Auxiliary lemmas

In document Reasoning with a Network of Aligned Ontologies (Page 32-42)

6.2 Only-if-direction

8.1.1 Auxiliary lemmas

Informally, we could be stated as follows: the function ibuilt before preserves the structure

of roles.

Lemma 4 For each R ∈Ri, i(Ii(R)) = {hIA(aW), IA(aW0)i | hW, W0, i : Ri ∈ Φ}.

Proof: The proof of this lemma is decomposed into two steps, corresponding to both inclu- sions of the two sets.

( ⊆ ): Let (x, x0) ∈ i(Ii(R)). There exists (e, e0) ∈ Ii(R) such that x = i(e) and x0 =

i(e0). It follows that e ∈ Ii(∃R.>). So, thanks to item 9 of Def. 19, and because Iiis a model

of bOΩi, there exists W, W0∈ Ω such that e ∈ Ii(CW,WR 0) and similarly, e0 ∈ Ii(CR

W0,W). This

implies that bOΩi 6|= CW,WR 0 v ∀R¬CWR−0,W, which means that hW, W0, Ri ∈ ΦΩ. Additionally,

thanks to item 8a of Def. 19, e ∈ Ii(CW>) and e0 ∈ Ii(CW>0), so i(e) = IA(aW) and

i(e0) = IA(aW0). This establishes the first inclusion.

( ⊇ ): Let hW, W0, i : Ri ∈ ΦΩ. According to item 8b of Def. 19, we have CW,WR 0(βW,WR 0) and

CW,WR 0 v ∃R.CR −

W0,W, which implies that there exists x ∈ Ii(CR

W0,W) such that hIi(βRW,W0), xi ∈

Ii(R). Moreover, by item 8a of Def. 19, we have Ii(βW,WR 0) ∈ Ii(CW>) and x ∈ Ii(CW>0), so by

definition of i, i(Ii(βW,WR 0)) = IA(aW) and i(x) = IA(aW0). Therefore, hIA(aW), IA(aW0)i ∈

i(Ii(R)). This establishes the other inclusion. 

8.1.2 Satisfaction of the system.

Cross-ontology role subsumption (i : R ↔ j : S ∈ Av ij =⇒ i(Ii(R)) ⊆ j(Ij(S))). Let us

assume that i : R ↔ j : S ∈ Av ij. Let hx, x0i ∈ i(Ii(R)). From Lem. 4, it follows that

there exist W, W0 ∈ Ω such that x = IA(aW), x0 = IA(aW0) and hW, W0, i : Ri ∈ Φ. By

the definition of bAΩ, we deduce that hIA(aW), IA(aW0)i ∈ IA(i : R). Moreover, due to the

presence of the role subsumption i : R ↔ j : S, the axiom i : R v j : S is satisfied by Iv A.

Therefore hIA(aW), IA(aW0)i ∈ IA(j : S) and so hW, W0, j : Si ∈ Φ. Hence, thanks to

Lem. 4, we obtain hIA(aW), IA(aW0)i ∈ j(Ij(S)). Therefore hx, x0i ∈ j(Ij(S)) and more

generally, i(Ii(R)) ⊆ j(Ij(S)).

8.2

Only-if-direction

The new version of this part of the proof is organised as follows: 1. Build the configurations according to the model hI, i:

(a) build the global configuration Ω; (done)

(b) build the role configuration ΦΩ with respect to Ω;

(c) build the local configurations Ωi with respect to Ω; (done) (d) prove that Ωiis indeed a local configuration; (done)

(e) build the local role configurations ΦΩi with respect to Ωi and ΦΩ;

(f) prove that ΦΩi is indeed a local role configuration.

2. Prove that bAΩis consistent:

(a) build an interpretation IA of bAΩ;

(b) show that IA|= bAΩ.

3. Prove that { bOΩi} is consistent for all i ∈ O:

(b) show that Ii|= bOΩi.

Before proceeding with this part of the proof, we first extend the function ϕ to also treat the interpretation of roles.

ϕ : C ∪ R −→ 2∆∪ 2∆× ∆

i : C 7−→ i(Ii(C)) if C ∈Ci

i : R 7−→ i(Ii(R)) if R ∈Ri

This will help simplifying the notations.

