Complexity of Subsumption in the EL Family of Description Logics: Acyclic and Cyclic TBoxes

Complexity of Subsumption in the EL Family of Description Logics: Acyclic and Cyclic TBoxes

Christoph Haase

and Carsten Lutz

Abstract. We perform an exhaustive study of the complexity of subsumption in theELfamily of lightweight description logics w.r.t.

acyclic and cyclic TBoxes. It turns out that there are interesting mem- bers of this family for which subsumption w.r.t. cyclic TBoxes is tractable, whereas it is EXPTIME-complete w.r.t. general TBoxes.

For other extensions that are intractable w.r.t. general TBoxes, we establish intractability already for acyclic and cyclic TBoxes.

1 MOTIVATION

Description logics (DLs) are a popular family of KR languages that can be used for the formulation of and reasoning about ontolo- gies [5]. Traditionally, the DL research community has strived for identifying more and more expressive DLs for which reasoning is still decidable. In recent years, however, there have been two lines of development that have led to significant popularity also of DLs with limited expressive power. First, a number of novel and useful lightweight DLs with tractable reasoning problems has been iden- tified, see e.g. [3, 8]. And second, many large-scale ontologies that are formulated in such lightweight DLs have emerged from practical applications. Prominent examples include the Systematized Nomen- clature of Medicine, Clinical Terms (SNOMED CT), which under- lies the systematized medical terminology used in the health systems of the US, the UK, and other countries [19]; and the gene ontology (GO), which aims at consistent descriptions of gene products in dif- ferent databases [20].

In this paper, we are concerned with theELfamily of lightweight DLs, which consists of the basic DLELand its extensions. Mem- bers of this family underly many large-scale ontologies including SNOMED CT and GO. The DL counterpart of an ontology is called a TBox, and the most important reasoning task in DLs is subsumption.

In particular, computing subsumption allows to classify the concepts defined in the TBox/ontology according to their generality [5]. In the DL literature, different kinds of TBoxes have been considered.

In decreasing order of expressive power, the most common ones are general TBoxes, (potentially) cyclic TBoxes, and acyclic TBoxes.

For the EL family, the complexity of subsumption w.r.t. general TBoxes has exhaustively been analyzed in [3] and its recent succes- sor [4]. In all of the considered cases, subsumption is either tractable or EXPTIME-complete. However, the study of general TBoxes does not reflect common practice of ontology design, as most ontologies from practical applications correspond to cyclic or acyclic TBoxes.

For example, SNOMED CT and GO both correspond to so-called acyclicTBoxes. Since cyclic and acyclic TBoxes are often prefer- able in terms of computational complexity [7, 14], the question arises

1University of Oxford, UK, christoph.haase@comlab.ox.ac.uk

2TU Dresden, Germany, lutz@tcs.inf.tu-dresden.de

whether there are useful extensions ofELfor which reasoning w.r.t.

such TBoxes is computationally cheaper than reasoning w.r.t. general TBoxes.

The goal of the current paper isto analyse the computational com- plexity of subsumption in theELfamily of description logics w.r.t.

acyclic TBoxes and cyclic TBoxes, with a special emphasis on the border of tractability. In our analysis, we omit extensions ofEL for which tractability w.r.t. general TBoxes has already been estab- lished. Our results exhibit a more varied complexity landscape than in the case of general TBoxes: we identify cases in which reason- ing is tractable, co-NP-complete, PSPACE-complete, and EXPTIME- complete. Notably, we identify two maximal extensions ofELfor which subsumption w.r.t. cyclic TBoxes is tractable, whereas it is EXPTIME-complete w.r.t. general TBoxes. In particular, these exten- sions include primitive negation and at-least restrictions. They also include concrete domains, but fortunately do not require the strong convexity condition that was needed in the case of general TBoxes to guarantee tractability [3]. For other extensions ofELsuch as inverse roles and functional roles, we show intractability results already w.r.t.

acyclic TBoxes. Compared to the case of general TBoxes, it is often necessary to develop new approaches to lower bound proofs. We also show that the union of the two identified tractable fragments is not tractable. Detailed proofs are provided in [10].

