Defeasible Inclusions in Low-Complexity Dls: Preliminary Notes

Defeasible Inclusions in Low-Complexity DLs: Preliminary Notes

P. A. Bonatti M. Faella L. Sauro [email protected] [email protected] [email protected] Department of Physics University of Naples “Federico II”

Abstract bodies of semantic web knowledge. The latter is interesting because it is spontaneously adopted in major biomedical on- We analyze the complexity of reasoning with cir- tologies. It is interesting to investigate whether the syntactic cumscribed low-complexity DLs such as DL-lite EL restrictions obeyed by such logics decrease the complexity of and the family, under suitable restrictions on reasoning also in a nonmonotonic context. the use of abnormality predicates. We prove that In this paper, we identify less complex circumscribed DLs in circumscribed DL-liteR complexity drops from by (i) using the constructs supported by DL-liteR and by the NExpNP to the second level of the polynomial hier- EL family, and (ii) restricting the use of abnormality predi- archy. In EL, reasoning remains ExpTime-hard, in cates by hiding them into “defeasible” inclusion axioms, sim- general. However, by restricting the possible occur- ilar to those adopted by [Straccia, 1993]. The latter restriction rences of existential restrictions, we obtain mem- p p is also expected to make the formalism easier to use. Under bership in Σ and Π for an extension of EL. 2 2 such restrictions, we prove that (i) satisfiability checking for circumscribed knowledge bases (KB) is equivalent to classi- 1 Introduction cal KB satisfiability, and hence in P (sometimes even triv- ⊥ The ample literature on nonmonotonic extensions of descrip- ial) for the logics we consider here: DL-liteR, EL, and EL ; (ii) concept satisfiability, instance checking, and subsumption tion logics (DLs) witnesses a long-standing interest for this ⊥ topic (for some early approaches see [Brewka, 1987; Straccia, over circumscribed DL-liteR and left local EL KBs remain 1993; Baader and Hollunder, 1995]). Recently, fresh motiva- within the second level of the polynomial hierarchy; (iii) the ⊥ tions came from the construction of ontologies for biomed- same reasoning tasks for circumscribed EL KBs, unfortu- ical domains (cf. [Rector, 2004; Stevens et al., 2007]) and nately, remain ExpTime-hard. from the use of description logics as policy languages [Us- Further related approaches are [Cadoli et al., 1990; Strac- zok et al., 2004; Kagal et al., 2003; Tonti et al., 2003] where cia, 1993].In[Cadoli et al., 1990], a fragment of ALE under nonmonotonic reasoning is needed to properly encode default minimal entailment (an instance of circumscription where all policies and authorization inheritance (cf. [Bonatti and Sama- predicates are minimized with the same priority) is proved p rati, 2003]). Several recent works [Donini et al., 1998; 1997; to belong to Π2. Our approach adopts different DLs and 2002; Bonatti et al., 2006; Giordano et al., 2008] improved more general forms of circumscription, supporting priorities our understanding of the complexity of nonmonotonic de- as well as fixed and variable predicates. In [Straccia, 1993] scription logics based on default logic, autoepistemic logic, the underlying nonmonotonic logic is a prioritized version of and circumscription. Unfortunately, nonmonotonic DLs are default logic. The paper contains NP-hardness results for ex- typically very complex. For example, reasoning with cir- tremely simplified DLs. NP cumscribed ALC knowledge bases is NExp -hard [Bonatti The rest of the paper is organized as follows: In Section 2, et al., 2006], and a tableaux calculus for reasoning with au- we recall the basics of DLs. Section 3 introduces the special- toepistemic knowledge bases is in 3-ExpTime [Donini et al., ized circumscription framework we adopt here. After some 2002]. Besides such complexity results, it turns out that some auxiliary results (Section 4), sections 5 and 6 illustrate the re- theoretical properties that are