Dynamic Bayesian Ontology Languages

Dynamic Bayesian Ontology Languages

İsmail İlkan Ceylan
Theoretical Computer Science
TU Dresden, Germany
ceylan@tcs.inf.tu-dresden.de &Rafael Peñaloza
KRDB Research Centre
Free University of Bozen-Bolzano, Italy
rafael.penaloza@unibz.it
Abstract

Many formalisms combining ontology languages with uncertainty, usually in the form of probabilities, have been studied over the years. Most of these formalisms, however, assume that the probabilistic structure of the knowledge remains static over time. We present a general approach for extending ontology languages to handle time-evolving uncertainty represented by a dynamic Bayesian network. We show how reasoning in the original language and dynamic Bayesian inferences can be exploited for effective reasoning in our framework.

Dynamic Bayesian Ontology Languages


İsmail İlkan Ceylan Theoretical Computer Science TU Dresden, Germany ceylan@tcs.inf.tu-dresden.de                        Rafael Peñaloza KRDB Research Centre Free University of Bozen-Bolzano, Italy rafael.penaloza@unibz.it

Copyright © 2019, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.

Introduction

Description Logics (DLs) (?) are a well-known family of knowledge representation formalisms that have been successfully employed for encoding the knowledge of many application domains. In DLs, knowledge is represented through a finite set of axioms, usually called an ontology or knowledge base. In essence, these axioms are atomic pieces of knowledge that provide explicit information of the domain. When mixed together in an ontology, these axioms may imply some additional knowledge that is not explicitly encoded. Reasoning is the act of making this implicit knowledge explicit through an entailment relation.

Some of the largest and best-maintained DL ontologies represent knowledge from the bio-medical domains. For instance, the NCBO Bioportal111http://bioportal.bioontology.org/ contains 420 ontologies of various sizes. In the bio-medical fields it is very common to have only uncertain knowledge. The certainty that an expert has on an atomic piece of knowledge may have arisen from a statistical test, or from possibly imprecise measurements, for example. It thus becomes relevant to extend DLs to represent and reason with uncertainty.

The need for probabilistic extensions of DLs has been observed for over two decades already. To cover it, many different formalisms have been introduced (????). The differences in these logics range from the underlying classical DL used, to the semantics, to the assumptions made on the probabilistic component. One of the main issues that these logics need to handle is the representation of joint probabilities, in particular when the different axioms are not required to be probabilistically independent. A recent approach solves this issue by dividing the ontology into contexts, which intuitively represent axioms that must appear together. The probabilistic knowledge is expressed through a Bayesian network that encodes the joint probability distribution of these contexts. Although originally developed as an extension of the DL  (?), the framework has been extended to arbitrary ontology languages with a monotone entailment relation (?).

One common feature of the probabilistic extensions of DLs existing in the literature is that they consider the probability distribution to be static. For many applications, this assumption does not hold: the probability of a person to have gray hair increases as time passes, as does the probability of a computer component to fail. To the best of our knowledge, there is so far no extension of DLs that can handle evolving probabilities effectively.

In this paper, we describe a general approach for extending ontology languages to handle evolving probabilities. By extension, our method covers all DLs, but is not limited to them. The main idea is to adapt the formalism from (?) to use dynamic Bayesian networks (DBNs) (?) as the underlying uncertainty structure to compactly encode evolving probability distributions.

Given an arbitrary ontology language , we define its dynamic Bayesian extension . We show that reasoning in can be seamlessly divided into the probabilistic computation over the DBN, and the logical component with its underlying entailment relation. In order to reduce the number of entailment checks, we compile a so-called context formula, which encodes all contexts in which a given consequence holds.

Related to our work are relational BNs (?) and their extensions. In contrast to relational BNs, we provide a tight coupling between the logical formalism and the DBN, which allows us to describe evolving probabilities while keeping the intuitive representations of each individual component. Additionally, restricting the logical formalism to specific ontology languages provides an opportunity for finding effective reasoning algorithms.

Bayesian Ontology Languages

To remain as general as possible, we do not fix a specific logic, but consider an arbitrary ontology language consisting of two infinite sets and of axioms and consequences, respectively, and a class of finite sets of axioms, called ontologies, such that if , then for all . The language is associated to a class of interpretations and an entailment relation . An interpretation is a model of the ontology () if for all . entails () if every model of entails . Notice that the entailment relation is monotonic; i.e., if and , then . Any standard description logic (DL) (?) is an ontology language of this kind; consequences in these languages are e.g. concept unsatisfiability, concept subsumption, or query entailment. However, many other ontology languages of varying expressivity and complexity properties exist. For the rest of this paper, is an arbitrary but fixed ontology language, with axioms , ontologies , consequences , and interpretations .