2 DESCRIPTION LOGICS

The two types of expressions in a DL areconceptsandroles, which are built inductively starting from infinite setsNCandNRofconcept namesandrole names, and applyingconcept constructorsandrole constructors. The basic description logicELprovides the concept constructors top (>), conjunction (CuD) and existential restriction (∃r.C), and no role constructors. Here and in what follows, we de- note the elements ofNCwithAandB, the elements ofNRwithr ands, and concepts withCandD. The semantics of concepts and roles is given in terms of aninterpretationI= (∆^I,·^I), with∆^Ia non-empty set called thedomainand·^I theinterpretation function, which maps everyA∈NCto a subsetA^Iof∆^Iand every role name rto binary relationr^Iof over∆^I.

Extensions ofELare characterized by the additional concept and role constructors that they offer. Figure 1 lists all relevant construc- tors, concept constructors in the upper part and role constructors in the lower part. The left column gives the syntax, and the right col- umn shows how to inductively extend interpretations to composite concepts and roles. In the presence of role constructors, composite roles can be used inside existential restrictions. Inatleast restrictions (≥n r) andatmost restrictions(≤n r) , we usento denote a non- negative integer. The concrete domain constructorp(f1, . . . , fk)de-

Syntax Semantics

> ∆^I

¬C ∆^I\C^I

CuD C^I∩D^I

CtD C^I∪D^I

(≤n r) {x|#{y|(x, y)∈r^I} ≤n}

(≥n r) {x|#{y|(x, y)∈r^I} ≥n}

∃r.C {x| ∃y: (x, y)∈r^I∧y∈C^I}

∀r.C {x| ∀y: (x, y)∈r^I→y∈C^I} p(f1, . . . , fk) {x| ∃d1, . . . , dk:f₁^I(x) =d1∧. . .∧

f_k^I(x) =dk∧(d1, . . . , dk)∈p^D}

r∩s r^I∩s^I

r∪s r^I∪s^I

r⁻ {(x, y)|(y, x)∈r^I}

r⁺ S

i>0(r^I)ⁱ

Figure 1. Syntax and semantics of concept and role constructors.

serves further explanation, to be given below. To denote extensions ofEL, we use the symbol of the added constructors in superscript.

For example, EL^t,∪,−denotes the extension of ELwithconcept disjunction(CtD),role disjunction(r∪s), andinverse roles(r⁻).

The concrete domain constructor permits reference to concrete data objects such as strings and integers. It provides the interface to aconcrete domainD= (∆D,ΦD), which consists of a domain

∆Dand a set of predicatesΦD[13]. Eachp∈ΦDis associated with a fixed aritynand a fixed extensionp^D ⊆∆ⁿD. In the presence of a concrete domainD, we assume that there is an infinite setNF of feature namesdisjoint fromNRandNC. In Figure 1 and in general, f1, . . . , fkare fromNFandp∈ΦD. An interpretationImaps every f ∈NFto a partial functionf^Ifrom∆^I to∆D. We useEL(D)to denote the extension ofELwith the concrete domainD.

In this paper, aTBoxT is a finite set ofconcept definitionsA≡C, whereA∈NCandCis a concept. We require that the left-hand side of all concept definitions in a TBox are unique. A concept nameA∈ NCisdefinedif it occurs on the left-hand side of a concept definition inT, andprimitiveotherwise. A TBoxT isacyclicif there are no concept definitionsA1 ≡C1, . . . , Ak ≡ Ck ∈ T such thatAi+1

occurs inCifor1≤i≤k, whereAk+1:=A1. An interpretationI is amodelofT iffA^I =C^Ifor allA≡C∈ T.

The main reasoning task considered in this paper is subsumption.

A conceptCissubsumedby a conceptDw.r.t. a TBoxT, written T |= C v D, ifC^I ⊆D^I for all modelsIofT. IfT is empty or missing, we simply writeC vD. Sometimes, we also consider satisfiability of concepts. A conceptCissatisfiablew.r.t. a TBoxT if there is a model ofT such thatC^I6=∅. For many extensions ofEL, satisfiability is trivial because there are no unsatisfiable concepts.

3 TRACTABLE EXTENSIONS

We identify two extensions of EL for which subsumption w.r.t.

TBoxes is tractable:EL^∪,(¬)(D)andEL^≥,∪. This should be con- trasted with the results in [3] which imply that subsumption w.r.t.

general TBoxes is EXPTIME-complete in both extensions. In Sec- tion 4.1, we show that taking the union of the two extensions results in intractability already w.r.t. acyclic TBoxes.

