An Automata-based algorithm for description logics around SRIQ

An Automata-based Algorithm for Description Logics around SRIQ

Magdalena Ortiz

Institute of Information Systems, Vienna University of Technology [email protected]

Abstract. In this paper we use automata-theoretic techniques to tightly characterize the worst-case complexity of the knowledge base satisfiability problem for the very expres- sive Description Logics (DLs)ALCQIb⁺regandSRIQ. The logicALCQIb⁺regextends ALCwith qualified number restrictions, inverse roles, safe Boolean role expressions, regular expressions over roles, and concepts of the form∃P.Selfin the style ofSRIQ, a well known DL closely related to the new Semantic Web Standard OWL 2. By reduc- ing its knowledge base satisfiability problem to the emptiness test of an automaton on infinite trees, we show that all these additions do not increase the worst case complex- ity ofALCand provide a decision procedure for one of the most expressive DLs that have been shown to be decidable in exponential time. We also close the open question of the precise complexity of reasoning inSRIQ, exploiting a reduction from theSRIQ knowledge base satisfiability problem into theALCQIb⁺regone.

1 Introduction

Description Logics [1] are a well-established branch of logics for knowledge representation and reasoning, and the premier formalisms for modeling ontologies that describe application domains in terms of concepts (classes of objects) and roles (binary relationships between classes). They have gained increasing attention in areas like data and information integration, peer-to-peer data management and ontology-based data access, as well as in the Semantic Web, where they provide the basis for the standard Web Ontology Languages (OWL) [6].

Recent research in DLs has usually focused on the logics of the so-calledSHfamily. In particular, the DLSHIQis closely related to OWL-Lite and extends the ‘basic’ALC (the minimal propositionally closed DL capable of capturing EXPTIME-hard problems) with in- verse roles and number restrictions (counting), as well as with role inclusions and transitive roles. The DL known asSHOIQ, underlying OWL-DL, further extendsSHIQwith nomi- nals.SHIQandSHOIQare EXPTIMEand NEXPTIMEcomplete, respectively.

Recently,SHIQandSHOIQwere enhanced with regular role hierarchies in which the composition of a chain of roles may imply another role. This and other smaller features were included in their extensions known asSRIQandSROIQrespectively; the latter underlies the new OWL 2 standard. For reasoning in them, adaptations of the tableaux algorithms for SHIQandSHOIQwere proposed [7, 8], where in a pre-processing stage, the implications between roles given by the role hierarchy are captured in a set of finite state automata, at the price of exponential blow-up in the size of the representation of the KB. It was recently shown that this exponential blow-up is unavoidable, and thatSRIQandSROIQare 2EXPTIME

and 2NEXPTIMEhard respectively. Even in the light of these results, no tight upper bounds emerged from [7, 8], as they build on tableaux algorithms that are known not to be optimal in the worst case. For example, the algorithm in [10] and its extension in [7] may require

⋆This work was partially supported by the Austrian Science Fund (FWF) grant P20840 and the Mexi- can National Council for Science and Technology (CONACYT) grant 187697.

non-deterministic double exponential time even in the restricted case ofSHIQ KBs. For SROIQ, an optimal 2NEXPTIMEupper bound could be established by reducing it (with an exponential blow-up) to the two variable fragment of First Order Logic with counting, but the tight complexity ofSRIQremained open [11].

In this paper, we introduce an extension of the logicALCQIbreg previously considered in [2, 4]. LikeSHIQ,SRIQand other ‘main stream’ DLs,ALCQIbregextendsALCwith inverses and counting. The main difference is that, instead of role inclusions and other axioms asserting properties of roles,ALCQIbregsupports regular expressions over roles. They allow ALCQIbreg to simulate the role hierarchies ofSHIQ andSRIQ and, together with the presence of (safe) Boolean role constructors, capture most of the features ofSRIQ. Since all remaining features are related to the∃R.Selfconstructor, we add it toALCQIbregand obtain its extensionALCQIb⁺_reg, which is (at least) as expressive asSRIQ(although less succinct).

As we show in this paper, theALCQIb⁺_regknowledge base satisfiability problem allows for an elegant reduction to the emptiness test of an automaton on infinite trees. The reduction builds on [4] and yields a worst-case optimal, single exponential time decision procedure. We also present a reduction of theSRIQknowledge base satisfiability problem toALCQIb⁺_reg that exponentially increases the size of the input knowledge base; this yields the first 2EXP- TIMEdecision procedure forSRIQand a tight characterization of its worst case complexity.

The paper is organized as follows. In Section 2, after introducing the syntax and semantics ofALCQIb⁺_regknowledge bases, we define their syntactic closure, introduce a normal form for them and a canonical form for their models. These allow us to extend the automata algo- rithm of [4] toALCQIb⁺_regin Section 3. In section 4 we show howSRIQcan be reduced to ALCQIb⁺_reg. Final discussion and conclusions are given in Section 5.

2 The DL ALCQI b

reg

We introduce the syntax and semantics of the DL ALCQIb⁺_reg, a natural extension of ALCQIbreg[2, 4] with concepts of the form∃S.Selfand Boolean role inclusion axioms.

Definition 1 (ALCQIb⁺_regconcepts and roles). We consider fixed, countably infinite alpha- bets of concept namesC(also called atomic concepts), role namesRand individual names I. We assume that the setCcontains the special concepts⊤(top) and⊥(bottom), whileR contains the top (universal) roleTand the bottom (empty) roleB.

According to the following syntax, we define (ALCQIb⁺_reg) conceptsC,C^′, atomic roles Q, simple rolesS,S^′, and rolesR,R^′, whereA∈C,P ∈RandP 6=T.

C, C^′−→A| ¬C|C⊓C^′ |C⊔C^′ | ∀R.C| ∃R.C|>n S.C|6n S.C| ∃S.Self Q−→P|P⁻

S, S^′−→Q|S∩S^′|S∪S^′|S∩ ¬S^′ R, R^′−→T|S|R∪R^′ |R◦R^′|R^∗|id(C)

We useS\S^′as a shortcut forS∩ ¬S^′. AnALCQIb⁺_reg expression is a concept or a role.

Subconcepts, subroles and subexpressions are defined in the natural way.

Definition 2 (Knowledge Bases). An (ALCQIb⁺_reg) assertion is of the formC(a),S(a, b), ora6= b, whereCis a concept,S a simple role anda, b ∈ I. An (ALCQIb⁺_reg) ABox is a set of assertions. An (ALCQIb⁺_reg) concept inclusion axiom (CIA) is an expressionC⊑C^′ for arbitrary conceptsCandC^′. An Boolean role inclusion axiom (BRIA) is an expression S⊑S^′for simple rolesSandS^′. AnALCQIb⁺_regTBox is a set of CIAs and BRIAs.