As an example language we use the DL  (?), which we briefly introduce here. Given two disjoint sets and , concepts are built by the grammar rule where and . axioms and consequences are expressions of the form , where and are concepts. An interpretation is a pair where is a non-empty set and maps every to and every to . This function is extended to concepts by , , and . This interpretation entails the axiom (or consequence) iff .

The Bayesian ontology language extends by associating each axiom in an ontology with a context, which intuitively describes the situation in which the axiom is required to hold. The knowledge of which context applies is uncertain, and expressed through a Bayesian network (?).

Briefly, a Bayesian network (?) is a pair , where is a finite directed acyclic graph (DAG) whose nodes represent Boolean random variables, and contains, for every , a conditional probability distribution of given its parents . Every variable is conditionally independent of its non-descendants given its parents. Thus, the BN defines a unique joint probability distribution (JPD) over :

Let be a finite set of variables. A -context is a consistent set of literals over . A -axiom is an expression of the form where is an axiom and is a -context. A -ontology is a finite set of -axioms, such that . A knowledge base (KB) over is a pair where is a BN over and is a -ontology.

We briefly illustrate these notions over the language , an extension of the DL , in the following Example.

Example 1.

Consider the KB where

and is the BN shown in Figure 1.

Figure 1: The BN over the variables

This KB represents a computer failure scenario, where stands for a critical situation, represents the memory failing, and the CPU failing.

The contextual semantics is defined by extending interpretations to evaluate also the variables from . A -interpretation is a pair where and is a propositional interpretation over the variables . The -interpretation is a model of (), where , iff 222We use to distinguish propositional entailment from . or .

It is a model of the -ontology iff it is a model of all the -axioms in . It entails if . The intuition behind this semantics is that an axiom is evaluated to true by all models provided it is in the right context.

Given a -ontology , every propositional interpretation, or world, on defines an ontology Consider the KB provided in Example 1: The world defines the ontology

Intuitively, a contextual ontology is a compact representation of exponentially many ontologies from ; one for each world . The uncertainty in is expressed by the BN , which is interpreted using multiple world semantics.

Definition 2 (probabilistic interpretation).

A probabilistic interpretation is a pair , where is a set of -interpretations and is a probability distribution over such that only for finitely many interpretations . It is a model of the -ontology if every is a model of . is consistent with the BN if for every valuation of the variables in it holds that

The probabilistic interpretation is a model of the KB iff it is a model of and consistent with .

The fundamental reasoning task in , probabilistic entailment, consists in finding the probability of observing a consequence ; that is, the probability of being at a context where holds.

Definition 3 (probabilistic entailment).

Let , and a KB. The probability of w.r.t. the probabilistic interpretation is The probability of w.r.t.  is

It has been shown that to compute the conditional probability of a consequence , it suffices to test, for each world , whether entails  (?).

Proposition 4.

Let be a KB and . Then .

This means that reasoning in can be reduced to exponentially many entailment tests in the classical language . For some logics, this exponential enumeration of worlds can be avoided (?). However, this depends on the properties of the ontological language and its entailment relation, and cannot be guaranteed for arbitrary languages.

Another relevant problem is to compute the probability of a consequence given some partial information about the context. Given a context , the conditional probability is defined via the rule , where

For simplicity, in the rest of this paper we consider only unconditional consequences. However, it should be noted that all results can be transferred to the conditional case.

Example 5.

Consider again the KB from Example 1 and the consequence . We are interested in finding the probability of the computer to fail, i.e. . This can be computed by enumerating all worlds for which , which yields the probability 0.238.

As seen, it is possible to extend any ontological language to allow for probabilistic reasoning based on a Bayesian network. We now further extend this formalism to be able to cope with controlled updates of the probabilities over time.

Dynamic Bayesian Ontology Languages

With , one is able to represent and reason about the uncertainty of the current context, and the consequences that follow from it. In that setting, the joint probability distribution of the contexts, expressed by the BN, is known and fixed. In some applications, see especially (?) this probability distribution may change over time. For example, as the components of a computer age, their probability of failing increases. The new probability depends on how likely it was for the component to fail previously, and the ageing factors to which it is exposed. We now extend to handle these cases, by considering dynamic BNs as the underlying formalism for managing uncertainty over contexts.