(C1) LT(B)⊆LT(A)

(C2) For each∃rB.B⁰∈ET(B)there is∃rA.A⁰∈ET(A) such thatrA⊆rBand(A⁰, B⁰)∈S

(C3) ConD(A)impliesConD(B)

Figure 2. EL^∪,(¬)(D): Conditions for adding(A, B)toS.

3.1 Role Disjunction, Primitive Negation, and Concrete Domains

We show that subsumption inEL^∪,(¬)(D)w.r.t. (acyclic and cyclic) TBoxes is tractable. The superscript·^(¬)indicatesprimitivenegation, i.e., negation can only be applied to concept names. The following is an example of anEL^∪,(¬)(D)-TBox, wherehas ageis a feature, and

≥13and≤19are unary predicates of the concrete domainD:

Parent ≡ Humanu ∃(has child∪has adopted).>

Mother ≡ ParentuFemaleu ¬Male

Teenager ≡ Humanu ≥13(has age)u ≤19(has age) To guarantee tractability, we require the concrete domainDto sat- isfy a standard condition. Namely, we requireDto bep-admissibile, i.e., satisfiability of and implication betweenconcrete domain ex- pressionsof the formp1(v₁¹, . . . , v_n¹₁)∧ · · · ∧pm(v₁^m, . . . , v^m_nm)are decidable in polynomial time, where thevⁱjare variables that range over∆D. In [3], it is shown that a much stronger condition is re- quired to achieve tractability inEL(D)with general TBoxes. This condition isconvexity, which requires that if a concrete domain atom p(v1, . . . , vn)implies a disjunction of such atoms, then it implies one of the disjuncts. For our result, there is no need to impose con- vexity.

When deciding subsumption, we only consider concept names in- stead of composite concepts. This is sufficient sinceT |=C v D iffT⁰|=AvB, whereT⁰:=T ∪ {A≡C, B≡D}andAandB do not occur inT.

The subsumption algorithm requires the input TBoxT to be in the followingnormal form. In eachA≡C∈ T,Cis of the form

1≤i≤k

u

u

1≤i≤`∃ri.Biu

u

1≤i≤mpi(f1ⁱ, . . . , fnⁱ_i) where theLiare primitive literals, i.e., possibly negated primitive concept names; theriare of the formr1∪. . .∪rn; and theBiare defined concept names. In the following, we refer to the set of literals occurring inC withLT(A), to the set of existential restrictions as ET(A), and define the following concrete domain expression, which for simplicity uses features as variables:

ConD(A) :=p1(f₁¹, . . . , f_n¹₁)∧ · · · ∧pm(f₁^m, . . . , f_n^m_m).

To ease notation, we confuse a roleri =r1∪. . .∪rnwith the set {r1, . . . , rn}.

It is easy to see how to adapt the algorithm given in [2] to convert anEL^∪,(¬)(D)-TBox into normal form in quadratic time. During the normalization, we check for unsatisfiable concepts. This is easy since a defined concept nameAwithA≡C∈ T is unsatisfiable w.r.t.T iff one of the following three conditions holds: (i) there is a primitive conceptPwith{P,¬P} ∈LT(A); (ii)ConD(A)is unsatisfiable;

or (iii) there is an∃r.B∈ET(A)withBunsatisfiable.

Suppose we want to decide whetherAis subsumed byBw.r.t. a TBoxT in normal form. IfAis unsatisfiable, the algorithm answers

(C1) PT(B)⊆PT(A)

(C2) For each∃rB.B⁰ ∈ET(B)there is∃rA.A⁰ ∈ET(A) such thatrA⊆rBand(A⁰, B⁰)∈S

(C3) For each(≥m r)∈NT(B), there is(≥n r)∈NT(A) such thatn≥m.

Figure 3. EL^≥,∪: Conditions for adding(A, B)toS.

“yes”. Otherwise and ifBis unsatisfiable, it answers “no”. IfAand Bare both satisfiable, it computes a binary relationSon the defined concept names ofT. The relationS is initialized with the identity relation and then completed by exhaustively adding pairs(A, B)for which the conditions in Figure 2 are satisfied.

