The construction of sufficiently tree-like models

We start with some preliminary definitions on morphism, neighbourhoods and bisimulations.

Morphisms. A homomorphism from an interpretation I to an interpretation J is a func-tionh:I → J satisfying for all concept namesAand all role namesrthe following properties:

if d∈ A^I then h(d) ∈ A^J and if (d, d⁰)∈ r^I then h(d),h(d⁰)

∈ r^J. An isomorphism is a bijectionfsuch that bothfandf⁻¹ are homomorphisms.

Neighbourhoods. For a given interpretationIand an elementd∈∆^Iwe denote withSucc_I(d) the set of role successors of d, i.e. the set S

r∈NR{d⁰ : (d, d⁰)∈ r^I}. Note that it is possible thatd∈Succ_I(d). Theforward neighbourhood (or simplyneighbourhood)N_I(d)ofdis the in-terpretationN_I(d) = (∆^NI(d),·^NI(d))such that∆^NI(d)=Succ_I(d)∪ {d},A^NI(d)=A^I∩∆^NI(d) for any concept nameA∈NC andr^NI(d)=r^I∩({d} ×∆^I)for any role namer∈NR.

The next definition introduces a notion of bisimulation tailored to normalizedALCSCCkbs.

Definition 27. Let I,J be interpretations with d ∈ I, d⁰ ∈ J. We say that d and d⁰ are forward-neighbourhood bisimilar (or simply bisimilar), denoted with d≡fb d⁰, if there exist a function f:N_I(d)→N_J(d⁰)(called bisimulation) satisfying the following conditions:

• f:N_I(d) _Succ

I(d)→N_J(d⁰) _Succ

J(d⁰) is a bijection, and

• For alld⁰ ∈ NI(d), for all concept names A∈NC and all role names r∈NR equivalences d⁰∈A^I ⇔f(d⁰)∈A^J and(d, d⁰)∈r^I ⇔(f(d),f(d⁰))∈r^J hold.

The following observation simplifies most of the forthcoming proofs. It can be either shown by a straightforward structural induction over the shape ofALCSCCconcepts or deduced from Proposition 2 from [5], where the notion ofALCQt–bisimulation was developed.

Observation 28. LetI |=K be a model of a normalizedALCSCCknowledge baseK. For any two domain elementsd, d⁰ ∈∆^I, if dandd⁰ are bisimilar then they satisfy the same ALCSCC concepts of depth at most one.

7.1.1 Forward-unravelings of finite models

For a finite interpretation I with ∆^I_named we denote those elements d∈∆^I for which a^I =d holds for some individual namea∈Ind_A.

Definition 29. LetIbe a finiteinterpretation. We define a forward-unravelingI^→= (∆^I→,·^I→) ofI as a (potentially infinite) interpretation satisfying the following conditions:

• ∆^I→ = (∆^I)⁺ \ ∆^I_named·∆^I_named·(∆^I)^∗

In words,∆^I→ consists of all nonempty sequences of elements from ∆^I except those, where the first two elements are named inI.

• For anya∈Ind_A, leta^I→ =a^I, i.e.ais interpreted by the one-element sequence consisting of the named elementa^I from I.⁷

• For concept namesA, we letA^I→ ={w|last(w)∈A^I}, where for a given elementw∈∆^I→ we uselast(w)to denote the last⁸ d∈∆^I in the sequencew.

• For role namesr, we letr^I→ =r^I∩(∆^I_named×∆^I_named)∪ {(w, wd)|(last(w), d)∈r^I}.

The notion of forward-unravelings differs only slightly from the classical notion of unraveling.

The only difference is that the sequences starting from two named individuals are excluded from the domain and that roles linking named individuals are assigned manually by the last item from Definition29. It is not surprising that forward-unravellings preserve satisfaction of ALCSCC Aboxes and Tboxes as well as conjunctive query non-entailment. The proof is standard and hinges on the fact that w∈ ∆^I→ and last(w)∈ ∆^I satisfy the same ALCSCC concepts. For CQ non-entailment it is enough to see thatlast(·)is a homomorphism fromI^→ toI.

