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A New Semantics for Overriding in Description Logics (Extended Abstract)∗

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A New Semantics for Overriding in Description Logics (Extended Abstract)∗ Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI-17) A New Semantics for Overriding in Description Logics (Extended abstract)∗ Piero A. Bonatti, Marco Faella Iliana M. Petrova Luigi Sauro Universita` degli studi di Napoli Federico II [email protected] Abstract Inheritance blocking: In several nonmonotonic logics a concept with exceptional properties inherits none of the de- Nonmonotonic inferences are not yet supported by fault properties of its superclasses. For instance penguins, Description Logic technology, although their po- that are exceptional birds because they do not fly, do not in- tential usefulness is widely recognized. Lack of herit any of the other default properties of birds, such as hav- support to nonmonotonic reasoning is due to a ing wings and feathers. number of issues related to expressiveness, compu- Undesired CWA effects: In some nonmonotonic DLs, an tational complexity, and optimizations. This work exceptional concept is shrinked to the individuals that explic- contributes to the practical support of nonmono- itly belong to it; it may even become inconsistent. Using the tonic reasoning in description logics by introduc- penguin example again, by default no penguins exist unless ing a new semantics designed to address knowl- explicitly stated otherwise. After asserting that the individual edge engineering needs. The formalism is validated Opus is a penguin, the concept Penguin becomes the single- through extensive comparison with the other non- ton {Opus}. monotonic DLs, and systematic scalability tests. Limited control on role ranges: In most nonmonotonic DLs, the knowledge engineer cannot specify whether a role 1 Introduction should range only over normal individuals or not. In the DLs based on default logic, rational closure, and variants thereof, Many modern applications of description logics (DLs, for default properties never apply to role fillers. short), such as biomedical ontologies and semantic web poli- Silent removal of unresolved conflicts: Very frequently, cies, provide fresh motivations for extending DLs with non- unresolved conflicts between nonmonotonic assertions are a monotonic inferences. Some recent examples stemming symptom of a gap in the axiomatization. The correct res- from the biomedical domain are discussed in [Rector, 2004; olution of such conflicts is typically domain dependent and Stevens et al., 2007]. There, the goal of supporting default should require human intervention (see [Bonatti et al., 2015a; attributes and exceptions is deemed important enough to look 2015b] for a detailed discussion of this issue). Most non- for alternative representation methods, based on classical rea- monotonic logics hide such conflicts (i.e. they do not have soning and ontology design patterns. However, these solu- any visible consequence) thereby hindering their identifica- tions do not scale to more complex examples with multiple tion – a necessary step for the validation and correction of exception dimensions, as discussed in [Rector, 2004]: The knowledge bases. number of additional concepts introduced by the patterns may Moreover, the computational complexity of nonmonotonic grow exponentially. Moreover, such auxiliary concepts are DLs is almost always higher than the complexity of the cor- defined using computationally expensive constructs such as responding classical DL, and the tractability of the OWL 2 disjunction. So, even if the given knowledge base belongs to profiles is not preserved. The lack of optimization techniques some low-complexity fragment (such as the OWL2 profiles), for nonmonotonic DL reasoning is a further obstacle to the its nonmonotonic extension is generally not tractable. practical application of these logics. Nonmonotonic DLs natively support default inferences and Given the above motivations (more extensively articulated exceptions. However none of the standard nonmonotonic se- in the full paper), in this work we have investigated a new N mantics produces exactly the set of expected consequences, family of nonmonotonic DLs, called DL , aimed at address- N and this can be verified on a range of rather simple examples. ing the above drawbacks. The comparison of DL with the Some of the major known drawbacks are the following: major nonmonotonic DLs is summarized in Table 1. Prelim- ∗This is an extended abstract of the paper [Bonatti et al., 2015a], inary performance tests have been reported in the full pa- integrated with the developments in [Bonatti and Sauro, 2017], con- per and have been later extended in [Bonatti et al., 2015b] ditionally accepted for publication on the AIJ. This research is cur- using additional optimization techniques that show unparal- rently being funded by the European Unions Horizon 2020 research leled scalability properties over large nonmonotonic knowl- and innovation programme under grant agreement N. 731601. edge bases, with more than 105 axioms. For instance, sub- 