In ontology engineering, it is essential to develop techniques for module extraction, ontology reuse and ontology comparison. Since it was introduced in DL-Lite [1], forgetting has become into a major method of extracting modules in DL ontologies. It has been applied in some other domains such as ontology comparison [2]. Given a (consistent) ontology K and a set of concepts and roles F , the result of forgetting is a new ontology K0 that does not contain symbols in F and preserves certain reasoning or query answering tasks. In existing approaches, forgetting is de ned for only consistent ontologies. However, ontologies can be error-prone and inconsistent. To our best knowledge, the problem of establishing a framework for inconsistent ontologies has not been investigated yet. In order to provide a de nition of forgetting for a possibly inconsistent DL ontology, we could rst repair it and then perform standard forgetting on the repaired ontology. However, there are some issues with this naive approach. In particular, it can give unwanted results for some ontologies. Example 1. Let K = hT ; Ai, where T = fA v D; B v C; A v :Bg and A = fA(a); B(a)g. Then K has two repairs (or maximal consistent components) fA(a)g and fB(a)g. The forgetting results of B in these repairs are K1 = hT 0; fA(a)gi, K2 = hT 0; fC(a)gi, respectively, where T 0 = fA v Dg. Under the semantics IAR for inconsistencytolerant querying, K 6j=IAR C(a), K 6j=AR C(a), and K j=CAR D(a). Apparently, these querying results are not satis ed in K1 or K2.
DL-lite An ontology in DL-lite is a pair K = hT ; Ai, where T is a set of inclusion axioms and A is a set of assertions. The concepts and roles of DL-litecore are de ned by
B ! A j 9R;
R ! P j P ;
C ! B j :B where A is a concept name and R is a role name.
An inclusion axiom in DL-litecore is of the form B v C. An assertion is of the form A(a) or R(a; b), where a and b are individuals.
The semantics of DL-lite is omitted here.
Inconsistency-tolerant query answering We consider four inconsistency-tolerant
semantics AR, IAR, CAR, ICAR [
AR IAR CAR ICAR
T [ A j= q if for each A0 2 repairT (A), T [ A0 j= q
T [ TA02repairT (A) A0 j= q for each A0 2 repairT (clcT (A)), T [ A0 j= q
T [ TA02repairT (clcT (A)) A0 j= q In this section, we introduce a de nition of forgetting for inconsistent KBs and present two useful properties. Our de nition is quite general and actually applies to any DLs.
The signature of a TBox T , denoted sig(T ) is the set of concept names and role names that occur in T . This notion extends to TBox axioms, ABox assertions, ABoxes and KBs. As with inconsistency-tolerant query answering approaches, we assume the TBox is consistent. For a DL L and a signature , TBox T 0 is a result of L-forgetting about in T , if T 0 j= i T j= for each TBox axiom in L over sig(T ) n . De nition 1. For a DL L, a (possible inconsistent) KB K = T [ A in L, a signature and an inconsistency-tolerant semantics , a consistent KB K0 = T 0 [ A0 is a result of -forgetting about in K if (1) sig(K0) sig(K) n , (2) T 0 is a result of L-forgetting about in T , and (3) for any BCQ q over sig(K) n , K0 j= q i K j= q.
A result of -forgetting is a consistent KB that has exactly the same logical consequences (i.e., TBox axioms and BCQs) as the initial KB (under semantics) over the remaining signature. The de nition does not suggest the existence of forgetting. In fact it is well known that L-forgetting does not always exist for basic DLs such as E L and ALC. Yet we show that -forgetting always exists for DL-Litecore. Theorem 1. For a KB K in DL-litecore and a signature , a result of -forgetting about in K exists for any inconsistency-tolerant semantics .
For DL-litecore KBs, it is e cient to compute their IAR and ICAR forgettings. Theorem 2. In DL-Litecore, the computation of -forgetting is in P for or ICAR with respect to data complexity. being IAR
The intuition behind this result is that forgetting under IAR and ICAR can be computed by removing all the con icts (i.e., facts that lead to inconsistency) from the KB and then perform the standard de nition of forgetting for consistent KBs. These two steps can be done in polynomial time. 4
Because of the linear property of axioms in DL-litecore, DL-litecore KBs can be stored
in directed binary graphs, thus some reasoning tasks can be naturally captured by
traversals of such graphs. We have implemented a prototype system with Neo4j1
for computing IAR concept forgetting in DL-litecore, using a graph-based algorithm.
Speci cally, given a DL-litecore KB K, and a set of concepts to forget from K,
we rst transfer K into a directed graph GK where each node of GK represents a
general concept or an individual of K, and each edge of GK represents an inclusion or
membership assertion of K [
:D 9R
C B A b a
D 9S :D 9R
C B A b a
D 9S
We use a modi ed version of LUBM with auto-generated inconsistent ABox data
[
Some experimental results are shown in Table 4, where Anm denotes an ABox containing the data of n universities and inconsistencies added with probability of m.
For the largest ABox with up to 2 million assertions, our algorithm can still solve the problem in around one minute. In all cases, the size of the resulting KB is considerably larger than the original one, which is not surprising because a large number of concepts related to facts are forgotten. We can also see that the ratio of inconsistencies contributes little to the cost of computation.
We plan to adapt the proposed approach to some other non-standard reasoning
supports for ontologies, such as query explanation (abduction) [
This work was partially supported by the Australian Research Council (ARC) under grant DP130102302.