Cristina Coppola Distances and Closeness Measures in Information Spaces

The notion of metric space plays a basic role in several researches addressed to process information. Indeed the objects we will investigate are represented by points and the distance is a measure of " dissimilarity " between objects. Now, the question that arises is if such a notion is the better one in a context in which we are not able to obtain complete information about the considered objects. This thesis is devoted to face this question, by giving suitable axioms extending the usual ones for metric spaces and by considering regions in a suitable space, instead of the points. This idea originates from A. N. Whitehead's researches, aimed to define a geometry without the concept of point as primitive (see [46], [47] and [48]) and from a metrical version of these researches, proposed by G. Gerla (see, for example [23], [24]). Indeed, we can re-interpret the regions as " incomplete pieces of information " and the diameter of a region as a measure of the vagueness of the available information: the bigger it is, the more there is uncertainty. Points (having zero-diameter) represent complete information. Another idea examined in this thesis is the possibility of referring to the " logical " notion of closeness instead of the one of distance. Indeed, there is a duality between these concepts, that is easily understandable: when comparing objects accordingly to their properties, we can use both a measure of how they are " similar " and a measure of how they are " dissimilar " ; the smaller the distance is, the bigger the closeness is. We investigate the notion of closeness in the fuzzy domain, examining similarities and fuzzy orders. More precisely, the thesis is structured as follows. In Chapter 1 we first give some necessary basic notions in multi-valued logics. Then we give some information about the metric structures we will start from and Distance and closeness measures in information spaces 6 we show some already known dualities between the metric notions and the fuzzy relations. In Chapter 2 we propose an approach to establish a link between point-free geometry and the categorical approach to fuzzy sets theory (as proposed by Höhle in [28]). In particular, starting from the definition of pointless metric spaces, we introduce the pointless ultrametric spaces. Then we define the semimetric spaces, the semisimilarities on some spaces, and we verify the relations between these two kind …