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Proceedings of the Twenty-Second International Joint Conference on

On the Complexity of Dealing with Inconsistency in Description

Riccardo Rosati Dipartimento di Informatica e Sistemistica Sapienza Universita` di Roma, Italy

Abstract ful conclusions from inconsistent ontologies, and can be the formal basis for an automated treatment of inconsistency. We study the problem of dealing with inconsistency The inconsistency-tolerant for DL ontologies in Description Logic (DL) ontologies. We consider that we consider are based on repairing (i.e., modifying) the inconsistency-tolerant semantics recently proposed extensional knowledge (i.e., the ABox A) while keeping the in the literature, called AR-semantics and CAR- intensional knowledge (i.e., the TBox T ) untouched. Specifi- semantics, which are based on repairing (i.e., modi- cally, such semantics are based on the notion of ABox repair. fying) in a minimal way the extensional knowledge Intuitively, given a DL KB K = T , A, a repair AR for K (ABox) while keeping the intensional knowledge is an ABox such that the KB T , AR is satisfiable under the (TBox) untouched. We study instance checking classical semantics, and AR “minimally” differs from A. No- and conjunctive query entailment under the above tice that in general not a single but several repairs may exist, inconsistency-tolerant semantics for a wide spec- depending on the particular minimality criteria adopted. trum of DLs, ranging from tractable ones (EL)to SHIQ In particular, we consider inconsistency-tolerant seman- very expressive ones ( ), showing that rea- tics recently proposed in [Lembo et al., 2010], called AR- soning under the above semantics is inherently in- semantics and CAR-semantics, for which reasoning has been tractable, even for very simple DLs. To the aim of studied in the context of the Description of the DL-Lite overcoming such a high computational complexity family. The notion of ABox repair in the AR-semantics is a of reasoning, we study sound approximations of the very simple and natural one: a repair is a maximal subset above semantics. Surprisingly, our computational of the ABox that is consistent with the TBox. The CAR- analysis shows that reasoning under the approx- semantics is a variant of the AR-semantics that is based on a imated semantics is intractable even for tractable notion of “equivalence under consistency” of ABoxes incon- DLs. Finally, we identify suitable language restric- sistent with respect to a given TBox. tions of such DLs allowing for tractable reasoning We study the main forms of extensional reasoning in DL under inconsistency-tolerant semantics. ontologies (instance checking and conjunctive query entail- ment) under the above inconsistency-tolerant semantics for a wide spectrum of DLs, ranging from tractable ones (EL)to 1 Introduction very expressive ones (SHIQ). Our results confirm the results In this paper we study the problem of dealing with inconsis- obtained in [Lembo et al., 2010] for DL-Lite and show that tency in Description Logic (DL) ontologies. This problem reasoning under the above semantics is inherently intractable, is becoming of increasing importance in practical applica- even for very simple DLs. So, to the aim of overcoming the tions of ontologies. In fact, the size of ontologies used by high computational complexity of reasoning, we also con- real applications is scaling up, and ontologies are increasingly sider sound approximations of the previous semantics. In merged and integrated into larger ontologies: the probabil- particular, we focus on semantics (called IAR-semantics and ity of introducing inconsistency in these activities is conse- ICAR-semantics in [Lembo et al., 2010]) that consider as a quently getting higher, and dealing with inconsistency is be- repair, in each of the previous semantics, the ABox corre- coming a practical issue in -based systems (see e.g. sponding to the intersection of all the repairs. Surprisingly, [Hogan et al., 2010]). our computational analysis shows that, differently from the A