7For convenience, we will not syntactically distinguish elements from ∆^I and one-element sequences from∆^I→; in particular this means∆^I⊆∆^I→.

8We definefirst(w)analogously.

Lemma 30. For any normalized ABoxAand any finite interpretationI, ifI |=Aholds, then alsoI^→|=Aholds.

Proof. Take an arbitrary normalized ABoxAas well as arbitrary finite interpretation I. As-sume that I |=Aholds. Note that ∆^I_named = ∆^I_named^→ holds since we agreed that we will not syntactically distinguish elements from∆^I and one-element sequences. First, see that satisfac-tion of assersatisfac-tions of the formA(a)∈ Ais guaranteed due to the third point of Definition29and the fact that the propertylast(w) =wholds for anyw∈∆^I_named^→ . Second, we can conclude that any assertion of the formr(a, b)∈ Ais also satisfied inI^→, due to the last item of Definition29, more precisely the fact thatr^I∩(∆^I_named×∆^I_named)⊆r^I→ holds. HenceI^→|=A.

An important step towards proving that forward unravelings preserve normalizedALCSCC TBoxes is to show that any sequencew ∈∆^I→ is forward-bisimilar to last(w)∈ ∆^I, i.e, the element from whichworiginated.

Lemma 31. Let K = (A,T,R) be a normalized ALCSCC knowledge base and let I be its arbitrary finite model. Then for all domain elementsd∈∆^I and all sequences w∈∆^I→ the implicationd=last(w)⇒d≡fbw holds.

Proof. We define a functionf:NI^→(w)→NI(d), which maps the neighbourhood ofw inI^→ to the neighbourhood of din I, asf(x) = last(x). The definition offis sound, since last(d)is defined uniquely for each sequence from(∆^I)⁺. Moreover, see thatf⁻¹:N_I(d)→N_I→(w)is defined asf⁻¹(x) =x for named individuals and f⁻¹(x) =wxotherwise, which is also sound due to the second and the last item of Definition29.

We will first show thatf:N_I^→(w) _Succ

I→(w)→N_I(d) _Succ

I(d) is a bijection. One can show it by proving that equationsf◦f⁻¹=id=f⁻¹◦fhold, whereidis the identity function and◦is a function-composition operator. Take an arbitrary elementwfrom N_I→(w)and assume that bothw, w⁰are named. Thenw=last(w)andw⁰ =last(w⁰)(since we identify named individuals with one-element sequences) and the following equations hold:

f(f⁻¹(w⁰)) =f(w⁰) =last(w⁰) =w⁰=f⁻¹(w⁰) =f⁻¹(last(w⁰)) =f⁻¹(f(w⁰)).

Now assume that one ofw, w⁰ is not named. Thenw⁰ is in the formw⁰=weand the presented equationsf◦f⁻¹=id=f⁻¹◦fhold again, as it is written below:

f(f⁻¹(e)) =f(we) =last(we) =eandwe=f⁻¹(e) =f⁻¹(last(we)) =f⁻¹(f(we)).

Hence f restricted to role successors of w is a bijection. Note that for any atomic concept A we know that w ∈ A^I→ holds iff d ∈ A^I holds, due to the third item of Definition 29 (and sinced=last(w) =f(w)). Thus, the only thing which remains to be done is to show that for allw⁰∈N_I^→(w)the equivalence(w, w⁰)∈r^I→⇔(f(w),f(w⁰))∈r^I→ holds.

Let us fix an arbitrary neighbourw⁰ofw, i.e., a domain elementw∈∆^I→ s.t.(w, w⁰)∈r^I→ holds for some role namer. Letd⁰=last(w⁰) =f(w⁰)be the corresponding element in∆^I.

We distinguish two cases.

• w, w⁰ are not named.

