Dynamic Bayesian Ontology Languages

Dynamic Bayesian Ontology Languages

˙Ismail ˙Ilkan Ceylan

Theoretical Computer Science TU Dresden, Germany

ceylan@tcs.inf.tu-dresden.de

Rafael Pe ˜naloza

KRDB Research Centre Free University of Bozen-Bolzano, Italy

rafael.penaloza@unibz.it

Abstract

Many formalisms combining ontology languages with uncer- tainty, usually in the form of probabilities, have been stud- ied over the years. Most of these formalisms, however, as- sume that the probabilistic structure of the knowledge re- mains static over time. We present a general approach for extending ontology languages to handle time-evolving un- certainty 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.

Introduction

Description Logics (DLs) (Baaderet al. 2007) are a well- known family of knowledge representation formalisms that have been successfully employed for encoding the knowl- edge of many application domains. In DLs, knowledge is represented through a finite set of axioms, usually called an ontologyor knowledge base. In essence, these axioms are atomic pieces of knowledge that provide explicit informa- tion 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 Bioportal¹ contains 420 ontologies of various sizes. In the bio-medical fields it is very common to have only uncertain knowledge. The certainty that an ex- pert has on an atomic piece of knowledge may have arisen from a statistical test, or from possibly imprecise measure- ments, 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 (Jaeger 1994;

Lukasiewicz and Straccia 2008; Lutz and Schr¨oder 2010;

Klinov and Parsia 2011). The differences in these logics range from the underlying classical DL used, to the seman- tics, to the assumptions made on the probabilistic compo- nent. One of the main issues that these logics need to handle Copyright c2015, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.

1http://bioportal.bioontology.org/

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 divid- ing the ontology into contexts, which intuitively represent axioms that must appear together. The probabilistic knowl- edge is expressed through a Bayesian network that encodes the joint probability distribution of these contexts. Although originally developed as an extension of the DL EL (Cey- lan and Pe˜naloza 2014b), the framework has been extended to arbitrary ontology languages with a monotone entailment relation (Ceylan and Pe˜naloza 2014a).

One common feature of the probabilistic extensions of DLs existing in the literature is that they consider the prob- ability 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 extend- ing 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 (Cey- lan and Pe˜naloza 2014a) to usedynamicBayesian networks (DBNs) (Murphy 2002) as the underlying uncertainty struc- ture to compactly encode evolving probability distributions.

Given an arbitrary ontology languageL, we define its dy- namic Bayesian extensionDBL. We show that reasoning in DBLcan be seamlessly divided into the probabilistic com- putation over the DBN, and the logical component with its underlying entailment relation. In order to reduce the num- ber of entailment checks, we compile a so-called context formula, which encodes all contexts in which a given conse- quence holds.

Related to our work are relational BNs (Jaeger 1997) and their extensions. In contrast to relational BNs, we pro- vide a tight coupling between the logical formalism and the DBN, which allows us to describe evolving probabilities while keeping the intuitive representations of each individ- ual component. Additionally, restricting the logical formal- ism 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 arbitraryontology languageLconsist- ing of two infinite setsAandCofaxiomsandconsequences, respectively, and a classO ⊆ ℘fin(A)of finite sets of ax- ioms, calledontologies, such that ifO ∈O, thenO⁰∈Ofor allO⁰ ⊆ O. The languageLis associated to a classIofin- terpretationsand anentailment relation|=⊆I×(A∪C). An interpretationI ∈Iis amodelof the ontologyO(I |=O) if I |= α for allα ∈ O. O entails c ∈ C (O |= c) if every model ofOentailsc. Notice that the entailment rela- tion is monotonic; i.e., ifO |= candO ⊆ O⁰ ∈ O, then O⁰ |=c. Any standard description logic (DL) (Baaderet al.

2007) 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,Lis an arbitrary but fixed ontology language, with axiomsA, ontologiesO, consequencesC, and interpretationsI.

