We propose an ABox repair approach for temporal DL-Lite (TDL-Lite) Knowledge Bases (KBs) [2,3] considering the data part of the KB as the source of inconsistency over time. As a TDL-Lite language, we here consider the TNDL-Lite fragment [11] allowing LTL future temporal operators [9] interpreted over N and we restrict ourselves to specifying functionality on roles and using the two future operators 3F and 2F . Our goal is twofold: 1) detect data inconsistencies and 2) propose a data temporal repair. For the inconsistency detection, we propose an equi-satis able reduction approach from TDL-Lite to DL-Lite which allows using DL reasoners that can return the precise set of inconsistent assertions. Thereafter, we propose a method for computing the best temporal repair based on the allowed rigid predicates and the time order of assertions. Let NC; NI and NR be countable sets of concept, individual names and roles respectively. NR is the union NG [ NL where NG and NL are countable and disjoint sets of global and local role names, respectively. TDL-Lite basic concepts B, concepts C,(temporal) concepts D, and roles R, are formed according to the following grammar: R ::= L j L j G j G ; B ::= ? j > j A j 9R; C ::= B j :C j C1 u C2; D ::= C j 3F D j 2F D j :D j D1 u D2: where L 2 NL, G 2 NG, A 2 NC. We call disjointness, inclusions of the form C u D v ?. We also add the ability to specify functional roles (f unct R) and rigid predicates which are elements from the set of rigid concepts NRC NC or of rigid roles NG and for all X 2 NRC [ NG and i; j 2 N, XIi = XIj (denoted simply by XI ).
? Extended version published as arXiv:2108.12149 ?? Copyright © 2021 for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0). (a) a case where the recent assertion is removed (b) a case where the rigid assertion is removed
The translation creates fresh concepts Ci and roles Ri in the DL-Lite KB denoting respectively the interpretation Ii of C and R at time point i. Now, the translation T y of a TBox and Ay of an ABox is as follows:
T y =f Ay =
m ^ ^ (tr(C; i; m) v tr(D; i; m)); CvD2T i=0
m ^ ^ tr(C; i; m) v uim tr(C; i; m) ; C2NRC i=0
m ^ ^ tr(R; i; m) v uim tr(R; i; m) ; R2NG i=0
m ^ ^(F unct(Ri)) g F unct(R) i=0
^ An(a) ^ ^ Rn(a; b):
nA(a)2A nR(a;b)2A
(1)
(2)
(3)
(4)
(5)
Inconsistency Detection. We start by checking the satis ability of the translated
TBox T y and then we check the consistency of Ay with T y using the reasoner
Pellet4. If Ay is inconsistent, the explanation support of Pellet points in each
explanation a minimal subset Im = fai; aj ; :::g of inconsistent assertions in
conict with a constraint ck such as a disjointness inclusion, a rigid predicate or a
functional role. We report those explanations in an intuitive way by building an
inconsistency graph IG which is similar to the con ict-hypergraph used to
represent constraint violations in databases [
Temporal ABox Cleaning in TDL-Lite Maximal Repair Semantics. An extreme repair solution would be to simply remove all the detected inconsistent assertions from the ABox. This would certainly not meet the principal of minimal set of changes that restore consistency. In other words, a maximal repair A0 should be a maximal consistent subset of A, obtained by removing a minimal set of inconsistent assertions (aka the Minimal Unsatis able Set MUS). In the inconsistency graph, this corresponds to remove only a single vertex from each hyperedge given that hyperedges are minimal subsets of inconsistent assertions. Moreover, the maximal repair is obtained by removing vertices in the intersection of hyperedges i.e. assertions who are involved in more than one con ict (violated constraint). In practice, the maximal repair algorithm removes from each hyperedge the vertex with the highest Inconsistency degree Id. Formally, we de ne an inconsistency degree Id(ai) of a vertex ai as the number of hyperedges in which ai belongs. When vertices have the same Id, one is randomly removed. Hence, this may result in several possible MUSs for the same IG which are all minimal and then make the repair maximal. wt(ai; ti) = ti + ti if ai 2 NRC [ NG otherwise Best Temporal Repair Semantics. Clearly, not just any random maximal repair is useful or interesting in the temporal setting. For instance, repairs that remove mostly recent assertions might be unwanted. Our aim is to guide the repair algorithm when removing assertions with the same Id, by chosing to remove assertions having the lowest temporal weight. To do so, we assign a temporal weight wt for each inconsistent assertion ai associated with the timestamp ti as follows: where is a predicate lifespan. If ai is not an instance of a rigid predicate, = 0. The intuition behind using for each rigid concept Ci 2 NRC and global role Gi 2 NG is to set a duration after which its instances are weakened.
Figure 1 shows that the maximal repair in cases (1a) and (1b) could be the same because they share the same Id. However, it is easy to see on a timeline that it is better to remove Minor(John; 2) in (1a) and Adult(John; 0) in (1b). The notion of temporal weight is intended to capture situations where a maximal repair is temporally better than another. Moreover, could be xed using a data driven approach or a guided user approach.
Conclusions. This work is a rst exploration of repairing the ABox w.r.t a
TBox de ned over Temporal DL-Lite. The temporal language considered so far
is the TNDL-Lite with which we can express and check several useful types of
temporal constraints, such as de ning temporal concepts in GCIs and rigid
predicates. More generally, we plan to investigate in practice repairing KBs based
on multiple temporal DL-Lite logics which are First Order rewritable [