Finite Model Reasoning in Horn-SHIQ

Finite Model Reasoning in Horn-SHIQ

Yazm´ın Iba˜nez-Garc´ıa¹, Carsten Lutz², and Thomas Schneider²

1 KRDB Research Centre, Free Univ. of Bozen-Bolzano, Italy,{ibanezgarcia@inf.unibz.it}

2 Univ. Bremen, Germany,{clu, tschneider}@informatik.uni-bremen.de

Abstract. Finite model reasoning in expressive DLs such asALCQIandSHIQ requires non-trivial algorithmic approaches that are substantially differerent from algorithms used for reasoning about unrestricted models. In contrast, finite model reasoning in the inexpressive fragmentDL-LiteF of ALCQI andSHIQis algorithmically rather simple: using a TBox completion procedure that reverses certain terminological cycles, one can reduce finite subsumption to unrestricted subsumption. In this paper, we show that this useful technique extends all the way to the popular and much more expressive Horn-SHIQfragment ofSHIQ.

1 Introduction

Description logics (DLs) that include inverse roles and some form of counting such as functionality restrictions lack the finite model property (FMP) and, consequently, reasoning w.r.t. the class of finite models (finite model reasoning) does not coincide with reasoning w.r.t. the class of all models (unrestricted reasoning). On the one hand, this distinction is becoming increasingly important since DLs are nowadays regularly used in database applications, where models are generally assumed to be finite. On the other hand, finite model reasoning is rarely used in practice, mainly because for many popular DLs that lack the FMP, no algorithmic approaches to finite model reasoning are known that lend themselves towards efficient implementation.

Typical examples include the expressive DLsALCQIandSHIQ, which are both a fragment of the OWL2 DL ontology language. While finite model reasoning inALCQI andSHIQare known to have the same complexity as unrestricted reasoning, namely EXPTIME-complete [9], the algorithmic approaches to the two cases are rather different.

For unrestricted reasoning, there is a wide range of applicable algorithms such as tableau and resolution calculi, which often perform rather well in practical implementations. For finite model reasoning, all known approaches rely on the construction of some system of inequalities [3,9], and then solve this system over the integers; the crux is that the system of inequalities is of exponential size in the best case, and consequently it is far from obvious how to come up with efficient implementations. Note that the same is true for the two-variable fragment of first-order logic with counting quantifiers (C2), into which DLs such asALCQI andSHIQcan be embedded [12,13], that is, all known approaches to finite model reasoning in C2 rely on solving (at least) exponentially large systems of inequalities.

Interestingly, the situation is quite different on the other end of the expressive power spectrum.DL-Lite_F is a very inexpressive DL that is used in database applications, but lacks the FMP because it still includes inverse roles and functionality restrictions.

Building on a technique that was developed in a database context by Cosmadakis, Kanellakis, and Vardi to decide the implication of inclusion dependencies and functional dependencies in the finite [4], Rosati has shown that finite model reasoning inDL-Lite_F can be reduced (in polynomial time) to unrestricted reasoning inDL-Lite_F [14]. In fact, the reduction is conceptually simple and relies on completing the TBox by finding certain cyclic inclusions and ‘reversing’ them. For example, the cycle

∃r⁻v ∃s ∃s⁻ v ∃r (functr⁻) (functs⁻)

that consists of existential restrictions in the ‘forward direction’ and functionality state- ments in the ‘backwards direction’ would lead to the addition of the reversed cycle

∃sv ∃r⁻ ∃rv ∃s⁻ (functr) (functs).

As a consequence, finite model reasoning in DL-LiteFdoes not require any new algorith- mic techniques and can be implemented as efficiently as unrestricted reasoning. Given that DL-LiteFis a very small fragment ofALCQIandSHIQ, these observations raise the question whether the cycle reversion technique extends also to larger fragments of these DLs. In particular,DL-LiteFis a ‘Horn DL’, and such logics are well-known to be algorithmically more well-behaved than non-Horn DLs such asALCQIandSHIQ.

Maybe this is the reason why cycle reversion works forDL-Lite_F?

In this paper, we show that the cycle reversion technique extends all the way to the expressive Horn-SHIQfragment ofSHIQ, which is rather popular in database applications [6,11,5,2] and properly extendsDL-LiteFand other relevant Horn fragments such asELIF. In particular, we show that finite satisfiability in Horn-SHIQcan be reduced to unrestricted satisfiability in Horn-SHIQ by completing the TBox with reversed cycles in the style of Cosmadakis et al. and of Rosati. While the reduction technique is essentially the same as forDL-LiteF, the construction of a finite model in the correctness proof is much more subtle and demanding. Another crucial difference to the DL-LiteFcase is that, when completing Horn-SHIQTBoxes, the cycles that have to be considered can be of exponential length, and thus the reduction is not polynomial. On first glance, this of course casts a doubt on the practical relevance of the proposed reduction.

