Gianpiero Cattaneo

Università degli Studi di Milano-Bicocca | UNIMIB · Department of Informatics, Systems and Communication (DISCo)

This unique collection of research papers offers a comprehensive and up-to-date guide to algebraic approaches to rough sets and reasoning with vagueness. It bridges important gaps, outlines intriguing future research directions, and connects algebraic approaches to rough sets with those for other forms of approximate reasoning. In addition, the boo...

In this chapter we are interested to study the structures arising from pairs of elements from a partially ordered set (poset) which share some orthogonality between them, the so-called orthopairs, with respect to a unary operation of De Morgan complementation (or in the case of a lattice interpreted as De Morgan negation).

This chapter deals with the abstract approach to rough sets theory through the equational notion of closure operator in the context of lattice theory, with the associated notion of internal operator as not-closure-not. The involved lattice structures are not necessarily distributive to allow the development of rough theories in the so-called logica...

The theoretical study of Genetic Algorithms and the dynamics induced by their genetic operators is a research field with a long history and many different approaches. In this paper we complete a recently presented approach to model one-point crossover using pretopologies (or Čechtopologies) in two ways. First, we extend it to the case of n-points c...

Genetic algorithms (GAs) are an optimization technique that has been successfully used on many real-world problems. There exist different approaches to their theoretical study. In this paper we complete a recently presented approach to model one-point crossover using pretopologies (or Cech topologies) in two ways. First, we extend it to the case of...

In a recent paper M. Lopez-Suarez, I. Neri, and L. Gammaitoni (LNG) present a concrete realization of the Boolean Or irreversible gate, but contrary to the standard Landauer principle, with an arbitrary small dissipation of energy. A Popperian good falsification! In this paper we discuss a theoretical description of the LNG gate which is indeed a 3...

We present a unique framework for connecting different topics: hypergraphs from one side and Formal Concept Analysis and Rough Set Theory from the other. This is done through the formal equivalence among Boolean information tables, formal contexts and hypergraphs. Links with generic (i.e., not Boolean) information tables are established, through so...

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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 B...

This paper reviews the well-known formalisations for ice and sand piles, based on a finite sequence of non-negative integers and its recent extension to signed partitions, i.e. sequences of a non-negative and a non-positive part of integers, both non increasing. The ice pile model can be interpreted as a discrete time dynamical system under the act...

We consider the unitary and the anti--unitary operator realizations of two important genuine quantum gates that transform elements of the computational basis of into superpositions: the square root of the identity and the square root of the negation.

This paper reviews the well-known formalisations for ice and sand piles, based on a finite sequence of non-negative integers and its recent extension to signed partitions, i.e. sequences of a non-negative and a non-positive part of integers, both non increasing. The ice pile model can be interpreted as a discrete time dynamical system under the act...

We analyze the dynamical behavior of the usual one dimensional sand pile model which actually describes the physical situation in which the pile is submitted to the uniform blow of a unidirectional wind. In the first step the Lagrangian formalism is investigated, showing that the stationary action principle does not select in a unique way the path...

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 a...

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 sy...

Some relevant algebraic structures involved by the so–called Intuitionistic Fuzzy Sets (IFS) are discussed, with a wide description of their relevant properties especially from the point of view of the algebraic semantic of a logical system. Algebraic comparison with analogous structures involving usual Fuzzy Sets are discussed.

In this paper we begin the study the dynamical behavior of non-uniform cellular automata and compare it to the behavior of "classical" cellular automata. In particular we focus on surjectivity and equicontinuity.

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...

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...

The basic notions of posets and lattices are introduced with a wide presentation of examples. In particular, distributive and modular lattices are treated. With respect to the distributive behavior, we discuss Boolean algebras with the associated orthocomplementation mapping. General orthoalgebras are also investigated with application to the quant...

Some approaches to the entropy of coverings, generalizing the case of the partition entropy, and some definitions of orderings and quasi–orderings of coverings as extensions to the covering context of various formulations of the standard order on partitions. Unfortunately, we here show that the problem of anti–monotonicity of non–pointwise entropy...

A hierarchy of closure operators on the abstract context of lattice structures is investigated, and compared to the abstract approach to rough approximation spaces. In particular, the Tarski, the Kuratowski and the Halmos closures are treated, with the corresponding models of covering, topological and partition spaces.

A hierarchy of closure operators on the abstract context of lattice structures is investigated, and compared to the abstract approach to rough approximation spaces. In particular, the Tarski, the Kuratowski and the Halmos closures are treated, with the corresponding models of covering, topological and partition spaces.