An (ALCQIb⁺reg) knowledge base (KB) is a pairK=hA,T iwhereT is anALCQIb⁺reg

TBox andAis a non-emptyALCQIb⁺reg ABox.We denote byC_Kthe set of atomic concepts occurring inK, byR_Kthe set of roles names occurring inK, byR_Kthe setR_K∪ {P⁻|P ∈ R_K}, and byI_K the individuals inK.

Definition 3 (Semantics). An interpretationI = (∆^I,·^I)consists of a non-empty domain

∆^I and a valuation function ·^I that maps each individuala∈Ito an element a^I ∈ ∆^I, each concept nameA∈Cto a setA^I ⊆ ∆^I, and each role nameP∈Rto a set of pairs R^I ⊆∆^I×∆^I, in such a way that⊤^I =∆^I,⊥^I =∅,T^I =∆^I×∆^IandB^I =∅. The function·^Iis inductively extended to all concepts and roles as follows:

(¬C)^I=∆^I\C^I (P⁻)^I={(y, x)|(x, y)∈P^I} (C⊓C^′)^I=C^I∩C^′I (R∩R^′)^I=R^I∩R^′I

(C⊔C^′)^I=C^I∪C^′I (R∪R^′)^I=R^I∪R^′I (∀R.C)^I={x| ∀y.(x, y)∈R^I→y∈C^I} (¬R)^I= (∆^I×∆^I)\R^I (∃R.C)^I={x| ∃y.(x, y)∈R^I∧y∈C^I} (R◦R^′)^I=R^I◦R^′I (>n S.C)^I={x| |{y|(x, y)∈S^I∧y∈C^I}| ≥n} (R^∗)^I= (R^I)^∗

(6n S.C)^I={x| |{y|(x, y)∈S^I∧y∈C^I}| ≤n} (id(C))^I={(x, x)|x∈C^I} (∃S.Self)^I={x|(x, x)∈S^I}

where∩,∪and\are overridden to denote the standard set-theoretic operations,◦to denote composition and·^∗to denote the reflexive transitive closure of a binary relation.

I satisfies an assertionα, denotedI |=α, ifa^I ∈A^I whenα=A(a);ha^I, b^Ii ∈P^I whenα = P(a, b); anda^I 6= b^I whenα = a6≈b.I is a model of an ABox A, denoted I |=Aif it satisfies every assertion inA.Isatisfies a CIA or BRIAE ⊑E^′ ifE^I ⊆E^′I.I is a model of a TBoxT, denotedI |=T if it satisfies every CIA and BRIA inT. Finally,Iis a model ofK=hT,Ai, denotedI |=K, ifI |=T andI |=A. Given a KBK, Knowledge base satisfiability is the problem of deciding whether there exists anIsuch thatI |=K.

2.1 Syntactic Closure

We define the (syntactic) closure of a concept, which contains all the concepts and simple roles that are relevant for deciding its satisfiability. It is analogous to the well-known Fischer- Ladner closure of PDL, and it is exploited by the automata construction in Section 3.

In this section, we consider concepts and roles in a DL that we call ALCQIB⁺_reg; it is very similar toALCQIb⁺_regbut supports arbitrary role negation¬Sinstead of role difference S∩¬S^′. If anALCQIB⁺_regexpression is equivalent to anALCQIb⁺_regone, we call it safe. In- tuitively, sinceALCQIB⁺_regdoes not impose safety, it allows for a more flexible manipulation of Boolean role expressions and a simpler notion of syntactic closure.

In what follows, we use≷to denote either>or6. For a role nameP ∈C, we callP⁻the inverse ofPandPthe inverse ofP⁻; the inverse of an atomic roleQis denotedInv(Q). For a simple roleS,Inv(S)denotes the role obtained by replacing each atomic roleQoccurring in Sby its inverseInv(Q). For brevity, we assume in this subsection that⊔and∀are expressed by means⊓,∃and using¬, and that\and∪are expressed using∩and¬. As usual,Cand C^′,SandS^′andRandR^′respectively stand for concepts, simple roles and arbitrary roles.

Definition 4. The closureCl(D)of anALCQIB⁺_regconceptDis defined as the smallest set ofALCQIB⁺_regexpressions such thatD∈Cl(D)and:

if C∈Cl(D) then ¬C∈Cl(D) (ifCis not of the form¬C^′) if ¬C∈Cl(D) then C∈Cl(D)

if C⊓C^′∈Cl(D) then C, C^′∈Cl(D) if ∃R.C∈Cl(D) then C∈Cl(D)

if ∃(R∪R^′).C∈Cl(D) then ∃R.C,∃R^′.C∈Cl(D) if ∃(R◦R^′).C∈Cl(D) then ∃R.∃R^′.C∈Cl(D) if ∃R^∗.C∈Cl(D) then ∃R.∃R^∗.C∈Cl(D) if ∃id(C).C^′∈Cl(D) then C, C^′∈Cl(D) if ∃S.C∈Cl(D) then S∈Cl(D) if ≷n S.C∈Cl(D) then S, C∈Cl(D) if ∃S.Self∈Cl(D) then S∈Cl(D) if S∩S^′ ∈Cl(D) then S, S^′∈Cl(D)

if ¬S∈Cl(D) then S∈Cl(D)

if S∈Cl(D) then ¬S∈Cl(D) (ifSis not of the form¬S^′) if S∈Cl(D) then Inv(S)∈Cl(D)

Note that |Cl(D)| is linear in the length of D and that Cl(D) may contain non safe ALCQIB⁺regexpressions even whenDis anALCQIb⁺_regconcept.

2.2 Normalizing Knowledge Bases

We present now some simple reductions that allow us to transform a KB into an equiva- lent one with a more restricted syntactic structure. We considerALCQIb⁺_reg KBs as well as ALCQIB⁺_regones, which are defined in the natural way.

First of all, we consider the special expressions⊤,⊥,BandT, and show that they can be expressed without the need of special symbols. Then we show how every KB can be trans- formed into an extensionally reduced one, where all terminological information is expressed by CIAs. Finally, we consider the well known negation normal form of KBs.

Universal role and special expressions. The special concepts⊤and⊥can be simulated via any concept namesC⊤ andC⊥ not occurring inKby adding, e.g., CIAs C⊔ ¬C ⊑ C⊤,

¬C⊤ ⊑ C⊥and¬C⊥ ⊑C⊤for any conceptC. Further, using⊤and⊥, the empty roleB can also be easily simulated by a fresh role nameP_Bby adding an axiom⊤ ⊑ ∀P_B.⊥. Note that the above holds for every DL containingALC.