Still, we are confident that our approach will lead to implementable algorithms for finite model reasoning in Horn-SHIQ. Specifically, the state-of-the-art of efficient reasoning in Horn description logics is to use so-called consequence based calculi, as introduced for Horn-SHIQin [7] and implemented for example in the reasoners CEL, CB, and ELK [1,7,8]. Instead of first completing the TBox and then handing over the completed TBox to a reasoner, it seems well possible to integrate the reversion of cycles directly as an inference rule into such a calculus. This avoids the detection of cycles by uninformed, brute-force search, and instead searches for cycles in the consequences that have already been computed by the calculus, anyway. Since the efficiency of consequence-based calculi are largely due to the fact that, for typical inputs, the set of derived consequences is relatively small, we expect that this will work well in practical applications. For now, though, we leave it as future work to work out the details of such a calculus.

Some proof details are deferred to the appendix of the long version of the paper, to be found at http://www.informatik.uni-bremen.de/tdki/research/papers.html

2 Preliminaries

The original definition of Horn-SHIQis based on a notion of polarity and somewhat unwieldy [6]; alternative definitions have been proposed later, see for example [10]. For brevity, we directly introduce Horn-SHIQTBoxes in a certain normal form similar to the one used in [7].

LetN_CandN_Rbe countably infinite and disjoint sets of concept names and role names. AHorn-SHIQTBoxT is a set ofconcept inclusions (CIs)that can take the following forms:

KvA Kv ⊥ Kv ∃r.K⁰ Kv ∀r.K⁰ Kv(61r K⁰) Kv(>n r K⁰) whereKandK⁰denote a conjunction of concept names,Aa concept name,ra (po- tentially inverse) role, andn≥1. Throughout the paper, we will deliberately confuse conjunctions of concept names and sets of concept names. The empty conjunction is abbreviated to>.

It was observed in [7] that, for the purposes of deciding unrestricted satisfiability, the above form can be assumed without loss of generality; that is, every Horn-SHIQTBox T conformant with the original definition in [6] can be converted in polynomial time into a TBoxT⁰in the above form such that for all concept namesAinT, we have that Ais satisfiable w.r.t.T iffAis satisfiable w.r.t.T⁰. It is straightforward to verify that all necessary transformations, such as coding out role hierarchies and transitive roles do not rely on unrestricted models to be available and thus, the introduced TBox normal form can be assumed w.l.o.g. also for finite satisfiability.

The semantics for Horn-SHIQis given as usual in terms of interpretationsI. For a given TBoxT and a concept inclusionCvD, we writeT |=CvDifI |=CvD for all modelsI ofT, andT |=finC vDif the same holds for all finite models. We recall that, in Horn-SHIQ, (un)satisfiability and subsumption can be mutually reduced to each other in polynomial time, and that this also holds in the finite. The following examples show that, in Horn-SHIQ, finite and unrestricted reasoning do not coincide.

Example 1. LetT ={Av ∃r.B, Bv ∃r.B, Bv(61r⁻ >), AuBv ⊥ }. Then Ais satisfiable w.r.t.T, but not finitely satisfiable. In fact, whend∈A^Iin some model I ofT, then the CIB v ∃r.Band functionality assertion onr⁻enforces an infinite chainr(d, d₁), r(d₁, d₂), . . .withd∈A^I,d6∈B^Iandd₂, d₃,· · · ∈B^I.

LetT⁰ ={A1v ∃r.A2, A2v ∃r.(A1uB),> v(61r⁻>)}. The reader might want to convince herself thatT⁰6|=A1vB, butT⁰|=finA1vB.

Eliminating At-Least Restrictions

The usual normal form for Horn-SHIQdoes not comprise at-least restrictions, that is, CIs of the formKv(>n r K⁰)are not allowed. This is achieved by replacing each such CI with

Kv ∃r.Bi, Bi vK⁰, BiuBj v ⊥ 1≤i < j ≤n (1) where eachBiis a fresh concept name. If infinite models are admitted, it is quite easy to see that this translation preserves the satisfiability of all concept names inT, exploiting

the tree model property of Horn-SHIQ. For finite satisfiability, the same construction works, but a more refined argument is needed to show this.

Proposition 2. LetT⁰ be obtained fromT by replacing the CIKv(>n r K⁰)with the CIs(1), and letAbe a concept name fromT. ThenAis finitely satisfiable w.r.t.T iff Ais finitely satisfiable w.r.t.T⁰.

Proof.The “if” direction is trivial since every model ofT⁰is also a model ofT. For the

“only if” direction, letIbe a finite model ofT withA^I 6=∅. We construct a finite model J ofT⁰withA^J 6=∅by takingncopies ofIand ‘rewiring’ all role edges across the concept namesB_ican be interpreted in a non-conflicting way.