traditional approach (analogous to data cleaning in case of DL-Lite, reasoning under the approximated semantics ) is to manually (or semi-automatically) repair the is in general intractable even for tractable DLs. inconsistency, by suitably modifying the ontology. This Finally, we study tractable cases, i.e., suitable restric- might be not be practical or feasible in many contexts, e.g., tions of such DLs that allow for tractable reasoning un- when dealing with (very) large ontologies, or integrating der inconsistency-tolerant semantics. In particular, we iden- different ontologies. An alternative approach is to define tify a fragment of the DL EL⊥ for which reasoning under inconsistency-tolerant semantics, which, differently from the an inconsistency-tolerant semantics (in particular, the IAR- classical FOL semantics of DLs, are able to derive meaning- semantics) is computationally not harder than reasoning un-

1057 DL and role expressions TBox and P (z,z), where A ia a concept name, P is a role name C A |⊥|C  C |∃P C z z EL ::= 1 2 . C  C and , are terms. Instance checking (IC) is a restricted form ⊥ R ::= P 1 2 of UCQ entailment, corresponding to the case when the UCQ C ::= A | C1  C2 |¬C |∃P .C is an ABox assertion (i.e., a ground atom). Notice that all ALC C1  C2 R ::= P the results we achieve about Boolean UCQ entailment can be C  C C A |¬C | C  C | ≥ nRC 1 2 easily extended in the standard way to the presence of free SHIQ ::= 1 2 ( ) R  R R P | P − 1 2 [ ] ::= R variables in queries (see e.g. Glimm et al., 2008 ). Trans( ) In this paper, we will consider data complexity (i.e., the Figure 1: Abstract of the DLs studied in the paper. complexity with respect to the size of the ABox) and com- bined complexity (i.e., the complexity with respect to the size of the whole input) of UCQ entailment and instance der the standard DL semantics. checking. Besides the usual complexity classes NP, coNP, p Π2,EXPTIME,2-EXPTIME, we will mention the following 2 Preliminaries classes: • p O n p O n The DLs mainly considered in this paper are the following: Δ2[ (log )] (resp., Δ3[ (log )]) is the class of prob- lems that can be solved in polynomial time using •EL p [Baader et al., 2005], a prominent tractable DL. O(log n) calls to an NP-oracle (resp., a Σ -oracle) (see EL EL 2 Here, we consider ⊥, a slight extension of al- e.g. [Gottlob, 1995]); ⊥ lowing for the empty concept (and hence for incon- • DP is the class of problems corresponding to the con- p p sistency in KBs); junction of a problem in Σ and a problem in Π .An •ALC 2 2 , a very well-known DL which corresponds to mul- example of a problem complete for this class is 3-CNF- K [ ] timodal logic n Baader et al., 2003 ; SAT+UNSAT: given two 3-CNF formulas φ1,φ2, decide •SHIQ[ ] Glimm et al., 2008 , a very expressive DL whether φ1 is satisfiable and φ2 is unsatisfiable. Notice which constitutes the basis of the OWL family of DLs that DP = BH2(NP), i.e., DP is the second level of the adopted as standard languages for ontology specification Boolean hierarchy over NP; p in the . • BH2(Σ ) is the class of problems corresponding to the 2 p p In every DL, concept expressions and role expressions can conjunction of a problem in Σ2 and a problem in Π2. be constructed starting from concept and role names. Such An example of a problem complete for this class is 2- expressions are used to define axioms. Typical axioms are QBF-SAT+UNSAT: given two 2-QBF formulas φ1,φ2, concept inclusions, role inclusions, and instance assertions. decide whether φ1 is not valid and φ2 is valid. C  C A concept inclusion is an expression of the form 1 2, Finally, the following table recalls known results on the C C where 1 and 2 are concept expressions. Similarly, a role complexity of instance checking and UCQ entailment under R  R R inclusion is an expression of the form 1 2, where 1 standard semantics in the DLs considered in this paper (all R and 2 are role expressions. An instance (ABox) assertion is problems are complete w.r.t. the class reported). an expression of the form A(a) or P (a, b), where A is a con- cept name, P is