The universal role can be expressed inALCQIB⁺regas¬B, but this is not a safe role and is hence disallowed inALCQIb⁺reg KBs. In fact, in the absence of theTsymbol, there is no ALCQIb⁺_reg role expression that is always equivalent toT. However, for each input KBK, there exists a role expressionUsuch that the resulting KB has a modelIwithU^I =∆^I×∆^I wheneverKis satisfiable. This is similar to what occurs in other expressive DLs (e.g.,SHIQ) and sufficient for the problems we consider.

Indeed, since every satisfiable ALCQIb⁺_reg concept C has a connected model, we can transform C into an equisatisfiable concept replacing each occurrence of T by U = (S

Q∈RCQ)^∗, whereR_Cdenotes the set of all atomic roles occurring inC. In the presence of ABoxes, we additionally need to ensure that all pairs of ABox individuals are in the extension of the role expression simulatingT. Hence, given a KBK = hA,T i, we can eliminate the universal roleTas follows:

- For a fresh role nameAU, an assertionAU(a, b)is added toAfor every paira, b∈I_K. - Each occurrence ofTinT is replaced by the roleU = (AU ∪S

Q∈RKQ)^∗.

Extensionally reduced KBs. Let KBK =hA,T ibe anALCQIB⁺_reg KB. Of the transfor- mations below, the one w.r.t. to the ABox is well known. Here we use a similar technique to rewrite by means of CIAs all terminological information in the∃R.Selfconcepts and BRIAs.

Please note that all the transformations below involve only simple roles, hence the result- ing KB complies to the allowed syntax. Also, since they do not introduce any non-safe role expressions, ifKis anALCQIb⁺_regKB, it will remain inALCQIb⁺_regafter the reduction.

– ABox reduction.Kis extensionally reduced w.r.t. the ABox if only assertions of the forms A(a),P(a, b)anda6=bwithA∈CandP ∈Roccur inA.

The ABox reduction ofKis the KBΩABox(K)obtained as follows:

1. Each assertion of the form C(a) inAwithC 6∈ C is replaced with an assertion AC(a)for a freshAC∈Cand an axiomAC⊑Cis added toT.

2. Each assertionS(a, b)inAwithS 6∈Ris replaced with an assertionPS(a, b)for a freshPS ∈Rand an axiomPS ⊑Sis added toT.

– Self concepts reduction.K is extensionally reduced w.r.t. Self concepts ifP ∈ Rfor every concept of the form∃P.Selfoccurring in it.

The Self concept reduction ofKis the KBΩSelf(K)obtained by replacing each concept of the form∃S.SelfwithS6∈Roccurring inT with a concept∃PS.Selffor a freshPS ∈R and an adding an axiomPS⊑StoT.

– BRIA reduction.Kis extensionally reduced w.r.t. BRIAs if all axioms inT are CIAs.

The BRIA reduction ofKis the KBΩBRIA(K)obtained by replacing inT each BRIA of the formS⊑S^′by a CIA∃(S\S^′).⊤ ⊑ ⊥.

– KB reduction.Kis extensionally reduced if it is extensionally reduced w.r.t. the ABox, Self concepts and BRIAs.

The reduction ofKis the KBΩ(K) =ΩBRIA(ΩSelf(ΩABox(K))).

The above reductions preserve the semantics of the knowledge baseK, andΩ(K)addi- tionally constrains the interpretation of some concept names not occurring inK. If we consider the standard first order translation ofKandΩ(K), then the latter is a conservative extension of the former. With this observation, the proof of the following proposition is straightforward:

Proposition 1. For a givenALCQIB⁺_reg KBK,Ω(K)can be obtained in linear time. For every interpretationI,I |=Ω(K)impliesI |=KandI |=KimpliesI^′ |=Ω(K), whereI^′ denotes the interpretation s.t. E^I′ =E^Ifor everyE ∈C_K∪R_K,(AC)^I′ =C^Ifor every AC∈C_Ω(K)\C_K, and(PS)^I′ =S^Ifor everyPS∈R_Ω(K)\R_K.

Thus deciding the satisfiability ofKcan be reduced (in linear time) to used deciding the satisfiability ofΩ(K), and we can restrict or attention to extensionally reduced knowledge bases. Note that we can also considerΩ(K)to decide the entailment of any sentence over the language ofK; this is useful, e.g., for query answering [3].

Negation Normal Form. Finally, we transform KBs to negation normal form (NNF).

AnALCQIB⁺regroleRis in NNF form ifQis atomic for every subrole¬QofR. Sim- ilarly, a conceptC is in negation normal form (NNF) ifA is atomic for every subconcept

¬A ofC, and all roles occurring in C are in NNF. A knowledge base is in NNF if only expressions in NNF occur in it. For an expressionE,nnf(E)denotes the equivalent expres- sion in NNF obtained fromE using the standard transformations. For a conceptD, we let Cl^nnf(D) ={nnf(E)|E∈Cl(D)}. For a KBK,nnf(K)denotes the KB obtained fromK by replacing each expressionEinKbynnf(E).

Proposition 2. For everyALCQIB⁺_regexpressionE,nnf(E)can be obtained in linear time, and ifEis anALCQIb⁺_regexpression, then so isnnf(E). Further, for every KBKand every interpretationI,I |=KiffI |=nnf(K).

AnALCQIb⁺_reg KBK =hA,T iis normal if it is extensionally reduced and in NNF, it does not contain⊤,⊥,TandB, and each concept occurring inAalso occurs inT.¹

1This is w.l.o.g. as we can add toT, e.g.,A⊑ ⊤for eachA∈Cthat occurs inAbut not inT.

2.3 Canonical Models and Trees

We have seen that, to decide KB satisfiability, we only need to consider KBs with a restricted syntax. Now we consider some semantic properties by which the shape of the considered interpretations can also be conveniently restricted.

In what follows, we consider onlyALCQIb⁺_regKBs; the results of this section do not hold forALCQIB⁺_reg in general.²Like many DLs,ALCQIb⁺_reg has some form of the tree model property: every satisfiable TBoxT (or similarly, every satisfiable conceptC) has a model that can be seen as a tree with possible additional loops at some nodes, say a ‘quasi-tree’. A satisfiableALCQIb⁺_regKBK=hA,T ihas a ‘quasi-forest’ shaped canonical model, in which each ABox individual is the root of a quasi-tree shaped model ofT.

Definition 5. LetK=hT,Aibe anALCQIb⁺_regKB, and let1≤n≤ |I_K|,k≥0. An inter- pretationI = (∆^I,·^I)forKis called a canonical interpretation (withnroots and branching degreek) if:

(1) eachi·x∈∆^Ihasi∈ {1, . . . , n}andx∈ {1, . . . , k}^∗; ifx=εtheniis called a root.

