these pieces of knowledge within the domain ontology. Two more individuals are created to represent the sources: - #human for class http://DS-Ontology.owl#Reporting_Source - #radar for class http://DS-Ontology.owl#Reporting_Source The following figure illustrates through a non-formal ontological schema, how the instances are linked together. Once the uncertainty contained in the information has been represented, reasoning processes have to be conducted to fuse the different observation and eventually decide of the instance with the most likelihood. This section has to be viewed as the chronological steps that are realized by the system in order to reason on the uncertain pieces of information represented through the DS-Ontology.
One Uncertain_concept instance of the DS-Ontology groups a set of candidate instances together (either individuals or properties). From this set of instances, we want to determine automatically a consistent frame of discernment, according to the Dempster-Shafer theory. The underlying assumptions of the theory are: an exhaustive frame of discernment and the exclusivity of elements of Ω (see section 2.1). In this paper, we have already managed the first constraint within the Open World assumption of ontologies in the modelling of the DS-Ontology. The second constraint of the frame of discernment is the exclusivity of its elements. This implies that each singleton hypothesis (i.e. the elements of Ω) are disjoint. In other words, if H1 and H2 are two singletons, we cannot have H1 ⊂ H2 or even H1 ∩ H2 ≠ ∅. In the instantiation example, #fireTruck and #car individuals are semantically “included” in #landVehicle. As there is an inclusion, #fireTruck and #car individuals have also a non-null intersection with the #landVehicle individual. Moreover, #fireTruck has a non-null intersection with #car. Indeed, these two individuals are sharing many
The semantic inclusion is quite straightforward to determine. Indeed, in case the instances are property assertions, for example if a property P1 has for ancestor P2, then we say that P1 is included in P2. Otherwise, in case the instances are individuals and they have zero or the same properties (or some included property), then there is an inclusion. In all other cases, the inclusion does not hold.
Concerning semantic intersection, things go a little further. First of all, logically, if
two instances have already a semantic inclusion, then they also have a non-null
semantic intersection. In all other cases, we will consider that two instances have a
non-null intersection when their semantic similarity is exceeding a certain threshold.
More specifically, our similarity measure is a global function, which combines
existing similarity measure defined in literature. As for individuals, it is a mixture of
similarity measure of their respective types and similarity measures concerning their
relations. Wu & Palmer similarity measure [
2 * depth(C ) conSim(C1, C 2) = depthC (C1) + depthC (C 2) . characteristics in common: they are both land vehicles and their main colors are in both cases red.
To deal with this second constraint, we take into account the explicit and inferred semantics of the domain ontology to generate the discernment frame. The granularity of the set of candidate instances affects the generation of the discernment frame. The semantic will help us determining the inclusion of hypotheses as well as the semantic similarity between instances. The whole set of candidate instances will help us fixing a threshold for semantic distances. (7) (8) The other combined similarity measures count the number of identical properties versus the number of different properties related to the two individuals. This is calculated both for object properties and datatype properties. On equation (8), I1 and I2 are the two individuals for which the global semantic similarity measure is calculated. For object properties (respectively for datatype properties), nbProp(I) is the number of object properties (resp. of datatype properties) of individual I. nbPropComm(I1,I2) is the number of common properties - identical predicate and related individual or value - for the two individuals I1 and I2. These three similarity measures focusing on the similarity of the types of individuals and their characteristics (through the datatype and object properties) are combined through a weighted mean.
propSim(I1, I 2) = 2 * nbPropComm(I1, I 2) .
nbProp(I1) + nbProp(I 2) Once the cross-similarity measure of the set of all candidate instances is calculated, the threshold is fixed through a clustering method. The threshold is thus varying according to all the computed semantic similarities. This process permits to adapt the granularity of the set of candidate instances. It translates our general impression that the concept of a compact car is closer to the concept of minivan than of a plane’s; however the concept of a compact car is closer to the concept of plane than of a book’s. In the first case, the intersection should be brought by the pair (compact car, minivan), whereas in the latter, it should be brought by the pair of (compact car, plane). It should be noted that, in both cases, the concepts of compact car and plane have the same semantic similarity. As a consequence, the semantic intersection is seen as a Boolean condition on the similarity measure exceeding the threshold. Finally, we consider the evidential set inclusion (respectively intersection) as equivalent to the semantic inclusion (respectively intersection). In case of our scenario, the intersection and inclusion are graphically represented on the figure below.