Dynamic BNs (DBNs) (??) extend BNs to provide a compact representation of evolving joint probability distributions for a fixed set of random variables. The update of the JPD is expressed through a two-slice BN, which expresses the probabilities at the next point in time, given the current context.

Definition 6 (Dbn).

Let be a finite set of Boolean random variables. A two-slice BN (TBN) over is a pair , where is a DAG containing no edges between elements of , , and contains, for every a conditional probability distribution of given its parents . A dynamic Bayesian network (DBN) over is a pair where is a BN over , and is a TBN over .

A TBN over is depicted in Figure 2.

Figure 2: The TBN over the variables

The set of nodes of the graph can be thought of as containing two disjoint copies of the random variables in . Then, the probability distribution at time depends on the distribution at time . In the following we will use and for , to denote the variables in at time .

As standard in BNs, the graph structure of a TBN encodes the conditional dependencies among the nodes: every node is independent of all its non-descendants given its parents. Thus, for a TBN , the conditional probability distribution at time given time is

We further assume the Markov property: the probability of the future state is independent from the past, given the present state.

In addition to the TBN, a DBN contains a BN that encodes the JPD of at the beginning of the evolution. Thus, the DBN defines, for every , the unique probability distribution

Intuitively, the distribution at time is defined by unraveling the DBN starting from , using the two-slice structure of until copies of have been created. This produces a new BN encoding the distribution over time of the different variables. Figure 3 depicts the unraveling to of the DBN where and are the networks depicted in Figures 1 and 2, respectively.

Figure 3: Three step unraveling of

The conditional probability tables of each node given its parents (not depicted) are those of for the nodes in , and of for nodes in . Notice that has copies of each random variable in . For a given , we call the BN obtained from the unraveling of the DBN to time , and eliminating all variables not in . In particular, we have that .

The dynamic Bayesian ontology language is very similar to , except that the probability distribution of the contexts evolves accordingly to a DBN.

Definition 7 ( Kb).

A knowledge base (KB) is a pair where is a DBN over and is a -ontology. Let be a KB over . A timed probabilistic interpretation is an infinite sequence of probabilistic interpretations. is a model of if for every , is a model of the KB .

In a nutshell, a DBN can be thought of as a compact representation of an infinite sequence of BNs over . Following this idea, a KB expresses an infinite sequence of KBs, where the ontological component remains unchanged, and only the probability distribution of the contexts evolves over time. A timed probabilistic interpretation simply interprets each of these KBs, at the corresponding point in time. To be a model of a KB, must then be a model of all the associated KBs.

Before describing the reasoning tasks for and methods for solving them, we show how the computation of all the contexts that entail a consequence can be reduced to the enumeration of the worlds satisfying a propositional formula.

Compiling Contextual Knowledge

From Proposition 4, we see that reasoning in can be reduced to checking, for every world , whether . This reduces probabilistic reasoning to a sequence of standard entailment tests over the original language . However, each of these entailments might be very expensive. For example, in the very expressive DL , deciding an entailment is already NExpTime-hard (?). Rather than repeating this reasoning step for every world, it makes sense to try to identify the relevant worlds a priori. We do this through the computation of a context formula.

Definition 8 (context formula).

Let be a -ontology, and . A context formula for w.r.t.  is a propositional formula such that for every interpretation of the variables in , it holds that iff .

The idea behind this formula is that, for finding whether , it suffices to check whether the valuation satisfies . This test requires only linear time on the length of the context formula. The context formula can be seen as a generalization of the pinpointing formula (?) and the boundary (?), defined originally for classical ontology languages.

Example 9.

Consider again the -ontology from Example 1. The formula is a context formula for w.r.t. . In fact, the valuation satisfies this formula.

Clearly, computing the context formula must be at least as hard as deciding an entailment in : if we label every axiom in a classical -ontology with the same propositional variable , then the boundary formula of w.r.t. this -ontology is iff . On the other hand, the algorithm used for deciding the entailment relation can usually be modified to compute the context formula. Using arguments similar to those developed for axiom pinpointing (??), it can be shown that for most description logics, computing the context formula is not harder, in terms of computational complexity, than standard reasoning. In particular this holds for any arbitrary ontology language whose entailment relation is ExpTime-hard. This formula can also be compiled into a more efficient data structure like binary decision diagrams (?). Intuitively, this means that we can compute this formula using the same amount of resources needed for only one entailment test, and then use it for verifying whether the sub-ontology defined by a world entails the consequence in an efficient way.

