Statistical Relational AI: Papers from the AAAI-14 Workshop Reasoning in the Description Logic BEL Using Bayesian Networks Ismail˙ Ilkan˙ Ceylan⇤ Rafael Penaloza˜ † Theoretical Computer Science Theoretical Computer Science, TU Dresden, Germany TU Dresden, Germany Center for Advancing Electronics Dresden [email protected] [email protected] Abstract 2 Proof Structures in EL is a light-weight DL that allows for polynomial-time rea- We study the problem of reasoning in the probabilistic De- EL scription Logic . Using a novel structure, we show that soning. It is based on concepts and roles, corresponding to probabilistic reasoningBEL in this logic can be reduced in poly- unary and binary predicates from first-order logic, respec- nomial time to standard inferences over a Bayesian network. tively. Formally, let NC and NR be disjoint sets of concept This reduction provides tight complexity bounds for proba- names and role names, respectively. concepts are defined bilistic reasoning in . through the syntactic rule C ::= A EL C C r.C, where BEL | > | u | 9 A NC and r NR. The2 semantics2 of is given in terms of an interpre- 1 Introduction EL tation =(∆I , I ) where ∆I is a non-empty domain Description Logics (DLs) (Baader et al. 2007) are a fam- I · and I is an interpretation function that maps every con- ily of knowledge representation formalisms tailored towards · cept name A to a set AI ∆I and every role name the representation of terminological knowledge in a formal ✓ r to a set of binary relations rI ∆I ∆I . The manner. In their classical form, DLs are unable to handle ✓ ⇥ interpretation function I is extended to concepts by the inherent uncertainty of many application domains. To · EL defining I := ∆I , (C D)I := CI DI , and overcome this issue, several probabilistic extensions of DLs > u \ ( r.C)I := d ∆I e ∆I :(d, e) rI e CI . have been proposed. The choice of a specific probabilistic 9The knowledge{ 2 of a| 9 domain2 is represented2 ^ through2 a} set DL over others depends on the intended application; these of axioms restricting the interpretation of the concepts. logics differ in their logical expressivity, their semantics, and their independence assumptions. Definition 1 (TBox). A general concept inclusion (GCI) is an expression of the form C D, where C, D are con- Recently, the DL (Ceylan and Penaloza˜ 2014) was v introduced as a meansBEL of describing certain knowledge that cepts. A TBox is a finite set of GCIs. The signature of (sig( )) is the setT of concept and role names appearing in T. depends on an uncertain context, expressed by a Bayesian T T network (BN). An interesting property of this logic is that An interpretation satisfies the GCI C D iff CI DI ; is a model of theI TBox iff it satisfiesv all the GCIs✓ in . reasoning can be decoupled between the logical part and I T T the BN inferences. However, despite the logical component The main reasoning service in is subsumption check- of this logic being decidable in polynomial time, the best ing, i.e., deciding the sub-conceptEL relations between given known algorithm for probabilistic reasoning in runs in BEL concepts based on their semantic definitions. A concept exponential time. C is subsumed by D w.r.t. the TBox ( = C D) T T| v In this paper we use a novel structure, called the proof iff CI DI for all models of . It has been structure, to reduce probabilistic reasoning for a ✓ I T BEL shown that subsumption can be decided in in poly- knowledge base to probabilistic inferences in a BN. In a nut- nomial time by a completion algorithm (Baader,EL Brandt, shell, a proof structure describes the class of contexts that and Lutz 2005). This algorithm requires the TBox to entail the wanted consequence. A BN can be constructed to be in normal form; i.e., where all GCIs are one of the compute the probability of these contexts, which yields the forms A B A B C A r.B r.B A. It probability of the entailment. Since this reduction can be is well knownv that| everyu v TBox| canv9 be transformed| 9 v into an done in polynomial time, it provides tight upper bounds