It is easily seen that the algorithm runs in time polynomial w.r.t.

the size of the input TBox. LetS0, . . . , Snbe the sequence of rela- tions that it produces. To show soundness, it suffices to prove that if (A, B)∈Si,i≤n, thenT |=AvB. This is straightforward by induction oni. To prove completeness, we have to exhibit a model IofT withA^I \B^I 6=∅. Such a model is constructed in a two- step process. First, we start with an instance ofA, and then “apply”

the concept definitions in the TBox as implications from left to right, constructing a potentially infinite, tree-shaped interpretation. In the second step, we apply the concept definitions from right to left, fill- ing up the interpretation of defined concepts. Both steps involve some careful bookkeeping which ensures that the constructed instance of Ais not an instance ofB.

Theorem 1 Subsumption inEL^∪,(¬)(D)w.r.t. TBoxes is inPTIME. This result still holds if we additionally allow role conjunction (r∩s) and require that composite roles are in disjunctive normal form (with- out DNF, subsumption becomes co-NP-hard).It is worth mentioning that, in the presence of general TBoxes, extendingELwith each sin- gle one of (i) primitive negation, (ii) role disjunction, and (iii)any non-convex concrete domain results in EXPTIME-hardness [3]. Note that convexity of a concrete domain is a rather strong restriction, and it is pleasant that we do not need it to achieve tractability. We point out that it should be possible to enhance the expressive power of EL^∪,(¬)(D)by enriching it with additional constructors of the DL EL⁺⁺[3]. Examples include nominals and transitive roles.

3.2 Role Disjunction and At-Least Restrictions

In EL^≥,∪, we allow role disjunction only in existential restric- tions, but not in number restrictions. To show that subsumption w.r.t.

TBoxes is tractable, we use a variation of the algorithm in the previ- ous section. In the following, we only list the differences. A TBox is innormal formif, in eachA≡C∈ T,Cis of the form

1≤i≤k

u

u

1≤i≤`∃ri.Biu

u

1≤i≤m(≥nisi)

where the Pi are primitive concept names, theri are of the form r1∪. . .∪rn, theBiare defined concept names, and thesiare role names. We usePT(A)to refer to the set of primitive concept names occurring inC,ET(A)is as in the previous section, andNT(A)is the set of number restrictions inC. The conditions for adding a pair (A, B)to the relationSare given in Figure 3.

Theorem 2 Subsumption inEL^≥,∪w.r.t. TBoxes is inPTIME.

In the extension ofELwith only at-least restrictions(≥n r), sub- sumption w.r.t. general TBoxes is EXPTIME-complete [3]. As we will show in Section 4.3, EL extended with at-most restrictions (≤n r)is intractable already w.r.t. acyclic TBoxes.

4 INTRACTABLE EXTENSIONS

We identify extensions ofELfor which subsumption is intractable w.r.t. acyclic and cyclic TBoxes.

4.1 Primitive Negation and At-Least Restrictions

We show that taking the union of the DLsEL^∪,(¬)(D)andEL^≥,∪

from Sections 3.1 and 3.2 results in intractability. To this end, we considerEL^≥,(¬)and show that subsumption w.r.t. the empty TBox is CO-NP-complete. It is easy to establish the lower bound also forEL^≥(D) as long as there are two conceptsp(f1, . . . , fn) and p⁰(f1⁰, . . . , fm⁰ )that are mutually exclusive. This is the case for most practically useful concrete domainsD.

For the lower bound, we reduce 3-colorability of graphs to non- subsumption. Given an undirected graphG = (V, E), reserve one concept namePvfor each nodev ∈ V, and a single role namer.

Then,Gis 3-colorable iffCG6v(≥4r), where CG:=

u

v∈V∃r.

„

Pvu

u

{v,w}∈E¬Pw

«

Intuitively, if d ∈ C_G^I \(≥4r)^I, then d has at most three r- successors, each describing one of the three colors. The use of primi- tive negation inCGensures that no two adjacent nodes have the same color.

A matching upper bound can be derived from theCO-NP-upper bound for subsumption inALU N, which has the concept construc- tors top, bottom (⊥), value restriction (∀r.C), conjunction, disjunc- tion, primitive negation, number restrictions, and unqualified exis- tential restriction [11]. Given twoEL^≥,(¬)-conceptsC, D, we have CvDiff¬D v ¬C. It remains to observe that bringing¬Cand

¬Dinto negation normal form yields twoALU N-concepts.

