Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015) Efficient Paraconsistent Reasoning with Ontologies and Rules∗ Tobias Kaminski and Matthias Knorr and Joao˜ Leite NOVA LINCS Departamento de Informatica´ Universidade NOVA de Lisboa 2829-516 Caparica, Portugal Abstract Krotzsch¨ et al., 2011], others follow a hybrid combination of ontologies with nonmonotonic rules, either providing a Description Logic (DL) based ontologies and non- modular approach where rules and ontologies use their own monotonic rules provide complementary features semantics, and allowing limited interaction between them whose combination is crucial in many applications. [Eiter et al., 2008], or defining a unifying framework for both In hybrid knowledge bases (KBs), which combine components [Motik and Rosati, 2010; Knorr et al., 2011]. both formalisms, for large real-world applications, Equipped with semantics that are faithful to their constituting often integrating knowledge originating from dif- parts, these proposals allow for the specification of so-called ferent sources, inconsistencies can easily occur. hybrid knowledge bases (hybrid KBs) either from scratch, These commonly trivialize standard reasoning and benefiting from the added expressivity, or by combining ex- prevent us from drawing any meaningful conclu- isting ontologies and rule bases. sions. When restoring consistency by changing the The complex interactions between the ontology component KB is not possible, paraconsistent reasoning offers and the rule component of these hybrid KBs – even more so an alternative by allowing us to obtain meaningful when they result from combining existing ontologies and rule conclusions from its consistent part. bases independently developed – can easily lead to contra- In this paper, we address the problem of efficiently dictions, which, under classical semantics, trivialize standard obtaining meaningful conclusions from (possibly reasoning and prevent us from drawing any meaningful con- inconsistent) hybrid KBs. To this end, we de- clusions, ultimately rendering these hybrid KBs useless. fine two paraconsistent semantics for hybrid KBs One way to deal with this problem is to employ some which, beyond their differentiating properties, are method based on belief revision (e.g. [Leite, 2003; Osorio faithful to well-known paraconsistent semantics as and Cuevas, 2007; Slota and Leite, 2012a; 2014; Delgrande well as the non-paraconsistent logic they extend, et al. , 2013] for LPs, [Flouris et al., 2008; Calvanese et al., and tractable if reasoning in the DL component is. 2010; Kharlamov et al., 2013] for DLs, and [Slota et al., 2011; Slota and Leite, 2012b] for hybrid KBs) to regain con- 1 Introduction sistency so that standard reasoning services can be used, or some method based on repairing (e.g. [Arenas et al., 1999] In this paper, we address the problem of dealing with incon- for LPs and [Haase et al., 2005] for DLs) where hypotheti- sistent knowledge bases consisting of ontologies and non- cal belief revision is employed for consistent query answer- monotonic rules, following a paraconsistent reasoning ap- ing, without actually changing the KB. However, this is not proach with a focus on efficiency. always feasible e.g. because we may not have permission to Description Logics (DLs) and Logic Programs (LPs) pro- change the KB – as for instance in [Alberti et al., 2011] where vide different strengths when used for Knowledge Represen- the KB encodes laws and norms – or because the usual high tation and Reasoning. While DLs employ the Open World complexity of belief revision and repairing methods simply Assumption and are suited for defining ontologies, LPs adopt renders their application prohibitive. When these methods the Closed World Assumption and are able to express non- are not possible or not feasible, paraconsistent reasoning ser- monotonic rules with exceptions and preference orders. Com- vices, typically based on some many-valued logics, offer an bining features of both formalisms has been actively pursued alternative by being able to draw meaningful conclusions in over the last few years, resulting in different proposals with the presence of contradiction. Whereas paraconsistent rea- different levels of integration and complexity: while some soning has been extensively studied in the context of each extend DLs with rules [Horrocks and Patel-Schneider, 2004; individual component of hybrid KBs (see Related Work in ∗Partially supported by Fundac¸ao˜ para a Cienciaˆ e a Tecnologia Sect. 5), it is still a rather unexplored field in the context of hy- under project PTDC/EIA-CCO/121823/2010 and strategic project brid KBs. Notable exceptions are [Huang et al., 2011; 2014; PEst/UID/CEC/04516/2013. M. Knorr was also supported by grant Fink, 2012], yet their computation is not tractable in general SFRH/BPD/86970/2012. even if reasoning in the DL component is. 3098 In this paper, we