(2) {ε} ∪∆^Iis prefix closed, i.e., ifx·c∈∆^I, thenx∈ {ε} ∪∆^I. (3) For eacha∈I_Kthere is exactly one rootjsuch thata^I=j.

(4) For each rootjthere is somea∈I_Kwitha^I=j.

(5) If(i·w, j·w^′)∈P^I for some atomic roleP and two rootsi, j, then eitherw=w^′ =ε, ori=jand one of the following holds: (i)w=w^′, (ii)w^′ =w·lor (iii)w=w^′·l, for some1≤l≤k.

InALCQIb⁺_reg, any TBoxT can be ‘internalized’ into an equivalent conceptCT, so that the satisfiability ofT can be established by obtaining a model ofCT [14].

Proposition 3. Consider an ALCQIbreg knowledge base K = hA,T i, and let CT =∀(S

R∈RKR)^∗.F

C₁⊑C₂∈T(¬C1⊔C2)

For every interpretationI = (∆^I,·^I),I |= T iff (CT)^I = ∆^I. Furthermore, ifI is a canonical interpretation, thenI |=T iffi∈(CT)^Ifor each rootiofI.

Now we can establish the canonical model property ofALCQIb⁺_reg; it can be shown by adapting corresponding proofs for related logics [17, 15]. Roughly, any model of a satisfiable ALCQIb⁺_regconceptDcan be unraveled into a model ofDthat is a quasi-tree and that has branching degreekD =|Cl(D)| ×n, wherenis the maximalnoccurring in a concept of the form>n S.CinCl^nnf(D), or1if there are no such concepts.

Theorem 1. Every satisfiableALCQIb⁺_reg KBK = hT,Aihas a canonical modelI such thatI is a canonical interpretation forKwith branchingkCT,I |=Aandi ∈ (CT)^I for each rootiofI.

By Theorem 1, which does not hold forALCQIB⁺_regin general, deciding the satisfiability of anALCQIb⁺reg KB boils down to deciding whether it has a canonical model. To decide the latter, we rely on a representation of canonical interpretations as infinite labeled trees and define an automaton that decides the existence of such trees.

2Theorem 1 fails already in the extension ofALCwith non-safe Boolean role expressions, see [12].

Definition 6. An (infinite) tree is a prefix-closed setTof words over the natural numbersIN.

The elements ofTare called nodes, the empty wordεis its root. For everyx∈T, the nodes x·cwithc ∈INare the successors ofx, andxis the predecessor of eachx·c. By convention, x·0 = x, and(x·i)·−1 = x. The branching degreed(x)of a nodexis the number of its successors. Ifd(x)≤kfor each nodexofT, thenT has branching degreek. An infinite path πofT is a prefix-closed setπ⊆T where for everyi ≥0there exists a unique nodex∈P with|x|=i. A labeled tree over an alphabetΣ(or simply aΣ-labeled tree) is a pair(T, V), whereTis a tree andV :T →Σmaps each node ofTto an element ofΣ.

In the following, we assume a (fixed) given normalALCQIb⁺_reg KBK = hA,T i. We denoteI_K byJ and assume an arbitrary, fixed, enumerationa1, . . . , amof the elements of J. We define PI={P ij |ai, aj ∈ J andP ∈R_K}and PS={PSelf |P ∈R_K}. We will define the tree representation of a canonical interpretationIforKas a labeled treeT_Iover the alphabetΣK= 2^CK∪RK∪J ∪{r}∪{d}∪PI∪PS.

To defineT_I, we first note that the domain of a canonical interpretation is almost a tree;

we only need to add a rootεto it. The extensions of the interpretations of individuals, concepts and roles can be represented as labels over the alphabetΣKin a straightforward way. Roughly, each element ofx∈∆^I, which is a node of the tree, is labeled with a setV(x)that contains:

(i) the atomic conceptsAsuch thatx∈A^I; (ii) the atomic roles that connect the predecessor ofxtox, and (iii) a special symbolP_Selffor eachPsuch thatx∈(∃P.Self)^I. The rootεand the roots ofI (which are at the first level of the treeT_I) are treated differently. The label of each rootiofI, apart from the atomic concepts to whichibelongs, also contains the name of the individuals inJ which it interprets, and it contains no basic roles. The relations between level one nodes are stored in the label of the rootε. The rootεdoes not represent any object in

∆^Iand is marked with a special identifierrand symbols of the formP ijindicating that the pair of individuals(ai, aj)belongs to the extension of the basic roleP. For simplicity, ifIhas nroots and|J |> n,|J | −ndummy children labeled{d}are added to the rootε, ensuring that it has exactly|J |children.

Definition 7. LetIbe a canonical interpretation forKwithnroots. The tree representation ofI is the labeled treeT_I = (T, V)over the alphabetΣK = 2^CK∪RK∪J ∪{r}∪{d}∪PI∪PS

defined as follows:

• T ={ε} ∪∆^I∪ {n+ 1, . . . , m}.

• V(ε) ={r} ∪ {P ij|ai, aj ∈ J, P ∈C_Kandha^I_i, a^I_ji ∈P^I},

• for each1≤i≤n,V(i) ={aj∈ J |a^I_j =i} ∪ {A∈C_K |a^I_j ∈A^Ianda^I_j =i},

• for eachn+ 1≤i≤m,V(i) ={d},

• for all other nodesi·xofT,V(i·x) ={A∈C_K |i·x∈A^I} ∪ {Q∈R_K |(i·w, i·x)∈ P^Iwherex=w·jfor somej,1≤j≤kCT } ∪ {P_Self|P ∈R_Kandi·x∈(∃P.Self)^I} Note that the branching degree ofT_Iis bounded by|J |at the root and bykCT at all other levels, so the tree has branching degreemax(kCT,|J |).

3 An Automata Algorithm for ALCQI b

reg

Automata on infinite trees allow for an elegant reduction of decision problems for temporal and program logics [5], and have been widely exploited for many variants of PDL, theµ- calculus and similar logics [16, 17]. In DLs they have been used for concept satisfiability [15]

and, to a more limited extent, for KB satisfiability [4].

3.1 Preliminaries: Automata on Infinite Trees

We consider two-way alternating tree automata over infinite trees, introduced in [16]. Let B(I)be the set of positive Boolean formulas built inductively from true, false, and atoms from a given setIapplying∧and∨. A setJ⊆Isatisfies a formulaϕ∈ B(I), if assigning true to the atoms inJand false to those inI\Jmakesϕtrue. A two-way alternating tree automaton (2ATA) running over infinite trees with branching degreek, is a tupleA =hΣ, Q, δ, q0, Fi, whereΣis the input alphabet;Qis a finite set of states;δ : Q×Σ → B([k]×Q), where [k] ={−1,0,1, . . . , k}, is the transition function;q0 ∈Qis the initial state; andF ⊆Qis the (B¨uchi) acceptance condition.

