On the Use of Automated Reasoning Systems in Ontology Integration

Abstract

Ontology Integration is a challenge in the field of Knowledge Engineering, whose solution is indispensable for the envisioned Semantic Web. Some approximations suffer from logical confidence, and others are hard to mechanize. In this paper a method - assisted by Automated Reasoning Systems - to solve a subproblem, the merging of ontologies, is presented. A case study of application is drawn from the field of Qualitative Spatial Reasoning.

Keywords

• Ontology merging
• lattice categorical theory
• formal ontology building

Introduction

Ontology building has been an aim of the Knowledge Representation (KR) community during the last decades and, with special emphasis, in the last years, with the project of Semantic Web (SW). Such a project aims to enrich the Web by machine processable information [ 5 ]. Ontologies help to assign metadata to information by means of the formal description of the elements which belong to the discourse universe. This point of view restricts the attention to projects more feasible than ontologies for commonsense reasoning, which need the representation of a huge portion of human experience, encompassing knowledge about spatial, physical, social, temporal, and psychological aspects [ 19 ].

Nevertheless, ontology building is not sufficent. Knowledge Engineering practice shows that the representation is not static. Ontologies must be maintained as any other component of information systems. Knowledge Reuse requires ontologies are extended, refined or integrated [ 28 ].

The acceptation of ontology representation languages as OWL1 has facilitated the proliferation of ontologies. Thus, the necessity of relating ontologies arises in order to take advantage of the knowledge jointly contributed by different ontologies. Basically, there exist three kinds of reconciliation of the knowledge represented by ontologies: merging, alineation and integration.

Partially financed by project 2C/040 belonging to Proyecto Minerva, plataforma de sevicios en movilidad Cartuja 93 1 http://www.w3.org/TR/owl-features/

The aim of this paper is to propose a formal definition of ontology merging, following ideas from [ 9 ]. The definition we propose is inspired in methods for cleaning Knowledge Databases referenced by ontologies [ 3, 11 ]. Likewise, a merging method assisted by Automated Reasoning Systems (ARS) is described. This method is based on the formal framework of ontological extensions presented in [ 9, 10 ].

For the sake of clarity, the method is illustrated by merging two spatial ontologies designed for Qualitative Spatial Reasoning (QSR). Two spatial ontologies will be merged. The first one is the Region Connection Calculus (RCC) [ 12 ]. The second one is a micro-ontology about the relative size of spatial entities, SIZE. The interest of this merging arises from human perception. For example, it is known that some topological relationships (i.e. A is properly contained in B) only are possible if the involved objects possess relatively suitable size. Thus, it is interesting the joint management of both ontologies. In this case, merging is advisable. Currently, we use RCC as meta-ontology in order to spatially interpret concepts associated to ontologies [ 8 ]. Spatial metaphora is a powerfull technique for user’s understanding of ontology-based information systems (see [ 15, 23 ]). In general, the mereology is a natural source for supporting metaproperties on concepts and relations in Ontological Engineering [ 18, 20 ].

The structure of the paper is as follows. Next section addresses the problem of ontological evolution, using as example the case of Qualitative Spatial Reasoning. Ontologies RCC and SIZE are described in section 3. In section 4, the notion of lattice categorical ontology is presented and practical features of this notion are justified. Next, section 5 contains the logical ground for the formalization of the ontology merging method and the running example is computed. The article ends with some notes on related work and future aims. 2

Evolution of ontologies: Extending or revising? The revision of an ontology may be considered, to some extent, from two points of view. On one hand, the task is similar to knowledge revision, thus the problem can be analysed by means of classic methods (see e.g. chapter 6 in [ 25 ]). On the other hand, ontology evolution should preserve some sort of backward compatibility, while possible. Therefore, the advantage of extension on revision is the feasibility of preserving ontological features of source theory. This option is only possible if the theory is robust [ 6 ]. A formalization of robustness given in [ 2 ] is presented in [ 9 ], and it will be described in section 4. A discussion of these aspects can be found in [ 10 ].