Since we agreed that∆^I_named^→ = ∆^I_namedholds, we infer thatd=f(w) =wandd⁰=f(w⁰) =w⁰. Thus we can use the last item of Definition29, namely the part stating thatr^I∩(∆^I_named×

∆^I_named) =r^I→∩(∆^I_named×∆^I_named)and conclude the mentioned property.

• At least one ofw, w⁰ is not named.

In this case, from the second part of the third item of Definition29we know thatw⁰is actually a sequence in the formw·e. But from the same definition as above,(w, w⁰) = (w, we)∈r^I→ holds if and only if(last(w), e) = (d, e)∈r^Iholds, which is exactly what we wanted to prove.

Since we have shown preservation (and non-preservation) of atomic concepts and roles byf and sincefis a bijection, we infer thatfis a bisimulation. Hencew≡fbdholds.

As an immediate consequence of Lemma31we obtain that any two sequencesw, w⁰ ∈∆^I→ having the same last element are forward-bisimilar, as stated below.

Lemma 32. For any finite interpretationI being a model of a normalizedALCSCCknowledge baseK and any sequencesw, w⁰∈∆^I→ with last(w) =last(w⁰), the propertyw≡fbw⁰ holds.

Proof. By applying Lemma31towandw⁰, we infer thatw≡fblast(w)andw⁰≡fblast(w⁰)holds.

Since the elementslast(w)andlast(w⁰)are equal, we conclude thatwis bisimilar tow⁰. Once we have shown that w≡fb last(w)for anyw∈∆^I→, we can employ this fact to show that forward-unraveling preserve satisfaction of normalized TBoxes.

Lemma 33. For any normalizedALCSCC TBoxT and any finite interpretationI, the impli-cationI |=T ⇒ I^→|=T holds.

Proof. Let w ∈ ∆^I→ be an arbitrary domain element from I^→ and let d = last(w) be the corresponding element from ∆^I. Let ε = C0 v C1 be an arbitrary GCI from the TBox T. Note thatC0, C1are not necessary atomic, but since we restricted our attention to normalized knowledge bases only, we can assume thatC0 andC1 are ALCSCCconcepts of depth at most one. Assume that w ∈ C₀^I→ holds. Then, to prove that I^→ |= ε holds, we need to show thatw∈C₁^I→ holds. Sincew≡fbdholds (by Lemma31), from Observation28we know thatd andwsatisfy the sameALCSCC concepts of depth at most one. Henced∈C₀^I. From the fact that I satisfies ε we infer thatd∈ C₁^I holds. Again, sincedand ware bisimilar, they satisfy the sameALCSCCconcepts of depth≤1and thusw∈ C₁^I→ holds too. Due to the fact thatw andεwere arbitrarily chosen, we conclude thatI^→|=T holds.

From the construction of forward unravelings one can immediately see that it also preserves non-entailment of conjunctive queries. Without loss of generality we can always assume that CQs contains only atomic concepts (e.g. by introducing a fresh nameA_C for each concept C and putting the GCIC≡A_C inside the TBox).

Lemma 34. For any finite interpretation I and any conjunctive query q, if I 6|= q holds thenI^→6|=q holds too.

Proof. Assume thatI 6|=qholds butI^→entailsq. Then there exists a matchπofqonI^→. Note thath(x) =last(x)is a homomorphismI^→ to I. Indeed, the preservation of atomic concepts by hcan be deduced from the third item of Definition 29, and the fact that if (d, d⁰) ∈ r^I→ holds then(h(d),h(d⁰))∈r^Iholds can be inferred from the last item of Definition29. However, in that caseπ⁰ withπ⁰(x) =h(π(x))would be a match of qonI, which contradicts the initial assumptionI 6|=q. ThusI^→6|=qholds.

7.1.2 Loosening of finite unravelings

Unraveling removes non-forest-shaped query matches, however,I^→ does not need to be finite even ifIis. To regain finiteness without re-introducing query matches, we are going to introduce the notion ofk-loosening.