The transition functionδ maps a state q ∈ Qand an input letter σ ∈ Σ to a positive Boolean formulaϕover atoms[k]×Q. Intuitively, ifδ(q, σ) =ϕ, then each atom(c, q^′)inϕ corresponds to a new copy of the automaton going in the direction given bycand starting in stateq^′. E.g., a transition of the formδ(q1, σ) = (1, q2)∧(1, q3)∨(−1, q1)∧(0, q3)indicates that if Ais in the stateq1and reads the nodexlabeled withσ, it proceeds by sending off either two copies, in the statesq2andq3respectively, to the first successor ofx, or one copy in the stateq1to the predecessor ofxand one copy in the stateq3toxitself.

Informally, a run of a 2ATAAover a labeled tree(T, V)is a labeled tree(Tr, r)in which each nodey∈Tris labeled by an elementr(y) = (x, q)∈T×Qand describes a copy ofA that is in the stateqand reads the nodexofT; the labels of adjacent nodes must satisfy the transition function ofA. The run is accepting if in every infinite path at least one state from F occurs infinitely often in the node labels.

Formally, given a 2ATAA = hΣ, Q, δ, q0, Fiand aΣ-labeled tree(T, V), a run ofA over(T, V)is aT×Q-labeled tree(Tr, r)satisfying:

• ε∈Trandr(ε) = (ε, q0).

• For each y∈Tr withr(y) = (x, q) there is a (possibly empty) setS = {(c1, q1), . . . , (cn, qn)} ⊆ [k]×Qsuch thatS satisfies δ(q, V(x))and for all1 ≤ i ≤ n, we have that y·i∈Tr,x·ciis defined andr(y·i) = (x·ci, qi).

A run(Tr, r)over(T, V)is accepting, if for each infinite pathπofTrthere is some state qf ∈F such that the set{y ∈π|r(y) = (x, qf)for somex∈T}is infinite. A labeled tree (T, V)is accepted byAif there is an accepting run ofAover(T, V), andL(A)denotes the set ofΣ-labeled trees accepted byA.

The non-emptiness problem for 2ATAs is to decide whether, given a 2ATAA, the setL(A) is nonempty. It was shown in [16] that non-emptiness ofA=hΣ, Q, δ, q0, Fiis decidable in time single exponential in|Q|and polynomial in|Σ|.

3.2 Constructing the Automaton

In the following, we assume a given, fixed, normalALCQIb⁺_regKBK=hA,T iand provide a construction of a 2ATA A_K fromK that accepts a given tree T iff it is the tree repre- sentation of a canonical model ofK. Similar automata constructions have been given in the literature. In particular, this one is a simple extension of the one in [4] to handle the∃S.Self concepts ofALCQIb⁺_reg. In turn, [4] extends to KB satisfiability a previous construction of an automaton that decidesALCQIbregconcept satisfiability [2]. Due to space constraints, we briefly describeA_K=hΣK, QK, δK, q0, FKiand focus on the extension fromALCQIbregto ALCQIb⁺_reg; we refer the reader to [3] for details.

The alphabet isΣK = 2^CK∪RK∪J ∪{r}∪{d}∪PI∪PSas given in Definition 7; note that the only difference w.r.t. [3] is the presence of the set PS.

Like in [3], the ‘basic’ states ofA_Kare the role and concept expressions in the closure of CT, which are used for checking whether a node of the tree is in the extension of an expression or marked with some symbol. There are also additional special states for verifying some more involved conditions like the satisfaction of the number restrictions, the correct representation of the individuals and the ABox assertions, etc.

More precisely, letS = J ∪PS∪ {c, d}. There is a state for each element of the the setClext=Cl^nnf(CT)∪ {s,¬s|s∈S}which we call theS-extended closure ofCT. The extended closure allows us to treat the individual names and special symbols in the node labels as well as the atomic expressions in a uniform way. Note that, in contrast to [3],Clextmay contain concepts of the form∃P.Self. The set of states isSK ={s0, s1} ∪Clext∪Snum∪ SArole∪SAquant∪SAnum∪SSelf, where the setsSAquantSnum,SAnumandSAroleare as in [3]. We use the states inSAquantof the formshj,∃S.Ciandhj,∀S.Cito respectively check whether concepts of the form∃Q.C and ∀Q.C are satisfied by a node representing some ABox individualaj. For checking the satisfaction of number restrictions we need the statesSnumof the formh≷n S.C, i, jithat store two countersiandj, plus the statesSAnum

of the formhm,≷nQC, i, ji, which also store the value of two counters but for a particular individualam. For checking whether two ABox individualsai, ajare related by a simple role S, we use states of the formSijinSArole. The only additional states w.r.t. to [3] are the ones in the setS_Self ={S_Self | Sa simple role inCl^nnf(CT)}, which we use for decomposing a simple role and checking whether it links an individual to itself. Finally, there is the initial state s0and another special states1. As in [3], both are used to verify the satisfaction of general conditions necessary for the input tree to represent a canonical model ofK, e.g., that the root is the only node labeledr, that its label correctly stores the role assertions in the ABox and that the ABox individuals are correctly represented by the level one nodes.

As in [2–4],FK={∀R^∗.C| ∀R^∗.C∈Cl^nnf(CT)}is the acceptance condition.

The transition functionδ : SK ×ΣK → B([k]×SK), wherek = max(kCT,|J |), is defined as follows. First, for each σin σ∈ΣK with r∈σ we define a transitionδ(s0, σ)

=F1 ∧ · · · ∧F8 from the initial state s0, which verifies (i) that the root is the only node containingrand that the ABox individuals are properly represented by the level one nodes (F1–F4); (ii) that all ABox assertions are satisfied (F5–F7); and (iii) that every non-dummy node at level one is the root of a tree representing a model ofCT (F8).F4requires a transition δ(s1, σ)from the other special states1for eachσ∈ΣK, which is necessary forF4to correctly ensure that the symbolronly occurs at the root and the symbols inJ only occur at the first level of the tree. These transitions are as in [3], to which the reader may refer for details.

