Davide Ciucci

Fuzzy Ontology, Fuzzy Description Logics and Fuzzy-OWL

The conceptual formalism supported by an ontology is not sufficient for handling vague information that is commonly found in many application domains. We describe how to introduce fuzziness in an ontology. To this aim we define a framework consisting of a fuzzy ontology based on Fuzzy Description Logic and Fuzzy---Owl.

Algebraic structures for rough sets

Using as example an incomplete information system with support a set of objects X, we discuss a possible algebraization of the concrete algebra of the power set of X through quasi BZ lattices. This structure enables us to define two rough approximations based on a similarity and on a preclusive relation, with the second one always better that the former. Then, we turn our attention to Pawlak rough sets and consider some of their possible algebraic structures. Finally, we will see that also Fuzzy Sets are a model of the same algebras. Particular attention is given to HW algebra which is a strong and rich structure able to characterize both rough sets and fuzzy sets.

A map of dependencies among three-valued logics

Three-valued logics arise in several fields of computer science, both inspired by concrete problems (such as in the management of the null value in databases) and theoretical considerations. Several three-valued logics have been defined. They differ by their choice of basic connectives, hence also from a syntactic and proof-theoretic point of view. Different interpretations of the third truth value have also been suggested. They often carry an epistemic flavor. In this work, relationships between logical connectives on three-valued functions are explored. Existing theorems of functional completeness have laid bare some of these links, based on specific connectives. However we try to draw a map of such relationships between conjunctions, negations and implications that extend Boolean ones. It turns out that all reasonable connectives can be defined from a few of them and so all known three-valued logics appear as a fragment of only one logic. These results can be instrumental when choosing, for each application context, the appropriate fragment where the basic connectives make full sense, based on the appropriate meaning of the third truth-value.

Fuzzy Ontology and Fuzzy-OWL in the KAON Project

In many application domains is necessary to reason with vague information, but, the conceptual formalism supported by a classical ontology is not sufficient for handling it. Thus, we have developed a suited plug-in of the KAON Project in order to introduce fuzziness in an ontology. This extension is based on corresponding fuzzy ontology and Fuzzy-OWL notions.

Automatic Labeling of Topics

An algorithm for the automatic labeling of topics accordingly to a hierarchy is presented. Its main ingredients are a set of similarity measures and a set of topic labeling rules. The labeling rules are specifically designed to find the most agreed labels between the given topic and the hierarchy. The hierarchy is obtained from the Google Directory service, extracted via an ad-hoc developed software procedure and expanded through the use of the OpenOffice English Thesaurus. The performance of the proposed algorithm is investigated by using a document corpus consisting of 33,801 documents and a dictionary consisting of 111,795 words. The results are encouraging, while particularly interesting and significant labeling cases emerged

Orthopairs: A Simple and Widely UsedWay to Model Uncertainty

The term orthopair is introduced to group under a unique definition different ways used to denote the same concept. Some orthopair models dealing with uncertainty are analyzed both from a mathematical and semantical point of view, outlining similarities and differences among them. Finally, lattice operations on orthopairs are studied and a survey on algebraic structures is provided.

Borderline vs. unknown

In this paper we compare the expressive power of elementary representation formats for vague, incomplete or conflicting information. These include Boolean valuation pairs introduced by Lawry and Gonzalez-Rodriguez, orthopairs of sets of variables, Boolean possibility and necessity measures, three-valued valuations, supervaluations. We make explicit their connections with strong Kleene logic and with Belnap logic of conflicting information. The formal similarities between 3-valued approaches to vagueness and formalisms that handle incomplete information often lead to a confusion between degrees of truth and degrees of uncertainty. Yet there are important differences that appear at the interpretive level: while truth-functional logics of vagueness are accepted by a part of the scientific community (even if questioned by supervaluationists), the truth-functionality assumption of three-valued calculi for handling incomplete information looks questionable, compared to the non-truth-functional approaches based on Boolean possibility-necessity pairs. This paper aims to clarify the similarities and differences between the two situations. We also study to what extent operations for comparing and merging information items in the form of orthopairs can be expressed by means of operations on valuation pairs, three-valued valuations and underlying possibility distributions. We explore the connections between several representations of imperfect information.In each case we compare the expressive power of these formalisms.In each case we study how to express aggregation operations.We demonstrate the formal similarities among these approaches.We point out the differences in interpretations between these approaches.

Lattices with Interior and Closure Operators and Abstract Approximation Spaces

The non---equational notion of abstract approximation space for roughness theory is introduced, and its relationship with the equational definition of lattice with Tarski interior and closure operations is studied. Their categorical isomorphism is proved, and the role of the Tarski interior and closure with an algebraic semantic of a S4---like model of modal logic is widely investigated. A hierarchy of three particular models of this approach to roughness based on a concrete universe is described, listed from the stronger model to the weaker one: (1) the partition spaces, (2) the topological spaces by open basis, and (3) the covering spaces.

Algebraic models of deviant modal operators based on de Morgan and Kleene lattices

An algebraic model of a kind of modal extension of de Morgan logic is described under the name MDS5 algebra. The main properties of this algebra can be summarized as follows: (1) it is based on a de Morgan lattice, rather than a Boolean algebra; (2) a modal necessity operator that satisfies the axioms N, K, T, and 5 (and as a consequence also B and 4) of modal logic is introduced; it allows one to introduce a modal possibility by the usual combination of necessity operation and de Morgan negation; (3) the necessity operator satisfies a distributivity principle over joins. The latter property cannot be meaningfully added to the standard Boolean algebraic models of S5 modal logic, since in this Boolean context both modalities collapse in the identity mapping. The consistency of this algebraic model is proved, showing that usual fuzzy set theory on a universe U can be equipped with a MDS5 structure that satisfies all the above points (1)-(3), without the trivialization of the modalities to the identity mapping. Further, the relationship between this new algebra and Heyting-Wajsberg algebras is investigated. Finally, the question of the role of these deviant modalities, as opposed to the usual non-distributive ones, in the scope of knowledge representation and approximation spaces is discussed.

Heyting Wajsberg Algebras as an Abstract Environment Linking Fuzzy and Rough Sets

Heyting Wajsberg (HW) algebras are introduced as algebraic models of a logic equipped with two implication connectives, the Heyting one linked to the intuitionistic logic and the Wajsberg one linked to the ?ukasiewicz approach to many-valued logic. On the basis of an HW algebra it is possible to obtain a de Morgan Brouwer-Zadeh (BZ) distributive lattice with respect to the partial order induced from the ?ukasiewicz implication. Modal-like operators are also defined generating a rough approximation space. It is shown that standard Pawlak approach to rough sets is a model of this structure.