Theorem 3 Subsumption inEL^≥,(¬)isCO-NP-complete.

4.2 Inverse Roles

In [1], it is shown that subsumption w.r.t. the empty TBox is tractable in (an extension of)EL⁻. We prove that, w.r.t. acyclic TBoxes, sub- sumption inEL⁻is PSPACE-complete. Since the upper bound fol- lows from PSPACE-completeness of subsumption inALCI [5], we concentrate on the lower bound.

We reduce validity of quantified Boolean formulas (QBFs). Let ϕ=Q1v1· · ·Qkvk.ψbe a QBF, whereQi∈ {∀,∃}for1≤i≤k.

W.l.o.g., we may assume thatψ =c1∧ · · · ∧cnis in conjunctive normal form. We construct an acyclic TBoxTϕand select two con- cept namesL0 and E0 such thatϕis valid iffTϕ |= L0 v E0. Intuitively, a model ofL0andTϕ is a binary tree of depthkthat is used to evaluateϕ. In the tree, a transition from a node at levelito its left successor corresponds to settingvi+1to false, and a transition to the right successor corresponds to settingvi+1to true. Thus, each node on levelicorresponds to a truth assignment to the variables v1, . . . , vi. InTϕ, we use a single role namerand the following con- cept names:

• L0, . . . , Lkrepresent the level of nodes in the tree model;

• Ci,j,1≤i≤nand1≤j≤k, represents truth of the clauseci

on leveljof the tree model;

• E0, . . . , Ekare used for evaluatingψ, and the index again refers to the level.

For1≤i≤k, we usePjto denote the conjunction of all concept namesCi,j,1≤i≤n, such thatvjoccurs positively inci; similarly, Njdenotes the conjunction of all concept namesCi,j,1≤i≤n, such thatvjoccurs negatively inci. Now, the TBoxTϕis as follows:

L0 ≡ ∃r.(L1uP1)u ∃r.(L1uN1)

· · ·

Lk−1 ≡ ∃r.(LkuPk)u ∃r.(LkuNk)

Ci,j ≡ ∃r⁻.Ci,j−1for1≤i≤nand1< j≤k Ek ≡ C1,ku · · · uCn,k

Ei ≡ ∃r.Ei+1for0≤i < kwhereQi+1=∃ Ei ≡ ∃r.(Pi+1uEi+1)u ∃r.(Ni+1uEi+1)

for0≤i < kwhereQi+1=∀ The definitions forL0, . . . , Lk−1build up the tree. The use ofP1and N1 in these definitions together with the definition ofCi,j sets the truth value of the clauseciaccording to a partial truth assignment of lengthj. Finally, the definitions ofE0, . . . , Ekevaluateϕaccording to its matrix formulaψand quantifier prefix. It can be checked that ϕis valid iffTϕ|=L0vE0.

Theorem 4 Subsumption inEL⁻w.r.t. acyclic TBoxes isPSPACE- complete.

We leave the case of cyclic TBoxes as an open problem. In this case, the lower bound from Theorem 4 is complemented only by the EXPTIMEupper bound for subsumption inEL⁻w.r.t. general TBoxes from [3].

4.3 Functional Roles

LetEL^fbeELextended with functional roles, i.e., there is a count- ably infinite subsetNF ⊆ NRsuch that all elements ofNF are in- terpreted as partial functions. It is shown in [3] that subsumption in EL^f w.r.t. general TBoxes is EXPTIME-complete. We show that it is co-NP-complete w.r.t. acyclic TBoxes and PSPACE-complete w.r.t.

cyclic ones.

We use EL^F to denote the variation of EL^f in whichall role names are interpreted as partial functions. It has been observed in [3]

that there is a close connection betweenEL^F andF L0, which pro- vides the concept constructors conjunction and value restriction. It is easy to exploit this connection to transfer the known co-NP-hardness (PSPACE-hardness) from subsumption inF L0w.r.t. acyclic (cyclic) TBoxes as proved in [16, 12] toEL^F. We omit details for brevity.