For a given finite interpretation I, we say that an element u ∈ ∆^I→ is k–blocked by its prefixw, if u=ww⁰ for somew⁰ of length longer thank, andw’s andu’s suffixes of lengthk coincide. The definition is depicted below. The definition is depicted below.

ww⁰ w w

ww⁰

size > k

k

We also say thatw isminimally k–blocked if it isk–blocked (by some prefix), but none of its prefixes isk–blocked. WithBl^[k]_I→ we denote the set of minimallyk–blocked elements inI^→. Definition 35. For a given finite interpretationIwe define itsk–looseningI^[k] = (∆^I[k],·^∆I[k]) as an interpretation obtained fromI^→by exhaustively selecting minimallyk–blocked elementsv fromBl^[k]_I→ (k–blocked by some w), removing all of descendants ofv and identifyingv and w.

More formally, we enumerate the set of minimallyk–blocked elements Bl^[k]_I→ ={v1, v2, . . . , vn} and define a sequence of auxiliary interpretationsJ0 =I, . . . ,Jn=I^[k], where the i–th inter-pretationJⁱ= (∆^Ji,·^Ji) for anyi >0 is defined as:

• ∆^Ji = ∆^Ji−1\ vi·(∆^I)^∗

• ∆namedJi = ∆namedJi−1 and for anya∈Ind_A the conditiona^Ji=a^Ji−1 is satisfied,

• A^Ji =A^Ji−1∩∆^Ji for any concept nameA∈NC

• r^Ji =r^Ji−1∩ ∆^Ji×∆^Ji

∪ {(w, v⁰_i)|(w, v_i)∈r^Ji−1}, for any role namer∈N_R, wherev_i⁰ is the elementk–blockingv_i inI^→.

> k w

v Bl^[k]_I→

Figure 1: A single step of the construction ofI^[k]. We first argue thatk–loosening of a finite interpretation is also finite.

Lemma 36. For any finite interpretationI, itsk–looseningI^[k] for any naturalk >0is finite.

Proof. Take an arbitrary finite I and observe that the branching of k–loosening is finite due to finiteness ofI and each element ofI^[k] has only finite number of successors (by pigeon-hole

principle the blocking eventually occurs on every branch of I^→). Hence by employing (the contraposition) of the König’s Lemma, we conclude thatI^[k] is finite.

Like unravelings, k-loosenings preserve satisfaction of normalized Aboxes and Tboxes, as well as CQ non-entailment. However, ERCBoxes might become violated in the construction.

We startfrom the ABox preservation.

Lemma 37. For any finite I and any normalized ABoxA and any natural k >0, the impli-cation ifI |=AthenI^[k] |=Aholds.

Proof. Assume that I |= A holds. Then, due to Lemma 30 we know that I^→ |= A holds.

Observe that ∆^I[k] is a subset of ∆^I→, due to the first item of Definition 35. Moreover the sets∆^I_named^→ and∆^I_named^[k] are equal, due to the second item of Definition35. Since thek–loosening construction does not affect the ABox part of I^→ (e.g. those elements are not k–blocked for anyk, see also the second item of Definition35) we conclude thatI^[k] is a model ofA.

Towards proving the TBox preservation ofk–loosening, we prepare a bisimulation argument.

Lemma 38. Let K = (A,T,R) be a normalized ALCSCC knowledge base and let I be its arbitrary finite model. Then anyw∈∆^I[k] is bisimilar tolast(w)∈∆^I.

Proof. Take an arbitrary domain element w = w_I[k] ∈ ∆^I[k] and, since ∆^I[k] ⊆ ∆^I→ holds (see: Definition35), let w_I^→ =w be the corresponding element from ∆^I→. To show that w andlast(w)are bisimilar, is sufficient prove thatw_I[k] ≡fbw_I^→ and use Lemma 31.

We proceed as follows. We define a function f : N_I[k](w) → N_I^→(w) as f(w⁰) = w⁰ for allw⁰∈N_I[k](w)∩N_I^→(w)andf(w⁰) =w·last(w⁰)otherwise (note that in this casew⁰is some of minimallyk–blocked elements).