As an example language we use the DLEL (Baader et al. 2005), which we briefly introduce here. Given two dis- joint sets N_C andN_R, EL conceptsare built by the gram- mar ruleC ::= A | > | CuC | ∃r.C whereA ∈ NC

andr ∈N_R. ELaxiomsandconsequencesare expressions of the form C v D, where C and D are concepts. An interpretation is a pair(∆^I,·^I)where∆^I is a non-empty set and ·^I maps every A ∈ NC to A^I ⊆ ∆^I and every r ∈ NR to r^I ⊆ ∆^I ×∆^I. This function is extended to concepts by >^I := ∆^I, (CuD)^I:=C^I∩D^I, and (∃r.C)^I:={d| ∃e∈C^I.(d, e)∈r^I}. This interpretation entailsthe axiom (or consequence)CvDiffC^I ⊆D^I.

The Bayesian ontology languageBLextendsLby associ- ating each axiom in an ontology with a context, which intu- itively describes the situation in which the axiom is required to hold. The knowledge of which context applies is uncer- tain, and expressed through a Bayesian network (Ceylan and Pe˜naloza 2014a).

Briefly, a Bayesian network (Darwiche 2009) is a pair B = (G,Φ), whereG = (V, E)is a finite directed acyclic graph (DAG) whose nodes represent Boolean random vari- ables, andΦcontains, for everyx∈V, a conditional proba- bility distributionPB(x|π(x))ofxgiven its parentsπ(x).

Every variable x ∈ V is conditionally independent of its non-descendants given its parents. Thus, the BNBdefines a unique joint probability distribution (JPD) overV:

P_B(V) = Y

x∈V

P_B(x|π(x)).

LetV be a finite set of variables. AV-contextis a con- sistent set of literals overV. AV-axiom is an expression of the formhα:κiwhereα ∈ A is an axiom and κis a V-context. AV-ontologyis a finite setOofV-axioms, such that{α | hα:κi ∈ O} ∈ O. ABLknowledge base(KB) overV is a pairK= (O,B)whereBis a BN overV andO is aV-ontology.

We briefly illustrate these notions over the languageBEL, an extension of the DLEL, in the following Example.

x

y z

x .4

z

x y .6

x ¬y .1

¬x y .3

¬x ¬y 1 y

x .5

¬x .1

Figure 1: The BNB1over the variablesV ={x, y, z}

Example 1. Consider theBELKBK₁= (B₁,O₁)where O1={ hCompv ∃use.Memu ∃use.CPU:∅i,

h∃use.FailMemvFailComp:{x}i, h∃use.FailCPUvFailComp:{x}i,

h∃use.FailMemu ∃use.FailCPUvFailComp:{¬x}i, hMemvFailMem:{y}i,hCPUvFailCPU:{z}i}, andB1is the BN shown in Figure 1.

This KB represents a computer failure scenario, wherex stands for a critical situation,yrepresents the memory fail- ing, andzthe CPU failing.

The contextual semantics is defined by extending in- terpretations to evaluate also the variables from V. A V-interpretation is a pair I = (I,V^I) where I ∈I and V^I is a propositional interpretation over the variables V. The V-interpretation I = (I,V^I) is a model of hα:κi (I|=hα:κi), whereα∈A, iff(V^I6|=pκ)²or(I |=α).

It is amodelof theV-ontologyOiff it is a model of all theV-axioms inO. Itentailsc∈CifI |=c. 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 V-ontology O, every propositional inter- pretation, or world, W on V defines an ontology OW :={α| hα:κi ∈O,W |=pκ}. Consider the KB K1

provided in Example 1: The worldW ={x,¬y, z}defines the ontology

O_W={Compv ∃use.Memu ∃use.CPU,

∃use.FailMemvFailComp,

∃use.FailCPUvFailComp,CPUvFailCPU}.

Intuitively, a contextual ontology is a compact representa- tion of exponentially many ontologies fromL; one for each worldW. The uncertainty inBLis expressed by the BNB, which is interpreted using multiple world semantics.

