A finite basis theorem for the description logic ${\cal ALC}$

We define the notion of rational closure in the context of Description Logics. We start from an extension of ALC with a typicality operator T allowing to express concepts of the form T(C), whose meaning is to select the "most normal" instances of a concept C. The semantics we consider is based on rational models and exploits a minimal models mechanism based on the minimization of the rank of domain elements. We show that this semantics captures exactly a notion of rational closure which is a natural extension to Description Logics of Lehmann and Magidor's one. We also extend the notion of rational closure to the ABox, by providing an EXPTIME algorithm for computing it that is sound and complete with respect to the minimal model semantics.

Studia Logica, 1978

Lecture Notes in Computer Science, 2012

We prove that any concept in any description logic that extends ALC with some features amongst I (inverse), Q k (quantified number restrictions with numbers bounded by a constant k), Self (local reflexivity of a role) can be learnt if the training information system is good enough. That is, there exists a learning algorithm such that, for every concept C of those logics, there exists a training information system consistent with C such that applying the learning algorithm to the system results in a concept equivalent to C.

Texts in Theoretical Computer Science an EATCS Series, 2007

Fuzzy Sets and Systems, 2021

The aim of the present paper is to prove that concept validity and positive satisfiability with an empty ontology in the Fuzzy Description Logic IALE, under standard product semantics and with respect to quasi-witnessed models, are decidable. In our framework we are not considering reasoning tasks over ontologies. The proof of our result consists in reducing the problem to a finitary consequence problem in propositional product logic with Monteiro-Baaz delta operator, which is known to be decidable. Product FDL and first order logic are known not to enjoy the finite model property, so we cannot restrict to finite interpretations. Thus, in order to obtain our result, we need to codify infinite interpretations using a finite number of propositional formulas. Such result was conjectured in [10], but the proof given was subsequently found incorrect. In the present work an improved reduction algorithm is proposed and a proof of the same result is provided.

ArXiv, 2019

We extend description logics (DLs) with non-monotonic reasoning features. We start by investigating a notion of defeasible subsumption in the spirit of defeasible conditionals as studied by Kraus, Lehmann and Magidor in the propositional case. In particular, we consider a natural and intuitive semantics for defeasible subsumption, and investigate KLM-style syntactic properties for both preferential and rational subsumption. Our contribution includes two representation results linking our semantic constructions to the set of preferential and rational properties considered. Besides showing that our semantics is appropriate, these results pave the way for more effective decision procedures for defeasible reasoning in DLs. Indeed, we also analyse the problem of non-monotonic reasoning in DLs at the level of entailment and present an algorithm for the computation of rational closure of a defeasible ontology. Importantly, our algorithm relies completely on classical entailment and shows t...

Fuzzy Sets and Systems, 2011

Fuzzy Description Logics (DLs) are a family of logics which allow the representation of (and the reasoning with) structured knowledge affected by vagueness. Although most of the not very expressive crisp DLs, such as ALC, enjoy the Finite Model Property (FMP), this is not the case once we move into the fuzzy case. In this paper we show that if we allow arbitrary knowledge bases, then the fuzzy DLs ALC under Lukasiewicz and Product fuzzy logics do not verify the FMP even if we restrict to witnessed models; in other words, finite satisfiability and witnessed satisfiability are different for arbitrary knowledge bases. The aim of this paper is to point out the failure of FMP because it affects several algorithms published in the literature for reasoning under fuzzy ALC.

Lecture Notes in Computer Science, 2006

We study the deterministic Horn fragment of ALC, which restricts the general Horn fragment of ALC only in that, the constructor ∀R.C is allowed in bodies of program clauses and queries only in the form ∀∃R.C, which is defined as ∀R.C ∃R.C. We present an algorithm that for a deterministic positive logic program P given as a TBox constructs a finite least pseudo-model I of P such that for every deterministic positive concept C, P |= C iff I validates C (and more strongly, iff I, τ |= C, where τ is the distinguished object of I and the satisfaction means τ is an instance of C w.r.t. I). Pseudo-interpretations are very similar to (traditional) interpretations, except that they have two interpretation functions for roles, one to deal with the constructor ∃R.C and the other to deal with ∀R.C. They are ordered by comparing the sets of validated positive concepts. Our algorithm runs in time 2 O(n) and returns a pseudo-interpretation of size 2 O(n). Our method is extendable for instance checking w.r.t. knowledge bases containing also an ABox in more expressive description logics.

Vietnam Journal of Computer Science

It is well known that any Boolean function in classical propositional calculus can be learned correctly if the training information system is good enough. In this paper, we extend that result for description logics. We prove that any concept in any description logic that extends ALC with some features amongst I (inverse roles), Q k (qualified number restrictions with numbers bounded by a constant k), and Self (local reflexivity of a role) can be learned correctly if the training information system (specified as a finite interpretation) is good enough. That is, there exists a learning algorithm such that, for every concept C of those logics, there exists a training information system such that applying the learning algorithm to it results in a concept equivalent to C. For this result, we introduce universal interpretations and bounded bisimulation in description logics and develop an appropriate learning algorithm. We also generalize common

arXiv (Cornell University), 2022

We present in this paper a reformulation of the usual set-theoretical semantics of the description logic ALC with general TBoxes by using categorical language. In this setting, ALC concepts are represented as objects, concept subsumptions as arrows, and memberships as logical quantifiers over objects and arrows of categories. Such a category-based semantics provides a more modular representation of the semantics of ALC. This feature allows us to define a sublogic of ALC by dropping the interaction between existential and universal restrictions, which would be responsible for an exponential complexity in space. Such a sublogic is undefinable in the usual set-theoretical semantics. We show that this sublogic is PSPACE by proposing a deterministic algorithm for checking concept satisfiability which runs in polynomial space.