a role name, and a, b are constant (individual) DL (problem) data complexity combined complexity names. We do not consider complex concept and role expres- EL, EL⊥ (IC) PTIME PTIME sions in instance assertions. A DL (KB) is a EL, EL⊥ (UCQ) PTIME NP pair T , A, where T , called the TBox, is a of concept and ALC (IC) coNP EXPTIME role inclusions, and A, called the ABox, is a set of instance ALC (UCQ) coNP EXPTIME assertions. SHIQ (IC) coNP EXPTIME The syntax of EL⊥, ALC and SHIQ is summarized in SHIQ (UCQ) coNP 2-EXPTIME Figure 1, in which the symbol A denotes a concept name and the symbol P denotes a role name (in addition to concept and 3 Inconsistency-tolerant semantics role inclusions, SHIQ also allows for TBox axioms of the R R In this section we recall the inconsistency-tolerant semantics form Trans( ), which state transitivity of the role ). 1 The semantics of DLs can be given through the well-known for DL ontologies defined in [Lembo et al., 2010]. We as- K T , A T translation ρfol of DL knowledge bases into FOL theories sume that, for a knowledge base = , is satisfiable, A T with counting quantifiers (see [Baader et al., 2003]). An in- whereas may be inconsistent with , i.e., the set of models K terpretation of K is a classical FOL interpretation for ρfol (K). of may be empty. A model ofaDLKBK = T , A is a FOL model of ρfol (K). AR-semantics The first notion of repair that we consider, We say that K is satisfiable if K has a model. We say that K called AR-repair, is a very natural one: a repair is a maximal entails a FOL sentence φ (and write K|= φ)ifφ is satisfied subset of the ABox that is consistent with the TBox. Thus, an in every model of K. AR-repair is obtained by throwing away from A a minimal In the following, we are interested in particular in UCQ set of assertions to make it consistent with T . entailment, i.e., the problem of deciding whether a KB entails Definition 1 Let K = T , A be a DL KB. An AR-repair of a (Boolean) of conjunctive queries (UCQ), i.e., a FOL K is a set A of membership assertions such that: (i)A ⊆A; sentence of the form ∃ y1.conj 1( y1) ∨···∨∃yn.conj n( yn) where y1,...,y n are terms (i.e., constants or variables), and 1Due to space limitations, we refer the reader to [Lembo et al., each conj i(yi) is a conjunction of atoms of the form A(z) 2010] for introductory examples illustrating these semantics.

1058 (ii) T , A is satisfiable; (iii) there does not exist A such 2010] that the AR-semantics is a sound approximation of the that A ⊂A ⊆Aand T , A is satisfiable. The set of CAR-semantics, i.e., for every KB K and every FOL sen- AR-repairs for K is denoted by AR-Rep(K). Moreover, we tence φ, K|=AR φ implies K|=CAR φ. say that a first-order sentence φ is AR-entailed by K, written   IAR-semantics and ICAR-semantics We then recall the K|=AR φ,ifT , A |= φ for every A ∈ AR-Rep(K). sound approximations of the AR-semantics and the CAR- semantics, called IAR-semantics and ICAR-semantics, re- CAR-semantics We start by formally introducing a notion of spectively [Lembo et al., 2010]. “equivalence under consistency” for inconsistent KBs. K T , A Given a KB K, let SK denote the signature of K, i.e., the Definition 3 Let = be a DL KB. The IAR- set of concept, role, and individual names occurring in K. repair (respectively, ICAR-repair) for K, denoted Given a signature S, we denote with HB(S) the Herbrand by IAR-Rep(K) (respectively, ICAR-Rep(K)) is de- K A Base of S, i.e. the set of ABox assertions (ground atoms) that fined as IAR-Rep()= A∈AR-Rep(K) (respectively, S K K A can be built over the signature . Then, given a KB = ICAR-Rep( )= A∈CAR-Rep(K) ). Moreover, we say T , A, we define the consistent logical consequences of K φ  that a first-order sentence is IAR-entailed (resp., ICAR- as the set clc(K)={α | α ∈ HB(SK ) and there exists A ⊆ K K| φ K| φ   entailed) by , and we write =IAR (resp., =ICAR ), A such that T , A is satisfiable and T , A |= α}. if T , IAR-Rep(K)|= φ (resp., T , ICAR-Rep(K)|= φ). Finally, we say that two KBs T , A and T , A are [ ] IAR consistently equivalent (C-equivalent) if clc(T , A)= It