Specifically, sinceI satisfiesK v (> n r K⁰)we can choose a functionsucc : K^I× {0, . . . , n−1} →∆^Isuch that the following conditions are satisfied:

– for alld∈K^Iandi < n:(d,succ(d, i))∈r^Iandsucc(d, i)∈(K⁰)^I; – for alld∈K^Iandi < j < n:succ(d, i)6=succ(d, j).

Then define the desired interpretationJ by setting

∆^J ={di|d∈∆^Iandi < n}

E^J ={d_i|d∈E^Iandi < n} for allE∈N_C\ {B₀, . . . , B_n−1} B_i^J ={di|d∈∆^I} for alli < n

s^J ={(di, ei)|(d, e)∈s^Iandi < n} for alls∈NR\ {r}

r^J ={(d_i, e_i)|(d, e)∈r^I, i < n,andd /∈K^Iore6=succ(d, j)for anyj}

∪ {(di, e_{(i+j) modn})|(d, e)∈r^I, i, j < n, ande=succ(d, j)}

It remains to verify thatJ is indeed a model ofT⁰. Clearly, the CIs in (1) are satisfied.

Moreover, it is not hard to verify that all concept inclusions inT are satisfied byJ, using the fact thatIis a model ofT and the construction ofJ. o From now on, we can thus safely assume that TBoxes do not contain at-least restrictions.

Note that the above translation is polynomial only if the numbersnin at-least restrictions are coded in unary. The same is of course true in unrestricted reasoning with Horn- SHIQ, where typically the same normal form is used.

3 Reduction to Unrestricted Satisfiability

We give a reduction of finite satisfiability to unrestricted satisfiability based on the completion of TBoxes with certain reversed cycles. LetT be a Horn-SHIQTBox.

Afinmod cycle inT is a sequenceK1, r1, K2, r2, . . . , rn−1, Kn,withK1, . . . , Kn

conjunctions of concept names andr1, . . . , rn−1(potentially inverse) roles that satisfies Kn=K1and

T |=Kiv ∃ri.Ki+1andT |=Ki+1v(61r⁻_i Ki) for1≤i < n.

Byreversinga finmod cycleK1, r1, K2, r2, . . . , r_n−1, Kn in a TBoxT, we mean to extendT with the concept inclusions

Kj+1v ∃r⁻_j.KjandKj v(61rjKj+1) for1≤j < n.

ThecompletionT_f of a TBoxT is obtained fromT by exhaustively reversing finmod cycles.

Example 3. The TBoxT⁰from Example 1 entails (in unrestricted models)

A1uBv ∃r.A2, A2v ∃r.(A1uB), A2v(61r⁻A1uB), A2uBv(61r⁻A1).

Thus,(A1uB), r, A2, r,(A1uB)is a finmod cycle inT⁰, which is reversed to A₂v ∃r⁻.(A₁uB), A₁uB v ∃r⁻.A₂, A₁uB v(61r A₂), A₁v(61r A₂uB).

Another finmod cycle inT⁰isA₁, r, A₂, r, A₁, reversed to

A₂v ∃r⁻.A₁, A₁v ∃r⁻.A₂, A₂v(61r A₁), A₁v(61r A₂).

Note thatT_f⁰ containsA1 v ∃r⁻.A2,A2 v ∃r.(A1uB), andA2 v (6 1 r A1).

ConsequentlyT_f⁰|=A1vB, in correspondence withT⁰ |=finA1vB.

The following result is the main result of this paper. It shows that TBox completion indeed provides a reduction from finite satisfiability to unrestricted satisfiability.

Theorem 4. LetT be a Horn-SHIQTBox andAa concept name. ThenAis finitely satisfiable w.r.t.T iffAis satisfiable w.r.t. the completionTfofT.

The “only if” direction of Theorem 4 is an immediate consequence of the observation that all CIs added by the TBox completion are actually entailed by the original TBox in finite models.

Lemma 5. LetK₁, r₁, . . . , r_n−1, K_na finmod cycle inT, then for every1 ≤i < n, T |=_finK_i+1v ∃r⁻_i .K_iandT |=_finK_iv(61r_iK_i+1).

Proof.We have to show that, ifK1, r1, . . . , r_n−1, Knis a finmod cycle inT andIis a finite model ofT, thenK_i^I⊆(61riKi+1)^IandK_i+1^I ⊆(∃r_i⁻.Ki)^Ifor1≤i < n.

