OWL ontologies have gained great popularity as a context modelling tool for intelligent environments due to their expressivity. However, they present some disadvantages when it is necessary to deal with uncertainty, which is common in our daily life and a ects the decisions that we take. To overcome this drawback, we have developed a novel framework to compute fact probabilities from the axioms in an OWL ontology. This proposal comprises the de nition and description of our probabilistic ontology. Our probabilistic ontology extends OWL 2 DL with a new layer to model uncertainty. With this work we aim to overcome OWL limitations to reason with uncertainty, developing a novel framework that can be used in intelligent environments.
In Ambient Intelligence applications, context can be de ned as any data which
can be employed to describe the state of an entity (a user, a relevant object, the
location, etc.) [
One of the most popular techniques for context modelling is OWL ontologies
[
OWL is the common way to encode description logics in real world. However,
when the domain information contains uncertainty, the employment of OWL
ontologies is less suitable [
As in other domains, uncertainty is also present in Ambient Intelligence [
For this reason, we present a novel approach to deal with uncertainty in intelligent environments. This work proposes a method to model uncertainty, that combines OWL ontologies with Bayesian networks. The rest of this article is organized as follows. The next section describes the problem that we address. Section 3 explains the semantics and syntax of our proposal. Section 4 gives an exemplary use case where our proposal is applied and describes how to model it. Finally, section 5 summarizes this work and addresses the future work. 2
In intelligent environments, the lack of information causes incomplete context information and it may be produced by several causes: { Sensors that have run out of batteries. Several sensors, such as wearable devices, depend on batteries to work. { Network problems. Sensors, actuators and computers involved in the environment sensing and monitoring are connected to local networks that can su er network failures. In these cases, the context information may be lost, although the sensors, actuators and computers are working properly. { Remote services' failures. Some systems rely on remote services to provide a functionality or to gather context information. { A system device stops working. Computer, sensors and actuators can su er software and hardware failures that hamper their proper operation.
When one of these issues occurs, the OWL reasoner will infer conclusions that are insu cient to attend the user's needs. Besides, taking into account that factors can improve several tasks carried in intelligent environments, such as ontology-based activity recognition. For instance, WatchingTVActivity can be de ned as an activity performed by a Person who is watching the television in a room:
(1)
If the user is watching the television and the system receives the values of all the sensors, then it is able to conclude that the user's current activity is of the type WatchingTVActivity. In contrast, if the value of the presence sensor is not available, then it is not possible to infer that the user is watching the television.
In addition, sometimes there is not a rule of thumb to classify an individual as a member of a class. For instance, we can classify the action that the system has to perform regarding the current activity of the user. Thus, we can de ne that the system should turn o the television, when the user is not watching it:
However, this concept de nition does not accurately model the reality. The action can ful l every condition expressed in the TurnO TV de nition, but the television should not be turned o . This situation may occur when the user goes to the toilet or answers a call phone in another room, among others.
In these cases in which the information of the domain comes with quantitative
uncertainty or vagueness, ontology languages are less suitable [
In our work, we are more interested in the uncertainty caused by the lack of information rather than the vague knowledge. For this reason, probabilistic approaches are more suitable to solve our problem. 3
Our proposal, called Turambar, combines a Bayesian network model with an
OWL 2 DL ontology in order to handle uncertainty. A Bayesian network [
Turambar is able to calculate the probability associated to a class, object property or data property assertions. These probabilistic assertions have only two constraints: { The class expression employed in the class assertion should be a class. { For positive and negative object property assertions, the object property expression should be an object property.