We extend the basic principles and results of conservative logic to include the main features of many-valued logics with a finite number of truth values. Different approaches to many-valued logics are examined in order to determine some possible functionally complete sets of logic connectives. As a result, we describe some possible finite-valued ga...

The abstract notion of rough approximation space is applied to the concrete cases of topological spaces with the particular situation of clopen–topologies generated by partitions, according to the Pawlak approach to rough set theory. In this partition context of a finite universe, typical of complete information systems, the probability space gener...

The standard approach to information entropy applied to partitions of a universe is equivalently formulated as the entropy of the corresponding crisp identity resolutions, interpreted as crisp granulations, by the corresponding characteristic functionals. Moreover, in this crisp context the co–entropy notion is introduced. The extension to the case...

Some quasi–ordering for coverings are here defined. Different definitions of entropies and co–entropies for coveringsare examined, also focusing on the distinction between the here called global and pointwise approaches to co–entropies. The entropies and co–entropies are defined both for coverings generated from an information system via a similar...

A partitioning approach to the problem of dealing with the entropy of incomplete information systems is explored. The aim is to keep into account the incompleteness and at the same time to obtain a probabilistic partition of the information system. For the resulting probabilistic partition, measures of entropy and co–entropy are defined, similarly...

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 o...

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 o...

In this paper we contribute to the terminological debate about Atanassov's use of the term “Intuitionistic” in defining his structure based on ortho-pairs of fuzzy sets. In particular, we stress that it is defined as “intuitionistic” a negation which from one side does not satisfy a standard property of the intuitionistic Brouwer negation (contradi...

Both in High Performance Computing and in Grid computing dynamic load balancing is becoming one of the most important features. In this paper, we present a novel load balancing model based on Lattice Boltzmann Cellular Automata. Using numerical simulations our model is compared to diffusion algorithms adopted on HPC load balancing and to agent-base...

A Cellular Automaton (CA) describing a predator–prey dynamics is proposed. This model is fully local, i.e., without any “spurious” Monte Carlo step during the movement phase. A particular attention has been addressed to the comparison of the obtained simulations with the discrete version of the Lotka–Volterra equations.

In this paper, it is remarked that BZ lattice structures can recover several theoretical approaches to rough sets, englobing their individual richness in a unique structure. Rough sets based on a similarity relation are also considered, showing that the BZ lattice approach turns out to be even more useful, since enables one to define another rough...

In this paper, it is remarked that BZ lattice structures can recover several theoretical approaches to rough sets, englobing their individual richness in a unique structure. Rough sets based on a similarity relation are also considered, showing that the BZ lattice approach turns out to be even more useful, since enables one to define another rough...

Generalized (i.e., positive operator valued) observables generated by a standard (i.e., projection valued) observable via suitable confidence functions are introduced in the context of finite dimensional Hilbert quantum mechanics. These generalized observables are considered as unsharp realizations of the unique physical magnitude associated to the...

In this paper we discuss the meaning of sensitivity and its implications in CA behavior. A new shift-invariant metric is given. The metric topology induced by this metric is perfect but not compact. Moreover we prove that the new space is “suitable” for the study of the dynamical behavior of CA. In this context sensitivity assumes a stronger meanin...

The main goal of this paper is the investigation of a relevant property which appears in the various definition of deterministic topological chaos for discrete time dynamical system: transitivity. Starting from the standard Devaney's notion of topological chaos based on regularity, transitivity, and sensitivity to the initial conditions, the critiq...

The categorical equivalence of three different approaches to roughness is discussed: the one based on the notion of abstract rough approximation spaces, the second one based on the abstract topological notions of interior and closure, and the third one based on a very weak form of BZ lattice.

An interesting procedure of fuzzi-fication of a probability measure is given, producing another prob-ability measure, whose rough ap-proximation from the bottom, by a sharp inner measure, and the top, by an outer measure, is discussed.

We study the subshift behavior of one dimensional cellular automata and we show how to associate to any subshift of finite type a cellular automaton which contains it. The relationships between some topological properties of subshifts and the behavior of the related languages are investigated. In particular we focus our attention to the notion of f...

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 tw...

Several algebraic structures (namely HW, BZMV(dM), Stonean MV and MV Delta 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.

