A Non-monotonic Description Logic of Typicality

Logics in Artificial Intelligence, 2008

In this paper we propose a nonmonotonic extension ALC\mathcal{ALC} for reasoning about prototypical properties and inheritance with exception. The logic ALC\mathcal{ALC} . The operator T is intended to select the “most normal” or “most typical” instances of a concept, so that knowledge bases may contain subsumption relations of the form“T(C) is subsumed by P”, expressing that typical C-members have the

Fundamenta Informaticae, 2009

We extend the Description Logic ALC with a "typicality" operator T that allows us to reason about the prototypical properties and inheritance with exceptions. The resulting logic is called ALC + T. The typicality operator is intended to select the "most normal" or "most typical" instances of a concept. In our framework, knowledge bases may then contain, in addition to ordinary ABoxes and TBoxes, subsumption relations of the form "T(C) is subsumed by P ", expressing that typical C-members have the property P . The semantics of a typicality operator is defined by a set of postulates that are strongly related to Kraus-Lehmann-Magidor axioms of preferential logic P. We first show that T enjoys a simple semantics provided by ordinary structures equipped with a preference relation. This allows us to obtain a modal interpretation of the typicality operator. We show that the satisfiability of an ALC+T knowledge base is decidable and it is precisely EXPTIME. We then present a tableau calculus for deciding satisfiability of ALC + T knowledge bases. Our calculus gives a (suboptimal) nondeterministic-exponential time decision procedure for ALC + T. We finally discuss how to extend ALC + T in order to infer defeasible properties of (explicit or implicit) individuals. We propose two alternatives: (i) a nonmonotonic completion of a knowledge base; (ii) a "minimal model" semantics for ALC + T whose intuition is that minimal models are those that maximise typical instances of concepts.

2009

We present an extension \(\mathcal{EL}^{+^\bot}{\bf {\rm T}}\) of the description logic \(\mathcal{EL}^{+^\bot}\) for reasoning about prototypical properties and inheritance with exceptions. \(\mathcal{EL}^{+^\bot}{\bf {\rm T}}\) is obtained by adding to \(\mathcal{EL}^{+^\bot}\) a typicality operator T, which is intended to select the “typical” instances of a concept. In \(\mathcal{EL}^{+^\bot}{\bf {\rm T}}\) knowledge bases may contain inclusions of the form “T(C) is subsumed by P”, expressing that typical C-members have the property P. We show that the problem of entailment in \(\mathcal{EL}^{+^\bot}{\bf {\rm T}}\) is in co-NP.

2007

We extend the Description Logic \(\mathcal{ALC}\) with a “typicality” operator T that allows us to reason about the prototypical properties and inheritance with exceptions. The resulting logic is called \(\mathcal{ALC}+{\bf T}\) . The typicality operator is intended to select the “most normal” or “most typical” instances of a concept. In our framework, knowledge bases may then contain, in addition to ordinary ABoxes and TBoxes, subsumption relations of the form “T(C) is subsumed by P”, expressing that typical C-members have the property P. The semantics of a typicality operator is defined by a set of postulates that are strongly related to Kraus-Lehmann-Magidor axioms of preferential logic P. We first show that T enjoys a simple semantics provided by ordinary structures equipped by a preference relation. This allows us to obtain a modal interpretation of the typicality operator. Using such a modal interpretation, we present a tableau calculus for deciding satisfiability of \(\mathcal{ALC}+{\bf T}\) knowledge bases. Our calculus gives a nondeterministic-exponential time decision procedure for satisfiability of \(\mathcal{ALC}+{\bf T}\) . We then extend \(\mathcal{ALC}+{\bf T}\) knowledge bases by a nonmonotonic completion that allows inferring defeasible properties of specific concept instances.