Definition 2 (probabilistic interpretation). Aprobabilistic interpretation is a pair P = (I, P_I), where I is a set of V-interpretations and P_I is a probability distribution over Isuch thatP_I(I) >0only for finitely many interpretations I ∈ I. It is amodelof theV-ontologyOif every I ∈ Iis a model ofO. P isconsistentwith the BNB if for every valuationWof the variables inV it holds that

X

I∈I,V^I=W

P_I(I) =P_B(W).

2We use|=pto distinguish propositional entailment from|=.

The probabilistic interpretation P is a model of the KB (B,O)iff it is a model ofOand consistent withB.

The fundamental reasoning task inBL, probabilistic en- tailment, consists in finding the probability of observing a consequencec; that is, the probability of being at a context wherecholds.

Definition 3(probabilistic entailment). Letc ∈ C, andK aBLKB. Theprobability ofcw.r.t. the probabilistic inter- pretationP = (I, P_I)isP_P(c) :=P

(I,W)∈I,I|=cP_I(I,W).

Theprobability ofcw.r.t.KisPK(c) := inf_P|=KPP(c).

It has been shown that to compute the conditional proba- bility of a consequencec, it suffices to test, for each world W, whetherO_W entailsc(Ceylan and Pe˜naloza 2014b).

Proposition 4. LetK = (B,O)be aBLKB and c ∈ C.

ThenPK(c) =P

OW|=cPB(W).

This means that reasoning inBLcan be reduced to expo- nentially many entailment tests in the classical languageL.

For some logics, this exponential enumeration of worlds can be avoided (Ceylan and Pe˜naloza 2014c). However, this de- pends 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 prob- ability of a consequence given some partial informa- tion about the context. Given a context κ, the con- ditional probability P_K(c|κ) is defined via the rule P_K(c, κ) =P_K(c|κ)P_B(κ), where

PK(c, κ) = X

OW|=c,W|=_pκ

PB(W).

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

Example 5. Consider again the KBK1 = (B1,O1)from Example 1 and the consequenceCompvFailComp. We are interested in finding the probability of the computer to fail, i.e.P_K1(CompvFailComp). This can be computed by enu- merating all worldsWfor whichOW |=CompvFailComp, 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 net- work. We now further extend this formalism to be able to cope with controlled updates of the probabilities over time.

Dynamic Bayesian Ontology Languages

WithBL, one is able to represent and reason about the uncer- tainty of the current context, and the consequences that fol- low 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 (Sadilek and Kautz 2010) this probability distribution may change over time. For ex- ample, as the components of a computer age, their probabil- ity 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 extendBLto

x

y

z t

x⁰

y⁰

z⁰

x⁰ x .4

¬x .4

y⁰

x y x⁰ .9

x y ¬x⁰ .5

x ¬y x⁰ .8

x ¬y ¬x⁰ .4

¬x y x⁰ .8

¬x y ¬x⁰ .4

¬x ¬y x⁰ .7

¬x ¬y ¬x⁰ .1

z⁰ z x⁰ y⁰ .7 z x⁰ ¬y⁰ .2 z ¬x⁰ y⁰ .4 z ¬x⁰ ¬y⁰ 1

¬z x⁰ y⁰ .6

¬z x⁰ ¬y⁰ .1

¬z ¬x⁰ y⁰ .3

¬z ¬x⁰ ¬y⁰ 1

t+ 1

Figure 2: The TBNB_→over the variablesV ={x, y, z}

handle these cases, by consideringdynamicBNs as the un- derlying formalism for managing uncertainty over contexts.