We first argue that fis a function. Since∆^I[k] ⊆∆^I→ holds, we infer that fis an identity function on the set N_I[k](w)∩N_I^→(w), thus well-defined. The problematic case is when w⁰ is not included in N_I[k](w)∩N_I^→(w). Observe that in this case w⁰ was identified, during the construction ofI^[k], with somek–blocked element v ∈Bl^[k]_I→, which originally was a successor ofw. It means that v wask–blocked by w⁰ and from the definition ofk–blocked elements we infer thatw⁰andvshare the same suffix of lengthk. Thusw⁰andvshare the same last element.

Sincevis a successor ofw, thenv=w·last(v) =w·last(w⁰). Hence the definition offis sound.

To see that f : N_I[k](w) _Succ

I[k](w) → N_I^→(w) _Succ

I→(w) is a bijection, we can restrict our attention only to the elements not included in the setN_I[k](w)∩N_I^→(w), since, as we already mentioned, on such setf is the identity function and thus, also a bijection. Observe thatfis injection for any w⁰ ∈N_I[k](w)\NI^→(w). Indeed, if there would be w⁰, w⁰⁰ satisfyingf(w⁰) = f(w⁰⁰), then it would imply that they originated from the same successor ofwinI^→(since they share the same suffix), which is clearly not possible. To see thatfis a surjection it is enough to see that for any successorw⁰=weofwinI^→the functionfis either identity (thusf(w⁰) =w⁰) orw⁰ was minimallyk–blocked and hance was identified with an element sharing the same last element. Hence,f(restricted to appropriate sets) is a bijection.

We will prove thatfis a bisimulation. In the first part we will prove the following statement:

∀A∈NC∀w⁰∈N_I[k](w)the equivalencew⁰ ∈A^I[k] ⇔f(w⁰)∈A^I→holds.

Take an arbitrary concept nameAand arbitrary domain elementw⁰ ∈N_I[k](w). If f(w⁰) =w⁰ then the above condition trivially holds. Assume thatf(w⁰)6=w⁰. Thenf(w⁰) =wlast(w⁰)and the preservation of concepts follows from Definition29.

In the second part we will prove:

∀r∈NR∀w⁰∈N_I[k](w)the equivalence(w, w⁰)∈r^I[k]⇔(f(w),f(w⁰))∈r^I→ holds.

Take an arbitrary role name r and arbitrary domain element w⁰ ∈ N_I[k](w). Once more, if f(w⁰) = w⁰ then the above condition trivially holds. Assume that f(w⁰) 6= w⁰. Then again f(w⁰) = wlast(w⁰) = v and v is minimally k–blocked by w⁰. From Definition 35 we know that(w, v)∈r^I→ iff(w, w⁰)∈r^I[k], which proves the statement about (non)preservation of roles during the construction ofI^[k].

We conclude thatfis a bisimulation and hencew_I[k]≡fbwI^→ holds.

The TBox preservation follows immediately from the previous lemma.

Lemma 39. For any finite I and any normalized TBox T and any natural k >0, the impli-cation ifI |=T thenI^[k] |=T holds.

Proof. Take an arbitrary finite interpretationI, a normalized TBoxT and a positive integerk.

Assume thatI |=T holds. To prove that each GCIεfrom T is also satisfied inI^[k], we apply the same reasoning as we already done for Lemma 33. Namely, it is sufficient to prove that thek–loosening construction is concept preserving but it can be concluded from Definition27 (of bisimulation) and from Lemma38.

Lemma 40. For any k∈N, ifI 6|=q thenI^[k]6|=q.

Proof. Assume thatI 6|=q, but I^[k] |=q. In this case there exists a match πof qonI^[k]. By using the same ideas as for Lemma34 we argue that in this case π⁰ with π⁰(x) = last(π(x)) would be a match ofq onI, which contradicts withI 6|=q. ThusI^→6|=q holds.