is immediate to see Lembo et al., 2010 that the -  AR clc(T , A ). semantics is a sound approximation of the -semantics, and that the ICAR-semantics is a sound approximation of the We argue that the notion of C-equivalence is very reason- CAR-semantics, while the converse in general does not hold. able in settings in which the ABox (or at least a part of it) has been “closed” with respect to the TBox, e.g., when (some or all) the ABox assertions that are entailed by the ABox and the 4 Lower bounds TBox have been added to the original ABox. This may hap- We start by considering data complexity. First, we provide 3 pen, for example, when the ABox is obtained by integrating lower bounds for instance checking in EL⊥. different (and locally consistent) sources, since some of these Theorem 1 Let K be an EL⊥ KB and let α be an ABox sources might have been locally closed with respect to some assertion. Then: (i) deciding whether K|=AR α is coNP- TBox axioms. hard w.r.t. data complexity; (ii) deciding whether K|=IAR α In settings where C-equivalence makes sense, the AR- is coNP-hard w.r.t. data complexity; (iii) deciding whether semantics is not suited to handle inconsistency. In fact, we K|=CAR α is DP-hard w.r.t. data complexity. would expect two C-equivalent ontologies to produce the same logical consequences under inconsistency-tolerant se- Then, we consider UCQ entailment, still in EL⊥. AR mantics. Unfortunately, the -semantics does not have Theorem 2 Let K be an EL⊥ KB and let Q be a UCQ. p this property. A simple example is the following: let Then: (i) deciding whether K|=CAR Q is Δ [O(log n)]- T {  , ⊥} 2 = student young student worker hard w.r.t. data complexity; (ii) deciding whether K|=ICAR and let A = {student(mary), worker(mary)}, A = p Q is Δ2[O(log n)]-hard w.r.t. data complexity. {student(mary), worker(mary), young(mary)}. It is im- mediate to verify that clc(K)=clc(K)=A, thus K and We then provide a lower bound for instance checking in   ALC K are C-equivalent, however K |=AR young(mary) while . K |=AR young(mary). Theorem 3 Let K be an ALC KB and let α be an ABox p To overcome the above problem, the CAR-semantics has assertion. Then: (i) deciding whether K|=AR α is Π2- been defined in [Lembo et al., 2010], through a modification hard w.r.t. data complexity; (ii) deciding whether K|=IAR 2 p of the AR-semantics. α is Π2-hard w.r.t. data complexity; (iii) deciding whether K| α p Definition 2 Let K = T , A be a DL KB. A CAR-repair =CAR is BH2(Σ2)-hard w.r.t. data complexity.   for K is a set A of membership assertions such that A is an We now turn our attention to UCQ entailment in ALC. AR-repair of T , clc(K). The set of CAR-repairs for K is Theorem 4 Let K be an ALC KB and let Q be a UCQ. denoted by CAR-Rep(T , A). Moreover, we say that a first- p Then: (i) deciding whether K|=CAR Q is Δ3[O(log n)]- order sentence φ is CAR-entailed by K, written K|=CAR φ,   hard w.r.t. data complexity; (ii) deciding whether K|=ICAR if T , A |= φ for every A ∈ CAR-Rep(K). p Q is Δ3[O(log n)]-hard w.r.t. data complexity. Going back to the previous example, it is immediate to see Finally, we consider combined complexity and provide that, since K and K are C-equivalent, the set of CAR-repairs lower bounds for UCQ entailment in EL⊥. (and hence the set of CAR-models) of K and K coincide. Theorem 5 Let K be an EL⊥ KB and let Q be a UCQ. Then: As the above example shows, there are sentences entailed p by a KB under CAR-semantics that are not entailed under (i) deciding whether K|=IAR Q is Δ2[O(log n)]-hard w.r.t. combined complexity; (ii) deciding whether K|=AR Q is AR-semantics. Conversely, it is shown in [Lembo et al., p Π2-hard w.r.t. combined complexity; (iii) deciding whether 2 p The definition provided here of the CAR-semantics is a slight K|=CAR Q is Π2-hard w.r.t. combined complexity. simplification of the one appearing in [Lembo et al., 2010]: this modification, however, does not affect any of the computational re- 3Due to lack of space, in the present version of the paper we omit sults presented in [Lembo et al., 2010]. the proofs of lower bounds.