very important for the imple- sults on DL-liteR and the EL family, respectively. Section 7 mentation of reasoning in “classical” DLs—such as the tree concludes the paper with a summary of the results and some model property for example— do not carry over to nonmono- directions for future work. tonic DLs. Independently from the works on nonmonotonic DLs, low- 2 Preliminaries complexity (monotonic) DLs of practical interest have been recently studied. Here we will focus on DL-liteR [Cal- In DLs, concepts are inductively defined with a set of con- vanese et al., 2005] and the EL family [Baader, 2003; structors, starting with a set NC of concept names, a set NR Baader et al., 2005], whose inferences are in PTIME. The of role names, and (possibly) a set NI of individual names (all former is motivated by efficient query processing over large countably infinite). We use the term predicates to refer to ele-

696 Name Syntax Semantics 3 Defeasible knowledge R− (R−)I = {(d, e) | (e, d) ∈ RI } inverse role A defeasible inclusion (DI) is an expression A n C whose nominal {a} {aI } intended meaning is: A’s elements are normally in C. I I DL negation ¬C Δ \ C A defeasible knowledge base (DKB) in a logic is a pair (S, D) S DL D conjunction C D CI ∩ DI where is a strong knowledge base, and is a I I I set of DIs A n C such that C is a DL concepts. existential ∃R.C {d ∈ Δ |∃(d, e) ∈ R : e ∈ C } restriction Example 3.1 The sentences: “in humans, the heart is usu- top I =ΔI ally located on the left-hand side of the body; in humans with ⊥ ⊥I = ∅ situs inversus, the heart is located on the right-hand side of bottom the body” [Rector, 2004; Stevens et al., 2007] can be formu- EL⊥ Figure 1: Syntax and semantics of some DL constructs lated with the following inclusions ∃ .∃ . ; N ∪ N A B Human n has heart has position Left ments of C R. Hereafter, letters and will range over Situs Inversus ∃has heart.∃has position.Right ; NC, P will range over NR, and a, b, c will range over NI. The ∃has heart.∃has position.Left concepts of the DLs dealt with in this paper are formed using ∃ .∃ . ⊥. the constructors shown in Figure 1. There, the inverse role has heart has position Right constructor is the only role constructor, whereas the remain- Intuitively, a model of (S, D) is a model of S that max- ing constructors are concept constructors. Letters C, D will R, S imizes the set of individuals satisfying the defeasible inclu- range over concepts and letters over (possibly inverse) sions in D, resolving conflicts by means of specificity when- roles. ever possible. The semantics of the above concepts is defined in terms of I =(ΔI , ·I ) ΔI In order to formalize this idea, we first have to specify how interpretations . The domain is a non-empty DIs are prioritized. We determine specificity based on clas- set of individuals and the interpretation function ·I maps each I I sically valid inclusions. For all DIs δ1 =(A1 n C1) and concept name A ∈ NC to a set A ⊆ Δ , each role name I I δ2 =(A2 n C2), we write r ∈ NR to a binary relation r on Δ , and each individual I I I name a ∈ NI to an individual a ∈ Δ . The extension of · δ1 ≺S δ2 iff A1 S A2 and A2 S A1 . to inverse roles and arbitrary concepts is inductively defined S as shown in the third column of Figure 1. An interpretation I For the sake of readability, the subscript will be omitted is called a model of a concept C if CI = ∅.IfI is a model when clear from context. of C, we also say that C is satisfied by I. Second, we have to specify how to deal with the predicates A (strong) knowledge base is a finite set of (i) concept in- occurring in the knowledge base: is their extension allowed clusions (CIs) C D where C and D are concepts, (ii) con- to vary in order to satisfy defeasible inclusions? A discussion cept assertions A(a) and role assertions P (a, b), where a, b of the effects of letting predicates vary vs. fixing their extension can be found in [Bonatti et al., 2006]; they conclude that are individual names, P ∈ NR, and A ∈ NC, (iii) role