For a given interpretation J, an anonymous cycle is simply a word w ∈ (∆^J)⁺·(∆^J \

∆^J_named)·(∆^J)⁺, where first and the last element are the same, and for any two consecutive elements di, di+1 of w there exists a role r witnessing (di, di+1)∈ r^J. The girth of J is the length of the smallest anonymous cycle inJ if such a cycle exists or∞otherwise. The main feature of thek–looseningI^[k] is that the girth ofI^[k] is at leastk, as proven below.

Lemma 41. For any k∈Nand any finite interpretationI, the girth ofI^[k] is at leastk.

Proof. We will prove inductively over immediate structures J0 =I^→,J1, . . . ,Jn =I^[k] pro-duced in Definition35that each of them have girth greater thank. Fori= 0it is clear thatJ0

has girth at leastk (actually its girth is ∞). Assume that for all i < mthe girth of each Ji

fori < mis at leastk. We will show that the girth of Imis at leastk.

For contradiction assume that the girth ofJmis smaller thank. We recall thatvmis them–

th minimallyk–blocked elements from Bl^[k]_I→ and v_m⁰ is the element k–blocking vm. Since Jm

was obtained from Jm−1 and the girth of Jm−1 is at least k then the only possibility of a anonymous cycle of length at least k to be present in Jm is to contain a freshly added edge between predecessors wof v_m and v⁰_m, namely (w, v_m⁰ )for some r∈N_R as a replacement for an original edge(w, v_m).

Letρbe an arbitrary shortest anonymous cycle inJ_m−1. As we already discussed it contains an edge(w, v_m)between some domain elementw. Henceρis in the form(w, v_m⁰ )ρ⁰ whereρ⁰ is some path fromv_m⁰ to w. But note that due the definition ofk–blocked element the distance betweenvmandv_m⁰ is at leastk. Henceρ⁰is of length at leastk. Thusρis not shorter thank, which contradict our initial assumption. Hence the girth ofJmis at least k, which allows us to conclude that the girth ofJn =I^[k] is also at leastk.

Once k is greater than the number of atoms in q (denoted with|q|), thek–loosening of a model is still “locally acyclic enough” so the query matches only in a “forest-shaped” manner. We will exploit this property when designing an algorithm for deciding conjunctive query entailment in Section7.2.

Lemma 42. For every conjunctive queryq, a positive integer k >|q| and a finite interpreta-tionI, the following equivalence I^→|=q⇔ I^[k]|=q holds.

Proof. Let suffs(w) be a function which for an input word w ∈ (∆^I)⁺ returns w if |w| ≤ s or its suffix of length s otherwise. Moreover let I_k^→ be the substructure ofI^→ with domain restricted to sequences of length at mostkonly. Note thath(w) =suffk(w)is a homomorphism fromI^→toI^→ (sincew≡fbsuffk(w), see the proof of Lemma33). Hence if there is a matchπ ofqin I^→, there is also a matchπ⁰ ofq inI_k^→. SinceI_k^→is a substructure of I^[k] (due to the definition of minimallyk–blocked elements and Definition35), henceπ⁰ is also a match inI^[k]. For the opposite way, that i.e.,I^[k]|=qimpliesI^→|=q, it is sufficient to show (sincek >|q|) that there is a homomorphism from any substructure of the size k of I^[k] to I^→. Take an arbitrary element w ∈ ∆^I[k] and take a interpretation Iw^[k] be an interpretation obtained by restricting the domain to elements reachable from w in at most k steps. More formally we define the setsR_i(w)of those elements reachable fromwin at mostisteps, i.e.R₀(w) ={w}, andRi(w) =R_i−1(w)∪ {v ∈∆^I[k] | ∃r∈NR (u, v)∈r^I[k]∧u∈R_i−1(w)} for all i >0. We set∆^I_w^[k]=Rk(w). First see thatIw^[k]is a tree-shaped. Indeed if it would contain an anonymous cycle of length at mostk it would contradict the fact that the girth of I^[k] is at least k (by Lemma41). Hence we take a homomorphismh:Iw^[k]→ I^→ defined ash(x) =suffk(x)and see that if there is a matchπofqinI^[k], thenπ⁰= (h◦π)would also be a match ofqin I^→. 7.1.3 Making ERCBoxes be satisfied again

We next consider how to adjust ak-loosening such that it again satisfies the initial ERCBox.

