Dynamic Bayesian Ontology Languages

Transcription

1 Dnamic Baesian Ontolog Languages İsmail İlkan Celan Theoretical Computer Science TU Dresden, German Rafael Peñaloa KRDB Research Centre Free Universit of Boen-Bolano, Ital Abstract Man formalisms combining ontolog languages with uncertaint, usuall in the form of probabilities, have been studied over the ears. Most of these formalisms, however, assume that the probabilistic structure of the knowledge remains static over time. We present a general approach for etending ontolog languages to handle time-evolving uncertaint represented b a dnamic Baesian network. We show how reasoning in the original language and dnamic Baesian inferences can be eploited for effective reasoning in our framework. Introduction Description Logics (DLs) (Baader et al. 2007) are a wellknown famil of knowledge representation formalisms that have been successfull emploed for encoding the knowledge of man application domains. In DLs, knowledge is represented through a finite set of aioms, usuall called an ontolog or knowledge base. In essence, these aioms are atomic pieces of knowledge that provide eplicit information of the domain. When mied together in an ontolog, these aioms ma impl some additional knowledge that is not eplicitl encoded. Reasoning is the act of making this implicit knowledge eplicit through an entailment relation. Some of the largest and best-maintained DL ontologies represent knowledge from the bio-medical domains. For instance, the NCBO Bioportal 1 contains 420 ontologies of various sies. In the bio-medical fields it is ver common to have onl uncertain knowledge. The certaint that an epert has on an atomic piece of knowledge ma have arisen from a statistical test, or from possibl imprecise measurements, for eample. It thus becomes relevant to etend DLs to represent and reason with uncertaint. The need for probabilistic etensions of DLs has been observed for over two decades alread. To cover it, man different formalisms have been introduced (Jaeger 1994; Lukasiewic and Straccia 2008; Lut and Schröder 2010; Klinov and Parsia 2011). The differences in these logics range from the underling 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 Copright c 2015, Association for the Advancement of Artificial Intelligence ( All rights reserved. 1 is the representation of joint probabilities, in particular when the different aioms are not required to be probabilisticall independent. A recent approach solves this issue b dividing the ontolog into contets, which intuitivel represent aioms that must appear together. The probabilistic knowledge is epressed through a Baesian network that encodes the joint probabilit distribution of these contets. Although originall developed as an etension of the DL EL (Celan and Peñaloa 2014b), the framework has been etended to arbitrar ontolog languages with a monotone entailment relation (Celan and Peñaloa 2014a). One common feature of the probabilistic etensions of DLs eisting in the literature is that the consider the probabilit distribution to be static. For man applications, this assumption does not hold: the probabilit of a person to have gra hair increases as time passes, as does the probabilit of a computer component to fail. To the best of our knowledge, there is so far no etension of DLs that can handle evolving probabilities effectivel. In this paper, we describe a general approach for etending ontolog languages to handle evolving probabilities. B etension, our method covers all DLs, but is not limited to them. The main idea is to adapt the formalism from (Celan and Peñaloa 2014a) to use dnamic Baesian networks (DBNs) (Murph 2002) as the underling uncertaint structure to compactl encode evolving probabilit distributions. Given an arbitrar ontolog language L, we define its dnamic Baesian etension DBL. We show that reasoning in DBL can be seamlessl divided into the probabilistic computation over the DBN, and the logical component with its underling entailment relation. In order to reduce the number of entailment checks, we compile a so-called contet formula, which encodes all contets in which a given consequence holds. Related to our work are relational BNs (Jaeger 1997) and their etensions. 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. Additionall, restricting the logical formalism to specific ontolog languages provides an opportunit for finding effective reasoning algorithms.