However, these limitations can be solved declaring a class equivalent to a class expression or an object property as the equivalent of an inverse object property. Examples of probabilistic assertions that can be calculated with Turambar are:
(3) (4) (5) The probabilistic object property assertion expressed in (3) states that John is in Bedroom1 with a probability of 0.7. On the other hand the probabilistic class assertion (4) describes that the Action1 is member of the class WatchingTVActivity with a probability of 0.2. Finally, the probabilistic data property assertion (5) de nes that the television, TV1, is on with a probability of 1.0. The probability associated to these assertions is calculated through Bayesian networks that describe how other property and class assertions in uence each other. In Turambar, the probabilistic relationships should be de ned by an expert. In other words, the Bayesian networks must be generated by hand, since learning a Bayesian network is out of the scope of this paper and it is not the goal of this work. 3.1
Turambar Functionality De nition The classes, object properties and data properties of the OWL 2 DL ontology involved in the probabilistic knowledge are connected to the random variables de ned in the Bayesian network model. For example, the OWL class WatchingTVActivity is connected to at least one random variable, in order to be able to calculate probabilistic assertions about that class. The set of data properties, object properties and classes that are linked to random variables is called Vprob and a member of Vprob, vprobi; such that vprobi 2 Vprob.
In Turambar, every random variable (RV) is associated to a Vprob and every RV's domain is composed of a set of functions that determine the values that a random variable can take, such as V al(RV ) = ff1; f2:::fng and fi 2 V al(RV ). These functions require a property or class and individual to calculate the probabilistic assertion, such as fi : a1; ex ! result where a1 is an OWL individual, ex, a class, data property or object property; result, a class assertion, object property assertion, data property assertion or void (no assertion). In the case, that every function in the domain of a random variable returns void, it means that the random variable is auxiliary. In contrast, if any fi in the domain of a random variable returns a probability associated to an assertion, then the random variable is called nal.
For instance, the data property lieOnBedTime is linked to a random variable named SleepTime whose domain is composed of two functions f1 that check if the user has been sleeping for less than 8 hours and f2 function that checks if the user has been sleeping for more than 8 hours. Both functions are not able to generate assertions, so the random variable SleepTime is auxiliary. By contrast, WatchingTVActivity class is linked to a random variable called WatchingTV whose domain comprises f3 function that checks if an individual is member of the class WatchingTVActivity (e.g. W atchingT V Activity(Activity1) 0:8) and the f4 function which checks if an individual is a member of the complement of WatchingTVActivity.
It is also important to remark that a vprobi can be referenced from several random variables. For example, the TurnO TV depends on the user's impairments, so if the blind user is deaf, it is more likely that the television needs to be turned o . Additionally, having an impairment also a ects to the probability of having another impairment: deaf people have a higher probability of also being mute. In this case, we can link hasImpairment object property with two random variables in order to model it.
Apart from the conditional probability distribution, nodes connected between them may have an associated node context. The context de nes how di erent random variables are related between them and the condition that must ful l. This context establishes an unequivocal relationship in which every individual involved in that relationship should be gatherer before calculating the probability of an assertion. If the relationship is not ful lled then the probabilistic query cannot be answered. For example, to estimate the probability for the TurnO TV, the reasoner needs to know who is the user and in which room he is. For this case the relationship may be the following one isIn(?user; ?room) ^ requiredBy(?action; ?user) ^ hasAppliance(?user; ?tv) , being ?user; ?action; ?tv and ?room variables. So, if we ask for the probability that Action1 is member of TurnO TV, such as Pr(TurnO TV(Action1)), then the rst step to calculate it is to check its context. If everything is right the evaluation of this relationship should return that the Action1 is required only by one user who is only in one room and has only one television. Otherwise, the probability cannot be calculated.
Our proposal can be viewed as a SROIQ(D) extension that includes a probabilistic function P r which maps role assertions and concept assertions to a value between 0 and 1. The sum of the probabilities obtained for a random variable is equal to 1. In contrast, the sum of probabilities for the set of assertions obtained for a vprobi may be di erent from 1. For instance, the object property hasImpairment is related to two random variables one to calculate the probability of being deaf and another one to calculate the probability of being mute. If both random variables have a domain with two functions, we can get four probabilistic assertions that sums 2 instead of 1, but the sum of probabilities obtained in one random variable is 1: { Random variable deaf's assertions: hasImpairment(J ohn; Deaf )0:8 and :hasImpairment(J ohn; Deaf )0:2. { Random variable mute's assertions: hasImpairment(J ohn; M ute)0:7 and :hasImpairment(J ohn; M ute)0:3.