4975 Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI-17) Table 1: Summary of comparisons with other nonmonotonic DLs Features CIRC DEF AEL TYP RAT PR DLN no inheritance X X X X X X blocking no CWA effects X X X X X fine-grained control X(1) X on role ranges detects inconsistent (2) X(3) X prototypes unique deductive X X X closure preserves tractability (5) (5) X(4) implicit specificity X X X X other priorities X X X CIRC, DEF, AEL, TYP, RAT, PR stand, respectively, for Circumscribed DLs [Bonatti et al., 2009; 2011], Default DLs [Baader and Hollunder, 1995a; 1995b], Autoepistemic DLs [Donini et al., 2002], DLs with Typicality [Giordano et al., 2009; 2013], DLs with Rational Closure [Casini and Straccia, 2010; Casini et al., 2013] and their combination with inheri- tance networks [Casini and Straccia, 2013], and Probabilistic DLs [Lukasiewicz, 2008]. (1) Context dependent, see Example 29 in the full paper. (2) Only direct conflicts, as in Example 10 in the full paper. (3) Inconsistency may propagate to the entire KB. (4) Subsumption and assertion checking only, which suffice for application examples. (5) Currently proved for EL (Giovanni Casini, personal communication). sumption/assertion query answering over nonmonotonic vari- The informal meaning of C ⊑n D is: “all standard in- ants of the Gene Ontology with classical role fillers ranges stances, by default, satisfy C ⊑ D, unless stated otherwise”, from 0.25 to 2.46 seconds. that is, unless some higher priority axioms entail that some standard instances satisfy C ⊓ ¬D; in that case, C ⊑n D is N 2 The Family DL overridden. The instances of any concept NE are required to satisfy all the DIs that are not overridden in NE. N N Let DL be any description logic. The language of DL is The priority relation over DIs is denoted by ≺. DL obtained by adding a new concept name NC for each DL solves automatically only the conflicts that can be settled us- concept C. The new concepts are called normality concepts, ing ≺. Any other conflict should be regarded as a representa- although the term standard would be more appropriate since tion error (cf. the discussion of silent conflict removal in the N DL – unlike typicality logics and rational closure – has not introduction) and shall be resolved by the knowledge engi- been designed to model normality (as formalized by the KLM neer, typically by adding specific DIs. Unresolved conflicts postulates, for example). yield inconsistent normality concepts, that can be detected by N queries of the form NC ⊑ ⊥. DL has a utilitarian purpose, related to McCarthy’s N conventions [McCarthy, 1986], aimed at making knowledge The priority relation is a parameter of DL . The full pa- bases more manageable, e.g. by reducing their size, improv- per considers both the specificity-based priority of rational ing modularity, and so on. We may want to assert that by closure and a simpler kind of specificity adopted by circum- default a drug has contraindication X not because this is nor- scribed DLs [Bonatti et al., 2015a], namely: δ1 ≺ δ2 iff mally so, but rather for mitigating the effects of potential hu- man errors [Rector, 2004; Stevens et al., 2007]: It is safer KB |≈ pre(δ1) ⊑ pre(δ2) and KB 6|≈ pre(δ2) ⊑ pre(δ1) . to signal more contraindications than missing some; accord- N The expression α means that α is a conse- ingly, with the above default assertion, forgetting to define KB |≈ DL quence of . Due to space limitations, we do not report the the contraindications of a new drug cannot result in a miss- KB N model-theoretic definition of |≈ and present only its reduc- ing contraindication. The goal of DL is supporting such a tion to classical reasoning [Bonatti et al., 2015a]. For all sub- variety of knowledge engineering needs in a scalable way. Σ 1 N sumptions and assertions α, KB |≈ α holds iff KB |= α, A DL knowledge base is a disjoint union KB = S ∪ D N where Σ is the set of normality concepts explicitly occurring Σ where S is a finite set of DL inclusions and assertions and in KB∪{α}, and KB is a classical knowledge base obtained D is a finite set of defeasible inclusions (DIs, for short) that as follows (recall that KB = S ∪ D): are expressions C ⊑n D where C is a DL concept and D N a DL concept. If δ = (C ⊑n D), then pre(δ) and con(δ) 1In this classical translation, normality concepts are treated like denote C and D, respectively. new concept names. 4976 Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI-17) DLN complexity Mamm lungs Mamm fins knowledge base ⊑n ∃ ⊑n ¬∃ DL subsumption and and concept SeaAnim fins Whales Mamm SeaAnim complexity assertion checking ⊑n ∃ ⊑ ⊓ consistency Dolphins ⊑ Mamm ⊓ SeaAnim ⊓ ∃ fins P P P ExpTime ExpTime ExpTime Table 3: The KB formalizing the example N2ExpTime PN2ExpTime PN2ExpTime N All results hold for specificity and other priority relations in 3 Behavior of DL C, where is the complexity of subsumption in P C DL A DI δ = (C ⊑n D) is roughly similar to a set of defaults stating, for each normality concept NE, that the instances of Table 2: Some complexity results NE satisfy C ⊑ D unless stated otherwise (equivalently, δNE holds by default).
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