inclusions (RIs) R R. An interpretation I satisfies (i) a CI the appropriate choice is application dependent. Here we let C D if CI ⊆ DI , (ii) an assertion C(a) if aI ∈ CI , (iii) roles vary to avoid undecidability problems (cf. [Bonatti et N an assertion R(a, b) if (aI ,bI ) ∈ rI , and (iv) a RI R R al., 2006]). The set of concept names C, on the contrary, F V RI ⊆ RI I can be arbitrarily partitioned into two sets and contain- iff . Then, is a model of a strong knowledge base ing fixed and varying predicates, respectively; we denote this S iff I satisfies all the elements of S. semantics with CircF . We write C S D iff for all models I of S, I satisfies F D ≺ C D The set , the DIs , and their ordering induce a strict . partial order over interpretations, defined below. As we move [ ] The logic DL-lightR Calvanese et al., 2005 restricts con- down the ordering we find interpretations that are more and cept inclusions to expressions CL CR, where more normal w.r.t. D. For all δ =(A n C) and all interpre- − CL ::= A |∃RR::= P | P tations I let the set of individuals satisfying δ be: CR ::= CL |¬CL sat (δ)={x ∈ ΔI | x ∈ AI x ∈ CI } . (as usual, ∃R abbreviates ∃R. ). I or EL The logic [Baader, 2003; Baader et al., 2005] restricts Definition 3.2 For all interpretations I and J , and all F ⊆ knowledge bases to assertions and concept inclusions built NC, let I

697 Deﬁnition 3.3 [Model] Let KB =(S, D) and F ⊆ NC.An Proof. A simple adaptation of a result for ALCIO [Bonatti interpretation I is a model of CircF (KB) iff I is a (classical) et al., 2006], taking role hierarchies into account. S J S J < I model of and for all models of , F . As a consequence, these logics preserve classical consis- Remark 3.4 This semantics is a special case of the circum- tency (because all

698 the corresponding restriction of I in the interpretation order- Claim: For all δ ∈D,ifsatJ (δ) ⊆ satJ (δ), then ing. It follows that I

699 It can be verified that T is satisfiable iff in some Theorem 6.4 Instance checking and subsumption are model of Circvar(KB) all UA are empty, which holds ExpTime-hard both in CircF (EL) and in Circfix(EL). The iff Circvar(KB) |= ∃R.U. Consequently, subsumption same holds in the restriction of EL not supporting . Circ (EL⊥) in var is ExpTime-hard. Concept consistency is simpler, instead. Call an interpre- a ∈ N T I I Similarly, for any given I, is satisfiable iff there tation I maximal iff for all A ∈ NC, A =Δ, and for I Circ (KB) aI ∈ (∃R.U)I I I I exists a model of var such that . P ∈ NR P =Δ × Δ ⊥ all , . It is not hard to verify that Therefore, instance checking in Circvar(EL ) is ExpTime- all ELHO concepts and all ELHO inclusions (both classical hard as well. and defeasible) are satisfied by all x ∈ ΔI , therefore maxi- Finally, add a fresh concept name B and all the inclusions CircF (KB) mal models are always models of , for all DKBs B ∃R.UA ⊥; call the new DKB KB . Note that T is sat- KB and all F ⊆ NC. As a consequence we have that concept isfiable iff in some model of Circvar(KB) all UA are empty, consistency is trivial: which holds iff B is satisfiable w.r.t. Circvar(KB ). Conse- ⊥ Theorem 6.5 For all EL concepts C, DKBs KB, and F ⊆ quently, concept satisfiability in Circvar(EL ) is ExpTime- NC, C is satisfied by some model of CircF (KB). hard. One of the causes of the complexity of instance checking Since Circvar is a special case of CircF , and by Theo- ⊥ and subsumption for Circfix(EL ) is the ability of inferring rem 4.3, the above theorem applies to CircF and Circfix, too: consequences from qualified existential restrictions ∃P.B. Corollary 6.3 For X = F, fix, concept satisfiability check- By limiting their occurrences, it is possible to reduce signifi- ⊥ ing, instance checking, and subsumption in CircX (EL ) are cantly the complexity of instance checking and subsumption ⊥ ExpTime-hard. These results still hold if ABoxes are empty for Circfix(EL ) knowledge bases. (i.e. assertions are not allowed). Definition 6.6 An EL⊥ knowledge base is left local (LL) The above proof can be adapted to CircF (EL). First we have to introduce a new concept name D representing and if its concepts inclusions are instances of the following translate each concept C into C∗ as follows: schemata: • C∗ = C if C is a concept name; A [n] BA[n] ∃P.B A1 A2 B ∃P B ∃P1 ∃P2.B • C∗ = A¯ if C is ¬A (for all A, A¯ is a new concept name); ∗ ∗ ⊥ • C = D ∃R.