Marco Cerami

On the (un)decidability of fuzzy description logics under Łukasiewicz t-norm

Recently there have been some unexpected results concerning Fuzzy Description Logics (FDLs) with General Concept Inclusions (GCIs). They show that, unlike the classical case, the DL ALC with GCIs does not have the finite model property under Lukasiewicz Logic or Product Logic, the proposed reasoning algorithms are neither correct nor complete and, specifically, knowledge base satisfiability is an undecidable problem for Product Logic. In this work, we show that knowledge base satisfiability is also an undecidable problem for Lukasiewicz Logic. We additionally provide a decision algorithm for acyclic ALC knowledge bases under Lukasiewicz Logic via a Mixed Integer Linear Programming (MILP) based procedure (note, however, that the decidability of this problem is already known). While similar MILP based algorithms have been proposed in the literature for acyclic ALC knowledge bases under Lukasiewicz Logic, none of them exhibit formal proofs of their correctness and completeness, which is the additional contribution here.

Finite-valued Lukasiewicz modal logic Is PSPACE-complete

It is well-known that satisfiability (and hence validity) in the minimal classical modal logic is a PSPACE-complete problem. In this paper we consider the satisfiability and validity problems (here they are not dual, although mutually reducible) for the minimal modal logic over a finite Lukasiewicz chain, and show that they also are PSPACE-complete. This result is also true when adding either the Delta operator or truth constants in the language, i.e. in all these cases it is PSPACE-complete.

From classical Description Logic to n-graded Fuzzy Description Logic

Description Logics (DLs) are knowledge representation languages built on the basis of classical logic. DLs allow the creation of knowledge bases and provide ways to reason on the contents of these bases. Fuzzy Description Logics (FDLs) are natural extensions of DLs for dealing with vague concepts, commonly present in real applications. Following the ideas of Hájek in [17] and García-Cerdaña et al. in [15] we develop a family of FDLs whose underlying logic is the fuzzy logic of a finite linearly ordered residuated lattice, that is, an n-graded fuzzy logic defined by a divisible finite t-norm over a finite chain. Moreover, the role of the constructor of implication in the languages for FDLs is discussed, and a hierarchy of AL-languages adapted to the behavior of the connectives in the fuzzy logics underlying these description languages is proposed. Finally, we deal with reasoning tasks within the framework of finitely valued DLs.

On finitely-valued Fuzzy Description Logics

This paper deals with finitely-valued fuzzy description languages from a logical point of view. From recent results in Mathematical Fuzzy Logic and following 44, we develop a Fuzzy Description Logic based on the fuzzy logic of a finite BL-chain. The constructors of the languages presented in this paper correspond to the connectives of that logic (containing an involutive negation, Monteiro-Baaz delta and hedges). The paper addresses the hierarchy of fuzzy attributive languages; knowledge bases and their reductions; reasoning tasks; and complexity. Our results regarding decidability together with a summary of the known results related to computational complexity are of particular interest. In Appendix B we also provide axiomatizations for expansions of the logic of a finite BL-chain considered in the paper.

On the implementation of a Fuzzy DL Solver over Infinite-Valued Product Logic with SMT Solvers

In this paper we explain the design and preliminary implementation of a solver for the positive satisfiability problem of concepts in a fuzzy description logic over the infinite-valued product logic. This very solver also answers 1-satisfiability in quasi-witnessed models. The solver works by first performing a direct reduction of the problem to a satisfiability problem of a quantifier free boolean formula with non-linear real arithmetic properties, and secondly solves the resulting formula with an SMT solver. We show that the satisfiability problem for such formulas is still a very challenging problem for even the most advanced SMT solvers, and so it represents an interesting problem for the community working on the theory and practice of SMT solvers.

On Finitely Valued Fuzzy Description Logics: The Łukasiewicz Case

Following the guidelines proposed by Hajek in [1], some proposals of research on Fuzzy Description Logics (FDLs) were given in [2]. One of them consists in the definition and development of a family of description languages, each one having as underlying fuzzy logic the expansion with an involutive negation and truth constants of the logic defined by a divisible finite t-norm. A general framework for finitely valued FDLs was presented in [3]. In the present paper we study the family of languages \(\mathcal{ALC}_{\textbf{\L}_n^c}\) based on the finitely valued Łukasiewicz logics with truth constants. In addition, we provide an interpretation of these FDLs into fuzzy multi-modal systems. We also deal with the corresponding reasoning tasks and their relationships, and we report some results on decidability and computational complexity.