8.2.1 Building configurations.

Global and local configuration can be defined exactly as in Sect. 6.2.1. The construction of global and local role configuration follows, according to the same notations as before.

Role configuration. For all W, W0∈ Ω and all R ∈R, hW, W0, Ri ∈ Ω if and only if

ϕ(R) ∩ (ϕ(CW) × ϕ(CW0)) 6= ∅

Local role configurations. For all w, w0 ∈ Ωi and all R ∈ Ri, hw, w0, Ri ∈ ΦΩi if and

only if Ii(R) ∩  Ii(CwCi) × Ii(CwCi0)  6= ∅

Local role configurations have to satisfy a specific constraint, so it must be proved that the previous construction is indeed a local role configuration.

Lemma 5 ΦΩi is a local role configuration with respect to Ωi and Φ.

Proof: Let assume that hw, w0, Ri ∈ ΦΩi. It implies that

ϕ(i : R) ∩   \ X∈wb ϕ(X) × \ X0∈cw0 ϕ(X0)  6= ∅

according to the following chain of implications: hw, w0, Ri ∈ ΦΩi =⇒Ii(R) ∩ (ϕ(CW) × ϕ(CW0)) 6= ∅ =⇒Ii(R) ∩ \ X∈w Ii(X) × \ X0∈w0 Ii(X0) ! 6= ∅ =⇒i(Ii(R)) ∩ \ X∈w i(Ii(X)) × \ X0∈w0 i(Ii(X0)) ! 6= ∅ =⇒ϕ(i : R) ∩   \ X∈wb ϕ(X) × \ X0∈cw0 ϕ(X0)  6= ∅

There exists a pair he, e0i in ϕ(i : R) ∩

  \ X∈wb ϕ(X) × \ X0∈cw0 ϕ(X0) 

. Let us name W the set

b

w∪{X ∈C \w | e ∈ ϕ(X)} and Wb 0the set cw0∪{X0C \

c

w0| e0∈ ϕ(X0)}. By definition,

b w ⊆ W , cw0⊆ W0, e ∈ ϕ(C

W) and e0∈ ϕ(CW0). This proves that ϕ(R) ∩ (ϕ(CW) × ϕ(CW0)) 6= ∅.

According to the construction of ΦΩ, this ensures that hW, W0, i : Ri ∈ Ω which establishes

the proof. 

8.2.2 Consistency of bAΩ.

Construction of a model of the alignment ontology. We only update the previous construction by adding the interpretation of roles.

4. For all R ∈R we define IA(R) = {hW, W0i ∈ Ω × Ω | hW, W0, Ri ∈ ΦΩ};

Checking satisfaction of axioms. We do not check the satisfaction of the axioms already present in the first version. So, these are the only necessary proofs.

4. Let assume there is a role correspondence i : R ↔ j : S ∈ Av ij. Therefore, i : R v

j : S ∈ bAΩ is an axiom in bAΩ. Let hW, W0i ∈ IA(i : R). By construction of IA,

hW, W0, i : Ri ∈ Φ

Ω. So by construction of ΦΩ, ϕ(i : S) ∩ (ϕ(CW) × ϕ(CW0)) 6= ∅. But

since hI, i |= i : R↔ j : S, ϕ(i : R) ⊆ ϕ(j : S). So, ϕ(j : R) ∩ (ϕ(Cv W) × ϕ(CW0)) 6= ∅.

Therefore, hW, W0, j : Si ∈ ΦΩ, so by definition of IA, hW, W0i ∈ IA(j : S).

5. If hW, W0, Ri ∈ Φ

Ω then {aW} v ∃R.{aW0} ∈ bA. It was proved above that

hW, W0, Ri ∈ ΦΩ implies hW, W0i ∈ IA(R) and since IA(aW) = W and IA(aW0) =

6. If hW, W0, Ri /∈ ΦΩ then {aW} v ∀R.¬{aW0} ∈ bA. When neither W nor W0 is in Ω,

the axiom is obviously satisfied by IA. Otherwise, IA(aW) = W and IA(aw0)) = W0,

and hW, W0i /∈ IA(R) by construction of IA. This clearly implies that IA|= {aW} v

∀R.¬{aW0}.