Since role inverses are not expressible inALCSCC, creating multiple copies of a single element and forward-linking them to other elements precisely in the same way as the original element, can be done without any harm to modelhood nor query-non-entailment. We formalize this intuition below.

Definition 43. For any interpretation I and any sets S ⊆ (∆^I ×N+) we define the S–

duplicationof I as the interpretation I+S = (∆^I_+S,·^I+S) with:

• ∆^I_+S = ∆^I∪ S

(v,n)∈S{vcpy⁽ⁱ⁾ |1≤i≤n},

• a^I+S =a^I for each individual namea∈Ind_A,

• For concept namesA∈NC and role names r∈NR we set:

– A^I+S =A^I∪ S

(v,n)∈S

n

vcpy⁽ⁱ⁾ |1≤i≤n∧v∈A^Io , and, – r^I+S =r^I∪ S

(v,n)∈S

n

(vcpy⁽ⁱ⁾, w)|1≤i≤n∧(v, w)∈r^Io .

As in the case of previous constructions, one can show that theS–duplication ofIpreserves satisfaction of ABoxes and TBoxes.

Lemma 44. For any finiteI and normalized ABoxAand normalized TBoxT, ifI |= (A,T), then for anyS ⊆(∆^I×N+), theS–duplicationI+S ofI is also a model of(A,T).

v vcpy⁽¹⁾

I

Figure 2: The interpretationI_+{(v,1)} obtained fromI by duplicating a nodev.

Proof. SinceIis a submodel ofI+S we conclude thatI+S |=A. To see thatS–duplication does not violate the TBoxT, it is sufficient to see that for any i∈N+ andv∈∆^I an elementv⁽ⁱ⁾cpy

is bisimilar tov(which follows immediately from Definition43). HenceI_+S |= (A,T).

Moreover a conjunctive queryqhas a match inI if and only if it has a match inI+S. Lemma 45. For any conjunctive queryq and any S ⊆(∆^I×N+) and any interpretationI, the equivalenceI |=q⇔ I+S|=qholds.

Proof. Without loss of generality we assume all concepts appearing inq are atomic. If I has a match πof q, then triviallyπ is also a match in I_+S (due to the fact that I is a submodel of I+S). For the second direction, assume that there is a query match π of q in I+S. Let us define h:I+S → I as h vcpy⁽ⁱ⁾

=v for freshly copied elements and ash(v) =v otherwise.

It is easy to see thath is a homomorphism, and hence h◦π is a match ofq in I. Thus the equivalenceI |=q⇔ I+S |=q holds.

From Lemma45and Lemma42we can immediately conclude:

Lemma 46. For any conjunctive query q, any positive integer k >|q| and any finite interpre-tationI the following equivalence holds:I^→|=q⇔ I_+S^[k] |=q.

Note that for any finite I being a model of a normalized K = (A,T,R) it could be the case thatI^[k] does not satisfy the ERCBoxRanymore. However, the inequalities fromRhave the convenient property that if a vector ~xcontaining the cardinalities of all atomic concepts’

extensions is a solution toR, then also a vector c·~x, i.e., the vector obtained by multiplying each entry of~xby a constant c, is a solution to R. Thus there is also a solution to Rin the shape(1 +|∆^I[k]|)·x~_I, wherex~_Iis the solution toRdescribing the atomic concept extensions’

cardinalities inI. SinceI^[k] preserves (non-)emptiness of all concepts fromI, we can simply

cardinalities inI. SinceI^[k] preserves (non-)emptiness of all concepts fromI, we can simply

Im Dokument Satisfiability and Query Answering in Description Logics with Global and Local Cardinality Constraints (Seite 25-35)