2 Baesian Ontolog Languages To remain as general as possible, we do not fi a specific logic, but consider an arbitrar ontolog language L consisting of two infinite sets A and C of aioms and consequences, respectivel, and a class O fin (A) of finite sets of aioms, called ontologies, such that if O O, then O O for all O O. The language L is associated to a class I of interpretations and an entailment relation = I (A C). An interpretation I I is a model of the ontolog O (I = O) if I = α for all α O. O entails c C (O = c) if ever model of O entails c. Notice that the entailment relation is monotonic; i.e., if O = c and O O O, then O = c. An standard description logic (DL) (Baader et al. 2007) is an ontolog language of this kind; consequences in these languages are e.g. concept unsatisfiabilit, concept subsumption, or quer entailment. However, man other ontolog languages of varing epressivit and compleit properties eist. For the rest of this paper, L is an arbitrar but fied ontolog language, with aioms A, ontologies O, consequences C, and interpretations I. As an eample language we use the DL EL (Baader et al. 2005), which we briefl introduce here. Given two disjoint sets N C and N R, EL concepts are built b the grammar rule C ::= A C C r.c where A N C and r N R. EL aioms and consequences are epressions of the form C D, where C and D are concepts. An interpretation is a pair ( I, I) where I is a non-empt set and I maps ever A N C to A I I and ever r N R to r I I I. This function is etended to concepts b I := I, (C D) I := C I D I, and ( r.c) I := {d e C I.(d, e) r I }. This interpretation entails the aiom (or consequence) C D iff C I D I. The Baesian ontolog language BL etends L b associating each aiom in an ontolog with a contet, which intuitivel describes the situation in which the aiom is required to hold. The knowledge of which contet applies is uncertain, and epressed through a Baesian network (Celan and Peñaloa 2014a). Briefl, a Baesian network (Darwiche 2009) is a pair B = (G, Φ), where G = (V, E) is a finite directed acclic graph (DAG) whose nodes represent Boolean random variables, and Φ contains, for ever V, a conditional probabilit distribution P B ( π()) of given its parents π(). Ever variable V is conditionall independent of its non-descendants given its parents. Thus, the BN B defines a unique joint probabilit distribution (JPD) over V : P B (V ) = V P B ( π()). Let V be a finite set of variables. A V -contet is a consistent set of literals over V. A V -aiom is an epression of the form α : κ where α A is an aiom and κ is a V -contet. A V -ontolog is a finite set O of V -aioms, such that {α α : κ O} O. A BL knowledge base (KB) over V is a pair K = (O, B) where B is a BN over V and O is a V -ontolog. We briefl illustrate these notions over the language BEL, an etension of the DL EL, in the following Eample Figure 1: The BN B 1 over the variables V = {,, } Eample 1. Consider the BEL KB K 1 = (B 1, O 1 ) where O 1 ={ Comp use.mem use.cpu :, use.failmem FailComp : {}, use.failcpu FailComp : {}, use.failmem use.failcpu FailComp:{ }, Mem FailMem : {}, CPU FailCPU : {} }, and B 1 is the BN shown in Figure 1. This KB represents a computer failure scenario, where stands for a critical situation, represents the memor failing, and the CPU failing. The contetual semantics is defined b etending interpretations 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 α : κ (I = α : κ ), where α A, iff (V I = p κ) 2 or (I = α). It is a model of the V -ontolog O iff it is a model of all the V -aioms in O. It entails c C if I = c. The intuition behind this semantics is that an aiom is evaluated to true b all models provided it is in the right contet. Given a V -ontolog O, ever propositional interpretation, or world, W on V defines an ontolog O W := {α α : κ O, W = p κ}. Consider the KB K 1 provided in Eample 1: The world W = {,, } defines the ontolog O W ={Comp use.mem use.cpu, use.failmem FailComp, use.failcpu FailComp, CPU FailCPU}. Intuitivel, a contetual ontolog is a compact representation of eponentiall man ontologies from L; one for each world W. The uncertaint in BL is epressed b the BN B, which is interpreted using multiple world semantics. Definition 2 (probabilistic interpretation). A probabilistic interpretation is a pair P = (I, P I ), where I is a set of V -interpretations and P I is a probabilit distribution over I such that P I (I) > 0 onl for finitel man interpretations I I. It is a model of the V -ontolog O if ever I I is a model of O. P is consistent with the BN B if for ever valuation W of the variables in V it holds that P I (I) = P B (W). I I, V I =W 2 We use = p to distinguish propositional entailment from =.