We first note that, by the semantics of Horn-SHIQ, we must have|K₁^I| ≤ · · · ≤ |K_n^I|, thus K_n = K₁ yields |K₁^I| = · · · = |K_n^I|. Fix some i with 1 ≤ i < n. Using

|K_i^I| = |K_i+1^I |,K_i^I ⊆ (∃ri.Ki+1)^I, andK_i+1^I ⊆ (61r⁻_i Ki)^I, it is now easy to verify thatK_i^I⊆(61r_iK_i+1)^IandK_i+1^I ⊆(∃r⁻_i .K_i)^I, as required. o The “if” direction of Theorem 4 is much more demanding to prove. It requires to construct a finite model ofAandT based on the assumption that there is a (possibly infinite) model ofAandTf. Such a construction is presented in the next section.

4 Constructing Finite Models

We show that the completionT_f ofT captures all finite entailments ofT, that is, we prove the “if” direction of Theorem 4 above.

Assume that the concept nameAis satisfiable w.r.t.Tf. LetCN(T)denote the set of concept names used inT. A subsett⊆CN(T)is atype forT if there is a (potentially infinite) modelI ofT and ad ∈∆^I such thattp_I(d) := {A ∈ CN(T) | d∈ A^I}

is identical witht. We useTP(T)to denote the set of all types forT. Our aim is to construct a finite modelI ofT_f (and thus also ofT) that realizes all types inTP(T).

Note that sinceAis satisfiable w.r.t.T_f, there is a typetforT withA∈t. Since this type is realized in the finite modelIofT that we construct, it follows thatAis finitely satisfiable w.r.t.T as desired.

Before we give details of the construction ofI, we introduce some relevant notation and preliminary results. For allt, t⁰∈TP(Tf)and rolesr, we write

– t→_rt⁰ifT_f |=tv ∃r.t⁰andt⁰is maximal with this property;

– t→¹_rt⁰ift→rt⁰andTf |=t⁰v(61r⁻t);

– t¹↔¹_rt⁰ift→¹_rt⁰andt⁰ →¹_r−t.

– t ⇒¹_r t⁰ if t →¹_r t⁰ and there are s ⊆ t and s⁰ ⊆ t⁰ such that s ¹↔¹_r s⁰, but T_f |=t⁰v ∃r⁻.tdoes not hold.

Atype partitionis a setP ⊆TP(T)that is minimal with the following conditions:

– Pis non-empty;

– ift∈Pandt¹↔¹_rt⁰, thent⁰∈P.

We setP ≺P⁰if there aret∈P andt⁰ ∈P⁰witht⁰ (t. We will later be referring to the strict partial order that is obtained by taking the transitive closure of≺, denoted by≺⁺. A proof of the following observation can be found in the appendix.

Lemma 6. ≺⁺is a strict partial order.

As a last bit of notation, ifλ=s¹↔¹_rs⁰, then we useλ⁻to denotes⁰¹↔¹_r− s.

4.1 Constructing the Model

We constructIby starting with an initial interpretation and then exhaustively applying fourcompletion stepsthat we denote with (c1) to (c4). While constructing the sequence, we will make sure that the following invariants are satisfied:

(i1) for eachd∈∆^I, we havetp_I(d)∈TP(Tf);

(i2) if(d, d⁰)∈r^I, thentp_I(d)→rtp_I(d⁰)ortp_I(d⁰)→_r⁻ tp_I(d);

(i3) ifT_f |=Kv(61r K⁰), thenI |=Kv(61r K⁰).

We shall prove in Section 4.2 that each of the steps (c1) to (c4) indeed preserves these invariants.

The initial interpretationI is defined by introducing an element for every type, intepreting the concept names in the obvious way, and interpreting all role names as empty:∆^I=TP(Tf);A^I={t∈TP(Tf)|A∈t};r^I=∅. The four completion steps are described in detail below. We prefer to apply rules with smaller numbers, that is, if completion steps(ci)and(cj)are both applicable andi<j, then we apply(ci)first.

(c1) Select ad∈∆^Isuch thattp_I(d)⇒¹_rt, andd /∈(∃r.t)^I.

Add a fresh domain elemente, and modify the extension of concept and role names such thattp_I(e) =tand(d, e)∈r^I.

(c2) Select a type partitionPthat is minimal w.r.t. the order≺⁺, aλ=s¹↔¹_rs⁰with s∈P, and an elementd∈∆^Isuch thatd∈s^Iandd /∈(∃r.s⁰)^I.

For eachs∈ P, setn_s=|{d∈∆^I |d∈s^I}|. Letn_max = max{n_s |s∈ P}.

Reserve fresh domain elements

∆:={ds,i|s∈Pandns< i≤nmax}.

For eachs∈P, define a set ofs-instances

Is={d∈∆^I|d∈s^I} ∪ {ds,i|ns< i≤nmax}.

To proceed, we treat eachλ=s¹↔¹_rs⁰ withs, s⁰ ∈P separately. Thus, fix such aλ. Define

Rλ:={(d, e)∈r^I |d∈s^Iande∈s^0I}.

