Fuzzy Similarities in Stars/Galaxies Classification Salvatore Sessa 1 Roberto Tagliaferri 2 Giuseppe Longo 3 Angelo Ciaramella 2 Antonino Staiano 2 1 Dipartimento di Costruzioni e Metodi Matematici in Architettura, Universit`a di Napoli, via Monteoliveto, 3 - 80134 Napoli, Italy E-mail: sessa@unina.it 2 Dipartimento di Matematica e Informatica, Universit`a di Salerno, Via Allende, 84081 - Baronissi, Salerno, Italy, and INFM Unit`a di Salerno . E-mail: (robtag, ciaram, staiano)@unisa.it. 3 Dipartimento di Scienze Fisiche, Universit`a di Napoli, Via Cintia, 80126 Napoli, Italy E-mail: longo@na.infn.it Abstract — By basing on the concept of fuzzy sim- ilarity with respect to a continuous triangular norm built via the well known method of ordinal sums, we propose a modification of an our previous al- gorithm which improves the performance of the Stars/Galaxies classification in astronomical data mining. This algorithm is implemented in Matlab 6 and we make also use of Fuzzy c-Means cluster- ing algorithm for constructing two prototypes with respect to which the fuzzy similarity is calculated. Keywords — Fuzzy Similarity, Neural Network I. Introduction The popularity of fuzzy logic comes mainly from many successful applications, where linguistic vari- ables are suitably transformed in fuzzy sets, com- bined via the conjunction and disjunction opera- tions by using a continuous triangular norm t : [0, 1] 2 -→ [0, 1] and the related dual conorm, respectively. As implication, the correspondent residuum -→ t is also used. For simplicity, in the sequel we write t-norm instead of triangular norm. It is well known that [1] a t-norm t is an associa- tive, commutative and monotone (in both variables) ope5ration such that t(x, 0) = 0 and t(x, 1) = x for any x ∈ [0, 1]. Throughout this paper, we use the standard notation xty instead of t(x, y), where x, y ∈ [0, 1] and we assume t to be continuous in classical sense. The related residuum operator -→ t is defined as (x -→ t y)= sup{z ∈ [0, 1] : xtz ≤ y}. The most used t-norm in fuzzy logic and the corre- spondent residua are the following [2]: G¨odel t-norm (minimum): x ∧ y = min{x, y} x → M y =  1 if x ≤ y y otherwise Goguen t-norm (product): xP y = xy x → P y =  1 if x ≤ y y x otherwise Lukasiewicz t-norm: xLy = max{0,x + y - 1} x → L y = min{1, 1 - x + y} These t-norms are evidently continuous. Later we shall also use the concept of bi-residuum (with respect the t-norm t) defined as (x ←→ t y) = (x -→ t y) ∧ (y -→ t x) For the above t-norms we have respectively: (x ←→ M y)=  1 if x = y min{x, y} otherwise (x ←→ P y)=  1 if x = y min{ y x , x y } otherwise (x ←→ L y) = 1 - max{x, y} + min{x, y} = = 1 -|x - y| In order to build an algebraic structure in which one considers a logic more general than the three logics (Lukasiewicz, Product and G¨odel logic) com- monly employed in fuzzy set theory (in narrow sense), Hajek [2] has invented the BL-algebras. A continuous t-norm induces a structure of BL-algebra over [0, 1], henceforth we assume such a structure just for formalizing the results of Section 2. In the context of BL-algebras [2], the concept of similar- ity is a direct generalization of the fuzzified equal- ity. Indeed, if X is a referential finite set and