Nevertheless, an ontological insertion is not interesting if it is not supported by a good theory about its relationship with the source theory, as well as a sound expansion of a representative class of models of the source theory to models of the new one (such class have to contain the intended models). For example, in the case of mereotopological reasoning, it can be necessary to show an intuitive topological interpretation of the new elements (and a re-interpretation of the older ones compatible with basic original principles), which should be formalized. This requirement is mandatory if one wishes to expand models of theory source to models of the new theory [ 11 ].

Therefore, one could consider ontology evolution as a sequence of extensions and revisions. As we have already commented, both methods are intimately tied. Likewise, above discussion is appliable for merging formal ontologies. Therefore, two ways for merging two ontologies O1 and O2 based on these ideas can be considered. A first one consists in repeatedly extending O1 by defining the terms of O2 (using the language of O1) producing this way a conservative extension of O1. Such definitions cannot exist in many cases, so it is necessary to consider a second method, based on the ontological insertion of the terms of O2 in O1 (possibly in parallel). For obtaining sound extensions, it is necessary to design axioms relating terms of both ontologies, for preserving basic features [ 9 ]. 3

Two examples of formal ontologies

We succintly present the main features of the ontologies used in the example. 3.1

RCC ontology

The Region Connection Calculus is a well-known topological approach to qualitative spatial representation and reasoning. For RCC, the spatial entities are non-empty regular sets (NERC)2 (A deep introduction to the theory can be found in [ 12 ], and the set of axioms appears in figure 1). The topological models of RCC have been investigated by N.M. Gotts in [ 16 ], where the author shows that every regular connected nontrivial topological space is a model of the theory (choosing as elements the NERCs); although it is also possible to study the models by means of a complete algebraic description [ 29 ]. Spatio-temporal reasoning has been studied in several papers [ 31, 27, 17 ].

DC(x, y) ↔ ¬C(x, y) (x disconnected from y) P (x, y) ↔ ∀z[C(z, x) → C(z, y)] (x part of y) P P (x, y) ↔ P (x, y) ∧ ¬P (y, x) (x proper part of y) EQ(x, y) ↔ P (x, y) ∧ P (y, x) (x identical with y) O(x, y) ↔ ∃z[P (z, x) ∧ P (z, y)] (x overlaps y) DR(x, y) ↔ ¬O(x, y) (x discrete from y) P O(x, y) ↔ O(x, y) ∧ ¬P (x, y) ∧ ¬P (y, x) (x partially overlaps y) EC(x, y) ↔ C(x, y) ∧ ¬O(x, y) (x externally connected to y) T P P (x, y) ↔ P P (x, y) ∧ ∃z[EC(z, x) ∧ EC(z, y)] (x tangential prop. part of y) N T P P (x, y) ↔ P P (x, y) ∧ ¬∃z[EC(z, x) ∧ EC(z, y)] (x non-tang. prop. part of y)

Fig. 1. Axioms of RCC 2 A set x of a topological space is regular if it agrees with the interior of its closure.

Fig. 2 illustrates the set of eight jointly exhaustive and pairwise disjoint (JEPD) relations called RCC8. If one considers RCC8 as a calculus, all possible unions of the basic relations are also considered.

a b a b a b a b a b a b a b a b DC(a,b) EC(a,b)

PO(a,b) TPP(a,b)

TPPi(a,b)

NTPP(a,b) NTPPi(a,b) EQ(a,b)

The set RCC relations enjoys the structure of lattice whose Hasse diagram is shown in figure 3, that coincides with their intended meanings. This structure is provable by RCC [ 11 ]. The ontology RCC is insufficient for reasoning with many spatial features. Therefore there appears the need of enriching RCC to reason with other features. The second ontology dealed in this paper is SIZE, the natural ontology about the relative size of spatial entities. Such an ontology has already been used jointly with RCC in [ 14 ], where the authors extend the study of constraint satisfaction problems on RCC8 done in [ 27 ].