We first note that it is a consequence of invariant (i3) that (∗) the relationR_λis functional and inverse functional.

In fact, let(d, e₁),(d, e₂)∈R_λ. Then(d, e₁),(d, e₂)∈ r^I,d∈s^I, ande₁, e₂ ∈ s^0I. Byλ, we haveT_f |= s v (6 1 r s⁰). Thus, (i3) yieldse₁ = e₂. Inverse functionality can be shown analogously.

LetR¹_λbe the domain ofRλ, and letR_λ²be the range. By (∗), we have|R_λ¹|=|R²_λ|.

By definition of∆, we have|Is|=|Is⁰|. Moreover,R¹_λ⊆IsandR²_λ⊆Is⁰. We can thus choose a bijectionπλbetweenIs\R¹_λandIs⁰\R²_λ. Now extendIas follows:

• add all domain elements in∆;

• extendr^Iwithπλ, for eachλ=s¹↔¹_rs⁰;

• extendr^Iwith the converse ofπ_λ, for eachλ=s¹↔¹_r− s⁰;

• interpret concept names so thattp_I(ds,i) =sfor allds,i∈∆.

(c3) Select ad∈∆^Isuch thattp_I(d)→¹_rtandd /∈(∃r.t)^I.

Add a fresh domain elemente, and modify the extension of concept and role names such thattp_I(e) =tand(d, e)∈r^I.

(c4) Select ad∈∆^Isuch thattp_I(d)→rtandd /∈(∃r.t)^I.

Add the edge(d, t)tor^I, wheretis the element for the typetintroduced in the initial interpretationI.

We briefly discuss the main effects of prioritizing the completion steps. It is important to prefer (c1) over (c2): together with the preference of type partitions that are minimal w.r.t.

≺⁺in (c2), this ensures that when (c2) is executed on type partitionP,λ=s¹↔¹_r− s⁰ withs, s⁰ ∈ P, andd∈ I_s\R¹_λ, then not onlytp_I(d)⊇s, but actuallytp_I(d) = s.

This central property, put as Lemma 7 below, is essential to guarantee preservation of invariants (i2) and (i3) by execution of (c2). Preferring (c1) and (c2) over (c3) ensures that, when (c3) is executed, then there are nos⊆tp_I(d)ands⁰ ⊆tsuch that s¹↔¹_rs⁰; and preferring (c3) over (c4) ensures that, when (c4) is executed, then we haveTf 6|=tp_I(d) v(6 1 r t). These statements are provided here only to help in understanding the model construction. A formal proof is omitted at this point, but it can be recovered from the proofs given in the subsequent sections.

4.2 Satisfaction of Invariants

Application of (c1) preserves all invariants. It is obvious that the invariants (i1) and (i2) are preserved with each single application of (c1). We have to show that the same is true for (i3). Assume that completion processedd∈∆^Iwithtp_I(d)⇒¹_rt, and that eis the fresh domain element added. Assume to the contrary of what is to be shown thatTf |=K v(61r K⁰)and there is ae⁰ ∈∆^I distinct fromesuch thatd∈K^I, (d, e⁰)∈r^I, ande, e⁰∈K^0I. According to (i2), we distinguish the following cases:

– tp_I(d)→rtp_I(e⁰)

ThenTf |=tp_I(d)v ∃r.tp_I(e⁰)andtp_I(e⁰)is maximal with this property. Since tp_I(d) →r t, we additionally haveTf |=tp_I(d)v ∃r.t. SinceK ⊆tp_I(d)and e, e⁰ ∈ K^0I impliesK⁰ ⊆ tp_I(e⁰)∩t, a simple semantic argument shows that Tf |=Kv ∃r.(tp_I(e⁰)∪t). The maximality oftp_I(e⁰)thus impliest⊆tp_I(e⁰), in contradiction to the fact thatd /∈(∃r.t)^Iwas true before the completion step.

– tp_I(e⁰)→_r−tp_I(d).

ThenTf |=tp_I(e⁰)v ∃r⁻.tp_I(d)and, additionally, we haveTf |=tp_I(d)v ∃r.t.

SinceK⊆tp_I(d)andK⁰⊆tp_I(e⁰)∩t, a simple semantic argument shows that Tf |=tp_I(e⁰)vt. Sincetp_I(e⁰)is a type forTf by (i1), it follows thatt⊆tp_I(e⁰).

This contradicts the fact thatd /∈(∃r.t)^Iwas true before the completion step.

Application of (c2) preserves all invariants. Invariant (i1) is clearly preserved by each single application of (c2). We have to prove that the same is true for (i2) and (i3).

First, we show that the following property holds:

Lemma 7. Ifλ=s¹↔¹_rs⁰andd∈Is\R_λ¹, thentp_I(d) =s.

