Tableau-based revision in SHIQ
Thinh Dong, Chan Le Duc, Philippe Bonnot, Myriam Lamolle
LIASD - EA4383, IUT of Montreuil, University of Paris8, France {dong, leduc, bonnot, lamolle}@iut.univ-paris8.fr
IntroductionThe problem of revising a description logic-based ontology (called DL ontology) is closely related to the problem of belief revision which has been widely discussed in the literature. Among early works on belief revision, the AGM theory (Alchourrónet al., 1985) introduced intuitive and plausible constraints (namely AGM postulates) which should be satisfied by any rational belief revision operator. However, it is not trivial to adapt belief revision operators to DLs because DLs have their own features (Flouriset al., 2005) (Qi and Yang, 2008). One main difficult for such revision is that DL ontologies often incur infinitely many models. To address this issue, we propose a finite set of finite structures, namely a setMT(O)of completion trees, for characterizing a possibly infinite set of models of an ontologyO. Then, we define a distance over a set of completion trees. This distance allows one to determine how far an ontology is from another one. Another problem our approach has to address is that there may not exist a revision ontology such that (i) it is expressible in the logic used for expressing initial ontologiesO,O0, and (ii) it admitsexactlya set of modelsMT(O,O0) computed fromMT(O)andMT(O0). For this reason, we borrow the notion ofmaximal approximation(De Giacomoet al., 2007) which allows us to build aminimalrevision ontology admittingMT(O,O0).
Construction of the revision ontology First, we define a novel tableau algorithm, namelyTA, for aSHIQontology without individuals by replacing expansionv-,u-, t,ch-rules by a new rule, namelysat-rule which chooses a subsetSfrom a setsub(O) including all sub-concepts of aSHIQontologyO. Note that all concepts in the form of conjunctions or of disjunctions are removed from sub(O)and replaced with their conjuncts and disjuncts. This can be performed by a functionFlat(C)that flattens con- junctions and disjunctions of a conceptCinto subsets of sub-concepts occurring inC.
For example,Flat(Au(∃R.BtC)) ={{A,∃R.B},{A, C}}.
sat-rule.Ifsat-rule has never been applied to a nodexthen we choose a subsetS ⊆ sub(O)such thatL(x)∪ S
CvD∈T
f(CvD)⊆Swheref(CvD)∈Flat(¬CtD)
for eachCvD∈ T, and setL(x) :=S∪S¯whereS¯={¬C|C∈sub(O)\S}
In this paper, a completion treefor O is a tree T = (V, E, L,x)b where V is a set of nodes with the root nodebx∈ V. Each nodex∈ V is labeled with a function L(x)⊆sub(O).Eis a set of edges and each edgehx, yi ∈Eis labeled with a function L(hx, yi)containing a set ofSHIQroles.
Thesat-rule that is applied to each node of a completion tree introduces a lot of non- determinisms. We need this “bad” behavior of the new tableau algorithm to control the generation process of completion trees in such a way that allows one to infer the ontol- ogy when knowing completion trees and its signature. We useMT(O)to denote the set of all completion trees which are generated by running the novel tableau algorithmTA for an ontologyO. Note thatTAdoes not necessarily terminate when a complete and
clash-free completion tree is built. It should terminate when all non-determinisms are considered. We can extend straightforwardlyMT(O)toMT(O0,sub(O))as follows.
The setMT(O0,sub(O))is built by the tableau algorithmTAforO0with an extra set of conceptssub(O)that is taken into account when applying thesat-rule. In this case one can import additional concepts into node labels of a completion tree forO0while re- specting the axioms ofO0. Importingsub(O)toMT(O0)ensures thatMT(O0,sub(O)) captures semantic constraints fromOwhich are compatible withO0.
Next, we introduce a distance between two completion treesTandT0which allows one to talk about the similarity between two ontologies. This distance is defined for two completion trees which are isomorphic, i.e., there is an isomorphismπthat maintains the successor relationship from two nodes of a completion tree to the two corresponding nodes of the other one via π. Note that we can always obtain such an isomorphism between two completion trees by adding empty nodes and edges to completion trees since node and edge labels are ignored in the definition of isomorphisms.
Definition 1 (Distance). Let T = hV, L, E,bxi and T0 = hV0, L0, E0,bx0i two com- pletion trees. Let Π(T, T0) be the set of all isomorphisms between T and T0. The distance between T and T0, denoted T M T0, is defined as follows: T M T0 =
min
π∈Π(T ,T0){max
x∈V(|L(x)ML0(π(x))|)+ max
hx,yi∈E(|L(hx, yi)ML0(hπ(x), π(y)i)|)}
We can check thatMis a distance over a set of isomorphic trees with the operatorM defined over two node or edge labels α, α0 as follows: L(α) M L0(α0) = (L(α)∪ L0(α0))\(L(α)∩L0(α0)). Based on this distance, we now define a set of completion trees a revision ontology of an ontologyOby anotherO0should admit.
Definition 2 (Revision operation).LetOandO0be two consistentSHIQontologies.
