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 exten- sion can be found in [Bonatti et al., 2006]; they conclude that are individual names, P ∈ NR, and A ∈ NC, (iii) role in- clusions (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 <D,F J iff: from the following constructs: ΔI =ΔJ C ::= A ||C1 C2 |∃P.C 1.