Dynamic BNs (DBNs) (Dean and Kanazawa 1989; Mur- phy 2002) extend BNs to provide a compact representa- tion 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). LetV be a finite set of Boolean ran- dom variables. A two-slice BN (TBN) over V is a pair (G,Φ), where G = (V ∪ V⁰, E) is a DAG containing no edges between elements of V, V⁰ = {x⁰ | x ∈ V}, and Φ contains, for every x⁰ ∈ V⁰ a conditional prob- ability distribution P(x⁰ | π(x⁰)) of x⁰ given its parents π(x⁰). Adynamic Bayesian network(DBN) overV is a pair D= (B1,B→)whereB1is a BN overV, andB→is a TBN overV.

A TBN overV ={x, y, z}is depicted in Figure 2. The set of nodes of the graph can be thought of as containing two disjoint copies of the random variables inV. Then, the prob- ability distribution at timet+ 1depends on the distribution at timet. In the following we will useV_tandx_tforx∈V, to denote the variables inV at timet.

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 TBNB, the conditional probability distribution at timet+ 1given timetis

P_B(V_t+1|V_t) = Y

x⁰∈V⁰

P_B(x⁰|π(x⁰)).

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 BNB₁that en- codes the JPD ofV at the beginning of the evolution. Thus, the DBND= (B₁,B_→)defines, for everyt≥1, the unique probability distribution

P_B(Vt) =P_B1(V1)

t

Y

i=2

Y

x∈V

P_B→(xi|π(xi)).

Intuitively, the distribution at timetis defined by unraveling the DBN starting fromB₁, using the two-slice structure of B→untiltcopies ofV have been created. This produces a

x1

z1

y1

x2

z2

y2

x3

z3

y3

Figure 3: Three step unravelingB_1:3of(B₁,B_→)

new BNB1:tencoding the distribution over time of the dif- ferent variables. Figure 3 depicts the unraveling to t = 3 of the DBN(B1,B_→)whereB1 andB_→ are the networks depicted in Figures 1 and 2, respectively. The conditional probability tables of each node given its parents (not de- picted) are those ofB₁for the nodes inV₁, and ofB_→for nodes inV2∪V3. Notice thatB1:thastcopies of each ran- dom variable inV. For a givent ≥ 1, we callBtthe BN obtained from the unravelingB1:tof the DBN to timet, and eliminating all variables not inVt. In particular, we have that P_Bt(V) =P_B1:t(V_t).

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

Definition 7(DBLKB). ADBLknowledge base (KB) is a pairK= (D,O)whereD= (B₁,B_→)is a DBN overV and Ois aV-ontology. LetK= (D,O)be aDBLKB overV. Atimed probabilistic interpretationis an infinite sequence P = (Pt)t≥1of probabilistic interpretations. Pis amodel ofKif for everyt,Ptis a model of theBLKB(Bt,O).

In a nutshell, a DBN can be thought of as a compact rep- resentation of an infinite sequence of BNsB1,B2, . . .over V. Following this idea, a DBL KB expresses an infinite sequence ofBLKBs, where the ontological component re- mains unchanged, and only the probability distribution of the contexts evolves over time. A timed probabilistic inter- pretationPsimply interprets each of theseBLKBs, at the corresponding point in time. To be a model of aDBLKB, Pmust then be a model of all the associatedBLKBs.

Before describing the reasoning tasks forDBLand meth- ods 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 for- mula.

Compiling Contextual Knowledge

From Proposition 4, we see that reasoning inBLcan be re- duced to checking, for every worldW, whetherO_W |= c.

This reduces probabilistic reasoning to a sequence of stan- dard entailment tests over the original languageL. However, each of these entailments might be very expensive. For ex- ample, in the very expressive DLSHOIQ, deciding an en- tailment is already NEXPTIME-hard (Tobies 2000). Rather than repeating this reasoning step for every world, it makes sense to try to identify the relevant worldsa priori. We do this through the computation of acontext formula.

Definition 8(context formula). LetObe aV-ontology, and c∈C. Acontext formulaforcw.r.t.Ois a propositional for- mulaφsuch that for every interpretationWof the variables inV, it holds thatO_W |=ciffW |=pφ.