3 The probabilistic interpretation P is a model of the KB (B, O) iff it is a model of O and consistent with B. The fundamental reasoning task in BL, probabilistic entailment, consists in finding the probabilit of observing a consequence c; that is, the probabilit of being at a contet where c holds. Definition 3 (probabilistic entailment). Let c C, and K a BL KB. The probabilit of c w.r.t. the probabilistic interpretation P = (I, P I ) is P P (c) := (I,W) I,I =c P I(I, W). The probabilit of c w.r.t. K is P K (c) := inf P =K P P (c). It has been shown that to compute the conditional probabilit of a consequence c, it suffices to test, for each world W, whether O W entails c (Celan and Peñaloa 2014b). Proposition 4. Let K = (B, O) be a BL KB and c C. Then P K (c) = O W =c P B(W). This means that reasoning in BL can be reduced to eponentiall man entailment tests in the classical language L. For some logics, this eponential enumeration of worlds can be avoided (Celan and Peñaloa 2014c). However, this depends on the properties of the ontological language and its entailment relation, and cannot be guaranteed for arbitrar languages. Another relevant problem is to compute the probabilit of a consequence given some partial information about the contet. Given a contet κ, the conditional probabilit P K (c κ) is defined via the rule P K (c, κ) = P K (c κ)p B (κ), where P K (c, κ) = P B (W). O W =c,w = pκ For simplicit, in the rest of this paper we consider onl unconditional consequences. However, it should be noted that all results can be transferred to the conditional case. Eample 5. Consider again the KB K 1 = (B 1, O 1 ) from Eample 1 and the consequence Comp FailComp. We are interested in finding the probabilit of the computer to fail, i.e. P K1 (Comp FailComp). This can be computed b enumerating all worlds W for which O W = Comp FailComp, which ields the probabilit As seen, it is possible to etend an ontological language to allow for probabilistic reasoning based on a Baesian network. We now further etend this formalism to be able to cope with controlled updates of the probabilities over time. Dnamic Baesian Ontolog Languages With BL, one is able to represent and reason about the uncertaint of the current contet, and the consequences that follow from it. In that setting, the joint probabilit distribution of the contets, epressed b the BN, is known and fied. In some applications, see especiall (Sadilek and Kaut 2010) this probabilit distribution ma change over time. For eample, as the components of a computer age, their probabilit of failing increases. The new probabilit depends on how likel it was for the component to fail previousl, and the ageing factors to which it is eposed. We now etend BL to t t Figure 2: The TBN B over the variables V = {,, } handle these cases, b considering dnamic BNs as the underling formalism for managing uncertaint over contets. Dnamic BNs (DBNs) (Dean and Kanaawa 1989; Murph 2002) etend BNs to provide a compact representation of evolving joint probabilit distributions for a fied set of random variables. The update of the JPD is epressed through a two-slice BN, which epresses the probabilities at the net point in time, given the current contet. Definition 6 (DBN). Let V be a finite set of Boolean random variables. A two-slice BN (TBN) over V is a pair (G, Φ), where G = (V V, E) is a DAG containing no edges between elements of V, V = { V }, and Φ contains, for ever V a conditional probabilit distribution P ( π( )) of given its parents π( ). A dnamic Baesian network (DBN) over V is a pair D = (B 1, B ) where B 1 is a BN over V, and B is a TBN over V. A TBN over V = {,, } 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 in V. Then, the probabilit distribution at time t + 1 depends on the distribution at time t. In the following we will use V t and t for V, to denote the variables in V at time t. As standard in BNs, the graph structure of a TBN encodes the conditional dependencies among the nodes: ever node is independent of all its non-descendants given its parents. Thus, for a TBN B, the conditional probabilit distribution at time t + 1 given time t is P B (V t+1 V t ) = V P B ( π( )). We further assume the Markov propert: the probabilit of the future state is independent from the past, given the present state. In addition to the TBN, a DBN contains a BN B 1 that encodes the JPD of V at the beginning of the evolution. Thus, the DBN D = (B 1, B ) defines, for ever t 1, the unique probabilit distribution P B (V t ) = P B1 (V 1 ) t P B ( i π( i )). i=2 V Intuitivel, the distribution at time t is defined b unraveling the DBN starting from B 1, using the two-slice structure of B until t copies of V have been created. This produces a