(C1 D) if C is ∃R.C1; where A and B can be concept names or ⊥.ALL EL con- ∗ ∗ ∗ • C = C1 C2 if C is C1 C2. cept is any concept that can occur in the above inclusions. ∗ ∗ EL Each C1 C2 in T is translated into C1 C2 . Note the similarity with the normal form of inclusions Then we extend the translated TBox with the follow- [Baader et al., 2005] that, however, would allow the more ing inclusions, where Bot (representing ⊥), all UA, and general inclusions ∃P.A B and ∃P1.A ∃P2.B . Bad R are new concept names and is a new role name: Lemma 6.7 Let KB be an LL EL⊥ knowledge base. For all I models I∈Circvar(KB) and x ∈ Δ there exists a model J I J A D (10) D n UA (18) J∈Circvar(KB) such that (i) Δ ⊆ Δ , (ii) x ∈ Δ , J A¯ D (11) D D (19) (iii) |Δ | is polinomial in the size of KB. UA D (12) D n ∃R.UA (20) Proof. The proof is analogous to the proof of Lemma 5.1. ¯ A A Bot (13) D n ∃R.Bot (21) Here we start with a slightly different set Δ. Choose a min- Δ ⊆ ΔI x aI A UA Bot (14) ∃R.UA Bad (22) imal set containing: (i) , (ii) all such that ¯ a ∈ NI ∩cl(KB), (iii) for each concept ∃P in cl(KB) satisfied A UA Bot (15) ∃R.Bot Bad (23) I I in I, a node yP such that for some z ∈∃P , (z,yP ) ∈ P D A n (16) and (iv) for each concept ∃P.B in cl(KB) satisfied in I,a D A¯ D(a) I I n (17) (ABox assertion) (24) node yP,B such that for some z ∈∃P.B , (z,yP,B) ∈ P I and yP,B ∈ B . Let KB be the resulting DKB. Finally set F = {D, Bot}. J J I ∗ Now define J as follows: (i) Δ =Δ, (ii) a = a (for Now (24) guarantees that D is nonempty; the translation (·) , J I a ∈ NI ∩cl(KB)), (iii) A = A ∩Δ (A ∈ NC ∩cl(KB)), and (10) and (11) make sure that by restricting to D any model I J I (iv) P = {(z,yP ) | z ∈ Δ and z ∈∃P }∪{(z,yP,B) | of CircF (KB) where all UA and Bot are empty one obtains a z ∈ Δ z ∈∃P.BI } P ∈ N model of T . Inclusion (19) gives (20) and (21) lowest priority. and ( R). These two DIs and (22)-(23) include D into Bad whenever The rest of the proof is similar to the proof of Lemma 5.1 the intended meaning of the atoms A¯ is violated. Then T and is omitted here for space limitations. We only remark I that the restriction to LL KBs is needed to ensure that J is a is satisfiable iff for some model I of CircF (KB), Bad = classical model of the CIs in KB. ∅. In turn, this happens iff CircF (KB) |= D Bad, and I ⊥ iff aI ∈ Bad . Then we have the desired reduction from Lemma 6.8 Let KB beaLLEL knowledge base. For all I EL¬A models I∈Circfix(KB) and x ∈ Δ there exists a model TBox satisfiability to the complement of subsumption J I J Circ (EL) J∈Circfix(KB) such that (i) Δ ⊆ Δ , (ii) x ∈ Δ , and instance checking in F . As a consequence, and J by Theorem 4.3: (iii) |Δ | is polinomial in the size of KB.

700 var F / fix [Baader et al., 2005] F. Baader, S. Brandt, and C. Lutz. Pushing the EL envelope. In IJCAI, pages 364–369. 2005. Concept sat. EL trivial up to ELHO (Thm 6.5) ⊥ [ ] EL ≥ ExpTime (Thm 6.2, Cor 6.3) Baader, 2003 F. Baader. The instance problem and the most spe- cific concept in the description logic EL w.r.t. terminological cy- EL⊥ ≤ Σp DL-liteR,LL 2 (Thm 5.3, Thm 6.9) cles with descriptive semantics. In Proc. of the 26th Annual Ger- Instance checking man Conference on AI, KI 2003, volume 2821 of Lecture Notes Subsumption EL P (*) ≥ ExpTime (Thm 6.4) in Computer Science, pages 64–78. Springer, 2003. EL⊥ ≥ ExpTime (Thm 6.2, Cor 6.3) [Bonatti and Samarati, 2003] P. A. Bonatti and P. Samarati. Logics EL⊥ ≤ Πp for authorization and security. In Logics for Emerging Applica- DL-liteR,LL 2 (Thm 5.3, Thm 6.9) tions of Databases, pages 277–323. Springer, 2003. (*) Classical up to ELHO (by Theorem 6.1) [Bonatti