A set of tree modelsMT(O,O0)of the revision ofObyO0is defined as follows:
MT(O,O0) ={T ∈MT(O0,sub(O))| ∃T0∈MT(O,sub(O0)),
∀T0 ∈MT(O,sub(O0)), T00∈MT(O0,sub(O)) :T MT0≤T0MT00}
Intuitively, MT(O0,sub(O)) includes completion trees from MT(O0) each node of which is consistently filled by an arbitrary set of concepts imported fromsub(O)such that each axiom of O0 remains satisfied. Among these completion trees, MT(O,O0) retains only those which are closest to completion trees fromMT(O,sub(O0))thanks to the operatorT MT0that characterizes the difference betweenTandT0. We consider the following example. Let O = {> v Au ∃R.(¬B)u ¬B} and O0 = {¬A v
∀R.B,¬B v Au ∀R.B}. By running the algorithm TA for O, we build the set MT(O,sub(O0))which contains a unique tree modelT1with nodes{a, b}and labels L(a) ={A,∃R.(¬B),¬B}, L(b) ={A,∃R.(¬B),¬B}, E={R(a, b)}. In the same way,MT(O0,sub(O))has 4 tree models one of which is T10 with nodes{a0, b0}and labelsL(a0) = {A,∃R.(¬B), B}, L(b0) = {¬B, A,∀R.B},{R(a0, b0)}. According to Definition 2, we haveT10 M T1 = 2that is minimal. Thus,MT(O,O0)contains a unique tree modelT10.
We obtain a strong result which states that the all AGM postulates rephrased (Qiet al., 2006) for DL ontologies in our setting hold. This result relies on a total pre-order over a set of all completion trees that can be devised from the distance according to
Definition 1. The main difference between the postulates presented by Qi et al. and those reformulated in our setting is that the set of modelsMod(O)of an ontologyO is replaced withMT(O). To illustrate this point, we consider a postulate by Qi et al.
(G2): If Mod(O)∩Mod(O0) 6= ∅ thenMod(O,O0) = Mod(O)∩Mod(O0); and our corresponding postulate: (P2)IfMT(O,sub(O0))∩MT(O0,sub(O)) 6= ∅ then MT(O,O0) =MT(O,sub(O0))∩MT(O0,sub(O)). A proof of(P2)can be obtained straightforwardly from the definition ofMT(O,sub(O0))andMT(O,O0).
By soundness and completeness of the tableau algorithm, we can show thatMod(O) is semantically equivalent toMT(O), i.e.,MT(O) |= αiffMod(O) |= αfor some axiom α. Moreover, it holds that Mod(O)∩Mod(O0) 6= ∅ iff MT(O,sub(O0))∩ MT(O0,sub(O)) 6= ∅. Therefore, as(G2) our postulate (P2)captures the fact that ifO ∪ O0 is consistent, then the revision ontology ofObyO0 should admit exactly shared models of O and O0. Such models are encapsulated in MT(O,sub(O0))∩ MT(O0,sub(O))by our setting.
Finally, our goal is to build fromMT(O,O0)a revision ontologyObthat admits ex- actlyMT(O,O0)as tree models. However, we can show that there may not exist such an ontologyObby reconsidering the example above withMT(O,O0) ={T10}. Assume that there exists an ontologyObwithsub(O) =b {A,¬A, B,¬B,∃R.(¬B),∀R.B}which admits the uniqueT10 as tree model. Due to the specific behavior of thesat-rule with sub(O), if we applyb TAtoObfor buildingMT(O), we must obtainb T1and another tree modelT20with one node{x},L(x) ={A,∀R.B, B}, which is a contradiction.
For this reason, we use the notion of maximal approximation (De Giacomoet al., 2007) to define an ontologyO∗ which satisfies the following conditions: (i)O∗ is ex- pressible inSHIQ, (ii) it admits tree models inMT(O,O0), and (iii) it is a “smallest”
ontology admittingMT(O,O0). Such an ontologyO∗, namelymaximal approximation, can be built from the node labels of all tree models inMT(O,O0).
Definition 3 (Revision ontology).LetOandO0be two consistentSHIQontologies with revision operationMT(O,O0) = {T1,· · ·, Tn}whereTi = hVi, Li, Ei,xbiifor 1≤i≤n. A revision ontologyO∗= (T,R)ofObyO0can be built from completion trees inMT(O,O0)as follows:Rincludes the role hierarchy ofO0and the one ofO;
T contains all axioms ofO0and the following axiom :> v G
1≤i≤n
(G
x∈Vi
( l
C∈Li(x)
C)).
Theorem 1. LetOandO0be two consistentSHIQontologies. The revision ontology O∗ofObyO0 is a maximal approximation fromMT(O,O0). Additionally, the size of O∗is bounded by a doubly exponential function in the size ofOandO0.
ConclusionThe main limitation of our approach is to omit individuals in ontologies.
However, our approach can be extended in order to deal with individuals by extend- ing the distance defined for completion trees to graphs. Another limitation is that the obtained revision ontology is very large. This exponential blow-up in size arises from doubly exponential size of completion trees. We believe that our procedure can be im- proved by using a method for compressing completion trees generated from tableau algorithms. Such a method has been proposed by Le Duc et al. (Le Ducet al., 2013).
AcknowledgementsThis work was partially supported by FUI project “Learning Café”.
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