1059 5 Upper bounds CAR1 (decides T , A |=CAR Q with Q UCQ) A = ∅; We start by providing upper bounds for the combined com- α ∈ HB T , A EL for each ( ) do plexity of instance checking in ⊥. if there exists A ⊆A T , A T , A| α Theorem 6 Let K be an EL⊥ KB and let α be an ABox as- such that satisfiable and = K| α then A = A ∪{α}; sertion. Then: (i) deciding whether =AR is in coNP  if T , A |=AR Q then return true else return false w.r.t. combined complexity; (ii) deciding whether K|=IAR α is in coNP w.r.t. combined complexity; (iii) deciding whether When the input KB is in EL⊥, the above algorithm can K|=CAR α is in DP w.r.t. combined complexity. be reduced to an NP tree [Gottlob, 1995], which proves the p Δ [O(log n)] upper bound. Proof. For case (i), we define the following Algorithm AR1: 2

ALGORITHM AR1 (decides T , A |=AR Q with Q UCQ) Theorem 8 Let K be an EL⊥ KB and let Q be a UCQ.  p if there exists A ⊆Asuch that Then: (i) deciding whether K|=AR Q is in Π2 w.r.t. com-  p 1) T , A  satisfiable and bined complexity; (ii) deciding whether K|=CAR Q is in Π2  2) A is maximal T -consistent subset of A and w.r.t. combined complexity; (iii) deciding whether K|=IAR Q  p 3) T , A  |= Q is in Δ [O(log n)] w.r.t. combined complexity; (iv) deciding 2 p then return false else return true whether K|=ICAR Q is in Δ2[O(log n)] w.r.t. combined It is easy to verify that the above algorithm is correct for every complexity. DL, i.e., it returns true iff T , A |=AR Q, and that, in the Proof. Case (i) follows from the correctness of Algorithm case of EL⊥ and when Q is an ABox assertion, the algorithm AR1 defined in the proof of Theorem 6. Case (ii) follows runs in coNP. from the correctness (for every DL) of the following Algo- For case (ii), we define the following Algorithm IAR1: rithm CAR2:

ALGORITHM IAR1 (decides T , A |=IAR Q with Q UCQ) ALGORITHM CAR2 (decides T , A |=CAR Q with Q UCQ)    A ⊆A A−A = {α1,...,αn} if there exists A ⊂ HB(T , A) such that if there exist (let ),  A1 ⊆A, ...,An ⊆Asuch that 1) for each α ∈A T , A |= Q α ∈ clc(T , A) and 1) and  2) for each i such that 1 ≤ i ≤ n, 2) for each α ∈ HB(T , A) −A  αi ∈Ai and Ai is a minimal T -inconsistent subset of A α ∈ clc(T , A) or T , A ∪{α} unsatisfiable and  then return false else return true 3) T , A  |= Q then return false else return true It is easy to verify that the above algorithm is correct for every α ∈ T , A α DL, i.e., it returns true iff T , A |=IAR Q, and that, in the and from the fact that deciding clc( ) (with ABox case of EL⊥ and when Q is an ABox assertion, the algorithm assertion) can be decided in NP. Case (iii) follows from the runs in coNP. correcness (for every DL) of the following Algorithm IAR2: For case (iii), we define the following algorithm ICAR1: ALGORITHM IAR2 (decides T , A |=IAR Q with Q UCQ) for each α ∈Ado ALGORITHM ICAR1 (decides T , A |=ICAR α)    if α does not belong to any minimal T -inconsistent subset of A if (a) for each A ⊆A, either T , A  unsatisfiable or T , A  |= α    then A = A ∪{α}; A {α ,...,α }⊆ T , A  or (b) there exist = 1 n HB( ), if T , A |= Q then return true else return false A1 ⊆A, ...,An ⊆Asuch that 1) α ∈A and 2) T , A unsatisfiable and Finally, Case (iv) follows from the correctness (for every 3) for every i s.t. 1 ≤ i ≤ n DL) of the following Algorithm ICAR2: T , A −{α } 3.1) i satisfiable and ALGORITHM ICAR2 (decides T , A |=ICAR Q with Q UCQ) 3.2) T , Ai satisfiable and 3.3) T , Ai|= αi A = ∅; then return false else return true for each α ∈ HB(T , A) do A ⊆A It is easy to verify that the above algorithm is correct for every if there exists such that T , A satisfiable and T , A|= α DL, i.e., it returns true iff T , A |=ICAR α, and that, in the   then A = A ∪{α}; case of EL⊥, the algorithm runs in DP.  if T , A |=IAR Q then return true else return false We now turn our attention to both data and combined com- and from the fact that, when the input KB is in EL⊥, the plexity of UCQ entailment in EL⊥. above algorithm can be reduced to an NP tree [Gottlob, 1995], p O n Theorem 7 Let K be an EL⊥ KB and let Q be a UCQ. Then: which proves the Δ2[ (log )] upper bound. (i) deciding whether K|=AR Q is in coNP w.r.t. data com- We then consider the combined complexity of UCQ entail- plexity; (ii) deciding whether K|=IAR Q is in coNP w.r.t. ment in ALC. data complexity; (iii) deciding whether K|=CAR Q is in p Theorem 9 Let K be an ALC KB and let Q be a UCQ. Then: Δ2[O(log n)] w.r.t. data complexity. (i) deciding whether K|=AR Q is in EXPTIME w.r.t. com- Proof. Case (i) follows from the correctness of Algorithm bined complexity; (ii) deciding whether K|=CAR Q is in AR1 defined in the proof of Theorem 6. Case (ii) follows EXPTIME w.r.t. combined complexity; (iii) deciding whether from the correctness of Algorithm IAR1 defined in the proof K|=IAR Q is in EXPTIME w.r.t. combined complexity; (iv) of Theorem 6. Finally, case (iii) follows from the correctness deciding whether K|=ICAR Q is in EXPTIME w.r.t. com- (for every DL) of the following Algorithm CAR1: bined complexity.

