The Thirty-Fourth AAAI Conference on Artificial Intelligence (AAAI-20) A Practical Approach to Forgetting in Description Logics with Nominals Yizheng Zhao,1 Renate A. Schmidt,2 Yuejie Wang,3 Xuanming Zhang,4 Hao Feng5 1National Key Laboratory for Novel Software Technology, Nanjing University, China 2Department of Computer Science, The University of Manchester, UK 3School of Electronics Engineering and Computer Science, Peking University, China 4School of Computer Science, University of Nottingham Ningbo China, China 5School of Knowledge Engineering, North China University of Science and Technology, China [email protected], [email protected], {kathywangyuejie, hao.feng0429}@gmail.com [email protected] Abstract ticular signature. Uniform interpolation is thus a weaker no- tion of forgetting than semantic forgetting; the results of This paper investigates the problem of forgetting in descrip- uniform interpolation (uniform interpolants), are in general tion logics with nominals. In particular, we develop a prac- weaker than those of semantic forgetting (semantic solu- tical method for forgetting concept and role names from on- tologies specified in the description logic ALCO, extending tions). This means for a specific language semantic solutions the basic ALC with nominals. The method always terminates, are always uniform interpolants, but not the other way round. and is sound in the sense that the forgetting solution com- Practical methods for computing uniform interpolants in- puted by the method has the same logical consequences with clude the approach of LETHE (Koopmann and Schmidt the original ontology. The method is so far the only approach 2013a; 2013b; 2014; 2015), the one developed by (Ludwig to deductive forgetting in description logics with nominals. and Konev 2014), and the one by (Zhao et al. 2019). LETHE An evaluation of a prototype implementation shows that the handles ALCH, SIF, SHQ-TBoxes, and ALC-ontologies. method achieves a significant speed-up and notably better The latter two handle ALC-TBoxes. success rates than the LETHE tool which performs deductive forgetting for ALC-ontologies. Compared to FAME, a seman- FAME (Zhao and Schmidt 2018) is presently the only se- tic forgetting tool for ALCOIH-ontologies, better success mantic forgetting tool for ontologies. It is also the only se- rates are attained. From the perspective of ontology engineer- mantic forgetting tool for description logics with nominals; ing this is very useful, as it provides ontology curators with a it handles ALCOIH-ontologies (Zhao and Schmidt 2016). powerful tool to produce views of ontologies. In this approach, the target language is not the same as the source language, in particular, it is more expressive in order Introduction to capture semantic solutions. This is however not satisfac- tory for users like SNOMED CT (Spackman 2000) who do Forgetting is an ontology engineering technique that seeks to not have the flexibility to easily switch to a more expres- produce views of ontologies. This is achieved by eliminating sive language, or are bound by the application, the available from ontologie a subset of their signature, namely the forget- support and tooling, to a specific language. Tracking logical ting signature, in such a way that all logical consequences up difference between two ontology versions is one application to the remaining signature are preserved. Forgetting is useful where the target language should coincide with the source for many ontology engineering tasks such as reuse (Wang language (Zhao et al. 2019). et al. 2014), alignment and merging (Wang et al. 2005), In this paper we consider catering for the situation where versioning (Klein and Fensel 2001), debugging (Ribeiro ALCO and Wassermann 2009), repair (Troquard et al. 2018), log- the description logic serves as both source and tar- get languages. One option is to attempt to approximate the ical difference (Konev, Walther, and Wolter 2008; 2009; ALCO ALCO Ludwig and Konev 2014; Zhao et al. 2019), and related semantic solutions for inputs to -ontologies. tasks (Bicarregui et al. 2001; Lang, Liberatore, and Mar- Actually there is current research concerned with reduction quis 2003; Ghilardi, Lutz, and Wolter 2006; Eiter et al. 2006; of expressiveness by approximation (Brandt et al. 2002). However, rather than going down that route, we develop a Grau and Motik 2012; Ludwig and Konev 2013). ALCO Forgetting is basically a non-standard reasoning problem novel practical forgetting approach for -ontologies. which can be defined deductively as the dual of uniform in- The approach is deductive but incorporates also techniques terpolation (Visser 1996; Lutz and Wolter 2011) or model- from semantic