2009

In this work we summarize our recent results on extending Description Logics for reasoning about prototypical properties and inheritance with exceptions. First, we focus our attention on the logic ALC. We present a nonmonotonic logic ALC + Tmin, which is built upon a monotonic logic ALC + T obtained by adding a typicality operator T to ALC. The operator T is intended to select the "most normal" or "most typical" instances of a concept, so that knowledge bases may contain subsumption relations of the form"T(C) is subsumed by P ", expressing that typical C-members have the property P . In order to perform nonmonotonic inferences, we define a "minimal model" semantics: the intuition is that preferred, or minimal models are those that maximise typical instances of concepts. By means of ALC +Tmin we are able to infer defeasible properties of (explicit or implicit) individuals. We also show that the satisfiability of an ALC + T-knowledge base is in EXPTIME, whereas deciding query entailment in ALC + Tmin is in co-NExp NP . We apply our approach based on the operator T also to the low complexity Description Logic EL + ⊥ . We propose an extension EL + ⊥ T and we show that the problem of entailment in EL + ⊥ T is in co-NP.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2011

In this paper we introduce a tableau calculus for a nonmonotonic extension of the low complexity Description Logic DL-Litecore of the DL-Lite family. The extension, called DL-LitecTmin, can be used to reason about typicality and defeasible properties. The calculus performs a two-phase computation to check whether a query is minimally entailed from the initial knowledge base. It is sound, complete and terminating. Furthermore, it is a decision procedure for DL-LitecTmin knowledge bases, whose complexity matches the known results for the logic, namely that entailment is in Π p 2 .

2004

We propose a rigorous definition of the notion of typicality, making use of the strict partial order naturally induced among the objects at hand by a given concept. This perspective enables us to transpose in the framework of object determination logic some well-known problems like contextual typicality or contextual inference. Then these problems can be treated with the help of the tools developed in the study of non monotonic logics. We show that it is possible to work in a logical formalism, using a new determination connector : the concept f g, when it exists, denotes the determination of the concept f by the concept g. Relatively to this connector, the relation of typical inference between concepts turns out to satisfy the important property of rational monotony.

Italian Conference on Theoretical Computer Science, 2017

We present RAT-OWL, a software system for reasoning about typicality in preferential Description Logics. It is implemented in the form of a Protégé 4.3 Plugin and it allows the user to reason in a nonmonotonic extension of Description Logics based on the notion of "rational closure". This logic extends standard Description Logics in order to express "typical" properties, that can be directly specified by means of a typicality operator T: a TBox can contain inclusions of the form T(C) D to represent that "typical Cs are also Ds". We show experimental results, indicating that the performances of RAT-OWL are promising.

The Twenty Sixth International Flairs Conference, 2013

The difference between typical instances and atypical instances in a natural categorization process has been introduced by E. Rosh and studied by cognitive psychology and AI. A lot of the knowledge representation systems are expressed in using fuzzy concepts but a degree of membership raises some problem for natural categorizations (especially to classification problems in anthropology, ethnology, archeology, linguistics but also in ontologies), but atypical instances of a concept cannot be apprehended adequately by different degrees from a prototype. Other formal approaches, as paraconsistent logics or non monotonic logics, conceptualize often atypical objects as exceptions. It had yet been developed an alternative way with the logics of determination of the objects (LDO). In this paper, we present the logics of typical and atypical (LTA) in order to give directly a logical approach of typicality / atypicality associated to a concept by a more common way than in LDO, in using only classes and not determination operators. It is introduced a distinction between predicative property and concept defined with its intension and its essence, a part of intension. A typical instance of a concept inherits all properties of intension; a typical instance inherits only properties of essence but it is a full member of the category associated to a concept and not a member with a weak degree of membership. In natural categorization, there are often instances (the exceptions) which do not inherit some properties of the essence; they cannot be considered as atypical instance and belong to the boundary of the category.

We introduce Propositional Typicality Logic (PTL), a logic for reasoning about typicality. We do so by enriching classical propositional logic with a typicality operator of which the intuition is to capture the most typical (or normal) situations in which a formula holds. The semantics is in terms of ranked models as studied in KLM-style preferential reasoning. This allows us to show that rational consequence relations can be embedded in our logic. Moreover we show that we can define consequence relations on the language of PTL itself, thereby moving beyond the propositional setting. Building on the existing link between propositional rational consequence and belief revision, we show that the same correspondence holds for rational consequence and belief revision on PTL. We investigate entailment for PTL, and propose two appropriate notions thereof.