1060 Proof. Case (i) follows from the correctness of Algorithm Proof. Case (i) follows from Algorithm AR1 defined in the AR1 defined in the proof of Theorem 6. Case (ii) follows proof of Theorem 6. Case (ii) follows from Algorithm CAR1 from the correctness of Algorithm CAR1 defined in the proof defined in the proof of Theorem 7. Case (iii) follows from of Theorem 7. Case (iii) follows from the correctness of Al- Algorithm IAR2 defined in the proof of Theorem 8. Finally, gorithm IAR2 defined in the proof of Theorem 8. Finally, case (iv) follows from Algorithm ICAR2 defined in the proof case (iv) follows from the correctness of Algorithm ICAR2 of Theorem 8. defined in the proof of Theorem 8. The theorems presented in the last two sections provide a Then, we provide upper bounds for the data complexity of complete picture of the complexity of reasoning in the DLs both instance checking and UCQ entailment in SHIQ. considered in this paper under the four inconsistency-tolerant Theorem 10 Let K be an SHIQ KB and let α be an ABox semantics. A summary of the complexity results obtained is p reported in Figure 2. All the results displayed are complete- assertion. Then, deciding whether K|=CAR α is in BH2(Σ2) w.r.t. data complexity. ness results: all the lower bounds are implied by theorems 1– 5 and by the lower bounds for reasoning under standard DL Proof. The proof follows from the correctness of Algorithm semantics reported in the preliminaries, while all the upper ICAR1 defined in the proof of Theorem 6. bounds are implied by theorems 6–13. Theorem 11 Let K be a SHIQ KB and let Q be a UCQ. p 6 Tractable cases Then: (i) deciding whether K|=AR Q is in Π w.r.t. data 2 p complexity; (ii) deciding whether K|=IAR Q is in Π2 w.r.t. We now consider a language restriction of EL⊥ whose aim is data complexity; (iii) deciding whether K|=CAR Q is in p to allow for tractable reasoning under inconsistency-tolerant Δ [O(log n)] w.r.t. data complexity; (iv) deciding whether IAR 3 p semantics, in particular, under the -semantics. The con- K|=ICAR Q is in Δ3[O(log n)] w.r.t. data complexity. dition is on the form of empty concept assertions, i.e., con- Proof. cept inclusions whose right-hand side is the empty concept Case (i) follows from the correctness of Algorithm ⊥ EL AR1 defined in the proof of Theorem 6. Case (ii) follows . Formally, an empty concept assertion in ⊥ is an inclu- sion of the form C1 ... Cn ⊥. from the correctness of Algorithm IAR2 defined in the proof EL T N of Theorem 8. Case (iii) follows from the correctness of Given an ⊥ TBox , we say that a concept name is relevant for empty in T if there exists an empty Algorithm CAR1 defined in the proof of Theorem 7. It δ N δ T| δ can be proved that, when the input KB is in SHIQ, Algo- concept assertion such that occurs in and = and p T | δ /N δ /N rithm CAR1 can be reduced to a Σ tree, which proves the = [ ], where [ ] is the empty concept assertion p 2 δ N Δ [O(log n)] upper bound. Analogously, case (iv) follows obtained from by replacing every occurrence of with . 3 Intuitively, concept names relevant for empty concepts in a from the correctness of Algorithm ICAR2 defined in Theo- T rem 8: when the input KB is in SHIQ, Algorithm ICAR2 TBox are those involved in the empty concept assertions p p entailed by T . The above definition filters out “redundant” can be reduced to a Σ2 tree, which proves the Δ3[O(log n)] upper bound. (i.e., non-minimal) empty concept assertions. Then, we introduce the notion of (non)-recursive concept Finally, we provide upper bounds for the combined com- name in T . A concept name N is recursive in T if T entails plexity of instance checking and UCQ entailment in SHIQ. an inclusion of the form C  N, where C contains at least an N Theorem 12 Let K be a SHIQ KB and let α be an ABox as- occurrence of nested into an existentially quantified con- C T sertion. Then: (i) deciding whether K|=AR α is in