Reasoning in

Rather than merely computing the probability of currently observing a consequence, we are interested in computing the probability of a consequence to follow after some fixed number of time steps .

Definition 10 (probabilistic entailment with time).

Let be a KB and . Given a timed interpretation and , the probability of at time w.r.t.  is . The probability of at time w.r.t.  is

We show that probabilistic entailment over a fixed time bound can be reduced to probabilistic entailment defined for BOLs by unravelling the DBN.

Lemma 11.

Let be a KB, , and . Then the probability of at time w.r.t.  is given by

Proof.

(Sketch) A timed model of is a sequence of probabilistic interpretations , where each is a model of the KB . We use this fact to show that

(1)
(2)
(3)

where (1) follows from Definition 10, (2) holds by definition, and (3) follows from Proposition 4. ∎

Lemma 11 provides a method for computing the probability of an entailment at a fixed time . One can first generate the BN , and then compute the probability w.r.t.  of all the worlds that entail . Moreover, using a context formula we can compile away the ontology and reduce reasoning to standard inferences in BNs, only.

Theorem 12.

Let be a KB, , a context formula for w.r.t. , and . Then the probability of at time w.r.t.  is given by .

Proof.

By Lemma 11 and the definition of a context formula, we have

which proves the result. ∎

This means that one can first compute a context formula for and then do probabilistic inferences over the DBN to detect the probability of satisfying at time . For this, we can exploit any existing DBN inference method. One option is to do variable elimination over the -step unraveled DBN , to compute . Assuming that is fixed, it suffices to make inferences (one for each world) over and the same number of propositional entailment tests over the context formula. If entailment in is already exponential, then computing the probability of at time is as hard as deciding entailments.

The previous argument only works assuming a fixed time point . Since it depends heavily on computing (e.g., via variable elimination), it does not scale well as increases. Other methods have been proposed for exploiting the recursive structure of the DBN. For instance, one can use the algorithm described in (?) that provides linear scalability over time. The main idea is to compile the structure into an arithmetic circuit (?) and use forward and backward message passing (?).

While computing the probability of a consequence at a fixed point in time is a relevant task, it is usually more important to know whether the consequence can be observed within a given time limit. In our computer example, we would be interested in finding the probability of the system failing within, say, the following twenty steps.

Abusing of the notation, we use the expression , , to denote that the consequence does not hold; i.e., iff . Thus, for example, is the probability of not holding at time . To find the probability of observing in the first time steps, one can alternatively compute the probability of not observing in any of those steps. Formally, for a timed interpretation and , we define

Definition 13 (time bounded probabilistic entailment).

The probability of observing in at most steps w.r.t. the KB is .

Just as before, given a constant , it is possible to compute by looking at the -step unraveling of . More precisely, to compute , it suffices to look at all the valuations of such that for all , it holds that . These valuations correspond to an evolution of the system where the consequence is not observed in the first steps. The probability of these valuations w.r.t.  then yields the probability of not observing this consequence. We thus get the following result.

Theorem 14.

Let be a KB, , a context formula for w.r.t. , and . Then

Proof (Sketch).

Using the pithy interpretations of the crisp ontologies , we can build a timed interpretation such that , in a way similar to Theorem 12 of (?). The existence of another timed interpretation such that contradicts the properties of the pithy interpretations. Thus, we obtain that

This means that the probability of observing a consequence within a fixed time-bound can be computed by simply computing the context formula and then performing probabilistic a posteriori computations over the unraveled BN. In our running example, the probability of observing a computer failure in the next 20 steps is simply

Thus, the computational complexity of reasoning is not affected by introducing the dynamic evolution of the BN, as long as the time bound is constant. Notice, however, that the number of possible valuations grows exponentially on the time bound . Thus, for large intervals, this approach becomes unfeasible.

By extending the time limit indefinitely, we can also find the probability of eventually observing the consequence (e.g., the probability of the system ever failing). The probability of eventually observing w.r.t.  is given by . Notice that is monotonic on and bounded by ; hence is well defined.

Observe that Theorem 14 cannot be used to compute the probability of eventually observing since one cannot necessarily predict the changes in probabilities of finding worlds that entail the consequence . Rather than considering these increasingly large BNs separately, we can exploit methods developed for probability distributions that evolve over time. This will also allow us to extract more information from KBs.

It is easy to see that every TBN defines a time-homogeneous Markov chain over a finite state space. More precisely, if is a TBN over , then is the Markov chain, where every valuation of the variables in is a state and the transition probability distribution given the current state is described by the BN obtained from adding as evidence to the first slice of . For example, the TBN from Figure 2 yields the conditional probability distribution given that was observed at time depicted in Figure 4.