4 Figure 3: Three step unraveling B 1:3 of (B 1, B ) new BN B 1:t encoding the distribution over time of the different variables. Figure 3 depicts the unraveling to t = 3 of the DBN (B 1, B ) where B 1 and B are the networks depicted in Figures 1 and 2, respectivel. The conditional probabilit tables of each node given its parents (not depicted) are those of B 1 for the nodes in V 1, and of B for nodes in V 2 V 3. Notice that B 1:t has t copies of each random variable in V. For a given t 1, we call B t the BN obtained from the unraveling B 1:t of the DBN to time t, and eliminating all variables not in V t. In particular, we have that P Bt (V ) = P B1:t (V t ). The dnamic Baesian ontolog language DBL is ver similar to BL, ecept that the probabilit distribution of the contets evolves accordingl to a DBN. Definition 7 (DBL KB). A DBL knowledge base (KB) is a pair K = (D, O) where D = (B 1, B ) is a DBN over V and O is a V -ontolog. Let K = (D, O) be a DBL KB over V. A timed probabilistic interpretation is an infinite sequence P = (P t ) t 1 of probabilistic interpretations. P is a model of K if for ever t, P t is a model of the BL KB (B t, O). In a nutshell, a DBN can be thought of as a compact representation of an infinite sequence of BNs B 1, B 2,... over V. Following this idea, a DBL KB epresses an infinite sequence of BL KBs, where the ontological component remains unchanged, and onl the probabilit distribution of the contets evolves over time. A timed probabilistic interpretation P simpl interprets each of these BL KBs, at the corresponding point in time. To be a model of a DBL KB, P must then be a model of all the associated BL KBs. Before describing the reasoning tasks for DBL and methods for solving them, we show how the computation of all the contets that entail a consequence can be reduced to the enumeration of the worlds satisfing a propositional formula. 2 Compiling Contetual Knowledge From Proposition 4, we see that reasoning in BL can be reduced to checking, for ever world W, whether O W = c. This reduces probabilistic reasoning to a sequence of standard entailment tests over the original language L. However, each of these entailments might be ver epensive. For eample, in the ver epressive DL SHOIQ, deciding an entailment is alread NEXPTIME-hard (Tobies 2000). Rather than repeating this reasoning step for ever world, it makes sense to tr to identif the relevant worlds a priori. We do this through the computation of a contet formula Definition 8 (contet formula). Let O be a V -ontolog, and c C. A contet formula for c w.r.t. O is a propositional formula φ such that for ever interpretation W of the variables in V, it holds that O W = c iff W = p φ. The idea behind this formula is that, for finding whether O W = c, it suffices to check whether the valuation W satisfies φ. This test requires onl linear time on the length of the contet formula. The contet formula can be seen as a generaliation of the pinpointing formula (Baader and Peñaloa 2010b) and the boundar (Baader et al. 2012), defined originall for classical ontolog languages. Eample 9. Consider again the V -ontolog O 1 from Eample 1. The formula φ 1 := ( ( )) ( ) is a contet formula for Comp FailComp w.r.t. O 1. In fact, the valuation {,, } satisfies this formula. Clearl, computing the contet formula must be at least as hard as deciding an entailment in L: if we label ever aiom in a classical L-ontolog O with the same propositional variable, then the boundar formula of c w.r.t. this {}-ontolog is iff O = c. On the other hand, the algorithm used for deciding the entailment relation can usuall be modified to compute the contet formula. Using arguments similar to those developed for aiom pinpointing (Baader and Peñaloa 