investigate efficient paraconsistent seman- sented by '[s=x].A closed MKNF formula contains no free tics for hybrid KBs. We adopt the base framework of [Motik variables. Formulas of the form K ' and not ' are called, and Rosati, 2010] because of its generality and tight integra- resp., K-atoms and not-atoms. Intuitively, K ' asks if ' is tion between the ontology and the rules – c.f. [Motik and known, while not' checks if ' does not hold, which allows Rosati, 2010] for a thorough discussion – under the semantics one to draw conclusions from the absence of information. of [Knorr et al., 2011] because of its computational proper- Hybrid KBs combine a set of MKNF rules and a DL on- ties. We extend such semantics with additional truth values tology O, which is translatable into a function-free first order to evaluate contradictory pieces of knowledge, following two formula with equality π(O) and for which checking of satis- common views on how to deal with contradictory knowledge fiability and instances are decidable [Baader et al., 2003]. An bases. According to one view, contradictions are dealt with MKNF rule r is of the given form where H, Ai, Bi are atoms: locally, in a minimally intrusive way, such that a new truth value is introduced to model inconsistencies, while consistent KH KA1;:::; KAn; notB1;:::; notBm: (1) pieces of information whose derivation depends on inconsis- K H is called the rule head, and the sets fK Aig and tent information are still considered to be true in the classi- fnotBjg are called the positive body and the negative body, cal sense. This view is adopted in paraconsistent semantics respectively. A rule r is positive if m = 0, and r is a fact if for DLs, e.g. [Maier et al., 2013], LPs, e.g. [Sakama, 1992; n = m = 0.A program P is a finite set of MKNF rules, Sakama and Inoue, 1995], and hybrid KBs [Huang et al., and positive if all rules in it are positive. A hybrid knowledge 2011; Fink, 2012]. The alternative view is to distinguish base (hybrid KB) K is a pair (O; P). Given such K = (O; P), truth which depends on the inconsistent part of a KB, from KG = (O; PG) is a ground hybrid KB where PG denotes the truth which is derivable without involving any contradictory grounding of P using all constants occurring in K as usual. knowledge. This view, commonly referred to as Suspicious As hybrid KBs can be translated into MKNF formulas using Reasoning, is adopted in paraconsistent semantics for LPs, π(O) and the match between ! and ⊃ for the MKNF rules, e.g. [Alferes et al., 1995; Sakama, 1992; Sakama and Inoue, their semantics is derived from that of MKNF formulas, and 1995] and hybrid KBs [Huang et al., 2014]. we abuse notation and refer to such translation π(K) by K. To We present solutions following both views through the def- achieve decidability of reasoning, hybrid KBs are restricted to inition of a five-valued and a six-valued paraconsistent se- be DL-safe, basically requiring that variables in rules appear mantics for hybrid KBs, the latter implementing Suspicious at least once in the positive body under a predicate which does Reasoning, both of which enjoy the following properties: not occur in O, thus limiting the applicability of constants in • Soundness w.r.t. the three-valued semantics for consis- rules to those in P. From now on, we only consider DL-safe tent hybrid KBs by [Knorr et al., 2011]; hybrid KBs as it can always be ensured [Ivanov et al., 2013]. • Faithfulness w.r.t. semantics for its base formalisms; Example 1. Consider the following simplified ground hybrid • Computability by means of a sound and complete fix- KB KG for assessing the risk of goods at a port. point algorithm; • Tractability when a tractable DL is used for the ontology. HasCertifiedSender v :IsMonitored (2) The paper is organized as follows: in Sect. 2, we present KIsMonitored(g) Krisk(g): (3) the formal background; in Sect. 3, we present both semantics, Krisk(g) notisLabelled(g): (4) starting with common parts, and proceeding with the five- KisLabelled(g) notrisk(g): (5) valued semantics in Sect. 3.1, and the six-valued one in Sect. KresolvedRisk(g) KIsMonitored(g): (6) 3.2; in Sect. 4, we investigate the properties of both semantics and discuss related work and conclude in Sect. 5. KHasCertifiedSender(g) (7) (4) and (5) state that good g is either a risk (r) or it is labeled 2 Preliminaries (iL). Any risk is monitored (IM) (3), thus a resolved risk (rR) (6). As g has a certified sender (HCS) (7), it can be In this section, we introduce hybrid knowledge bases, and proven by means of axiom (2) that it is not monitored. Thus, also recall the syntax of MKNF formulas, originating from g can be derived to be monitored and not monitored at the the logic of minimal knowledge and negation as failure same time if it is considered to be a risk, which can only be (MKNF) [Lifschitz, 1991], into which the former are embed- modeled by means of a paraconsistent evaluation.