The transitions that inductively decompose complex concepts while navigating the tree are as in [3]; so are the ones that decompose simple roles. Additionally, for each concept of the form ∃P.Self such that∃P.Self ∈ SK and for each σ ∈ ΣK there is a transition δ(∃P.Self, σ) = (0, PSelf). Special transitions are necessary to decompose the simple roles for each possible self loop and for each pair of ABox individuals. Thus we have for each σ∈ΣKand for eachs∈SSelf∪SArolea transitionδ(s, σ)as follows:

δ(S∩S^′Self, σ) = (0, SSelf)∧(0, S^′Self) δ(S∩S^′ij, σ) = (0, Sij)∧(0, S^′ij) δ(S∪S^′Self, σ) = (0, SSelf)∨(0, S^′Self) δ(S∪S^′ij, σ) = (0, Sij)∨(0, S^′ij) δ(Q\Q^′_Self, σ) = (0, QSelf)∧(0,¬Q^′_Self) δ(Q\Q^′ij, σ) = (0, Qij)∧(0,¬Q^′ij) Informally, given a tree representing an interpretation, let us call potential neighbors ofx all the nodes that could be related to a nodexvia a simple role. Ifxdoes not represent an ABox individual, then its potential neighbors arexitself, its predecessor and all its successors.

Ifxdoes represent an ABox individual, then instead of its predecessor (which is the dummy root), all the other nodes representing ABox individuals are potential neighbors ofx.

Further transitions in [3] verify whether a node satisfies an universal restriction∀S.C, an existential restriction∃S.Cor a number restriction≷nS.C, for a simple roleS; they navigate all the potential neighbors of a node and test which are reachable viaSand labelledC. We use similar transitions, but adapt them to take into account that, in anALCQIb⁺_reginterpretation, a node is a potential neighbor of itself. The transitions for the concepts of the form∀S.Cand

∃S.Care given below. The ones for the number restrictions (which use the states inSnumand SAnumto encode counters) are adapted similarly but omitted here due to space constraints.

Since the potential neighbors of a nodexare different depending on whether (i)xis a level one node, or (ii)xis a node at any other level, the transitions must differ accordingly.

Hence, for each concept of the form∃S.C or∀S.C inCl^nnf(CT)withS simple and each σ∈ΣK, ifσ∩(J ∪ {d}) =∅we define:

δ(∃S.C, σ) = ((0, SW Self)∧(0, C)) ∨ ((0,Inv(S))∧(−1, C)) ∨

1≤i≤kCT((i, S)∧(i, C))

δ(∀S.C, σ) = ((0,V nnf(¬SSelf))∨(0, C)) ∧ ((0,nnf(¬Inv(S)))∨(−1, C)) ∧

1≤i≤kCT((i,nnf(¬S))∨(i, C)) Otherwise, ifσ∩(J ∪ {d})6=∅, we have:

δ(∃S.C, σ) = ((0, SSelf)∧(0, C))∨W

a_j∈σ(−1,hj,∃S.Ci)∨

W

1≤i≤kCT((i, S)∧(i, C)) δ(∀S.C, σ) = ((0,nnf(¬S_Self))∨(0, C)) ∧ V

aj∈σ(−1,hj,∀S.Ci) ∧ V

1≤i≤kCT((i,nnf(¬S))∨(i, C))

The first disjunct/conjunct in these transitions, which verifies whether the current node reaches itself viaSand is labelledC, is not present in the corresponding transitions from [3].

In the last set of transitions (case (i) above, σ ∩(J ∪ {d}) 6= ∅), the second dis- junct/conjunct is responsible for sending a copy of the automaton to the root of the tree and moving to the special states inSAquant, in order to traverse all the ABox individuals which are potential neighbors of the current node. The latter is achieved as in [4], i.e., for eachσ∈ΣK

and eachhj,∃S.Ciorhj,∀S.CiinSAquant, there is a transition δ(hj,∃S.Ci, σ) = _

1≤i≤|J |

( _

1≤k≤|J |

((0, Sjk)∧(i, ak)∧(i, C)))

δ(hj,∀S.Ci, σ) = ^

1≤i≤|J |

( ^

1≤k≤|J |

((0,nnf(¬Sjk))∨(i,¬ak)∨(i, C)))

The above transitions decompose all concepts and roles until they reach states correspond- ing to atomic expressions or (possibly negated) special symbols in PI∪PS; it is then checked whether the expression is contained in the node labelσ. The transitions form the statess0and s1may also move to states corresponding to possibly negated symbols inJ ∪{r, d}, which are similarly checked at the node labels. Thus, for eachσ∈ΣK, eachs∈C_K∪R_K∪ J ∪ {r, d}

and eacht∈PI∪PS, there are transitions:

δ(s, σ) =

(true, ifs∈σ

false, ifs6∈σ δ(¬s, σ) =

(true, ifs6∈σ false, ifs∈σ δ(t, σ) =

(true, ift∈σorInv(t)∈σ

false, otherwise δ(¬t, σ) =

(true, ift6∈σandInv(t)6∈σ false, otherwise

whereInv(t) =Inv(P)ji ift=P ij andInv(t) = Inv(P)_Self if t=P_Self for each P∈R_K.

These transitions extend similar ones in [4] to include the symbols in PS.

By a simple adaptation of the proofs in [3], it can be easily verified that, given a canonical modelIofK, its tree representationT_Iis accepted byA_K. Furthermore, a modelIT ofK can be constructed from any labeled treeT= (T, V)accepted byA_K. The domain∆^IT is given by the nodesxinTwithai ∈ V(x)for some individualai, and the nodes inTthat are reachable from any suchxthrough the roles. The extensions of concepts and roles are determined by the labels of the nodes inT. The only difference w.r.t. [4] is that we also add a pair(x, x)toP^ITfor everyxwhose label containsPSelf.

This shows that the automatonA_Kcan be used to decide the satisfiability ofK.

Theorem 2. Kis satisfiable iff the set of trees accepted byA_Kis nonempty.

Under unary encoding of numbers in restrictions, the number of states ofA_Kis polynomial in the size ofK. SinceΣKis single exponential in the size ofK, we can exploit the results of [16] to obtain an EXPTIMEupper bound for satisfiability ofALCQIb⁺_regKBs. This is optimal, as a matching lower bound is known for much weaker DLs [1].

Corollary 1. Satisfiability ofALCQIb⁺_regknowledge bases is EXPTIME-complete.

4 Reasoning in SRIQ

The description logicSRIQwas introduced in [7] as an extension of the DLRIQ[9], which in turn extends the well knownSHIQ[10] underlying OWL-Lite. It has gained considerable attention recently due to its close relationship to the logicSROIQunderlying the new OWL 2 standard. The most prominent feature ofSRIQare complex role inclusion axioms of the formR1◦. . .◦Rn⊑R. Additionally, in anSRIQKB it is possible to explicitly state certain properties of roles like transitivity, (ir)reflexivity and disjointness. Some of these additions increase the expressivity of the logic, while other are just ‘syntactic sugar’ and are intended to be useful for ontology engineering. We now recall the definition ofSRIQ.