The probability of an assertion that exists in the OWL 2 DL ontology is always 1 although the data property, object properties or class is not member of Vprob. For example, if an assertion states that John is a Person (P erson(J ohn)) and we ask for the probability of this assertion, then its probability is 1, such as P erson(J ohn)1. However, if the data property, object properties or class is not member of Vprob and there is not an assertion in the OWL 2 DL ontology that states it, then the probability for that assertion is unknown. We consider that the probabilistic ontology is satis ed if the OWL 2 DL ontology is satis ed and the Bayesian network model is not in contradiction with the OWL ontology knowledge. 3.2
Turambar Ontology Speci cation In Turambar, a probabilistic ontology comprises an ordinary OWL 2 DL ontology and the dependency description ontology that de nes the Bayesian network model.
The ordinary OWL ontology imports the Turambar annotations ontology, which de nes the following annotations: { turambarOntology annotation de nes the URI of the dependency description ontology. { turambarClass annotation links OWL classes in the ordinary ontology to random variables in the dependency description ontology. { turambarProperty annotation connects OWL data properties and object properties in the ordinary ontology to random variables in the dependency description ontology.
We choose to separate the Bayesian network de nition from the ordinary ontology in order to isolate the probabilistic knowledge de nition from the OWL knowledge. We de ne isolation as the ability of exposing an ontology with an unique URI that locates the traditional ontology and the probabilistic one. So, given the URI of a probabilistic ontology, a compatible reasoner loads the ordinary ontology and the dependency description ontology it. In contrast, a traditional reasoner only loads the ordinary ontology. So, if the Turambar probabilistic ontology is loaded by a traditional reasoner, the traditional reasoner does not have access to the knowledge encoded in the dependency description ontology. In this way, we also want to promote the re-utilization of probabilistic ontologies as simple OWL 2 DL ontologies by traditional systems and the interoperability between our proposal and them.
On the other hand, the dependency description ontology de nes the probabilistic model employed to estimate the probabilistic assertions. To model that knowledge, it imports the Turambar ontology, which de nes the vocabulary to describe the probabilistic model. As the gure 1 shows, the main classes and properties in the Turambar ontology are the following ones: { Node class represents the nodes in Bayesian networks. Node instances are de ned as auxiliary random variables through the property isAuxiliar. The hasProbabilityDistibution object property links Node instances with their corresponding probability distributions and hasState object property associates Node instances with their domains. Furthermore, hasChildren object property and its inverse hasParent set the dependencies between Node instances. Finally, hasContext object property de nes the context for a node and hasVariable object property, the value of the variable that the node requires. { MetaNode is a special type of Node that is employed with non functional object properties and data properties. Its main functionality is to group several nodes that share a context and are related to the same property. For instance, in the case of the hasImpairment object property we can model a MetaNode with two nodes: Deaf and Mute. Both nodes share the same context but have di erent parents and states. The object property compriseNode identi es the nodes that share a context. { State class de nes the values of random variables' domain. In other words, it describes the functions which generate probabilistic assertions. These functions are expressed as a string through the data property stateCondition. { ProbabilityDistribution class represents a probability distribution. Probability distributions are given in form of conditional probability tables. Cells of the conditional probability table are associated to the instances of ProbabilityDistribution through hasProbability object property. { Probability class represents a cell of a conditional probability table, such P (x1 j x2; x3) = value, where x1; x2 and x3 are State individuals and value is the probability value. x1 State is assigned to an instance of Probability class through the hasValue object property and x2 and x3 conditions through hasCondition object property. Finally, the data property hasProbabilityValue sets the probability value for that cell. { Context class establishes the relationships between the nodes of a Bayesian network. Relationships between nodes are expressed as a SPARQL-DL query through the data property relationship. { Variable class represents the variables of the context. Their instances identify the