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1

The Lattice Structure of Equally Extended Signed Partitions : A generalization of the Brylawski approach to integer partitions with a p-n junction between two semiconductors model

Cattaneo G., Chiaselotti G., Gentile T., Oliverio P. A.

Fundamenta Informaticae

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2015

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Vol. 141, nr 1

1--36

EN

Signed partitions are used in order to describe a new discrete dynamical model whose configurations have fixed sum and whose evolution rules act in balancing from left and right on the configurations of the system. The resulting model can be considered as an extension to the case of signed partitions of the discrete dynamical system introduced by Brylawski in his classical paper concerning the dominance order of integer partitions. We provide a possible interpretation of our model as a simplified description of p − n junction between two semiconductors. We also show as our model can be embedded in a specific Brylawski dynamical system by means of the introduction of a new evolution rule.

2

An Investigation About Rough Set Theory: Some Foundational and Mathematical Aspects

Cattaneo G.

Fundamenta Informaticae

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2011

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Vol. 108, nr 3/4

197-221

EN

The approach to rough set theory is investigated from the foundational point of view, starting from the formal analysis of the mathematical structures generated by equivalence relations (the standard Pawlak approach to complete information systems) and then by tolerance or similarity relations (the approach to roughness by incomplete information systems). Making use of these approaches the “minimal” meta–theoretical principles which allow one to treat approximations in rough set theory are discussed and formulated in explicit way, in particular the conditions which can assure what can be rightly considered as lower and upper approximations of some approximable set. The hierarchy of approximation spaces generated in this way on a universe is then discussed.

3

Entropies and Co-Entropies of Coverings with Application to Incomplete Information Systems

Bianucci D., Cattaneo G., Ciucci D.

Fundamenta Informaticae

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2007

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Vol. 75, nr 1-4

77-105

EN

Different generalizations to the case of coverings of the standard approach to entropy applied to partitions of a finite universe X are explored. In the first approach any covering is represented by an identity resolution of fuzzy sets on X and a corresponding probability distribution with associated entropy is defined. A second approach is based on a probability distribution generated by the covering normalizing the standard counting measure. Finally, the extension to a generic covering of the Liang-Xu approach to entropy is investigated, both from the "global" and the "local" point of view. For each of these three possible entropies the complementary entropy (or co-entropy) is defined showing in particular that the Liang-Xu entropy is a co-entropy.

4

Algebraic Structures Related to Many Valued Logical Systems. Part 2, Equivalence Among some Widespread Structures

Cattaneo G., Ciucci D., Giuntini R., Konig M.

Fundamenta Informaticae

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2004

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Vol. 63, nr 4

357--373

EN

Several algebraic structures (namely HW, BZMVdM, Stonean MV and MVΔ algebras) related to many valued logical systems are considered and their equivalence is proved. Four propositional calculi whose Lindenbaum-Tarski algebra corresponds to the four equivalent algebraic structures are axiomatized and their semantical completeness is given.

5

Algebraic Structures Related to Many Valued Logical Systems. Part 1, Heyting Wajsberg Algebras

Cattaneo G., Ciucci D., Giuntini R., Konig M.

Fundamenta Informaticae

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2004

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Vol. 63, nr 4

331--355

EN

A bottom-up investigation of algebraic structures corresponding to many valued logical systems is made. Particular attention is given to the unit interval as a prototypical model of these kind of structures. At the top level of our construction, Heyting Wajsberg algebras are defined and studied. The peculiarity of this algebra is the presence of two implications as primitive operators. This characteristic is helpful in the study of abstract rough approximations.

6

Shadowed Sets and Related Algebraic Structures

Cattaneo G., Ciucci D.

Fundamenta Informaticae

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2003

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Vol. 55, nr 3,4

255-284

EN

BZMVdM algebras are introduced as an abstract environment to describe both shadowed and fuzzy sets. This structure is endowed with two unusual complementations: a fuzzy one \lnot and an intuitionistic one ~ . Further, we show how to define in any BZMVdM algebra the Boolean sub-algebra of exact elements and to give a rough approximation of fuzzy elements through a pair of exact elements using an interior and an exterior mapping. Then, we introduce the weaker notion of pre-BZMVdM algebra. This structure still have as models fuzzy and shadowed sets but with respect to a weaker notion of intuitionistic negation ~ a with a Î [0,1/2). In pre-BZMVdM algebras it is still possible to define an interior and an exterior mapping but, in this case, we have to distinguish between open and closed exact elements. Finally, we see how it is possible to define a-cuts and level fuzzy sets in the pre-BZMVdM algebraic context of fuzzy sets.

7

Chaotic Subshifts and Related Languages Applications to one-dimensional Cellular Automata

Cattaneo G., Dennunzio A., Margara L.

Fundamenta Informaticae

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2002

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Vol. 52, nr 1-3

39-80

EN

Subshift behaviors of one-dimensional (1D) bi-infinite Cellular Automata are studied. In particular the conditions under which subshifts generated by CA 1D dynamical systems exhibit some components of the chaotic behavior (in particular transitivity, topological mixing and strong transitivity) are investigated. A complete classification of all elementary (Boolean radius one) CA with respect to subshifts is given.