2010a; Kalanpur et al. 2007), it can be shown that for most description logics, computing the contet formula is not harder, in terms of computational compleit, than standard reasoning. In particular this holds for an arbitrar ontolog language whose entailment relation is EXPTIME-hard. This formula can also be compiled into a more efficient data structure like binar decision diagrams (Lee 1959). Intuitivel, this means that we can compute this formula using the same amount of resources needed for onl one entailment test, and then use it for verifing whether the sub-ontolog defined b a world entails the consequence in an efficient wa. Reasoning in DBL Rather than merel computing the probabilit of currentl observing a consequence, we are interested in computing the probabilit of a consequence to follow after some fied number of time steps t. Definition 10 (probabilistic entailment with time). Let K = (D, O) be a DBL KB and c C. Given a timed interpretation P and t 1, the probabilit of c at time t w.r.t. P is P P (c[t]) := P Pt (c). The probabilit of c at time t w.r.t. K is P K (c[t]) := inf P =K P P (c[t]). We show that probabilistic entailment over a fied time bound can be reduced to probabilistic entailment defined for BOLs b unravelling the DBN. Lemma 11. Let K = (D, O) be a DBL KB, c C, and t 1. Then the probabilit of c at time t w.r.t. K is given b P K (c[t]) = P Bt (W) O W =c Proof. (Sketch) A timed model P of K is a sequence of probabilistic interpretations P 1, P 2,..., where each P i is a

5 model of the BL KB K i := (B i, O). We use this fact to show that P K (c[t]) = inf P =K P P(c[t]) = inf P =K P P t (c) (1) = inf P Pt (c) = P Kt (c) (2) P t =K t = P Bt (W), (3) O W =c where (1) follows from Definition 10, (2) holds b definition, and (3) follows from Proposition 4. Lemma 11 provides a method for computing the probabilit of an entailment at a fied time t. One can first generate the BN B t, and then compute the probabilit w.r.t. B t of all the worlds that entail c. Moreover, using a contet formula we can compile awa the ontolog and reduce reasoning to standard inferences in BNs, onl. Theorem 12. Let K = (D, O) be a DBL KB, c C, φ a contet formula for c w.r.t. O, and t 1. Then the probabilit of c at time t w.r.t. K is given b P K (c[t]) = P Bt (φ). Proof. B Lemma 11 and the definition of a contet formula, we have P K (c[t]) = P Bt (W) = P Bt (W) = P Bt (φ), O W =c which proves the result. W = pφ This means that one can first compute a contet formula for c and then do probabilistic inferences over the DBN to detect the probabilit of satisfing φ at time t. For this, we can eploit an eisting DBN inference method. One option is to do variable elimination over the t-step unraveled DBN B 1:t, to compute B t. Assuming that t is fied, it suffices to make 2 V inferences (one for each world) over B t and the same number of propositional entailment tests over the contet formula. If entailment in L is alread eponential, then computing the probabilit of c at time t is as hard as deciding entailments. The previous argument onl works assuming a fied time point t. Since it depends heavil on computing B t (e.g., via variable elimination), it does not scale well as t increases. Other methods have been proposed for eploiting the recursive structure of the DBN. For instance, one can use the algorithm described in (Vlasselaer et al. 2014) that provides linear scalabilit over time. The main idea is to compile the structure into an arithmetic circuit (Darwiche 2009) and use forward and backward message passing (Murph 2002). While computing the probabilit of a consequence at a fied point in time t is a relevant task, it is usuall more important to know whether the consequence can be observed within a given time limit. In our computer eample, we would be interested in finding the probabilit of the sstem failing within, sa, the following twent steps. Abusing of the notation, we use the epression c, c C, to denote that the consequence c does not hold; i.e., I = c iff I = c. Thus, for eample, P K ( c[t]) is the probabilit of c not holding at time t. To find the probabilit of observing c in the first t time steps, one can alternativel compute the probabilit of not observing c in an of