It is widely recognized today that the management of imprecision and vagueness will yield more intelligent and realistic knowledge-based applications. Description Logics (DLs) are a family of knowledge representation languages that have gained considerable attention the last decade, mainly due to their decidability and the existence of empirically high performance of reasoning algorithms. In this paper, we extend the well known fuzzy ALC DL to the fuzzy SHIN DL, which extends the fuzzy ALC DL with transitive role axioms (S), inverse roles (I), role hierarchies (H) and number restrictions (N). We illustrate why transitive role axioms are difficult to handle in the presence of fuzzy interpretations and how to handle them properly. Then we extend these results by adding role hierarchies and finally number restrictions. The main contributions of the paper are the decidability proof of the fuzzy DL languages fuzzy-SI and fuzzy-SHIN, as well as decision procedures for the knowledge base satisfiability problem of the fuzzy-SI and fuzzy-SHIN. 1.

Abstract. We present and discuss some novel and somewhat surprising complexity results for a basic but significant fuzzy description logic (DL) which extends the classical ALC language. In particular we show that checking the consistency of a concept or a KB in fuzzy DLs has a complexity which jumps from linear-time to EXPTIME-complete, while the subsumption problem is always (at least) as hard as in crisp DLs. 1

Abstract. In this paper we introduce reasoning procedures for ALCQ + F, a fuzzy description logic with extended qualified quantification. The language allows for the definition of fuzzy quantifiers of the absolute and relative kind by means of piecewise linear functions on N and Q ∩ [0, 1] respectively. In order to reason about instances, the semantics of quantified expressions is defined based on recently developed measures of the cardinality of fuzzy sets. A procedure is described to calculate the fuzzy satisfiability of a fuzzy assertion, which is a very important reasoning task. The procedure considers several different cases and provides direct solutions for the most frequent types of fuzzy assertions. 1

By the compactness property of a logic we mean that nite satis - ability implies satis ability for a set of formulas. This property of the classical logic was successfully generalized to several fuzzy logics, including the Lukasiewicz logic. We prove that the product logic does not have the compactness property.

We study self-referential sentences of the type related to the Liar paradox. In particular, we consider the problem of assigning consistent fuzzy truth values to collections of self-referential sentences. We show that the problem can be reduced to the solution of a system of nonlinear equations. Furthermore, we prove that, under mild conditions, such a system always has a solution (i.e. a consistent truth value assignment) and that, for a particular implementation of logical “and”, “or” and “negation”, the “mid-point ” solution is always consistent. Next we turn to computational issues and present several truth-value assignment algorithms; we argue that these algorithms can be understood as generalized sequential reasoning. In an Appendix we present a large number of examples of self-referential collections (including the Liar and the Strengthened Liar), we formulate the corresponding truth value equations and solve them analytically and / or numerically.

In this work we propose a general approach for representing uncertainty measures in the framework of t-norm based logics. This approach is extended also to classes of measures like probability, possibility, necessity, lower and upper probability. We show that, under certain conditions, the logical consistency of a theory of uncertainty is tantamount to the coherence of a related assessment of rational values. Finally, we characterize the basic requirements that guarantee the compactness of coherent assessments. Keywords: T-norm based logics, Fuzzy measures, Coherence, Compactness.