Figure 4:

From this, we can derive the probability of observing at time given that it was observed at time , which is .

We extend the notions from Markov chains to TBNs in the obvious way. In particular, the TBN is irreducible if for every two worlds , the probability of eventually reaching given is greater than 0. It is aperiodic if for every world there is an such that for all , it holds that . A distribution over the worlds is stationary if holds for every world . It follows immediately that if is irreducible and aperiodic, then it has a unique stationary distribution (?).

Given a TBN over , let now be the set of all stationary distributions over the worlds of . For a world , define to be the smallest probability assigned by any stationary distribution of to . If , then we know that, regardless of the initial distribution, in the limit we will always be able to observe the world with a constant positive probability. In particular, this means that the probability of eventually observing equals 1. Notice moreover that this results is independent of the initial distribution used.

We can take this idea one step forward, and consider sets of worlds. For a propositional formula , let

In other words, expresses the minimum probability of satisfying in any stationary distribution of . From the arguments above, we obtain the following theorem.

Theorem 15.

Let be a KB over with , , and a context formula for w.r.t. . If , then .

In particular, if is irreducible and aperiodic, contains only one stationary distribution, which simplifies the computation of the function . Unfortunately, such a simple characterization of cannot be given when . In fact, in this case the result may depend strongly on the initial distribution.

Example 16.

Let , , and consider the TBN over defined by and . It is easy to see that any distribution over the valuations of is stationary. For every initial distribution , if where , then .

So far, our reasoning services have focused on predicting the outcome at future time steps, given the current knowledge of the system. Based on our model of evolving probabilities, the distribution at any time depends only on time , if it is known. However, for many applications it makes sense to consider evidence that is observed throughout several time steps. For instance, in our computer failure scenario, the DBN ensures that, if at some point a critical situation is observed ( is true), then the probability of observing a memory or CPU failure in the next step is higher. That is, the evolution of the probability distribution is affected by the observed value of the variable .

Suppose that we have observed over the first time steps that no critical situation has occurred, and we want to know the probability of a computer failure. Formally, let be a consistent set of literals over . We want to compute the probability of observing at time given the evidence . This is just a special case of bounded probabilistic entailment, where the worlds are not only restricted w.r.t. the context formula but also w.r.t. the evidence .

The efficiency of this approach depends strongly on the time bound , but also on the structure of the TBN . Recall that the complexity of reasoning in a BN depends on the tree-width of its underlying DAG (?). The unraveling of produces a new DAG whose tree-width might increase with each unraveling step, thus impacting the reasoning methods negatively.

Conclusions

We have introduced a general approach for extending ontology languages to handle time-evolving probabilities with the help of a DBN. Our framework can be instantiated to any language with a monotonic entailment relation including, but not limited to, all the members of the description logic family of knowledge representation formalisms.

Our approach extends on ideas originally introduced for static probabilistic reasoning. The essence of the method is to divide an ontology into different contexts, which are identified by a consistent set of propositional variables from a previously chosen finite set of variables . The probabilistic knowledge is expressed through a probability distribution over the valuations of which is encoded by a DBN.

Interestingly, our formalism allows for reasoning methods that exploit the properties of both, the ontological, and the probabilistic components. From the ontological point of view, we can use suplemental reasoning to produce a context formula that encodes all the possible worlds from which a wanted consequence can be derived. We can then use standard DBN methods to compute the probability of satisfying this formula.

This work represents first steps towards the development of a formalism combining well-known ontology languages with time-evolving probabilities. First of all, we have introduced only the most fundamental reasoning tasks. It is possible to think of many other problems like finding the most plausible explanation for an observed event, or computing the expected time until a consequence is derived, among many others.

Finally, the current methods developed for handling DBNs, although effective, are not adequate for our problems. To find out the probability of satisfying the context formula , we need compute the probability of each of the valuations that satisfy at different points in time. Even using methods that exploit the structure of the DBN directly, the information of the context formula is not considered. Additionally, with the most efficient methods to-date, it is unclear how to handle the evidence over time effectively. Dealing with these, and other related problems, will be the main focus of our future work.

Acknowledgments

İsmail İlkan Ceylan is supported by DFG within the Research Training Group “RoSI” (GRK 1907). Rafael Peñaloza was partially supported by DFG through the Cluster of Excellence ‘cfAED,’ while he was still affiliated with TU Dresden and the Center for Advancing Electronics Dresden, Germany.

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