EXPTIME cept expression, and there exists no concept such that C  N C C T T w.r.t. combined complexity; (ii) deciding whether K|=CAR α entails and is more specific than in (i.e., C  C is in EXPTIME w.r.t. combined complexity; (iii) deciding entails but not vice versa). Otherwise, we say that N T EL⊥nr EL⊥ whether K|=IAR α is in EXPTIME w.r.t. combined complex- is non-recursive in . We call ( with non- EL⊥ K = T , A ity; (iv) deciding whether K|=ICAR α is in EXPTIME w.r.t. recursive empty concepts) KB every KB combined complexity. such that every concept name that is relevant for empty con- cepts in T is non-recursive in T . It is easy to see that decid- Proof. Case (i) follows from the correctness of Algorithm ing whether an EL⊥ KB is an EL⊥nr KB can be done in time AR1 defined in the proof of Theorem 6. Case (ii) follows polynomial w.r.t. the size of the TBox. from the correctness of Algorithm CAR1 defined in the proof The language restriction imposed in EL⊥nr KBs allows us of Theorem 7. Case (iii) follows from the correctness of Al- to prove the following property, which is crucial for lowering gorithm IAR2 defined in the proof of Theorem 8. Finally, the complexity of reasoning under the IAR-semantics. case (iv) follows from the correctness of Algorithm ICAR2 defined in the proof of Theorem 8. Lemma 1 Let T , A be an EL⊥nr KB. The maximum car- dinality of a minimal T -inconsistent subset of A is bounded Theorem 13 Let K be a SHIQ KB and let Q be a UCQ. by the number of concept expressions occurring in T . K| Q Then: (i) deciding whether =AR is in 2-EXPTIME We are now ready to state the complexity of reasoning in w.r.t. combined complexity; (ii) deciding whether K|=CAR Q EL⊥nr under IAR-semantics. is in 2-EXPTIME w.r.t. combined complexity; (iii) deciding whether K|=IAR Q is in 2-EXPTIME w.r.t. combined com- Theorem 14 Let K be an EL⊥nr KB, let α be an ABox as- plexity; (iv) deciding whether K|=ICAR Q is in 2-EXPTIME sertion, and let Q be a UCQ. Then: (i) deciding whether w.r.t. combined complexity. K|=IAR α is PTIME-complete w.r.t. data complexity; (ii)

1061 data complexity combined complexity semantics AR CAR IAR ICAR AR CAR IAR ICAR EL⊥ (IC) coNP DP coNP DP coNP DP coNP DP p p p p p p EL⊥ (UCQ) coNP Δ2[O(log n)] coNP Δ2[O(log n)] Π2 Π2 Δ2[O(log n)] Δ2[O(log n)] p p p p ALC (IC) Π2 BH2(Σ2) Π2 BH2(Σ2) EXPTIME EXPTIME EXPTIME EXPTIME p p p p ALC (UCQ) Π2 Δ3[O(log n)] Π2 Δ3[O(log n)] EXPTIME EXPTIME EXPTIME EXPTIME p p p p SHIQ (IC) Π2 BH2(Σ2) Π2 BH2(Σ2) EXPTIME EXPTIME EXPTIME EXPTIME p p p p SHIQ (UCQ) Π2 Δ3[O(log n)] Π2 Δ3[O(log n)] 2-EXPTIME 2-EXPTIME 2-EXPTIME 2-EXPTIME

Figure 2: Complexity of UCQ entailment over DL KBs under inconsistency-tolerant semantics. deciding whether K|=IAR α is PTIME-complete w.r.t. com- setting have a very special form, which has never been con- bined complexity; (iii) deciding whether K|=IAR Q is sidered in the research in belief revision and update. There- PTIME-complete w.r.t. data complexity; (iv) deciding whether fore, (to the best of our knowledge) there are no available K|=IAR Q is NP-complete w.r.t. combined complexity. results about reasoning in the present setting. Proof. The present work can be continued along several lines. For all the four cases, the upper bounds are an im- For instance, it would be very interesting to see whether mediate consequence of Lemma 1 and of the correctness of the present approach can be extended to handle more gen- Algorithm IAR2 in the proof of Theorem 8 (observe that eral forms of ABoxes (e.g., ABoxes with variables and/or Lemma 1 implies that there are only polynomially many po- T non-atomic concept expressions). Also, it would be very im- tential minimal -inconsistent subsets w.r.t. the size of the portant for practical purposes to identify further DL sublan- ABox), while the lower bounds are immediately implied by guages allowing for tractable reasoning under the IAR and the complexity