The SIZE relationships are: LS(x, y) (x have less size than y) and its inverse, LSi(x, y); the relation LSE (x have less or equal size than y) and its inverse LSEi(x, y); and SS(x, y) (x and y have the same size).

Some axioms of SIZE are shown in figure 4 (rigth). The SIZE ontology suffices to prove that the relations form the lattice depicted in figure 4 (left). ∀x, y(LS(x, y) ∨ LSi(x, y) ∨ SS(x, y)) ∀x, y(LS(x, y) → ¬(LSi(x, y) ∨ SS(x, y)) ∀x, y(LSE(x, y) ↔ LS(x, y) ∨ SS(x, y)) ∀x, y(SS(x, y) ↔ SS(y, x)) ∀x, y(LS(x, y) ↔ LSi(y, x))

Note that SIZE does not deal with the qualitative size of regions. In this case, adjectifs (unary predicates) have to be used [ 21 ]. The articulation of two ontologies describes the existent relationship between terms of both ontologies (see [ 4 ] for a formalization of such notion). The articulation of RCC and SIZE is depicted in figure 5. Note that the articulation is ⎧ T P P E := ⎨⎪⎪ N T P P

T P P i ⎪⎪⎩ N T P P

LS LS LSi LSi

EQ SS LSi LS

SS DC DC DC

EC EC EC

P O P O P O

EQ T P P i T P P

N T P P i N T P P simple; in fact, it can be considered as a sublattice of that of RCC (see fig. 6).

In this work the merging of RCC and SIZE as running example is computed, preserving articulation and satisfying a minimality condition (the size of concept lattice) more feasible to achieve than categorical minimality [ 4 ]. Lastly, note that SIZE is independent from space dimension. Thus it is also valid for working with temporal intervals, then one might think of combining SIZE with Allen’s ontology about temporal intervals [ 1 ]. 4

Extension of lattice-categorical theories In this section we remember formal definitions and properties, formalizing two notions: robustness (lattice categoricity) and extension (lattice categorical extension). Our thesis states that one can change logical completeness by lattice categoricity to make easier the design of feasible methods for extending a theory. A formalization of these ideas can be found in [ 9, 10 ].

Consider a first order language, let R = {R1, . . . , Rn} be a (finite) set of concept symbols (or relations symbols with the same arity n) and let T be a theory3. Given M |= T , we consider the structure L(M, R), in the language 3 In the general case, one can consider definable concepts/relations in T .

PPi 16 14 C 15 DR 13 O 9 PP 10PPi 0 1PO 2NTPP 3TPP 4EQ 5TPPi N6TPPi 7EC 8DC LSE

LSEi LS

SS

LSi LR = { , ⊥, ≤} + {r1, . . . , rn}, whose universe is the interpretations in M of the relations (interpreting ri as RiM ), is M n, ⊥ is ∅ and ≤ is the subset relation. Notice that, in general, L(M, R) does not have lattice structure. Nevertheless, this requisite is assumed in the remainder of the paper. The Formal Concept Analysis FCA [ 13 ], produce such structures. In any case, it is easy to satisfy it by extending by definitions of new relations.

From the logic point of view, a finite lattice L on a set A = {a1, . . . an} can be categorically axiomatized by the set of axioms composed by: 1. A set ΘR of formulas that contains the axioms of lattice theory 2. The domain closure axiom (d.c.a.), 3. Unique names axiom (u.n.a.), And additionally, a set of (dis)equations ER which characterizes the Hasse diagram of the lattice.

When L(M, R) is a lattice, the relationship with the self model M is based on two facts. The first one is that the lattice L can be characterized by a finite set of equations EL, plus ΘR. The second observation is that there exists a natural translation Π of such equations into formulas in the First Order Logic language in such way that if E is a set of lattice equations characterizing L(M, R) (so L(M, R) |= E), it holds that M |= Π(E). Note that Π(E) is consistent with T . Definition 1. Let E be a LR-theory. We say that E is a lattice skeleton for a theory T if the following conditions hold: 12.. ETh+erΘe Rexihsatss Monl|y=oTnesumcohdtehl a(tmLo(dMulo,Ris)o|m=oErp+hisΘmR)..