The idea behind this formula is that, for finding whether O_W |=c, it suffices to check whether the valuationWsatis- fiesφ. This test requires only linear time on the length of the context formula. The context formula can be seen as a gen- eralization of the pinpointing formula (Baader and Pe˜naloza 2010b) and the boundary (Baaderet al.2012), defined orig- inally for classical ontology languages.

Example 9. Consider again theV-ontologyO1from Exam- ple 1. The formulaφ1:= (x∧(y∨z))∨(¬x∧y∧z)is a context formula forCompvFailCompw.r.t.O₁. In fact, the valuation{x,¬y, z}satisfies this formula.

Clearly, computing the context formula must be at least as hard as deciding an entailment in L: if we label every axiom in a classical L-ontologyO with the same proposi- tional variablex, then the boundary formula ofcw.r.t. this {x}-ontology isxiffO |= c. On the other hand, the al- gorithm used for deciding the entailment relation can usu- ally be modified to compute the context formula. Using arguments similar to those developed for axiom pinpoint- ing (Baader and Pe˜naloza 2010a; Kalyanpuret al. 2007), it can be shown that for most description logics, comput- ing the context formula is not harder, in terms of computa- tional complexity, than standard reasoning. In particular this holds for any arbitrary ontology language whose entailment relation is EXPTIME-hard. This formula can also be com- piled into a more efficient data structure like binary decision diagrams (Lee 1959). 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 ver- ifying whether the sub-ontology defined by a world entails the consequence in an efficient way.

Reasoning in DBL

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 stepst.

Definition 10 (probabilistic entailment with time). Let K= (D,O)be aDBLKB and c ∈ C. Given a timed in- terpretationPandt≥1, theprobability ofcat timetw.r.t.

PisP_P(c[t]) :=P_Pt(c). Theprobability ofcat timetw.r.t.

KisP_K(c[t]) := inf_P|=KPP(c[t]).

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. LetK = (D,O)be aDBL KB,c ∈ C, and t≥1. Then the probability ofcat timetw.r.t.Kis given by

P_K(c[t]) = X

OW|=c

P_Bt(W)

Proof. (Sketch) A timed modelP of K is a sequence of probabilistic interpretationsP1,P2, . . ., where eachPi is a

model of theBL KBKi := (Bi,O). We use this fact to show that

PK(c[t]) = inf

P|=KPP(c[t]) = inf

P|=KPP_t(c) (1)

= inf

P_t|=K_tPPt(c) =PKt(c) (2)

= X

OW|=c

PB_t(W), (3)

where (1) follows from Definition 10, (2) holds by defini- tion, and (3) follows from Proposition 4.

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

Theorem 12. LetK = (D,O)be aDBLKB,c ∈ C,φa context formula forcw.r.t.O, andt≥1. Then the probabil- ity ofcat timetw.r.t.Kis given byP_K(c[t]) =P_Bt(φ).

Proof. By Lemma 11 and the definition of a context for- mula, we have

P_K(c[t]) = X

OW|=c

P_Bt(W) = X

W|=_pφ

P_Bt(W) =P_Bt(φ),

which proves the result.

This means that one can first compute a context formula forcand then do probabilistic inferences over the DBN to detect the probability of satisfyingφat timet. For this, we can exploit any existing DBN inference method. One option is to do variable elimination over thet-step unraveled DBN B1:t, to compute Bt. Assuming thatt is fixed, it suffices to make 2^|V| inferences (one for each world) over Btand the same number of propositional entailment tests over the context formula. If entailment inLis already exponential, then computing the probability ofc at time tis as hard as deciding entailments.

The previous argument only works assuming a fixed time pointt. Since it depends heavily on computingBt(e.g., via variable elimination), it does not scale well astincreases.

Other methods have been proposed for exploiting the recur- sive structure of the DBN. For instance, one can use the al- gorithm described in (Vlasselaeret al. 2014) that provides linear scalability over time. The main idea is to compile the structure into an arithmetic circuit (Darwiche 2009) and use forward and backward message passing (Murphy 2002).