of reasoning under standard DL semantics in the ICAR semantics. EL (notice that EL is a sublogic of EL⊥nr ). Acknowledgments This research has been partially sup- The above theorem shows the nice computational be- ported by the EU funded Project ACSI (Artifact-Centric Ser- haviour of reasoning under IAR semantics in EL⊥nr , which vice Interoperation). is not harder than reasoning under standard DL semantics. This case seems very important not only from the theoretical References viewpoint, but also for practical applications, since it identi- [Baader et al., 2003] Franz Baader, Diego Calvanese, Deborah fies a DL allowing for tractable automated treatment of incon- McGuinness, Daniele Nardi, and Peter F. Patel-Schneider, edi- sistency. Conversely, it can be shown that reasoning under the tors. The Description Logic Handbook: Theory, Implementation ICAR semantics is still intractable in EL⊥nr . Informally, the and Applications. Cambridge University Press, 2003. EL reason is that the restriction imposed by ⊥nr only affects [Baader et al., 2005] Franz Baader, Sebastian Brandt, and Carsten the cardinality of minimal T -inconsistent subsets of A, while Lutz. Pushing the EL envelope. In Proc. of IJCAI 2005. it leaves unbounded the cardinality of the minimal subsets of [Chomicki, 2007] Jan Chomicki. Consistent query answering: Five A entailing an ABox assertion. Therefore, it is still NP-hard easy pieces. In Proc. of ICDT 2007. T , A to decide whether an ABox assertion belongs to clc( ). [Dantsin et al., 2001] Evgeny Dantsin, Thomas Eiter, Georg Gott- lob, and Andrei Voronkov. Complexity and expressive power of 7 Related work and conclusions logic programming. ACM Computing Surveys, 33(3), 2001. [Eiter and Gottlob, 1992] T. Eiter and G. Gottlob. On the complex- Inconsistency-tolerance has been studied in various forms in ity of propositional knowledge base revision, updates and coun- several areas of Artificial Intelligence and databases. While terfactuals. Artificial Intelligence, 57:227–270, 1992. several recent papers have dealt with different forms of incon- [Glimm et al., 2008] Birte Glimm, , Carsten Lutz, and sistency in DL ontologies (focusing especially on the TBox, Uli Sattler. Conjunctive query answering for the description logic see e.g. [Qi and Du, 2009]), the approach considered in this SHIQ. J. of Artificial Intelligence Research, 31:151–198, 2008. paper (based on instance-level repair only) is novel for DLs, [Gottlob, 1995] Georg Gottlob. NP trees and Carnap’s . and is actually inspired by the work on consistent query an- J. of the ACM, 42(2):421–457, 1995. swering (CQA) in databases (see [Chomicki, 2007] for a sur- [Hogan et al., 2010] Aidan Hogan, Andreas Harth, Alexandre Pas- vey). Moreover, the form of inconsistency-tolerance con- sant, Stefan Decker, and Axel Polleres. Weaving the pedantic sidered in this paper corresponds to a form of belief revi- web. In Proc. of 3rd Int. Workshop on on the Web sion [Eiter and Gottlob, 1992]: more precisely, with respect (LDOW2010), 2010. to the knowledge base revision framework, the ABox corre- [Lembo et al., 2010] Domenico Lembo, Maurizio Lenzerini, Ric- sponds to the initial knowledge, while the TBox represents cardo Rosati, Marco Ruzzi, and Domenico Fabio Savo. the new information. Based on such a correspondence, the Inconsistency-tolerant semantics for description logics. In Proc. semantic studies on belief revision are indeed very relevant of RR 2010, 2010. for the present setting (e.g., the IAR-semantics and ICAR- [Qi and Du, 2009] Guilin Qi and Jianfeng Du. Model-based revi- semantics are deeply connected to the well-known WIDTIO sion operators for terminologies in description logics. In Proc. of semantics of belief revision [Eiter and Gottlob, 1992]): how- IJCAI 2009, pages 891–897, 2009. ever, the kind of theories and formulas considered in the DL

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