those steps. Formall, for a timed interpretation P and t N, we define P P (c[1 : t]) := 1 P P ( c[1],..., c[t]). Definition 13 (time bounded probabilistic entailment). The probabilit of observing c in at most t steps w.r.t. the DBL KB K is P K (c[1 : t]) := inf P =K P P (c[1 : t]). Just as before, given a constant t 1, it is possible to compute P K (c[1 : t]) b looking at the t-step unraveling of D. More precisel, to compute P P ( c[1],..., c[t]), it suffices to look at all the valuations W of t i=1 V i such that for all i, 1 i t, it holds that O W(i) = c. These valuations correspond to an evolution of the sstem where the consequence c is not observed in the first t steps. The probabilit of these valuations w.r.t. B 1:t then ields the probabilit 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 contet formula for c w.r.t. O, and t 1. Then P K (c[1 : t]) = W i.w(i) = P pφ B 1:t (W). Proof (Sketch). Using the pith interpretations of the crisp ontologies O W(i), we can build a timed interpretation P 0 such that P P0 (c[1 : t]) = W i.w(i) = P pφ B 1:t (W), in a wa similar to Theorem 12 of (Celan and Peñaloa 2014b). The eistence of another timed interpretation P such that P P (c[1 : t]) < P P0 (c[1 : t]) contradicts the properties of the pith interpretations. Thus, we obtain that P P0 (c[1 : t]) = inf P =K P P (c[1 : t]) = P K (c[1 : t]). This means that the probabilit of observing a consequence within a fied time-bound t can be computed b simpl computing the contet formula and then performing probabilistic a posteriori computations over the unraveled BN. In our running eample, the probabilit of observing a computer failure in the net 20 steps is simpl P K (Comp FailComp[1 : 20]) = P Bi (φ). W i.w(i) = pφ Thus, the computational compleit of reasoning is not affected b introducing the dnamic evolution of the BN, as long as the time bound is constant. Notice, however, that the number of possible valuations grows eponentiall on the time bound t. Thus, for large intervals, this approach becomes unfeasible. B etending the time limit indefinitel, we can also find the probabilit of eventuall observing the consequence c (e.g., the probabilit of the sstem ever failing). The probabilit of eventuall observing c w.r.t. K is given b P K (c[ ]) := lim t P K (c[1 : t]). Notice that P K (c[1 : t]) is monotonic on t and bounded b 1; hence P K (c[ ]) is well defined. Observe that Theorem 14 cannot be used to compute the probabilit of eventuall observing c since one cannot necessaril predict the changes in probabilities of finding worlds that entail the consequence c. Rather than considering these

6 Figure 4: P (V {,, } t ) increasingl large BNs separatel, we can eploit methods developed for probabilit distributions that evolve over time. This will also allow us to etract more information from DBL KBs. It is eas to see that ever TBN defines a timehomogeneous Markov chain over a finite state space. More precisel, if B is a TBN over V, then M B is the Markov chain, where ever valuation W of the variables in V is a state and the transition probabilit distribution given the current state W is described b the BN obtained from adding W as evidence to the first slice of B. For eample, the TBN B from Figure 2 ields the conditional probabilit distribution given that {,, } was observed at time t depicted in Figure 4. From this, we can derive the probabilit of observing {,, } at time t + 1 given that it was observed at time t, which is P ({,, } t+1 {,, } t ) = We etend the notions from Markov chains to TBNs in the obvious wa. In particular, the TBN B is irreducible if for ever two worlds V, W, the probabilit of eventuall reaching W given V is greater than 0. It is aperiodic if for ever world W there is an n W such that for all n n W, it holds that P (W n W) > 0. A distribution P W over the worlds is stationar if W P (V W)P W (W) = P W (V) holds for ever world V. It follows immediatel that if B is irreducible and aperiodic, then it has a unique stationar distribution (Harris 1956). Given a TBN B over V, let now B be the set of all stationar distributions over the worlds of V. For a world W, define δ B (W) := min P B P (W) to be the smallest probabilit assigned b an stationar distribution of B