To show that (i2) is preserved by step (c2), consider an arbitrary pair(d, d⁰)∈r^Ithat has been added in a step (c2). Henceπ_λ(d) =d⁰, i.e., there is someλ=s¹↔¹_rs⁰such thatd∈I_s\R¹_λandd⁰∈I_s⁰\R²_λ. From Lemma 7, we obtaintp_I(d) =s. Analogously, consideringλ⁰ =s^{0 1}↔¹_r− sand the factd⁰ ∈I_s⁰\R²_λ =I_s⁰\R¹_λ0, we obtain from Lemma 7 thattp_I(d⁰) = s⁰. Consequently,s ¹↔¹_r s⁰ yieldstp_I(d)→_r tp_I(d⁰)and tp_I(d⁰)→_r− tp_I(d).

We now show that (i3) is preserved by step (c2). LetTf |=Kv(61r K⁰), and assume to the contrary of what is to be shown that, after some application of (c2), there are(d, d1),(d, d2)∈r^Iwithd∈K^I,d1, d2∈K^0I, andd16=d2. We distinguish the following cases:

– (d, d1)was added by an application of (c2),(d, d2)was not. By the former, there is λ=s¹↔¹_rs⁰such thatd∈Is\R¹_λ,d1∈Is⁰\R²_λ, and(d, d1)∈πλ. By Lemma 7, d∈Is\R¹_λyieldstp_I(d) =s. Moreover,d1 ∈Is⁰\R²_λimpliesd1∈Is⁰\R¹_λ−, and thus another application of Lemma 7 yieldstp_I(d1) =s⁰.

Since(d, d2)was not added by step (c2), (i2) gives the following subcases:

• tp_I(d)→rtp_I(d2). ThusTf |=tp_I(d)v ∃r.tp_I(d2)andtp_I(d2)is maximal with this property. Sincetp_I(d) = sand byλ,Tf |= tp_I(d) v ∃r.tp_I(d2).

Using the facts thatTf |=Kv(61r K⁰),K ⊆tp_I(d) =s,K⁰⊆tp_I(d2),

andK⁰⊆tp_I(d₁) =s⁰, an easy semantic argument shows thatT_f |=tp_I(d)v

∃r.(tp_I(d₂)∪s⁰). The maximality oftp_I(d₂)thus yieldss⁰⊆tp_I(d₂). Thus, d∈(∃r.s⁰)^Iwas true before step (c2) was applied, which is a contradiction to d /∈R¹_λ.

• tp_I(d2)→r⁻ tp_I(d). ThenTf |=tp_I(d2)v ∃r⁻.s. Byλ, we haveTf |=sv

∃r.s⁰. SinceK ⊆ s,K ⊆tp_I(d₂),K ⊆ tp_I(d₁) = s⁰, andTf |= K v(6 1 r K⁰), a simple semantic argument shows thats⁰ ⊆ tp_I(d₂). This again means thatd∈(∃r.s⁰)^Iwas true before step (c2) was applied, in contradiction tod /∈R¹_λ.

– both(d, d1)and(d, d2)were added by an application of (c2). Then there areλ1and λ2, such that, fori∈ {1,2}, we have

λ_i=s_i¹↔¹_rs⁰_i, (d, d_i)∈π_λi, d∈I_si\R¹_λ

i, d_i∈I_s⁰

i\R_λ²

i. Applying Lemma 7 toλi andd ∈ Is_i \R¹_λ

i yieldssi = tp_I(d), fori ∈ {1,2}.

Consequently,s1=s2. We next shows⁰₁=s⁰₂, thusλ1=λ2.

For uniformity, we usesto denotes1ands2. Fromλi, we obtainTf |=sv ∃r.si

andsi is maximal with this property, for i ∈ {1,2}. Note thatdi ∈ I_s⁰

i \R²_λ

i

impliesd_i∈I_s⁰

i\R¹

λ⁻_i . Applications of Lemma 7 toλ⁻_i andd_i∈I_s⁰

i\R¹

λ⁻_i yield tp_I(di) =si. Using the facts thatTf |=sv ∃r.sifori∈ {1,2},K⊆tp_I(d) =s, K⁰ ⊆tp_I(di) =sifori∈ {1,2}, andTf |=K v(61r K⁰), an easy semantic argument shows thatTf |= sv ∃r.(s1∪s2). The maximality ofs1ands2thus impliess1=s2as desired.

Hence,λ1 = λ2 and(d, d1),(d, d2) ∈ πλ₁. Sinceπλ₁ is a bijection, we obtain d1=d2, a contradiction.