Proposition 1. Suppose that E is a set of equations in the language LR such that E + ΘR has an only model. The following conditions are equivalent: 1. E is a lattice skeleton of T . 2. T + Π(E) is consistent.

Corollary 1. Every consistent theory has a lattice skeleton.

The theory that we preserve throughout is the translation of RCC8 language of ERCC (In figure 7 it is described using Propositional Description Logic notation)4. Note that such theory has a limited syntactic complexity; one can say that this theory has exogenous character, as countepart to the endogenous character of their definitions in RCC. That is, it is a theory on relations without explicit reference to their definitions with respect to elements of the universe of the model.

≡ C D DR ≡ EC

C ≡ O O ≡ P O P i ≡ EQ

P ≡ EQ P P i ≡ T P P i P P ≡ T P P

DC EC

P P P i P P

P i N T P P i N T P P

P O N T P P

T P P

EQ

T P P i N T P P i

EC ¬P ¬P i ¬DR ¬T P P ¬P i ¬DR ¬P i ¬DR ¬P P i ¬DR ¬N T P P i ¬DR ¬DR ¬DC Definition 2. T is called a lattice categorical (l.c.) theory if any two lattice skeletons of T are equivalent modulo ΘR.

In other words, if one assumes that the only relations are in R, then T can prove the intended lattice structure induced by R.

Proposition 2. The following conditions are equivalent: 1. T is l.c. 2. If E is any lattice skeleton for T , then T Π(E).

That is, T is a l.c. if E0 + ΘR ≡ E1 + ΘR for all E0, E1 lattice skeletons of T . Since a l.c. theory T has an only nonisomorphic lattice associated to R, we denote this lattice by L(T , R), to emphasize the lattice categoricity5. As it will shown in the next section, RCC is l.c. 4 The last two formulas are ommited because their translations are tautologies. 5 T is a l.c. theory if ET + ΘR is a strictly categorical theory (that is, given any two models of ET + ΘR, there exists one only isomorphism between them). Theorem 1. Let T be a consistent theory. There exists T an extension of T which is lattice categorical.

A pair (T , E) where T is lattice categorical and E is a lattice skeleton for T , called lattice categorical core (l.c.c.). Thus, (T , E) is a l.c.c. if T + Π(E) is a l.c. theory. The pair (RCC, ERCC ) that it will describe in next section is a l.c.c.

To simplify the discourse, in the remainder of the paper a l.c.c. will be simply called ontology.

Definition 3. Given two ontologies O1 = (T1, E1) and O2 = (T2, E2) with respect to sets of concepts/relations R1 and R2 respectively, we will say that O2 is a lattice categorical extension of O1 (denoted by O1 →lc O2) if R1 ⊆ R2 and E2 |= E1. 5

Merging Lattice categorical ontologies In this section a proposal of ontology merging is presented. To simplify, suppose O1 and O2 are two ontologies with disjoint signatures. We assume that E is a set of formulas relating concepts of both ontologies.

Definition 4. An l.c. ontology O = (T , E) is a lattice categorical merging of O1 and O2 with respect to E if O1 →lc O, O2 →lc O , L(T , R) |= E , and it is the lattice of minimum size with that property. In the case that articulation Oa exists, then the following diagram is conmutative, the interpretation of Oa in O does not depend on the intermediate ontology:

Oa lc lc

O1 O2 lc lc

O

The following method for l.c. merging is based on the one for lattice categorical extensions introduced in [ 9 ]: 1. First stage: Joint skeletons E1 and E2 with E ; let E0 := E1 ∪ E2 ∪ E 2. Find, with MACE4, the lattices that models E0.