While computing the probability of a consequence at a fixed point in timetis a relevant task, it is usually more im- portant 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∼c,c∈C, to denote that the consequencecdoes not hold; i.e.,I |=∼c iffI 6|=c. Thus, for example,PK(∼c[t])is the probability

ofcnot holding at timet. To find the probability of observ- ingcin the firstttime steps, one can alternatively compute the probability of not observingcin any of those steps. For- mally, for a timed interpretationPandt∈N, we define

PP(c[1 :t]) := 1−PP(∼c[1], . . . ,∼c[t]).

Definition 13(time bounded probabilistic entailment). The probability of observingcin at mosttstepsw.r.t. theDBL KBKisPK(c[1 :t]) := inf_P|=KPP(c[1 :t]).

Just as before, given a constant t ≥ 1, it is possible to compute P_K(c[1 : t]) by looking at thet-step unraveling ofD. More precisely, to computePP(∼c[1], . . . ,∼c[t]), it suffices to look at all the valuationsWofSt

i=1V_isuch that for alli,1 ≤i ≤t, it holds thatO_W(i) |=∼c. These val- uations correspond to an evolution of the system where the consequencecis not observed in the firsttsteps. The prob- ability of these valuations w.r.t.B1:tthen yields the proba- bility of not observing this consequence. We thus get the following result.

Theorem 14. Let K = (D,O) be a DBL KB, c ∈ C, φ a context formula for c w.r.t. O, and t ≥ 1. Then PK(c[1 :t]) =P

W|∃i.W(i)|=_pφPB_1:t(W).

Proof (Sketch). Using the pithy interpretations of the crisp ontologies O_W(i), we can build a timed interpretationP0

such that PP₀(c[1 : t]) = P

W|∃i.W(i)|=_pφPB_1:t(W), in a way similar to Theorem 12 of (Ceylan and Pe˜naloza 2014b). The existence of another timed interpretation P such that PP(c[1 : t]) < PP₀(c[1 : t]) contradicts the properties of the pithy interpretations. Thus, we obtain that PP0(c[1 :t]) = inf_P|=KPP(c[1 :t]) =PK(c[1 :t]).

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

P_K(CompvFailComp[1 : 20]) = X

W|∃i.W(i)|=_pφ

P_Bi(φ).

Thus, the computational complexity of reasoning is not af- fected 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 t. Thus, for large intervals, this approach becomes unfeasible.

By extending the time limit indefinitely, we can also find the probability ofeventuallyobserving the consequence c (e.g., the probability of the system ever failing). The probability of eventually observing c w.r.t. K is given by P_K(c[∞]) := lim_t→∞P_K(c[1 :t]). Notice thatP_K(c[1 :t]) is monotonic on t and bounded by 1; henceP_K(c[∞]) is well defined.

Observe that Theorem 14 cannot be used to compute the probability ofeventuallyobservingcsince one cannot neces- sarily predict the changes in probabilities of finding worlds that entail the consequencec. Rather than considering these

x

y z

x .4

z

x y .7

x ¬y .2

¬x y .4

¬x ¬y 1 y

x .9

¬x .5

Figure 4:P(V | {x, y, z}t)

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 DBLKBs.

It is easy to see that every TBN defines a time- homogeneous Markov chain over a finite state space. More precisely, ifB is a TBN overV, then M_B is the Markov chain, where every valuationW of the variables in V is a state and the transition probability distribution given the cur- rent stateWis described by the BN obtained from addingW as evidence to the first slice ofB. For example, the TBNB→

from Figure 2 yields the conditional probability distribution given that{x, y, z}was observed at timetdepicted in Fig- ure 4. From this, we can derive the probability of observing {x, y, z}at timet+ 1given that it was observed at timet, which isP({x, y, z}t+1| {x, y, z}t) = 0.252.

