Not using background knowledge

Let us look at the basisBi in a little more detail. Bi is the set of all GCIs of the formC →Cⁱⁱ where the left-hand sidesC of these GCIs are contained in the set Λi. Λi is the set of all conjunctions over the elements of

Mi =Np∪ {∃r.Xⁱ |r∈Nr, X ⊆∆i}.

Therefore it is natural to look at contexts, that are induced by some subset of Mi.

The number of i-intents Xⁱ can be exponential in the size of the model. Hence also Mi can become exponentially large. Therefore we want to avoid having to compute the whole set before starting the process. Instead more attributes should be computed on the fly. An idea is to instantiate Algorithm 1 in the following way. We are given a finite model i. We initiate Algorithm 1 with the context

K0 = (∆i,Np, I0). The context^K0 is obtained as the induced context byiand the set of primitive concept names Np. In every stepkof the algorithm the following things happen.

• A new premise Pk is found.

• Pk is a subset of Mk.

• Algorithm 1 asks for a new context with an extended attribute set Mk+1. Mk+1 is obtained by adding to Mk all attributes of the form ∃r.(d

Pk)ⁱⁱ where r is some role name. (This is done only, if no equivalent concept description is already present in the attribute set.)

• ^K_k+1 is computed as the induced context by the new attribute setMk+1and i.

Algorithm 3 Exploration for computing a basis for a given model i

1: Input: model i= (∆i,·ⁱ)

2: // Initialisation

3: M0 =Np,^K0 = the context induced by M0 and i

4: Π0 =∅, P0 =∅,k = 0

5: while Pk 6=null do

6: Πk+1 = Πk∪ {Pk}

7: Mk+1=Mk∪ {∃r.(d

Pk)ⁱⁱ|r∈ Nr}

8: k=k+ 1

9: if Mk =M_k−1 =Pk then

10: Pk =null

11: else

12: Pk = lectically next set of attributes that is closed with respect to{Pj → P_j^′′k |1≤j < k}

13: end if

14: end while

For Algorithm 1 we know that it terminates if there is some n ∈ ^N such that starting from the n-th step of the algorithm no more new attributes are being added. This is guaranteed by the fact that there are only finitely manyi-intents.

Therefore only finitely many attributes of the form ∃r.(d

Pk)ⁱⁱ can be added.

Assume that Algorithm 3 terminates after the n-th iteration of the while-loop.

Lemma 5 proves that then the set {Pk →P_k^′′n |Pk ∈Πn} is sound and complete for ^Kn. We are, however, not interested in a basis for this context, but in a basis for the underlying modeli. The rest of this section will be dealing with the proof that the corresponding set of ELgfp-GCIs

{l

Pk→(l

Pk)ⁱⁱ|Pk ∈Πn}

is a basis for i. As a subset of Bi it is obviously finite and sound. The harder part shall be proving completeness.

The first thing to prove is that our algorithm is capable of finding all elements of M =N_p∪ {∃r.Xⁱ |r∈Nr, X ⊆∆i}.

We show that for every role name r ∈ Nr the set Mn is closed with respect to the following mapping: U ⊆Mn is mapped to the concept description∃r.(d

U)ⁱⁱ. In a later step we prove that every concept description from Mi can be obtained from Np by repeatedly applying one of these mappings. This means, that once the algorithm has terminated, every concept description from Λican be expressed in terms of Mn.

Lemma 8 Assume that Algorithm 3 terminates after the n-th iteration. Let Mn

be the final set of attributes. Then for every role name r ∈ Nr and every subset U ⊆Mn there is a concept description C ∈Mn such that C ≡ ∃r.(d

U)ⁱⁱ.

Proof: V = prKn((d

U)ⁱⁱ) is an intent in ^Kn by Lemma 20. As an intent it is also quasi-closed. Therefore Lemma 5 shows that there is some Pk ∈Πn such that Pk ⊆ V and P_k^′′n = V^′′n = V. In the k-th iteration of the algorithm the attributes ∃r.(d

Pk)ⁱⁱare added to the set of attributes. ∃r.(d

Pk)ⁱⁱ ≡ ∃r.(d U)ⁱⁱ is a consequence of Lemma 21. We have thus proved that for every set U ⊆Mn

and for every r∈ Nr there is a descriptionC ∈Mn such thatC ≡ ∃r.(d

U)ⁱⁱ.

Lemma 9 Let O be a set of concept descriptions with the following properties

• For every role name r ∈ Nr and every subset U ⊆O there is some C ∈ O such that C ≡ ∃r.Uⁱⁱ.

• O contains Np.

Then for every role name r ∈ Nr and every set X ⊆ ∆i there is some concept description C∈O such that C ≡ ∃r.Xⁱ.

Proof: LetX ⊆∆i be some set of objects. We prove that there is some concept description C ∈Mn such thatC ≡ ∃r.Xⁱ by induction over the depth of Xⁱ. We say that an intent Xⁱ has depth d if there is an acyclic concept description D of role depth d such that Xⁱ ≡ Dⁱⁱ. By a result from [7] such an acyclic concept description Dexists for every concept description C, i. e. everyi-intent has finite depth.

Base case: LetXⁱbe ani-intent of depth 0. This means thatXⁱcan be expressed as

Xⁱ ≡ l

P∈P

P

for some set of primitive concept names P ⊆ Np. Since Np ⊆ O it follows from the hypothesis that for every r ∈ Nr there is some concept description C ∈ O such that

C ≡ ∃r.(l

P)ⁱⁱ≡ ∃r.Xⁱ.