Since the described approach is not very illuminating regarding the source of intractability, however, we give a dedicated proof of co- NP-hardness of subsumption inEL^F w.r.t. acyclic TBoxes using a reduction from 3-SAT tonon-subsumption.

Letϕ=c1∧. . .∧ckbe a 3-formula in the propositional variables p1, . . . , pnand withcj=`<sup>j</sup><sub>1</sub>∨`^j₂∨`^j₃for1≤j≤k. We construct a TBoxTϕand select concept namesAϕandB1such thatϕis sat- isfiable iffTϕ6|=AϕvB1. In the reduction, we use two role names r0andr1to represent falsity and truth of variables. More precisely, a pathrv₁· · ·rv_nwithrv_i ∈ {r0, r1}corresponds to the valuation pi 7→ vi,1 ≤ i ≤ n. Additionally, we use a number of auxiliary

concept names. The TBoxTϕis as follows:

A^j_i ≡ 8

<

:

∃r0.A^j_i+1 ifpi∈ {`<sup>j</sup><sub>1</sub>, `^j₂, `^j₃}

∃r1.A^j_i+1 if¬pi∈ {`<sup>j</sup><sub>1</sub>, `^j₂, `^j₃}

∃r0.A^j_i+1u ∃r1.A^j_i+1 otherwise A^j_n+1 ≡ >

Aϕ ≡

u

1≤j≤kA^j₁

Bi ≡ ∃r0.Bi+1u ∃r1.Bi+1 Bn+1≡ >

IfIis a model ofTϕandd∈(A^j₁)^I,1≤j≤k, thendis the root of a tree inIwhose edges are labelled withr0andr1and whose paths are the valuations that make the clausecjfalse. Due to functionality ofr0andr1, eachd ∈A^I_ϕis thus the root of a (single) tree whose paths are precisely the valuations that makeanyclause inϕfalse.

Finally,d∈B₁^Imeans thatdis the root of a full binary tree of depth nwhose paths describeallvaluations. It follows thatϕis satisfiable iffTϕ6|=AϕvB1.

To prove matching upper bounds forEL^f, we exploit the fact that, due to theF L0-connection, subsumption inEL^F is easily shown to be inCO-NP w.r.t. acyclic TBoxes and in PSPACEw.r.t. cyclic ones.

We give an algorithm for subsumption inEL^f that uses subsump- tion inEL^F as a subprocedure. Like the algorithms in Section 3, it computes a binary relationS on the set of defined concept names by repeatedly adding pairs(A, B)such that the input TBox entails A v B. The algorithm works for both acyclic and cyclic TBoxes, giving us the desired upper bound in both cases.

We assume the input TBoxT to be in the same normal form as described in Section 3.2, but without concepts of the form(≥n r).

LetS be a binary relation on the defined concept names inT. For every concept∃r.Aoccurring inT withr /∈ NF, introduce a fresh concept nameXr,Asuch thatXr,A=Xr⁰,A⁰iffr=r⁰,(A, A⁰)∈ S, and(A⁰, A)∈S. Now let theEL^F-TBoxTSbe obtained fromT by (i) replacing every concept∃r.Awherer /∈NF withXr,A, and (ii) for each∃r.AinT withr /∈NF, adding the concept definition

Xr.A≡Xr,B₁u · · · uXr,B_nuZr,A

whereB1, . . . , Bn are all concept names with(A, Bi) ∈ S and (Bi, A)∈/S; andZr,Ais a fresh concept name. The algorithm starts withSas the identity relation and then exhaustively performs the fol- lowing step: add(A, B)toSifTS|=AvB. It returns “yes” if the input concepts form a pair inS, and “no” otherwise. Additionally, we can show that subsumption inEL^f without TBoxes is in PTIME

by a reduction to subsumption inEL.

Theorem 5 Subsumption in EL^f is in PTIME, CO-NP-complete w.r.t. acyclic TBoxes andPSPACE-complete w.r.t. cyclic TBoxes.

It is not hard to see that the lower bounds carry over toEL^≤.

4.4 Booleans

We consider extensions ofELwith Boolean constructors, starting with negation. SinceEL^¬is a notational variant ofALC, we obtain the following from the results in [17, 18].