Definition 8 (SRIQknowledge bases). LetR=R∪ {R⁻ |R ∈ R}. EachR ∈ Ris a (SRIQ) role. As usual,P⁻is the inverse ofP,Pthe inverse ofP⁻, andInv(R)denotes the inverse of the roleR. A (SRIQ) role inclusion axiom (SRIA) is an expression of the form R1◦. . .◦Rn⊑R, wheren≥1and{R1, . . . , Rn, R} ⊆R.

A setRhof SRIAs is regular if there exists a partial order≺onRsuch thatInv(R)≺R^′ iffR ≺R^′for everyR, R^′ ∈ R, and such that every SRIA inRhis of one of the following forms:

(i) R◦R⊑R, or (ii) Inv(R)⊑R, or

(iii) R1◦. . .◦Rn⊑RandRi≺Rfor each1≤i≤n, or (iv) R◦R1◦. . .◦Rn⊑RandRi≺Rfor each1≤i≤n, or

(v) R1◦. . .◦Rn◦R⊑RandRi≺Rfor each1≤i≤n.

Given a set Rh of SRIAs, letR_R

h denote the set of roles occurring inRh. The set of simple roles inRhis the minimal setSR(Rh)⊆ R_R

h containing each roleRsuch that (i) there are no SRIAs of the formR1◦. . .◦Rn⊑RinRh, and (ii) ifn= 1andR1∈SR(Rh) for every SRIA of the formR1◦. . .◦Rn⊑RinRh, thenR∈SR(Rh).

A role assertion is an expression of the formSym(R),Ref(R),Irr(R), orDis(R, R^′), where R, R^′ ∈R.³An RBoxRis a finite set of SRIAs and role assertions such thatR ∈SR(Rh) for each R occurring in a role assertion of the form Ref(R),Irr(R) or Dis(R, R^′)in R, whereRhdenotes the set of SRIAs inR. We say thatRis regular ifRhis regular, and define SR(R) =SR(Rh).

Similarly as inALCQIb⁺_reg, (SRIQ) conceptsC, C^′obey the following syntax C, C^′−→A| ¬C|C⊓C^′ |C⊔C^′| ∀R.C| ∃R.C|>n S.C|6n S.C| ∃S.Self, whereA∈ CandR, S ∈ R. A (SRIQ) concept inclusion axiom (SCIA) is an expression C⊑C^′for arbitrarySRIQconceptsCandC^′. A (SRIQ) TBox is a set of SCIAs. For an RBoxR, a conceptCisR-simple ifS∈SR(Rh)for eachSoccurring in a subconcept ofC of the form>n S.C^′,6n S.C^′or∃S.Self; a TBoxT isR-simple if all concepts occurring in the SCIAs ofT areR-simple.

An (SRIQ) assertion is an expressionC(a),R(a, b),¬S(a, b)ora 6= b, whereC is a SRIQconcept,S, RareSRIQroles anda, b ∈ I. An (SRIQ) ABox is a set ofSRIQ assertions. An ABoxAisR-simple for an RBoxRifS∈SR(Rh)for eachSoccurring in an assertion of the form¬S(a, b).

An (SRIQ) knowledge base is a tupleK = hA,T,RiwhereRis a regularSRIQ RBox,Aa non-emptyR-simpleSRIQABox andT anR-simpleSRIQTBox.

The semantics ofSRIQTBoxes and ABoxes is defined as forALCQIb⁺_reg. An interpre- tationI satisfies a role assertionSym(R),Ref(R)orIrr(R)ifR^I is a symmetric, reflexive or irreflexive relation, respectively;I satisfies Dis(R, R^′) if the relations are disjoint, i.e., R^I∩R^′I =∅;Isatisfies a SRIAR1◦. . .◦Rn⊑RifR₁^I◦. . .◦R^I_n⊆R^I, where again we override the symbol◦and use it to denote binary role composition. An interpretationI is a model of an RBox, in symbolsI |=R, if it satisfies all SRIAs and role assertions inR. Mod- elhood of a KB is restricted in the natural way to the models of the RBox, i.e.,I |=hA,T,Ri iffI |=A,I |=T andI |=R.

4.1 ReducingSRIQtoALCQIb⁺

reg

The restriction to regular RBoxes, which is crucial for the decidability ofSRIQ, ensures that all implications between roles can be captured by a set of regular expressions. This will allow us to reduce reasoning inSRIQto reasoning inALCQIb⁺_reg, and to show that the algorithm we have presented is also an optimal decision procedure forSRIQKB satisfiability.

IfRis a regular RBox, then for every roleR∈ Rthere is a regular expressionρRsuch that a wordw=R1◦. . .◦RnoverRis in the language ofρRiffw^I ⊆R^I in every model IofR. This was shown in [9] forRIQand its generalization toSRIQis straightforward.⁴ Proposition 4. Given a regular RBoxR, for eachR ∈ Rthere is a regular expressionρR

such thatR^I= (ρR)^Ifor every modelIofR.

Reducing TBoxes. Each regular expressionρR is anALCQIb⁺_reg role. Further, if a roleS is simple inR, thenρS is the regular expression S

(S^′⊑S)∈RS^′, which is in fact a simple ALCQIb⁺_reg role. As a consequence, if we replaceRbyρR is anySRIQ concept, the re- sulting expression is anALCQIb⁺_regconcept. This allows for a natural translation ofSRIQ concepts and TBoxes intoALCQIb⁺_reg.

3In [7] also role assertions of the formTra(R), asserting thatRis transitive, are allowed. We omit them since this can be equivalently expressed with an SRIAR◦R⊑R.

4It follows also from [7] and the equivalence of finite state automata and regular expressions.

Definition 9. LetRbe a regular RBox. For each R ∈ R,ρR denotes a (fixed, arbitrary) regular expression such thatR^I = (ρR)^Ifor every modelIofR. For anySRIQconceptC, we denote byC^R theALCQIb⁺regconcept that results from replacing each roleRinCwith ρR. Similarly, for anySRIQTBoxT, we defineT^R ={C^R⊑D^R|C⊑D∈ T }.

Proposition 5. LetRbe a regularSRIQRBox andIa model ofR. ThenR^I= (ρR)^Ifor everySRIQroleRandC^I = (C^R)^Ifor everySRIQconceptC, andI |=T iffI |=T^R. Reducing ABoxes. Now we consider ABoxes. First, we remove the negative role assertions, which are not allowed inALCQIb⁺_reg, as follows.

Definition 10. Given an SRIQ ABox A, let A^neg contain all negative role assertions

¬S(a, b), and letA^posbe obtained by replacing each assertion¬S(a, b)inA^negbyP¬S(a, b), whereP¬Sis a fresh role name. Further, letR^Abe the set ofALCQIb⁺_regBRIAs containing P¬S∩S⊑Bfor each¬S(a, b)inA.

If an ABox contains an assertion in which a non-simple roleRoccurs,ρR is not a sim- pleALCQIb⁺_reg role and hence we can not replaceR with it. Instead, we explicitly add as assertions all relations between individuals that are implied by the RBox.