Application of (c3) and (c4) preserves all invariants. It is obvious that the invariants (i1) and (i2) are preserved with each single application of (c3) or (c4). To show that the same is true for (i3), we can use the same proof as for (c1) because the assumptions of (c3) and (c4) differ from that of (c1) in weakeningtp_I(d)⇒¹_rttotp_I(d)→¹_rtand tp_I(d)→_rt, respectively, which is sufficient for that proof.

4.3 Termination of Model Construction

We show that the constructed interpretationIis indeed finite.

Proposition 8. ∆^Iis finite.

Proof.To analyze the termination of the construction ofI, we associate a certain directed treeT = (V, E)with the modelIthat makes explicit the way in whichIwas constructed.

Note that only the completion steps (c1) to (c3) introduce new domain elements and that (c1) and (c3) introduce a single new element with each application whereas (c2) introduces a whole (finite) set of fresh elements. Also note that each application of a completion step is triggered by a single domain elementdfor which some existential restriction is not yet satisfied. Now, the treeTis defined as follows:

– V consists of all subsets of∆^Ithat were introduced together by a single application of one of the completion steps (c1) to (c3); additionally, the set of all elements in the initial interpretationIis a node inV (in fact, the root node);

– the edge(v, v⁰)is included inEif the elements inv⁰were introduced by an applica- tion of a completion step to an elementdofv. We call this element thetriggerofv⁰ and denote it withdv⁰.

To show that∆^Iis finite, it clearly suffices to show thatV is finite. The outdegree ofT is finite since every rule application introduces only finitely many elements. By K¨onig’s Lemma, it thus remains to show thatT is of finite depth. We first note that an easy analysis of the completion steps (c1) to (c3) reveals the following property:

(∗) if(v1, v2),(v2, v3)∈E, then there ared0, . . . , dk∈v2and rolesr0, . . . , r_k−1s.t.

• d0=dv₂ ∈v1,d1, . . . , dk∈v2, anddk=dv₃ ∈v2;

• tp_I(d_i)→¹_r

i tp_I(d_i+1)for alli < k.

Now assume towards a contradiction that the depth ofT is larger than2|TP(Tf)|+ 1 and choose a concrete pathv1· · ·vnwithv1the root ofTandn >2|TP(Tf)|+ 1. This path gives rise to a corresponding sequence of triggersdv₁, . . . , dv_n. Since the length of this sequence exceeds2|TP(Tf)|, there must bei, j with1 ≤i < j ≤nand such thattp_I(d_vi) =tp_I(d_vj)andj > i+ 1. By applying (∗) multiple times, we obtain a sequence of domain elementsd₀, . . . , d_kand rolesr₀, . . . , r_k−1such that

1. d0=dv_i∈vi−1,d1∈vi, anddp=dv_j ∈vj−1; 2. tp_I(d`)→<sup>1</sup><sub>r`</sub> tp_I(d`+1)for` < k.

3. d0, . . . , dkcontains all elementsdv_i, . . . , dv_j.

Sincetp_I(dv_i) =tp_I(dv_j)and by Point 2,tp_I(d0), r0, . . . , rk−1,tp_I(dk)is a finmod cycle inTf. Since all finmod cycles inTf have been reversed, we have

tp_I(d0)¹↔¹_r

0tp_I(d1)¹↔¹_r

1 · · ·¹↔¹_rn−1tp_I(dn). (†) In the appendix, we prove the following claim:

Claim.If step (c1) or step (c3) is triggered byd∈∆^I and generates a new element e∈∆^I, then there is no rolersuch thattp_I(d)¹↔¹_rtp_I(e).

Sinced_vi ∈v_i−1andd₁∈v_i,d₁was generated by the application of a completion step triggered byd₀. By(†)and the claim, this completion step must be (c2). By definition of (c2) and(†), all elementsd₁, . . . , d_khave been introduced by exactly this application of (c2). This leads to a contradiction: we haved₁ ∈ v_i and d_k ∈ v_j−1, and since j > i+ 1,v_i6=v_j−1. Consequently,d₁andd_kwere introduced by different applications

of completion steps. o

4.4 Correctness of Model Construction

To complete the proof of the “if” direction of Theorem 4, it remains to show the following.

Proposition 9. Iis a model ofTf.

Proof.We show that for every axiomK vC ∈ T_f, we have thatI |=K vC. We distinguish the following cases:

– C = A. Letd ∈ K^I. Then K ⊆ tp_I(d)and by (i1) tp_I(d) ∈ TP(Tf). Since Tf |=KvA, this yieldsA∈tp_I(d)and thusd∈A^I.

– C = ⊥. Follows from (i1). Indeed since for every d ∈ ∆^I,tp(d) ∈ TP(Tf), K^I =∅.