We extend the notions from Markov chains to TBNs in the obvious way. In particular, the TBN B is irreducible if for every two worldsV,W, the probability of eventually reachingWgivenV is greater than 0. It isaperiodicif for every worldW there is ann_W such that for all n ≥ n_W, it holds thatP(Wn | W) >0. A distributionPW over the worlds isstationaryifP

WP(V | W)PW(W) = PW(V) holds for every worldV. It follows immediately that ifB is irreducible and aperiodic, then it has a unique stationary distribution (Harris 1956).

Given a TBNBoverV, let now∆_Bbe the set of all sta- tionary distributions over the worlds ofV. For a worldW, defineδ_B(W) := minP∈∆BP(W)to be the smallest prob- ability assigned by any stationary distribution ofBtoW. If δ_B(W)>0, then we know that, regardless of the initial dis- tribution, in the limit we will always be able to observe the worldWwith a constant positive probability. In particular, this means that the probability of eventually observing W 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 formulaφ, let

δ_B(φ) := min

P∈∆B

X

W|=_pφ

P(W).

In other words,δ_B(φ)expresses the minimum probability of satisfying φin any stationary distribution ofB. From the arguments above, we obtain the following theorem.

Theorem 15. LetK = (D,O)be aDBLKB overV with

D = (B1,B→),c ∈ C, andφa context formula forcw.r.t.

O. Ifδ_B→(φ)>0, thenP_K(c[∞]) = 1.

In particular, ifB_→is irreducible and aperiodic,∆_Bcon- tains only one stationary distribution, which simplifies the computation of the function δ. Unfortunately, such a sim- ple characterization of P_K(c[∞]) cannot be given when δB→(φ) = 0. In fact, in this case the result may depend strongly on the initial distribution.

Example 16. LetV = {x},O2 = {hAvB :{x}i}, and consider the TBNB⁰_→ overV defined by P(x⁰ | x) = 1 andP(x⁰ | ¬x) = 0. It is easy to see that any distribu- tion over the valuations ofV is stationary. For every initial distributionB, ifK = (D,O2)whereD = (B,B⁰_→), then P_K(AvB[∞]) =P_B(x).

So far, our reasoning services have focused on predicting the outcome at future time steps, given the current knowl- edge of the system. Based on our model of evolving prob- abilities, the distribution at any timet+ 1depends only on time t, if it is known. However, for many applications it makes sense to consider evidence that is observed through- out several time steps. For instance, in our computer fail- ure scenario, the DBNB_→ensures that, if at some point a critical situation is observed (xis 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 variablex.

Suppose that we have observed over the firstttime steps that no critical situation has occurred, and we want to know the probability of a computer failure. Formally, letEbe a consistent set of literals overSt

i=1Vi. We want to compute the probabilityP_K(c[t] | E)of observingcat timetgiven the evidenceE. This is just a special case of bounded prob- abilistic entailment, where the worlds are not only restricted w.r.t. the context formula but also w.r.t. the evidenceE.

The efficiency of this approach depends strongly on the time boundt, but also on the structure of the TBNB→. Re- call that the complexity of reasoning in a BN depends on the tree-width of its underlying DAG (Panet al.1998). The unraveling of B_→ 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 on- tology 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 includ- ing, 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 variablesV. The probabilis- tic knowledge is expressed through a probability distribution over the valuations ofV which is encoded by a DBN.

Interestingly, our formalism allows for reasoning meth- ods 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 stan- dard 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 intro- duced only the most fundamental reasoning tasks. It is pos- sible 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 prob- lems. To find out the probability of satisfying the context formulaφ, we need compute the probability of each of the valuations that satisfyφat different points in time. Even us- ing 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

˙Ismail ˙Ilkan Ceylan is supported by DFG within the Research Training Group “RoSI” (GRK 1907). Rafael Pe˜naloza was partially supported by DFG through the Clus- ter of Excellence ‘cfAED,’ while he was still affiliated with TU Dresden and the Center for Advancing Electronics Dres- den, Germany.

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