SPARQL-DL variables de ned in the context SPARQL-DL query. The variableName data property establishes the name of the variable. For example, if the context has been de ned with the following SPARQL-DL expression: select ?a ?b where f PropertyValue(p:livesIn, ?a, ?b)g , then we should create two instances of Variable with the variableName property value of a and b, respectively. { Plugin class de nes a library that provides some functions that are referenced by State class instances and are not included as member of the Turambar core. The core functions are the following ones: (i) numbers and strings comparison, (ii) ranges of number and string comparison, (iii) individual instances comparison, (iv) boolean comparison, (v) class memberships checking and (vi) the void assertion to de ne the probability that no assertion involves an individual. Only i, iii and iv are able to generate probabilistic assertions. Every function, except the void function, has their inverse function to check if that value has been asserted as false. 4
We can classify probabilistic approaches to deal with uncertainty in two groups:
probabilistic description logics approaches and probabilistic web ontology
languages [
Fig. 1. Classes and properties de ned by the Turambar ontology
In the rst group, P-CLASSIC [
In contrast, probabilistic web ontology languages combine OWL with
probabilistic formalisms based on Bayesian networks. Since our proposal falls under
this group, we will review in depth the most important works in this category:
BayesOWL, OntoBayes and PR-OWL. The BayesOWL [
In contrast to BayesOWL, OntoBayes [
PR-OWL [
The last version of PR-OWL [
Our proposal is focused on computing the probability of data properties assertions, object properties assertions and class assertions. This issue is only covered by PR-OWL, because BayesOWL only takes into account class membership and OntoBayes, object and data properties.
In addition, we pretend to o er a way to keep the uncertainty information isolated as much as possible from the traditional ontology. With this policy, we want to ease the reutilization of our probabilistic ontologies by traditional systems that do not o er support for uncertainty and the interoperability between them. Furthermore, we aim to avoid that traditional reasoners load unnecessary information about the probabilistic knowledge that they do not need. Thus, if we load the Turambar probabilistic ontology located in http://www.example.org/ont.owl, traditional OWL reasoners load only the knowledge de ned in the ordinary OWL ontology and do not have access to the probabilistic knowledge. In contrast, Turambar reasoner is able to load the ordinary OWL ontology and the dependency description ontology. The Turambar reasoner needs to access to the ordinary OWL ontology to answer traditional OWL queries and to nd the evidences of the Bayesian networks de ned in the dependency description ontology. It is also important to clarify that a class or property can have deterministic assertions and probabilistic assertions without duplicating them due to the links between Bayesian networks' nodes and OWL classes and properties through turambarClass and turambarProperty, respectively. Thanks to this feature, a Turambar ontology has a unique URI that allows it to be used as an ordinary OWL 2 DL ontology without loading the probabilistic knowledge. This characteristic is not o ered by other approaches as far as we know.
Another di erence with other approaches is that we have taken into account the extensibility of our approach through plug-ins to increase the basis functionalities. We believe that it is necessary to o er a straightforward, transparent and standard mechanism to extend reasoner functionality in order to cover heterogeneous domains' needs.
However, our approach has the shortcoming of assuming a simple
attributevalue representation in comparison to PR-OWL. That means that each
probabilistic query involves reasoning about the same xed number of nodes, with
only the evidence values changing from query to query. To solve this drawback,
we can opt to employ situation speci c Bayesian networks [
In this work we have presented a proposal to deal with uncertainty in intelligent environments. Its main features are: a) it isolates the probabilistic information de nition from traditional ontologies, b) it can be extended easily and c) it is oriented to intelligent environments.
As ongoing work, we are developing an extension to SPARQL-DL [
As future work, we plan to create a graphical tool to ease the creation of probabilistic ontologies in order to promote its adoption. On the other hand, we plan to extend its expressivity and evaluate new and better ways to de ne the probabilistic description ontology in order to improve its maintainability. In addition, we are studying a formalism that allows us the de nition of custom function for state evaluation that was independent of the programming language employed.