8.2.3 Consistency of bOΩi.

The proof of consistency of bOΩi will necessitate two additional results. They assert the

non-emptiness of sets in which bR

w,w0 and βW,WR 0 will be interpreted.

The next lemma will be used to define an interpretation of the individual βR W,W0.

Lemma 6 If hW, W0, i : Ri ∈ ΦΩthen there exists

he, e0i ∈   \ X∈Ci\W |i ∆i\ Ii(X)  ×   \ X0C i\W0|i ∆i\ Ii(X0)  

such that he, e0i ∈ Ii(R).

Before proceeding with the proof, we have to give a meaning to the specific set \

X∈∅

∆i\ Ii(X).

By convention, we assume that this set is equal to ∆i, and we also assume that the set

\

X∈Ci\W |i

∆\ ϕ(i : X) is equal to ∆.

Proof: Let us assume that hW, W0, i : Ri ∈ ΦΩ. We can devise the following implications:

hW, W0, i : Ri ∈ ΦΩ =⇒ϕ(i : R) ∩ (ϕ(CW) × ϕ(CW0)) 6= ∅ =⇒ϕ(i : R) ∩   \ X∈C \W ¯ ϕ(X) × \ X0C \W0 ¯ ϕ(X0)  6= ∅ =⇒i(Ii(R)) ∩   \ X∈Ci\W |i ∆\ ϕ(i : X) × \ X0C i\W0|i ∆\ ϕ(i : X0)  6= ∅ =⇒Ii(R) ∩   \ X∈Ci\W |i ∆i\ Ii(X) × \ X0C i\W0|i ∆i\ Ii(X0)  6= ∅  This last lemma will be used to define an interpretation of the individual bR

w,w0.

Proof: By contruction of ΦΩi, hw, w0, Ri ∈ ΦΩi =⇒Ii(R) ∩  Ii(CwCi) × Ii(CwCi0)  6= ∅ =⇒Ii(CwCi) ∩ Ii(∃R.CwC0i) 6= ∅ 

Construction of a model of the extended ontology. We build the model of bOΩi by

extending the construction already started in Sect. 6.2.3.

8. for all bRw,w0 added by Def. 19 to bOΩi with assertion (CwCiu ∃R.CwCi0)(bRw,w0), we define

I0

i(bRw,w0) = x for any x ∈ Ii(CwCi) ∩ Ii(∃R.CwCi0) (which exists, according to Lem. 7);

9. for all CR

W,W0 added by Def. 19 to bOΩi, we define Ii0(CW,WR 0) as the set

{e ∈ \ X∈Ci\W |i ∆i\ Ii(X) | ∃e0∈ \ X0C i\W0|i ∆i\ Ii(X0) s.t. he, e0i ∈ Ii(R)} 10. for all βR

W,W0added by Def. 19 to bOΩiwith assertion CW,WR 0(βW,WR 0), we define Ii0(βW,WR 0) =

x for any x ∈ Ii0(CW,WR 0) (which exists according to Lem. 6).

Checking satisfaction of axioms. As for the consistency of the alignment ontology, we only present the proofs of satisfaction of the new axioms.

7. Satisfaction of (CCi

w u ∃R.CwCi0)(bw). Let hw, w0, Ri ∈ ΦΩi. By Lem. 7 and the con-

struction of Ii0, we have Ii0(bw) ∈ Ii0(CwCiu ∃R.CwCi0).

8. Satisfaction of CCi

w v ∀R.¬CwC0i. Let hw, w0, Ri /∈ ΦΩi. This is a direct consequence of

the construction of ΦΩi.