et al., 2006] P. A. Bonatti, C. Lutz, and F. Wolter. De- scription logics with circumscription. In Proc. of the Tenth Inter- Table 1: Summary of complexity results national Conference on Principles of Knowledge Representation and Reasoning, KR 2006, pages 400–410. AAAI Press, 2006. Proof. Similar to the proof of Lemma 5.2. Use a slightly [Brewka, 1987] G. Brewka. The logic of inheritance in frame sys- modified, minimal set Δ ⊆ ΔI containing: (i) x, (ii) all tems. In IJCAI, pages 483–488, 1987. I a such that a ∈ NI ∩ cl(KB), (iii) for each concept ∃P in [Cadoli et al., 1990] M. Cadoli, F.M. Donini, and M. Schaerf. I cl(KB) satisfied in I, a node yP such that for some z ∈∃P , Closed world reasoning in hybrid systems. In Proc. of ISMIS’90, I (z,yP ) ∈ P , (iv) for each concept ∃P.B in cl(KB) sat- pages 474–481. Elsevier, 1990. I isfied in I, a node yP,B such that for some z ∈∃P.B , [Calvanese et al., 2005] D. Calvanese, G. De Giacomo, D. Lembo, I I (z,yP,B) ∈ P and yP,B ∈ B and finally (v) for all M. Lenzerini, and R. Rosati. DL-Lite: Tractable description logics for ontologies. In Proc. of AAAI 2005, pages 602–607, 2005. inclusions C ∃R.B or C n ∃R.B in KB such that (C ∃R.B)I = ∅, a node z ∈ (C ∃R.B)I . The rest [Donini et al., 1997] F. M. Donini, D. Nardi, and R. Rosati. Au- of the proof is similar to the proof of Lemma 5.2 and omitted toepistemic description logics. In IJCAI, pages 136–141, 1997. here. [Donini et al., 1998] F. M. Donini, M. Lenzerini, D. Nardi, W. Nutt, As a consequence of the above lemmata we get: and A. Schaerf. An epistemic operator for description logics. Artif. Intell., 100(1-2):225–274, 1998. Theorem 6.9 Concept consistency over circumscribed LL ⊥ p [Donini et al., 2002] F. M. Donini, D. Nardi, and R. Rosati. De- EL DKBs is in Σ2. Subsumption and instance checking over ⊥ p scription logics of minimal knowledge and negation as failure. circumscribed LL EL DKBs are in Π2. ACM Trans. Comput. Log., 3(2):177–225, 2002. [Giordano et al., 2008] L. Giordano, V. Gliozzi, N. Olivetti, and 7 Conclusions and further work G. Pozzato. Reasoning about typicality in preferential descrip- The complexity of circumscribed description logics can be tion logics. In Proc. of Logics in Artificial Intelligence, 11th Eu- significantly reduced by (i) restricting the underlying DL to ropean Conference, JELIA 2008, volume 5293 of Lecture Notes in Computer Science. Springer, 2008. DL-liteR and to suitable members of the EL family, and (ii) restricting nonmonotonic constructs to defeasible inclusions [Kagal et al., 2003] L. Kagal, T. Finin, and A. Joshi. A policy lan- A n C. KB satisfiability is equivalent to its classical ver- guage for a pervasive computing environment. In 4th IEEE Inter- sion (by Theorem 4.2) and hence it is within P (sometimes national Workshop on Policies for Distributed Systems and Net- even trivial) for the logics we investigated. The results for all works (POLICY 2003), Lake Como, Italy, June 2003. the other reasoning tasks are summarized in Table 1. Surpris- [Rector, 2004] Alan Rector. Defaults, context, and knowledge: Al- ingly, fixed predicates in conjunction with qualified existen- ternatives for OWL-indexed knowledge bases. In Proc. of the tial restrictions are powerful enough to keep the complexity of Pacific Symposium on Biocomputing, pages 226–237, 2004. instance checking and subsumption ExpTime-hard even for a [Stevens et al., 2007] R. Stevens, M. E. Aranguren, K. Wolsten- language like EL, which is not able to express any inconsis- croft, U. Sattler, N. Drummond, M. Horridge, and A. Rector. Us- ⊥ p ing OWL to model biological knowledge. International Journal tency. For DL-liteR and LL EL , complexity drops to Σ2 p of Man-Machine Studies, 65(7), 2007. and Π2, instead. We are currently sharpening our complexity bounds, and [Straccia, 1993] U. Straccia. Default inheritance reasoning in hy- extending them to more expressive logics, looking for alterna- brid KL-ONE-style logics. 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