Quantum computation has suggested new forms of quan-tum logic, called quantum computational logics ([CDCGL01]). The basic semantic idea is the following: the meaning of a sentence is identified with a quregister, representing a possible pure state of a compound phys-ical system, whose associated Hilbert space is an n-fold tensor product ⊗ n C 2 . T...

The problem of fluid compressibility in the ordinary approach to lattice Boltzmann (LB) according to the BGK method is analyzed. A new velocity is introduced besides the usual one, linked to this later by a dimensionless mass density. The LBGK method based on this second velocity, with a suitable second order equilibrium distribution (different fro...

We study two dynamical properties of linear D-dimensional cellular automata over namely, denseness of periodic points and topological mixing. For what concerns denseness of periodic points, we complete the work initiated in (Theoret. Comput. Sci. 174 (1997) 157, Theoret. Comput. Sci. 233 (1–2) (2000) 147, 14th Annual Symp. on Theoretical Aspects of...

Once defined a rough approximation map on the abstract notion of minimal BZ lattice, the collection of all rough approximations is took into account and its structure is analyzed with respect to lattice properties. Further, some nonstandard complementations are introduced on it. Finally, the results obtained in the abstract environment are applied...

One of the most interesting logical suggestions that arise from quantum-computation theory is to use the quantum-theoretical formalism in order to represent parallel reasoning ([3], [4]). As is well known, the unit of measurement in classical information theory is the bit: one bit measures the information quantity that can be either transmitted or...

We introduce some conservative gates for finite-valued logics which are able to realize all the main connectives of the many-valued logics of ukasiewicz, the MV-algebras of Chang and Brower–Zadeh algebras. After a brief exposition of the motivations for this work, the gates are defined and their properties are explored. Finally, a possible quantum...

Starting from an incomplete information system, we add some information in two different ways: by an increase in the number of known values and by an increase in the number of attributes. The behavior of the similarity and preclusive rough approximations are studied in both cases.

Once defined a rough approximation map on the abstract notion of minimal BZ lattice, the collection of all rough approximations is took into account and its structure is analyzed with respect to lattice properties. Further, some nonstandard complementations are introduced on it. Finally, the results obtained in the abstract environment are applied...

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 f...

Several algebraic structures (namely HW, BZMV dM, Stonean MV and MV Δ 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.

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 f...

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 ¬ 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...

After the introduction of shadowed sets and the investigation of their relation with fuzzy sets, we present BZMVdM algebras as an abstract environment for both shadowed and fuzzy sets. Then, we introduce the weaker notion of pre-BZMVdM algebra. This structure enables us to algebraically define a mapping from fuzzy sets to shadowed sets.

After the introduction of shadowed sets and the investigation of their relation with fuzzy sets, we present BZMVdM algebras as an abstract environment for both shadowed and fuzzy sets. Then, we introduce the weaker notion of pre-BZMVdM algebra. This structure enables us to algebraically define a mapping from fuzzy sets to shadowed sets.

Intuitionistic Fuzzy Sets Theory is based on a wrong nominalistic (terminological) assumption. It is defined as “intuitionistic” a negation which does not satisfy usual properties of the intuitionistic Brouwer negation, but it is called with this term only a particular generalized notion of negation which indeed corresponds to the deMorgan negation...

BZMV^{dM} 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 ¬ and an intuitionistic one ∼. Further, we show how to define in any BZMV^{dM} algebra the Boolean sub-algebra of exact elements and to give a rough approximation of fuzzy el...

Intuitionistic Fuzzy Sets (IFS) are defined as pairs of mutually orthogonal fuzzy sets. We discuss this approach from an algebraic point of view. As a result we characterize two implication operators on the collection of IFS, which on a particular subset of IFS behave as a ̷Lukasiewicz and a Gödel implication.

We extend the notion of conservativeness, given by Fredkin and Toffoli in 1982, to generic gates whose input and output lines may assume a finite number d of truth values. A physical interpretation of conservativeness in terms of conservation of the energy associated to the data used during the computation is given. Moreover, we define conservative...

The basic principles and results of conservative logic introduced by Fredkin and Toffoli in 1982, on the basis of a seminal paper of Landauer, are extended to d-valued logics, with a special attention to three-valued logics. Different approaches to d-valued logics are examined in order to determine some possible universal sets of logic primitives....

In the context of generalized rough sets, it is possible to introduce in an Information System two different rough approximations. These are induced, respectively, by a Similarity and a Preclusivity relation ([3,4]). It is possible to show that the last one is always better than the first one. Here, we present a quantitative analysis of the relati...