Definition 11. Let Aand Rbe a SRIQ ABox and aSRIQ RBox respectively. TheR- extension ofA, denotedA^R, is the minimal set of assertions containingAsuch that, if there are individualsa1, . . . , an+1∈I_Kand a SRIAR1◦. . .◦Rn⊑RinRsuch thatRi(ai, ai+1)∈ A^Rfor each1≤i≤n, thenR(a1, an+1)∈ A^R.

Reducing RBoxes. As for the RBox, the satisfaction of the SRIAs is ensured by the regular expressions added to the TBox and the assertions added to the ABox. We only have to ensure the satisfaction of the role assertions, which is easy inALCQIb⁺_reg.

Definition 12. LetRbe a regularSRIQRBox and letR^′ denote the set of role assertions inR. We denote byR^Bthe set of BRIAs and CIAs that contains:

• R⊑R^′for every SRIAR⊑R^′inRsuch thatR^′∈SR(R),

• R⊑Inv(R)andInv(R)⊑Rfor every assertionSym(R)inR^′,

• ⊤ ⊑ ∃R.Selffor every assertionRef(R)inR^′,

• ∃R.Self⊑ ⊥for every assertionIrr(R)inR^′, and

• R∩R^′⊑Bfor every assertionDis(R, R^′)inR^′.

Reducing KBs. Now we can reduce any givenSRIQKBK=hA,T,Riinto an equivalent ALCQIb⁺_regKBK=hA,T ias follows.

Definition 13. LetK=hA,T,Ribe aSRIQknowledge base. We denote byK^′ =hA^′,T^′i the followingALCQIb⁺_regKB:

• The ABoxA^′isA^R, whereA = (A \ A^neg)∪ A^pos.

• The TBoxT^′isT^R∪ R^A∪ R^B.

For any interpretationI,I |=KimpliesI |=K^′. The converse holds in a slightly weaker form. While a model ofK^′need not be closed under the RBox and it may not be a model ofK, the models ofK^′andKmay only differ in the interpretation of some ‘implied’ roles. Adding them to a model ofK^′, we obtain a model ofK.

For a modelIofK^′, the extension ofItoRis the interpretationI^Rsuch that∆^IR =∆^I; (A)^IR = (A)^I for every atomic conceptA; and for every atomic roleP occurring inR, if (x, y) ∈ (ρP)^I or(y, x) ∈ (ρP⁻)^I, then(x, y) ∈ (P)^IR. It can be easily verified that if I |= K^′, then I^R |= K. Hence, we can reduce the satisfiability problem for anySRIQ knowledge bases to satisfiability of anALCQIb⁺regone.

Proposition 6. LetK=hA,T,Ribe aSRIQknowledge base. ThenKis satisfiable iffK^′ is satisfiable.

As a consequence, the automata algorithm in Section 3 can be used to decide the satisfia- bility ofSRIQknowledge bases.

Furthermore, it is worst case optimal. All steps are clearly polynomial in the size ofA,T andexp(R). However, the size ofexp(R)can be exponential in the size ofR[9].

Theorem 3. The satisfiability of anSRIQKBK=hA,T,Riis decidable in time exponen- tial in the combined sizes ofT,Aandexp(R), and double exponential in the size ofK.

It was recently shown thatSRIQis 2EXPTIMEhard [11], hence our bound is optimal.

Corollary 2. The satisfiability ofSRIQknowledge bases is 2EXPTIME-complete.

Note that the blow-up in complexity w.r.t.ALCQIb⁺_regis due to the size ofexp(R), and that the algorithm is single exponential wheneverexp(R)is polynomial inR; this includes, for example, the so-called simple role hierarchies defined in [9]. Our algorithm compares well to theSRIQalgorithm in [7], which may require non-deterministic double exponential time even for such restricted cases.

5 Conclusions

We have shown that the DLALCQIb⁺_reg fully captures the DLSRIQ. Although the latter is exponentially more succinct, reasoning is also (provably) exponentially harder in the worst case. The syntax ofALCQIb⁺_regallows for additional constructions not supported bySRIQ, like Boolean role expressions and inclusions between them. Since these additional features can be useful for modeling KBs and for ontology engineering [13], it seems reasonable to considerALCQIb⁺_regas a suitable alternative forSRIQ. It would even be possible to mod- ify the syntax ofALCQIb⁺_reg and make it more similar toSRIQfrom the perspective of ontology engineering. For example, instead of using the regular expressions in concepts as inALCQIb⁺_reg, a special set of defined role namesrR could be used, and assertions of the formρR ⊑ rR added to the KB. Provided that the defined rolesrR are not used as atomic roles in complex role expressions, reasoning could be done with the techniques described here. The resulting logic would be a very simple and natural extension ofSRIQand would avoid the effort of recognizing whether a given role hierarchy is regular and of transforming it into a set of finite automata or regular expressions. Note that the language of the regular expressions supported in (such a variation of)ALCQIb⁺_reg could also be extended to be ex- ponentially more succinct (e.g. by allowing squaring of expressions), resulting in a logic that may be comparable toSRIQin terms of succinctness.

In order forALCQIb⁺_regto become a suitable alternative toSRIQin practice, the main challenge that remains ahead is to explore alternatives for practicable algorithms for this kind of logics, and to implement them in actual reasoners. Unfortunately, DLs that support transi- tive closure, or more generally, regular expressions over roles in the style of PDL, have largely

been displaced by the better known alternative of role hierarchies on which implemented rea- soners have focused. While it is often claimed that the development systems that support such logics is problematic, to our knowledge there have been no major attempts to do it. Regular expressions are nevertheless worth exploring, as they are well understood and widely used in computer science, and are likely to be useful for ontology engineering.

Another natural alternative is to extendSRIQwith more constructors fromALCQIb⁺_reg. In particular, one can consider SRIQb, the extension of SRIQ with safe Boolean role expressions; similar extensions of closely related DLs like SHIQ andSROIQ were al- ready considered in [13]. It is easy to verify that anySRIQb KB can be transformed into an ALCQIb⁺_reg one with the reduction above, and the results given forSRIQ extend to SRIQbas well. Our results also show thatSRIQandSRIQbcan be easily extended with a universal role (which is already supported inSROIQ) as long as it is considered to be non-simple. Note that relaxing this restriction and allowing the universal role in the Boolean role expressions ofSRIQbmakes reasoning NEXPTIMEhard, see [12].

Finally, we point out that with the techniques given here, the query answering problem forSRIQcan be reduced toALCQIb⁺_reg, showing that answering two-way positive regular path queries as defined in [4] overSRIQandSRIQbknowledge bases is decidable.

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