– C=∃r.K⁰. Letd∈K^I. Then we have thatK⊆tp_I(d). SinceT_f |=Kv ∃r.K⁰, we have thattp_I(d)→r t⁰for somet⁰ withK⁰ ⊆t⁰. Thus there is somed⁰ with K⁰ ⊆tp_I(d⁰)such that(d, d⁰)∈r^Iwas added by an application of (c1), (c2), (c3) or (c4). In fact, if no suchd⁰is added by (c1) to (c3), then the edge(d, d⁰)∈r^I will clearly be added by (c4).

– C=∀r.K⁰. Letd∈K^Iand(d, d⁰)∈r^I, We have thatK⊆tp_I(d). Further, by (i2), we can distinguish the following cases:

• tp_I(d) →r tp_I(d⁰). ThenTf |= tp_I(d) v ∃r.tp(d⁰)andtp(d⁰)is maximal with this property. Since T_f |= K v ∀r.K⁰, we have thatT_f |= tp_I(d) v

∃r.tp(d⁰)∪K⁰, and the maximality oftp(d⁰)yieldsK⁰ ⊆ tp(d⁰), and thus d⁰∈K^0I.

• tp_I(d⁰)→r⁻ tp_I(d). ThenTf |=tp_I(d⁰)v ∃r⁻.tp_I(d). Together withTf |= K v ∀r.K⁰, we obtainTf |=tp_I(d⁰)vK⁰. Sincetp_I(d⁰)∈TP(Tf)by (i1), we obtainK⁰ ⊆tp_I(d⁰)and thusd⁰∈K^0I.

– C= (61r K)⁰. Follows from (i3). o

5 Conclusion

We have presented a reduction from finite satisfiability in Horn-SHIQto unrestricted satisfiability, extending the technique introduced forDL-Lite_F in [14]. As discussed in the introduction, we believe that our technique is a more suitable basis for efficient implementation than the techniques for fullALCQIandSHIQbased on exponentially large systems of inequalities.

As future work, we plan to develop a consequence based calculus for finite satis- fiability in Horn-SHIQand to extend the results obtained in this paper from finite satisfiability to answering conjunctive queries over ABoxes, assuming finite models.

While we believe that the constructions given in this paper can be easily extended to instance query answering over ABoxes, the treatment of full conjunctive queries requires significant modification of the model construction.

References

1. F. Baader, C. Lutz, and B. Suntisrivaraporn. CEL – a polynomial-time reasoner for life science ontologies. InProc. of IJCAR-06, volume 4130 ofLNCS, pages 287–291. Springer, 2006.

2. M. Bienvenu, C. Lutz, and F. Wolter. First-order rewritability of atomic queries in Horn description logics. InProc. of IJCAI-13. IJCAI/AAAI, 2013.

3. D. Calvanese. Finite model reasoning in description logics. InProc. of KR-96, pages 292–303.

Morgan Kaufmann, 1996.

4. S.S. Cosmadakis, P.C. Kanellakis, and M.Y. Vardi. Polynomial-time implication problems for unary inclusion dependencies. J. ACM, 37(1):15–46, 1990.

5. T. Eiter, M. Ortiz, M. ˇSimkus, T.-K. Tran, and G. Xiao. Towards practical query answering for Horn-SHIQ. InProc. of DL-12, volume 846 ofCEUR-WS.org, 2012.

6. U. Hustadt, B. Motik, and U. Sattler. Reasoning in description logics by a reduction to disjunctive datalog.J. Autom. Reasoning, 39(3), 2007.

7. Y. Kazakov. Consequence-driven reasoning for HornSHIQontologies. InProc. of IJCAI-09, pages 2040–2045, 2009.

8. Y. Kazakov, M. Kr¨otzsch, and F. Simanˇc´ık. Unchain myELreasoner. InProc. of DL-11, volume 745 ofCEUR-WS.org, 2011.

9. C. Lutz, U. Sattler, and L. Tendera. The complexity of finite model reasoning in description logics. Information and Computation, 199:132–171, 2005.

10. C. Lutz and F. Wolter. Non-uniform data complexity of query answering in description logics.

InProc. of KR-12. AAAI Press, 2012.

11. M. Ortiz, S. Rudolph, and M. ˇSimkus. Query answering in the Horn fragments of the descrip- tion logicsSHOIQandSROIQ. InProc. of IJCAI-11, pages 1039–1044. IJCAI/AAAI, 2011.

12. L. Pacholski, W. Szwast, and L. Tendera. Complexity results for first-order two-variable logic with counting.SIAM J. Comput., 29(4):1083–1117, 2000.

13. I. Pratt-Hartmann. Complexity of the two-variable fragment with counting quantifiers.J. of Logic, Language and Information, 14(3):369–395, 2005.

14. R. Rosati. Finite model reasoning in DL-Lite. InProc. of ESWC-08, volume 5021 ofLNCS, pages 215–229. Springer, 2008.