Theorem 6 Satisfiability and subsumption in EL^¬ is PSPACE- complete without TBoxes and w.r.t. acyclic TBoxes, andEXPTIME- complete w.r.t. cyclic TBoxes.

Now for disjunction. It has been shown in [6] that subsumption in EL^tisCO-NP-complete without TBoxes. In order to establish lower

bounds for subsumption w.r.t. TBoxes, we reduce satisfiabilityin EL^¬ to non-subsumption in EL^t. AnEL^¬-TBoxT is in normal formif for eachA≡C∈ T,Cis of the form>,P,¬B,∃r.B, or B1uB2withP primitive andB, B1, B2 defined. It is straightfor- ward to show that anyEL^¬-TBoxT can be transformed into normal form in linear time such that all (non-)subsumptions are preserved.

Thus, letT = {A1 ≡ C1, . . . , An ≡ Cn}be anEL^¬-TBox in normal form. Since the proofs underlying Theorem 6 use only a sin- gle role name, we may assume w.l.o.g. thatT contains only a sin- gle role namer. We convertT into anEL^t-TBoxT⁰by introduc- ing fresh concept namesA1, . . . , An representing the negations of A1, . . . , Anand replacing everyA≡ ¬Aj ∈ T withA≡Ajand everyAi≡ ∃r.Aj∈ T with

Ai≡ ∃r.(Aju

u

1≤k≤n(AktAk)).

The additional conjunct ensures thatAiandAicover the domain. To additionally ensure that they are disjoint, we add toT⁰the concept definition

M ≡

t

t

| {z }

itimes

(AjuAj) ifT is acyclic M ≡ ∃r.Mt

t

In both cases,M is a fresh concept name. Then a defined concept nameAis satisfiable w.r.t.T iffT⁰6|=Au

u

1≤i≤n(AitAi)vM.

We obtain the following result.

Theorem 7 Subsumption inEL^tisPSPACE-complete w.r.t. acyclic TBoxes andEXPTIME-complete w.r.t. cyclic TBoxes.

4.5 Transitive Closure

We considerEL⁺, the extension of ELwith transitive closure of roles. Using a result by Miklau and Suciu on query containment in a fragment of XPath [15], it is easy to show that subsumption inEL⁺is co-NP-complete. By reusing the techniques from Miklau and Suciu’s lower bound proof, we can establish PSPACE-hardness (EXPTIME-hardness) of subsumption inEL⁺w.r.t. acyclic (cyclic) TBoxes. More precisely, this is achieved by a reduction of satisfiabil- ity inEL^¬to non-subsumption inEL⁺, similar to the one described in Section 4.4. A corresponding EXPTIMEupper bound for the case of cyclic TBoxes is obtained by a straightforward reduction to satis- fiability in propositional dynamic logic (PDL). For acyclic TBoxes, we obtain a PSPACEupper bound by a less straightforward reduction to subsumption inEL^tw.r.t. acyclic TBoxes, c.f. Theorem 7.

Theorem 8 Subsumption in EL⁺ is CO-NP-complete, PSPACE- complete w.r.t. acyclic TBoxes andEXPTIME-complete w.r.t. cyclic ones.

5 CONCLUSION

The complexity landscape for acyclic/cyclic TBoxes is much less uniform than for general TBoxes. For the case of general TBoxes, non-existence of a unique minimal model of a TBox (in the sense that it can be homomorphically embedded into any other model) was a sufficient (but not necessary) condition for intractability. This is not the case here: inEL^∪,(¬)(D)andEL^≥,∪, such models do not exist.

It is also interesting to note that we did not find a single case in which subsumption is tractable w.r.t. acyclic TBoxes, but intractable w.r.t.

cyclic ones.

ELwith no TBox acyclic cyclic

¬C PSPACE PSPACE EXPTIME

¬A PTIME PTIME PTIME

CtD CO-NP PSPACE EXPTIME

functionality PTIME CO-NP PSPACE

(≥n r) PTIME PTIME PTIME

p(f1, . . . , fk) PTIME PTIME PTIME

r∩s PTIME PTIME PTIME

r∪s PTIME PTIME PTIME

r⁻ PTIME PSPACE EXPTIME

r⁺ CO-NP PSPACE EXPTIME

Figure 4. Complexity of subsumption in extensions ofEL. Light gray cell background indicates membership in class, dark gray completeness for class.

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