to W. If δ B (W) > 0, then we know that, regardless of the initial distribution, in the limit we will alwas be able to observe the world W with a constant positive probabilit. In particular, this means that the probabilit of eventuall 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 formula φ, let δ B (φ) := min P (W). P B W = pφ In other words, δ B (φ) epresses the minimum probabilit of satisfing φ in an stationar distribution of B. From the arguments above, we obtain the following theorem. Theorem 15. Let K = (D, O) be a DBL KB over V with D = (B 1, B ), c C, and φ a contet formula for c w.r.t. O. If δ B (φ) > 0, then P K (c[ ]) = 1. In particular, if B is irreducible and aperiodic, B contains onl one stationar distribution, which simplifies the computation of the function δ. Unfortunatel, such a simple characteriation of P K (c[ ]) cannot be given when δ B (φ) = 0. In fact, in this case the result ma depend strongl on the initial distribution. Eample 16. Let V = {}, O 2 = { A B : {} }, and consider the TBN B over V defined b P ( ) = 1 and P ( ) = 0. It is eas to see that an distribution over the valuations of V is stationar. For ever initial distribution B, if K = (D, O 2 ) where D = (B, B ), then P K (A B[ ]) = P B (). So far, our reasoning services have focused on predicting the outcome at future time steps, given the current knowledge of the sstem. Based on our model of evolving probabilities, the distribution at an time t + 1 depends onl on time t, if it is known. However, for man applications it makes sense to consider evidence that is observed throughout several time steps. For instance, in our computer failure scenario, the DBN B ensures that, if at some point a critical situation is observed ( is true), then the probabilit of observing a memor or CPU failure in the net step is higher. That is, the evolution of the probabilit distribution is affected b the observed value of the variable. Suppose that we have observed over the first t time steps that no critical situation has occurred, and we want to know the probabilit of a computer failure. Formall, let E be a consistent set of literals over t i=1 V i. We want to compute the probabilit P K (c[t] E) of observing c at time t given the evidence E. This is just a special case of bounded probabilistic entailment, where the worlds are not onl restricted w.r.t. the contet formula but also w.r.t. the evidence E. The efficienc of this approach depends strongl on the time bound t, but also on the structure of the TBN B. Recall that the compleit of reasoning in a BN depends on the tree-width of its underling DAG (Pan et al. 1998). The unraveling of B produces a new DAG whose tree-width might increase with each unraveling step, thus impacting the reasoning methods negativel. Conclusions We have introduced a general approach for etending ontolog languages to handle time-evolving probabilities with the help of a DBN. Our framework can be instantiated to an language with a monotonic entailment relation including, but not limited to, all the members of the description logic famil of knowledge representation formalisms. Our approach etends on ideas originall introduced for static probabilistic reasoning. The essence of the method is to divide an ontolog into different contets, which are identified b a consistent set of propositional variables from a previousl chosen finite set of variables V. The probabilistic knowledge is epressed through a probabilit distribution over the valuations of V which is encoded b a DBN. Interestingl, our formalism allows for reasoning methods that eploit the properties of both, the ontological, and