We present the behavior of simple subshifts generated by 1D Elementary CA (ECA) with respect to some components of chaoticity as transitivity, topological mixing and strong transitivity. A classification of subshifts generated by ECA with respect to transitivity is given. In literature one can find several notions of topological transitivity. We di...

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 Lukasiewicz approach to many-valued logic. On the basis of an HW algebra it is possible to obtain a de Morgan Brouwer-Zadeh (BZ) distributive la...

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 elem...

A necessary and sufficient condition for a one-dimensional q-state n-input cellular automaton rule to be number-conserving is established. Two different forms of simpler and more visual representations of these rules are given, and their ...

It is shown that the unit interval of a von Neumann algebra is a Sum Brouwer–Zadeh algebra when equipped with another unary operation sending each element to the complement of its range projection. The main result of this Letter says that a von Neumann algebra is finite if and only if the corresponding Brouwer–Zadeh structure is de Morgan or, equiv...

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 Lukasiewicz approach to many-valued logic. On the basis of an HW algebra it is possible to obtain a de Morgan Brouwer-Zadeh (BZ) distributive la...

This volume contains the papers selected for presentation at the Third International Conference on Rough Sets and Current Trends in Computing (RSCTC 2002) held at Penn State Great Valley, Malvern, Pennsylvania, U.S.A., 14-16 October 2002.

A cellular automata (CA) model for fluid-dynamic simulations is presented as a strategic tool for monitoring physical properties which are difficult to be measured by an experiment. The use of an arbitrary number of rest particles let the FHP-N model recover the Galileian invariance, which was missing in the standard FHP model. The code has been im...

Three different cellular automata (CA) based approaches for computational fluid dynamics (CFD) simulations are presented. We will try to discuss the advantages and disadvantages of each model, giving some consideration of the complexity to implement each of them.

Two different generalizations of Brouwer–Zadeh posets (BZ posets) are introduced. The former (called pre-BZ poset) arises from topological spaces, whose standard power set orthocomplemented complete atomic lattice can be enriched by another complementation associating with any subset the set theoretical complement of its topological closure. This c...

The standard Brouwer–Zadeh poset (H) is the poset of all effect operators on a Hilbert space H, naturally equipped with two types of orthocomplementation. In developing the theory, the question occured if (when) (H) fulfils the de Morgan property with respect to both orthocomplementation operations. In Ref.3 the authors proved that it is the case p...

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 Devaney's definition of chaos iff it is topologically transitive, has dense per...

One of the authors (MM) would like to dedicate this paper to the late Michele Costato (1940-1997). He was a real teacher and a restless researcher. His scientific curiosity together with his respect for everybody's ideas and his deep interest for discussion made him the perfect participant in a research team and not only. NC could spend hours in el...

An unsharp quantum observable can be considered a realization of a sharp observable if and only if it is commutative. In this paper we describe an explicit procedure for reconstructing such a sharp observable and for establishing the probabilistic correlations between the sharp reconstruction and the given unsharp observable. © 2000 American Instit...

The algebraic structures arising in the axiomatic framework of unsharp quantummechanics based on effect operators on a Hilbert space are investigated. It isstressed that usually considered effect algebras neglect the unitary Brouwerianmap of complementation, and the main results based on this complementationare collected, showing the enrichment pro...

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 transitivi...

It is shown that every effect algebra can be represented as a pasting of a systemwhere each element is the range of an unsharp observable. To describe the rangeof an unsharp observable algebraically, the notion of a para-Booleanquasi-effect algebra is introduced. Some intrinsic compatibility conditions ensuringcommensurability of effects are studie...

This paper studies the state-effect-probability structure associated with thequantum mechanics of nonlinear (homogeneous, in general nonadditive) operatorson a Hilbert space. Its aim is twofold: to provide a concrete representation ofthe features of nonlinear quantum mechanics on a Hilbert space, and to showthat the properties of the nonlinear vers...

The natural algebraic structure of fuzzy sets suggests the introduction of an abstract algebraic structure called de Morgan BZMV-algebra (BZMVdM-algebra). We study this structure and sketch its main properties. A BZMVdM-algebra is a system endowed with a commutative and associative binary operator ⊕ and two unusual orthocomplementations: a Kleene o...

This article presents and compares various algebraic structures that arise in axiomatic unsharp quantum physics. We begin by stating some basic principles that such an algebraic structure should encompass. Following G. Mackey and G. Ludwig, we first consider a minimal state-effect-probability (minimal SEFP) structure. In order to include partial op...