forgetting, called Ackermann’s Lemma. The theoretically as semantic forgetting (Wang et al. 2014). Uni- method always terminates, and is sound in the sense that the form interpolation preserves logical consequences and se- forgetting solution computed by the method has the same mantic forgetting preserves semantic equivalence over a par- logical consequences with the original ontology. It is so far the only approach to deductive forgetting for description log- Copyright c 2020, Association for the Advancement of Artificial ics with nominals. An evaluation of the method over ALCO- Intelligence (www.aaai.org). All rights reserved. ontologies shows that it has similar speed as semantic for- 3073 getting with FAME, but with better success rates. An eval- Definition 1 says that the forgetting solution V has exactly uation over ALC ontologies shows that the method is both the same logical consequences with the original ontology O significantly faster than deductive forgetting of LETHE and in the remaining signature sig(O)\F. F is called the forget- exhibits better success rates. From the perspective of ontol- ting signature. V can be seen as a view of O for sig(O)\F. ogy engineering this is very useful, as it provides ontology Forgetting solutions (views) are unique up to logical equiv- curators with a powerful tool to produce views of ontologies. alence, that is, if both V and V are solutions of forgetting Proofs of all theorems and lemmas can be found in a long F from O, then they are logically equivalent, though their version of this paper, which can be downloaded via http:// representations may not be identical. www.cs.man.ac.uk/∼schmidt/publications/aaai20/. Normalization of ALCO-Ontologies ALCO-Ontologies and Forgetting Our method works with ALCO-ontologies in clausal nor- Let NC, NR and NI be pairwise disjoint and countably infi- nite sets of respectively concept names, role names and in- mal form. Clauses are obtained from corresponding axioms ALCO using the standard transformations based on logical equiv- dividual names (nominals). Concepts in have one of ¬∃r.C ≡∀r.¬C ¬∀r.C ≡∃r.¬C the following forms: alences such as , , and ¬¬C ≡ C. By incrementally applying the standard trans- |⊥|{a}|A |¬C | C D | C D |∃r.C |∀r.C, formations, any ALCO-ontology can be transformed into a set of clauses (Zhao 2018). where a ∈ NI, A ∈ NC, r ∈ NR, and both C and D denote arbitrary concepts in ALCO. Definition 2 (Clausal Normal Form). A literal in ALCO is An ALCO-ontology consists of a TBox and an ABox. A a concept of the form a, ¬a, A, ¬A, ∃r.C and ∀r.C, where C D TBox is a finite set of axioms of the form (concept a ∈ NI, A ∈ NC, r ∈ NR, and C is an arbitrary concept. A inclusions), where C and D are concepts. An ABox is a fi- clause in ALCO is a finite disjunction of literals. A clause is C(a) nite set of axioms of the form (concept assertions) and called an S-clause if it contains an S∈NC ∪ NR. the form r(a, b) (role assertions), where a, b ∈ NI, r ∈ NR, and C is a concept. In description logics with nominals, In the following we introduce two specialized normal form notions (based on clausal normal form), namely A- ABox assertions are superfluous, as they can be internalized r as concept inclusions via nominals, namely C(a) as a C reduced form and -reduced form. These two notions are and r(a, b) as a ∃r.b. Hence in this paper, we assume that important because they are used in the calculi of our method an ALCO-ontology is a finite set of concept inclusions. for concept name and role name elimination, respectively, The semantics of ALCO is defined in terms of an inter- described in detail in the next section. I I I pretation I = Δ , · , where Δ denotes the domain of Definition 3 (A-Reduced Form). Let N be a set of clauses, the interpretation, which is a non-empty set, and ·I denotes A ∈ (N ) A and let sigC . A clause is in -reduced form if it has the interpretation function, which assigns to every nominal CA C¬A C∃r.A C∃r.¬A C∀r.(AD) I I the form , , , , , a ∈ NI a singleton a ⊆ Δ , to every concept name A ∈ NC C ∀r.(¬A D) r ∈ C I I or , where NR is any role name, and a set A ⊆ Δ , and to every role name r ∈ NR a binary re- D A N A I I I I ( ) is a clause (concept) that does not contain . is in - lation r ⊆ Δ × Δ . The interpretation function · is in- reduced form if all A-clauses in N are in A-reduced form. ductively extended to concepts as follows: I I I I I I =Δ ⊥ = ∅ (¬C) =Δ \C The A-reduced form generalizes all basic forms of an A- (C D)I = CI ∩ DI (C D)I = CI ∪ DI clause in which a concept name could occur; a concept name I I I I could occur (either positively or negatively) at the top level (∃r.C) = {x ∈ Δ |∃y.(x, y) ∈ r ∧ y ∈ C } of a clause, or under an ∃-or∀-restriction.