If such lattices do not exist, ontologies are not compatible with respect to E0, then there is not merging (E0 is not satisfiable). If MACE4 outputs some model, follow in (3). 3. Second stage: User analysis of a lattice of minimum size. If any of the relationships of the lattice is not accepted by the user, refine E0 by adding new (dis)equations, to discard such lattice. Turn to the apply MACE4 to the refinement.

This stage is repeatedly applied until a model L (of minimum size k) is accepted by the user. On this way a skeleton S0 is obtained. 4. Third stage: achieving lattice categoricity. Refine S0 (by the addition of (dis)equations) until that the only model of size k is L. Let S be the resulting set of equations. 5. Fourth stage: Certification. Certify (with an automated reasoning system, OTTER, if it is necessary) that L is the only model of S of size k. This way it is accurated that O = (T1 ∪ T2 ∪ S, S) is a l.c. merging.

Theorem 2. The method described above outputs a l.c. merging of O1 and O2. 5.1

Merging RCC and SIZE

Suppose that it is intended to use SIZE on connected regions. Intuition says that it is unknown (a priori) the size relation between disconnected regions. Notice that this implies an ontological re-interpretation of SIZE relations. For example, LS(x, y) has to be understood as region x is connected with a region y of bigger size. That reinterpretation is compatible with articulation depicted in figure 6.

Starting with the SIZE skeleton:

It accepts the final lattice depicted in figure 8. It is important to note that the user may need to interpret some of the new nodes. For example, node 22 represents the relation distinct sizes (that is, ¬SS), and node 23 represents the relation x partially overlaps with a region y of distinct size. However, the use of new nodes can be unnecessary in practice; some nodes only represent algebraic operations on basic nodes (to satisfy lattice structure). Therefore, this lattice can be pruned in order to obtain a ontological simpler structure. x is connected with y but they have distinct size

15 DR 10 PPi 3 TPP 9 PP 2NTPP x 23 y In this paper a method for merging formal ontologies is presented. The method is concept-driven (the basic requirement is the categoricity of concept lattice), and logic-based. There are several approachs to solve the problem, but, in general, they do not make use of automated model finders to find an alternative merging. Therefore it is hard to compare these (not logic-based) methods with the method of this paper. For example, Prompt [ 26 ] facilitates the necessary tasks for merging two classes in a new class, locating possible equivalences. In our method, limited to concepts, the ARS induces the possible algebraic operations with the classes, while Prompt asks the user. Hcone [ 22 ] is, in general terms, a similar approach. However, it is based on the use of Wordnet to determine relations between concepts. In our case, the process starts with known relations and the remainder is induced by the method, always asking user’s approval. A similar situation occurs with ONION [ 24 ] which also uses linguistic features to compute the articulation.

The essential feature of the method of this paper, designed for formal ontologies, is that it builds an ontology with certifiable categoric features. Additionally, since alternative models are offered, the user is enforced to refine the knowledge for discarding some ones, obtaining in this way more information about the merging.

In [ 4 ], ontology merging by means of algebraic specifications is formalized. The merging is a colimit, when the ontologies are compatible. In our approach, the resulting common lattice categorical extension(s), of minimun size, can be considered as a practical variant of colimit notion, and the compatibility is determined by the existence of models. The feasibility of the method depends on the reasoning on finite structures, in constras to the abstract notion of colimit.

A pertinent question to ask is: what are the limits of applicability of the method? At first sight it seems that depends on the model finder. However, the formal method is designed for concrete domains. In such situations, users can recognize the soundness of ontologies depicted from MACE4’s results. In the case of merging a little ontology with a big size one , it could be interesting to design contextualized merging in order to simplify the merging process. In fact we have contextualized SIZE on connected regions. Finally, when one must select the best merging ontology from data (data-driven), the problem is different. In this case the user can not recognize new concepts. To solve this limitation it can be interesting to use cognitive entropy [ 10 ] to select the lattice in each step of the model. The ontological insertion of new data-driven concepts has been studied in [ 7 ].

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