Once the intersection and inclusion of candidate instances identified, we are able to set up a consistent frame of discernment. For this, we reframe the set of candidate instances into single or composite disjoint hypotheses.
In case of a discovered intersection between two candidate instances #inst1 and #inst2, #inst1 is reformulated as the union of two singletons {H1, Hinters} and #inst2 as {H2, Hinters}. In case of discovered inclusions between two candidate instances #inst1 and #inst2, where #inst1 is included in #inst2, #inst1 is represented by a single hypothesis {H1} and #inst2 by the union of hypotheses {H2, H1}. Single hypotheses, grouped together, constitute the frame of discernment. In fact, each initial candidate instance belongs to the power-set of the frame of discernment. Taking our scenario, each candidate instance can now be decomposed as such: - #aircraft = {H1} - #car = {H2, H3} - #fireTruck = { H3, H4} - #landVehicle = {H2, H3, H4, H5} Indeed, relying on Figure 4, #aircraft instance has no intersection nor inclusion; thus, it constitutes a single hypothesis within the frame of discernment. The non-null intersection, between #fireTruck and #car instances, has been modeled through a common and shared single hypothesis: H3. Finally, the inclusion brought by #landVehicle results in the union of the set of single hypotheses of #fireTruck and #car, in addition to its own singleton H5.
Once the discernment frame has been obtained, we can reformulate in the DempsterShafer formalism, the basic mass assignment of the scenario: • mradar({H2, H3, H4, H5}) = 0.6 ; mradar({H1}) = 0.1 ; mradar({H1, H2, H3, H4, H5}) = 0.3 • mhuman({H2, H3}) = 0.2 ; mhuman({H3, H4}) = 0.4 ; mhuman({H2, H3, H4, H5}) = 0.4 We are now able to apply directly the classical combination rules found in the Dempster-Shafer theory, and then go through the decision process. 5
During the last decade, approaches considering both uncertainty and the Semantic Web have been proposed. In this section, we mention some of them in order to position and compare our work. We consider their goal, underlying mathematical theory and processes.
Fuzzy and rough set theories aim to model vagueness and uncertainty. Regarding
fuzzy sets, classes are considered to have unsharp definitions. fuzzyDL approach [
Probabilistic adaptations or extensions (Pr-OWL [
Approaches in [13, 14 and 15] are more related to our chosen mathematical theory
as they directly deal with evidential theory. [
Other reports enlarging the state-of-the-art to all ontology languages can be found
in [
This paper proposes a solution in order to handle uncertainty within ontologies. Our approach is relying on current W3C standards. Modeling of uncertainty is realized through an imported pre-defined ontology: the DS-Ontology. Uncertain instantiation of the domain ontology is performed through the use of this imported DS-Ontology. The DS-Ontology relies on the theory of Dempster-Shafer, which manages uncertainty, as well as imprecision and ignorance. This paper has underlined some key issues that have to be dealt when implementing such parallelism between a formal mathematical theory to manage uncertainty and semantic world. The assumption of Open World in ontologies is one of these issues. Reasoning on uncertainty is made possible through an automatic generation of the frame of discernment. For that purpose, Boolean semantic operators, such as the intersection and inclusion, have been developed based on the semantic expressivity of the domain ontology. As a consequence, this paper provides a double and mutual contribution in the domains of the Semantic Web and of uncertain theories, which benefits clearly from the semantics of the hypotheses.
Further researches are also in discussion to extend the reasoning over the Boolean inclusion and intersection of candidate instances. Indeed, it could be interesting to keep the semantic similarity degree (which is a real ranging from 0 to 1) and use it instead of Boolean notions within the theory of Dempster-Shafer. This could be made by rearranging the basic measures of belief and plausibility and of the rules of combination.