9. Let W, W0 ∈ Ω and R ∈Ri.

(a) Satisfaction of CW,WR 0 v CW>. By construction of Ii0,

Ii0(CW,WR 0) ⊆ \ X∈Ci\w (∆i\ Ii(X)) = Ii0(C > W) (b) Satisfaction of CR W,W0 v ∃R.CR −

W0,W and CW,WR 0(βW,WR 0). Assume that hW, W0, i :

Ri ∈ ΦΩ. Let e ∈ Ii0(CW,WR 0). The construction of Ii0 implies that there exists

e0∈ \

X0C i\W0|i

there exists e ∈ \ X∈Ci\W |i ∆i\ Ii(X) with he, e0i ∈ Ii0(R−). So e0 ∈ Ii0(C R− W0,W). Therefore e ∈ I0(∃R.CR− W0,W). More generally, Ii0(CW,WR 0) ⊆ Ii0(∃R.CR − W0,W). The satisfaction of CR

W,W0(βRW,W0) is a direct consequence of Lem. 6 and the construc-

tion of Ii0.

(c) Satisfaction of CR

W,W0 v ∀R.¬CR −

W,W0. Let us assume that hW, W0, i : Ri /∈ ΦΩ.

So ϕ(R) ∩ (ϕ(CW) × ϕ(CW0)) = ∅. Let us consider he, e0i ∈ Ii0(R) such that

e ∈ I0

i(CW,WR 0) and e0 ∈ Ii0(CR −

W0,W). Since Ii0 |= CW,WR 0 v CW> (proved above), e ∈

I0

i(CW>) and Ii0 |= CR

W0,W v CW>0 implies that e0 ∈ Ii0(CW>0). So, by construction

of Ii0, hi(e), i(e0)i ∈ ϕ(i : R) ∩ (ϕ(CW) × ϕ(CW0)). This contradict the initial

assumption. Therefore, there is no such e and e0. 10. Satisfaction of ∃R.> v G

W,W0∈Ω

CW,WR 0. Let e ∈ Ii0(∃R.>), which means that there

exists e0 ∈ ∆i such that he, e0i ∈ Ii0(R). Using the function Θ defined in previous

section (Θ : ∆i → 2C such that for all e ∈ ∆i, Θ(e) = {X ∈ Ci | e ∈ Ii(X)), we

have e ∈ Ii(CΘ(e)Ci ) and e0 ∈ Ii(CΘ(eCi 0)). So he, e0i ∈ Ii(R) ∩



Ii(CwCi) × Ii(CwCi0)

 . This implies that hΘ(e), Θ(e0), RiΦΩi. By definition of Φi, there exists W, W0 ∈ Ω such

that Θ(e) ⊆ W |iand Θ(e0) ⊆ W0|i and hW, W0, i : Ri ∈ ΦΩ. By construction of Ii0 and

what precedes, we deduce that e ∈ Ii0(CR

W,W0). So finally, I0 |= ∃R.> v

G

W,W0∈Ω

CW,WR 0.



9

Conclusion and Future Work

We have introduced a new formalism for distributed reasoning with aligned descriptions logics. Although we do not pretend to replace or improve existing formalisms, we believe it has several advantages. In terms of paradigm, the differentiation of local and global reasoning is, in our opinion, more appropriate to the case of independently produced ontologies and ontology alignments. The reasoning task is not focused on one particular local knowledge. It is rather adapted to deducing knowledge about the overall system. In particular, it can be use to compose alignments. As a matter of technical quality, it has the advantage of being decidable iff local languages are decidable. Moreover, our reasoning procedure strictly encapsulate local reasoning so that it is never expected that the content of a local ontology is actually accessed by the global reasoner. As a result, we would advertise this formalism as an appropriate modular ontology language in the Semantic Web. Additionally, some inferences do not transfer from local to global level. Consequently, potential inconsistencies are semantically avoided.

Nevertheless, there are still important drawbacks that need to be filled. The most im- portant one is the algorithmic complexity of reasoning, which is far from reasonable with

the naive procedure we proposed. A real life implementation would inevitably require op- timization. Additionally, we have not yet investigated reasoning with concrete domains. Moreover, we do not allow concept-role correspondences, although these are well formalized in DDL [12].

As far as future research is concerned, we envisage the following directions: • implement a slightly optimized version of the algorithm;

• optimize and test it on real life examples;

• extend the theorem to description logics with concrete domains; • integrate the reasoner our modular ontology framework [11];

• define the notion of local reasoning with respect to to a distributed system; • enrich the alignment language with constructors of its own.

The last item is important to reason with an expressive alignment language as found in [10].

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