7 the probabilistic components. From the ontological point of view, we can use suplemental reasoning to produce a contet 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 probabilit of satisfing this formula. This work represents first steps towards the development of a formalism combining well-known ontolog languages with time-evolving probabilities. First of all, we have introduced onl the most fundamental reasoning tasks. It is possible to think of man other problems like finding the most plausible eplanation for an observed event, or computing the epected time until a consequence is derived, among man others. Finall, the current methods developed for handling DBNs, although effective, are not adequate for our problems. To find out the probabilit of satisfing the contet formula φ, we need compute the probabilit of each of the valuations that satisf φ at different points in time. Even using methods that eploit the structure of the DBN directl, the information of the contet formula is not considered. Additionall, with the most efficient methods to-date, it is unclear how to handle the evidence over time effectivel. Dealing with these, and other related problems, will be the main focus of our future work. Acknowledgments İsmail İlkan Celan is supported b DFG within the Research Training Group RoSI (GRK 1907). Rafael Peñaloa was partiall supported b DFG through the Cluster of Ecellence cfaed, while he was still affiliated with TU Dresden and the Center for Advancing Electronics Dresden, German. References Fran Baader and Rafael Peñaloa. Automata-based Aiom Pinpointing. J. of Automated Reasoning, 45(2):91 129, Fran Baader and Rafael Peñaloa. Aiom Pinpointing in General Tableau. J. of Logic and Computation, 20(1):5 34, Fran Baader, Sebastian Brandt, and Carsten Lut. Pushing the EL envelope. In Proc. of IJCAI 05, pages Morgan Kaufmann Publishers, Fran Baader, Diego Calvanese, Deborah L McGuinness, Daniele Nardi, and Peter F Patel-Schneider, editors. The Description Logic Handbook: Theor, Implementation, and Applications. Cambridge Universit Press, 2nd edition, Fran Baader, Martin Knechtel, and Rafael Peñaloa. Contet-Dependent Views to Aioms and Consequences of Semantic Web Ontologies. J. of Web Semantics, 12 13:22 40, İsmail İlkan Celan and Rafael Peñaloa. Baesian Description Logics. In Proc. of DL 14, volume 1193 of CEUR Workshop Proceedings. CEUR-WS, İsmail İlkan Celan and Rafael Peñaloa. The Baesian Description Logic BEL. In Proc. of IJCAR 14, volume 8562 of LNCS, pages Springer Verlag, İsmail İlkan Celan and Rafael Peñaloa. Tight Compleit Bounds for Reasoning in the Description Logic BEL. In Proc. of JELIA 14, volume 8761 of LNCS, pages Springer Verlag, Adnan Darwiche. Modeling and Reasoning with Baesian Networks. Cambridge Universit Press, Thoma Dean and Keiji Kanaawa. A model for reasoning about persistence and causation. Computational intelligence, pages , Theodore Edward Harris. The eistence of stationar measures for certain Markov processes. In Proc. of Berkele Smposium on Mathematical Statistics and Probabilit, volume 2. Universit of California Press, Manfred Jaeger. Probabilistic Reasoning in Terminological Logics. In Proc. of KR 94, pages AAAI Press, Manfred Jaeger. Relational Baesian Networks. In Proc. of UAI 97, pages Morgan Kaufmann Publishers, Adita Kalanpur, Bijan Parsia, Matthew Horridge, and Evren Sirin. Finding All Justifications of OWL DL Entailments. In Proc. of ISWC 07, volume 4825 of LNCS, pages Springer Verlag, Pavel Klinov and Bijan Parsia. Representing sampling distributions in P-SROIQ. In Proc. of URSW 11, volume 778 of CEUR Workshop Proceedings. CEUR-WS, Chang-Yeong Lee. Representation of Switching Circuits b Binar-Decision Programs. Bell Sstem Technical Journal, 38(4), Thomas Lukasiewic and Umberto Straccia. Managing uncertaint and vagueness in description logics for the Semantic Web. Web Semantics: Science, Services and Agents on the World Wide Web, 6(4): , November Carsten Lut and Lut Schröder. Probabilistic Description Logics for Subjective Uncertaint. In Proc. of KR 10, pages AAAI Press, Kevin Patrick Murph. Dnamic baesian networks: representation, inference and learning. PhD thesis, Universit of California, Berkele, Heping Pan, Daniel McMichael, and Marta Lendjel. Inference Algorithms in Baesian Networks and the Probanet Sstem. Digital Signal Processing, 8(4): , Adam Sadilek and Henr Kaut. Recogniing multi-agent activities from GPS data. In Proc. of AAAI 10, pages , Stephan Tobies. The Compleit of Reasoning with Cardinalit Restrictions and Nominals in Epressive Description Logics. J. of Artificial Intel. Research, 12: , Jonas Vlasselaer, Wannes Meert, Gu Van Den Broeck, and Luc De Raedt. Efficient Probabilistic Inference for Dnamic Relational Models. In Proc. of StarAI 14, volume WS of AAAI Workshops. AAAI Press, 2014.