Step case: Now assume that for every intentXⁱ of depth less thank and for every r ∈ Nr there is some CX ∈O with CX ≡ ∃r.Xⁱ. Let Yⁱ, Y ⊆∆i be some intent of role depth k. Then there must be some acyclic concept description D of role depth k such thatYⁱ ≡Dⁱⁱ. D can be written as

D≡ l

P∈PD

P ⊓ l

1≤l≤s

∃rl.El.

where PD ⊆ Np is a set of primitive concept names, rl ∈ Nr is a role name and El is an acyclic concept description of role depth less thankfor alll ∈ {1, . . . , s}.

Then From the induction hypothesis we know that there are concept descriptions Cl ∈ O, for all l ∈ {1, . . . , s} such that Cl ≡ ∃r.E_lⁱⁱ for all l ∈ {1, . . . , s}. Define

Lemma 9 and Lemma 8 prove that once Algorithm 3 terminates the final attribute setMn contains all attributes of the form∃r.Xⁱ forr∈ Nr,X ⊆∆i. This means that all concept descriptions from Λi can be expressed in terms of Mn (Recall that Λi is the set of premises of the sound an complete set of GCIs Bi presented in Section 2.3).

We are still trying to prove that {l

Pk→(l

Pk)ⁱⁱ|Pk ∈Πn}

with the sets Pk obtained in Algorithm 3 is complete fori. Since we have already proved completeness of B_i we try to deduce completeness of {d

Pk → (d Pk)ⁱⁱ | Pk ∈Πn} from completeness of Bi. The following result helps us to do this.

Lemma 10 Let n be the number of the iteration for which Algorithm 3 termi-nates. Let L ∈ Λi be a premise of the GCI set Bi. Then either L is an i-intent or there is some Pk∈Πn such that L⊑d

Pk, L6⊑(d Pk)ⁱⁱ.

Proof: Assume thatLis not ani-intent, i. e.L6=Lⁱⁱ. Then prKn(L)6= prKn(Lⁱⁱ) (This follows from Lemma 18 since both L and Lⁱⁱ are expressible in terms of Mn). From Lemma 18 we know that prKn(L)^′′n = prKn(Lⁱⁱ). Thus prKn(L) is not an intent of ^Kn. Since {P_k → P_k^′′n |1 ≤k ≤n} is complete for ^Kn by Lemma 5 there must be somek ∈ {1, . . . , n}such thatPk⊆prKn(L),P_k^′′n 6⊆prKn(L). Thus L ⊑ d

Pk, L 6⊑ d

P_k^′′n by Lemmas 16 and 18. Also from Lemma 18 we obtain dP_k^′′n ≡(d

Pk)ⁱⁱ and thus L6⊑d

P_kⁱⁱ.

The next result finally ties the previous ones together. It finally enables us to prove completeness of {d

Pk →(d

Pk)ⁱⁱ |Pk ∈Πn}.

Lemma 11 Let C be a set of EL_gfp-concept descriptions such that for every L∈ Λi it holds that either

• there is some C ∈ C such that L⊑C, L6⊑Cⁱⁱ, or

• L≡Lⁱⁱ.

Then {C →Cⁱⁱ |C ∈ C} is complete for i.

Proof: Assume that there is a concept descriptions L∈ Λi such that L →Lⁱⁱ does not follow from{C→Cⁱⁱ |C∈ C}. Since Λi is finite we can assume without loss of generality that L is minimal with this property. L cannot be an i-intent for then L → Lⁱⁱ ≡ L would follow trivially from any set of GCIs. Hence, the preconditions imply that there is some C ∈ C such that L ⊑C, L6⊑ Cⁱⁱ. Then L→L⊓Cⁱⁱfollows fromC →Cⁱⁱ. FurthermoreL⊓Cⁱⁱ⊏Lis strictly subsumed, because of L 6⊑ Cⁱⁱ. Also L⊓Cⁱⁱ ∈ Λi because of Lemma 4. We have assumed that L is minimal among all ELgfp-concept descriptions for which L→ Lⁱⁱ does not follow from {C → Cⁱⁱ | C ∈ C}. Therefore L ⊓Cⁱⁱ → (L⊓ Cⁱⁱ)ⁱⁱ ≡ Lⁱⁱ follows from {C → Cⁱⁱ | C ∈ C}. Thus both L → L⊓Cⁱⁱ and L⊓Cⁱⁱ → Lⁱⁱ follow from {C → Cⁱⁱ | C ∈ C}. This contradicts the assumption that L → Lⁱⁱ does not follow from {C → Cⁱⁱ | C ∈ C}. Therefore the assumption is false, i. e. for every L ∈ Λi the GCI L → Lⁱⁱ follows from {C → Cⁱⁱ | C ∈ C}. Since {L→Lⁱⁱ|L∈Λi} is complete, {C →Cⁱⁱ |C ∈ C} must also be complete.

Now let us look at the final context ^Kn once more. From Lemma 8 we know that for every role name r ∈ Nr and every subset U ⊆ Mn there is a concept description C ∈ Mn such that C ≡ ∃r.(d

U)ⁱⁱ. Hence we can apply Lemma 9 which yields that all concept descriptions from Λi can be expressed in terms of Mn. This is exploited in the proof of Lemma 10 which shows that every L∈Λi

is either an i-intent or there is some Pk ∈Πn such that L⊑ d

Pk, L 6⊑(d Pk)ⁱⁱ. Finally, Lemma 11 yields completeness of {d

Pk→(d

Pk)ⁱⁱ|Pk ∈Πn}.

Theorem 3 Assume that Algorithm 3 terminates after the nth iteration of the while-loop. Then the set of GCIs

{l

Pk→(l

Pk)ⁱⁱ|Pk ∈Πn} is complete for i.

Im Dokument Exploring finite models in the Description Logic EL_gfp (Seite 23-28)