Gianpiero Cattaneo

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.

Brouwer-Zadeh posets and three-valued Ł ukasiewicz posets

Abstract This paper is a study of the structure of Brouwer-Zadeh (or BZ-) poset, i.e. a poset endowed with two non-usual orthocomplementations. These two orthocomplementations allow definition of two unary operators which can be considered as algebraic counterparts of the necessity and possibility operators of modal logic. A construction of how to induce a three-valued BZ-poset from a BZ-poset is given. The examples of the BZ-lattice of all generalized characteristic functionals on a reference space and of the BZ-poset of all generalized orthogonal projections on a Hilbert space are dealt with.

Cellular automata in fuzzy backgrounds

Abstract The main purpose of this work is to understand some limitations introduced by the classical definitions of cellular automata (CA). To this end, we have defined a new model of CAs ( fuzzy CA s) which allows the observation of interesting “chaotic” properties of elementary CAs. To date neither a formal nor a precise definition of “chaos” in CAs exists; we believe that the proposed model provides a “sharper” tool to detect which properties can be associated to a “chaotic” behavior. We also define a measure ( rule entropy ) which gives information about the CAs dynamics solely on the basis of the rule table and provides theoretical explanations to some of the empirical observations.

Investigating topological chaos by elementary cellular automata dynamics

We apply the two different definitions of chaos given by Devaney and by Knudsen for general discrete time dynamical systems (DTDS) to the case of elementary cellular automata, i.e., 1-dimensional binary cellular automata with radius 1. A DTDS is chaotic according to the Devaneys definition of chaos iff it is topologically transitive, has dense periodic orbits, and it is sensitive to initial conditions. A DTDS is chaotic according to the Knudsens definition of chaos iff it has a dense orbit and it is sensitive to initial conditions. We enucleate an easy-to-check property (left or rightmost permutivity) of the local rule associated with a cellular automaton which is a sufficient condition for D-chaotic behavior. It turns out that this property is also necessary for the class of elementary cellular automata. Finally, we prove that the above mentioned property does not remain a necessary condition for chaoticity in the case of non elementary cellular automata.

Pattern growth in elementary cellular automata

A classification of elementary cellular automata (CA) based on their pattern growth is introduced. It is shown that this classification is effective, that is, there exists an algorithm to determine to which class a given CA belongs. This algorithm is based on the properties of the local rule of CAs, not requiring the observation of their evolution. Furthermore, necessary and sufficient conditions to detect all the elementary CAs exhibiting a shift-like behavior are given; these CAs have interesting dynamical properties and chaotic characteristics.

Information Entropy and Granulation Co---Entropy of Partitions and Coverings: A Summary

Some approaches to the covering information entropy and some definitions of orderings and quasi---orderings of coverings will be described, generalizing the case of the partition entropy and ordering. The aim is to extend to covering the general result of anti---tonicity (strictly decreasing monotonicity) of partition entropy. In particular an entropy in the case of incomplete information systems is discussed, with the expected anti-tonicity result, making use of a partial partition strategy in which the missing information is treated as a peculiar value of the system. On the other side, an approach to generate a partition from a covering is illustrated. In particular, if we have a covering ? coarser than another covering ? with respect to a certain quasi order relation on coverings, the induced partition ?(?) results to be coarser than ?(?) with respect to the standard partial ordering on partitions. Thus, one can compare the two coverings through the entropies of the induced partitions.

Ergodicity, transitivity, and regularity for linear cellular automata over Z m 1

We study the dynamical behavior of D-dimensional linear cellular automata over Zm. We provide an easy-to-check necessary and sufficient condition for a D-dimensional linear cellular automata over Zm to be ergodic and topologically transitive. As a byproduct, we get that for linear cellular automata ergodicity is equivalent to topological transitivity. Finally, we prove that for 1-dimensional linear cellular automata over Zm, regularity (denseness of periodic orbits) is equivalent to surjectivity.

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.

On the dynamical behavior of chaotic cellular automata

Abstract The shift (bi-infinite) cellular automaton is a chaotic dynamical system according to all the definitions of deterministic chaos given for discrete time dynamical systems (e.g., those given by Devaney [6] and by Knudsen [10]). The main motivation to this fact is that the temporal evolution of the shift cellular automaton under finite description of the initial state is unpredictable . Even tough rigorously proved according to widely accepted formal definitions of chaos, the chaoticity of the shift cellular automaton remains quite counterintuitive and in some sense unsatisfactory. The space-time patterns generated by a shift cellular automaton do not correspond to those one expects from a chaotic process. In this paper we propose a new definition of strong topological chaos for discrete time dynamical systems which fulfills the informal intuition of chaotic behavior that everyone has in mind. We prove that under this new definition, the bi-infinite shift is no more chaotic. Moreover, we put into relation the new definition of chaos and those given by Devaney and Knudsen. In the second part of this paper we focus our attention on the class of additive cellular automata (those based on additive local rules) and we prove that essential transformations [2] preserve the new definition of chaos given in the first part of this paper and many other aspects of their global qualitative dynamics.

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.