for equivalent one in normal form of linear size (Brandt 2004; the complexity of reasoning in . BEL Baader, Brandt, and Lutz 2005); for the rest of this paper, we assume that is a TBox in normal form and GCI denotes a ⇤Supported by DFG in the Research Training Group “RoSI” T (GRK 1907). normalized subsumption relation. In this paper, we are interested in deriving the GCIs in †Partially supported by DFG within the Cluster of Excellence normal form that follow from ; i.e. the normalised logical ‘cfAED’. T Copyright c 2014, Association for the Advancement of Artificial closure of . We introduce the deduction rules shown in Ta- Intelligence (www.aaai.org). All rights reserved. ble 1 to produceT the normalised logical closure of a TBox. 15 Table 1: Deduction rules for . EL Algorithm 1 Construction of the pruned proof structure Input: TBox Premises (S) Result (↵) Output: H =(T W, F ) 7! A B B C A C 1 , 1: V , E , i 0 2 hA v r.Bi h , vB ihC A v r.Ci 0 0 2: do T ; 3 hA v9r.Bi, hC v AihC v9r.Bi h v9 i h v ihv9 i 3: i i +1 4 r.A B , B C r.A C h9 v i h v ih9v i 4: Vi := Vi 1 ↵ S ↵,S Vi 1 5 r.A B , C A r.C B − [ { | 7! ✓ − } h9 v i h v ih9v i 5: Ei = (S, ↵) S ↵,S Vi 1 6 r.A B , B r.C A C { | 7! ✓ − } h9 v i h v9 ihv i 6: while Vi = Vi 1 or Ei = Ei 1 A r.B r.B C A C − − 7 , 7: W := (↵6 ,k) ↵ V ,6 0 k i hA v9B Ci h9C vX ihA vB i X k 8 , 8: E := {(S, (↵,k| )) 2(S, ↵) E, 0} k i 9 hA u B v Ci, hX v AihX u B v Ci { | 2 k } [ h u v i h v ihu v i 9: ( (↵,k 1) , (↵,k)) ↵ Vk 1, 0 k i 10 A B C , X B A X C − 10: return{({W, E)− } | 2 } 11 hX u Xv Ci h v ihX u Cv i h u v ihv i Each rule maps a set of premises to a GCI that is implicitly der to avoid cycles. In this case, nodes are pairs of GCIs and labels, where the latter indicates to which level the nodes be- encoded in the premises. It is easy to see that the sets of i premises cover all pairwise combinations of GCIs in normal long in the hypergraph. We write S = (↵,i) ↵ S to denote the i-labeled set of GCIs in S. Let{ n :=| sig2( }) 3, form and that the deduction rules produce the normalised | T | logical closure of a TBox. Moreover, the given deduction we start with the set W0 := (↵, 0) ↵ and define the levels 0 i<ninductively{ by | 2 T} rules introduce axioms only in normal form, and do not cre- i ate any new concept or role name. Hence, if n = sig( ) , Wi+1 := (↵,i+ 1) S Wi,S ↵ the logical closure of is computed after n3 rule| applica-T | { | ✓ 7! } [ (↵,i+ 1) (↵,i) Wi . tions, at most. T { | 2 } For each i, 0 i n, W contains all the consequences Later on we will associate a probability to the GCIs in the i TBox , and will be interested in computing the probability that can be derived by at most i applications of the deduction of a consequence.T It will then be useful to be able not only rules from Table 1. The unfolded proof structure of is the u n T to deduce the GCI, but also all the sub-TBoxes of from hypergraph H =(W ,F ), where W := i=0 Wi and T n T T T T which this GCI follows. Therefore, we store the traces of F := Fi, with T i=1 S the deduction rules using a directed hypergraph. i i FiS+1 := (S , (↵,i+ 1)) S Wi,S ↵ { | ✓ 7! } [ Definition 2. A directed hypergraph is a tuple H =(V,E) ( (↵,i) , (↵,i+ 1)) (↵,i) W where V is a non-empty set of vertices and E is a set of { { } | 2 i} directed hyper-edges of the form e =(S, v) where S V The following is a simple consequence of our constructions and v V .Apath from S to v in H is a sequence of hyper-✓ and Lemma 3. 2 edges (S1,v1), (S2,v2),...,(Sn,vn) such that vn = v and Theorem 4. Let be a TBox, and H =(V ,E ) and u T T T T Si S vj 0 <j<i for every i, 1 i n. In this H =(W ,F ) the proof structure and unfolded proof ✓ [ { | } T T case, the path has length n. structureT of , respectively. Then, T Given a TBox in normal form, we build the hypergraph 1. for all C D V and all , = C D iff H =(V ,E ),T where V is the set of all GCIs that follow there is av path from2 T (↵, 0) O↵ ✓ T toO|(C D,v n) in fromT andT ET = (S, ↵T) S ↵,S V where Hu , and { | 2 O} v T T { | 7! ✓ T } 7! T n 1 is the deduction relation defined in Table 1.