- I am Professor Emeritus at the University of Salerno and I am interested in Fuzzy Logic, Pointfree Geometry and Mathematics Education.edit
Si espongono ed analizzano due dimostrazioni del teorema di Cantor Bernstein
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ABSTRACT
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Let U be any complete lattice. Then, as observed in Chapter 4, in fuzzy logic it would be misleading to consider an initial valuation v: F→ U as a fuzzy subset of F As a matter of fact, for any formula α, the number v(α) is not the truth... more
Let U be any complete lattice. Then, as observed in Chapter 4, in fuzzy logic it would be misleading to consider an initial valuation v: F→ U as a fuzzy subset of F As a matter of fact, for any formula α, the number v(α) is not the truth value of α but a constraint on its actual truth value, namely a constraint like “the truth value of a is greater than or equal to v(α)”.
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The concepts of a decidable subset and a recursively enumerable subset are crucial for first order classical logic. In particular, they are basic tools for the proof of the famous limitative theorems about the undecidability and... more
The concepts of a decidable subset and a recursively enumerable subset are crucial for first order classical logic. In particular, they are basic tools for the proof of the famous limitative theorems about the undecidability and incompleteness of first order logic (see, for example, Shoenfield [1967]). Then, the question of a suitable extension of such concepts to fuzzy set theory arises. A first proposal in such a direction was made by E. S. Santos in an interesting series of papers. Indeed, Santos, starting from an idea of L. Zadeh (Zadeh [1968]), proposed the notions of fuzzy Turing machine, Markov normal fuzzy algorithm and fuzzy program. Santos proved that all these definitions determine the same notion of computability for fuzzy maps (see Santos [1970] and Santos [1976]). As in the classical case, a corresponding definition of recursively enumerable fuzzy subset is obtained by calling recursively enumerable any fuzzy subset which is the domain of a computable fuzzy map. Successively, a notion of recursive enumerability was proposed in Harkleroad [1984] where a fuzzy subset s is said to be recursively enumerable if the restriction of s to its support is a partial recursive function.
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Research Interests: Mathematics and Geometry
ABSTRACT Given a set X, we take into consideration the lattice F(X, ∗[0, 1]) of the nonstandard fuzzy subsets of X, that is the L-subsets with L equal to the unitary interval ∗[0, 1] of a nonstandard model of analysis. To show the... more
ABSTRACT Given a set X, we take into consideration the lattice F(X, ∗[0, 1]) of the nonstandard fuzzy subsets of X, that is the L-subsets with L equal to the unitary interval ∗[0, 1] of a nonstandard model of analysis. To show the appropriateness of such a concept, we give two examples of vague concepts, positive divergence for functions and vagueness for fuzzy sets, that are representable by suitable nonstandard fuzzy sets. One proves that they are not representable by Zadeh's fuzzy sets. Also, we observe that the same operations and relations defined for fuzzy sets are definable for nonstandard fuzzy sets. In particular, the complementation operation and the sharpening relation. Finally one proves that every L-subset with L totally ordered is a nonstandard fuzzy set.
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ABSTRACT In this paper we propose a general approach to the theory of fuzzy algebras, while the early existing papers deal with a particular type of fuzzy structures as fuzzy groups, fuzzy ideals, fuzzy vector spaces and so on.
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In the last years several research projects have been motivated by the problem of constructing the usual geometrical spaces by admitting “regions” and “inclusion” between regions as primitives and by defining the points as suitable... more
In the last years several research projects have been motivated by the problem of constructing the usual geometrical spaces by admitting “regions” and “inclusion” between regions as primitives and by defining the points as suitable sequences or classes of regions (for references see [2]).In this paper we propose and examine a system of axioms for the pointless space theory in which “regions”, “inclusion”, “distance” and “diameter” are assumed as primitives and the concept of point is derived. Such a system extends a system proposed by K. Weihrauch and U. Schreiber in [5].In the sequel R and N denote the set of real numbers and the set of natural numbers, and E is a Euclidean metric space. Moreover, if X is a subset of R, then ⋁X is the least upper bound and ⋀X the greatest lower bound of X.
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Generally speaking, all fuzzy logics considered so far arise from the fiizzification of notions having a metalogic character (see Section 10 in Chapter 3). By contrast, in this chapter we will investigate several fuzzy logics directly... more
Generally speaking, all fuzzy logics considered so far arise from the fiizzification of notions having a metalogic character (see Section 10 in Chapter 3). By contrast, in this chapter we will investigate several fuzzy logics directly arising from “fuzzy worlds”, i.e., worlds whose properties can be vague and therefore whose truth-value assignments can be graded. This leads to truth-functional semantics, i.e., semantics in which the truth value of a composite sentence is determined by the truth-values of its constituents. More precisely, we consider a language by starting from an infinite set VAR = {p 1, ... , p n , ...} of propositional variables, a set of logical connectives containing the usual connectives, namely ∧ , ∨ , ¬.
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... By following [CHA 88], in this section we will extend the concept of consequence relation. Let Fonn be a set whose elements are called fonnulas. Then a graded conclusion relation is any fuzzy relation g:P(Fonn) x Fonn ~u and we write... more
... By following [CHA 88], in this section we will extend the concept of consequence relation. Let Fonn be a set whose elements are called fonnulas. Then a graded conclusion relation is any fuzzy relation g:P(Fonn) x Fonn ~u and we write g(X 1-a) instead of g(X,a). ...
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... Expert Systems Cristina Coppola, Giangiacomo Gerla, and Tiziana Pacelli Dipartimento di Matematica e Informatica, Universit`a degli Studi di Salerno Via Ponte don Melillo, 84084 Fisciano (SA), Italy {ccoppola,gerla,tpacelli}@unisa.it... more
... Expert Systems Cristina Coppola, Giangiacomo Gerla, and Tiziana Pacelli Dipartimento di Matematica e Informatica, Universit`a degli Studi di Salerno Via Ponte don Melillo, 84084 Fisciano (SA), Italy {ccoppola,gerla,tpacelli}@unisa.it Summary. ...
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Traditional control techniques are possible only in the case of complete understanding of the physical nature of the problem and only after a suitable mathematical treatment leading to a usable model. This enables us to obtain a numerical... more
Traditional control techniques are possible only in the case of complete understanding of the physical nature of the problem and only after a suitable mathematical treatment leading to a usable model. This enables us to obtain a numerical function f whose intended meaning is that f(x) is the correct control given x. Unfortunately, this is not the case for a majority of real systems. Difficulties can arise, for instance, from poor understanding of the underlying phenomena (and therefore from a lack of theory), or from the complexity of the resulting mathematical model. In such cases fuzzy control, as devised in Zadeh [1965], [1975]a, [1975]b and in Mamdani [1981], is a very useful tool. To explain the idea, we can distinguish two phases in the building of a fuzzy controller.
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Several notions in crisp mathematics can be translated into the corresponding notions in fuzzy mathematics in a uniform way by Zadeh Extension Principle (see Zadeh [1975]b). So, it is natural to ask the following question: Given a crisp... more
Several notions in crisp mathematics can be translated into the corresponding notions in fuzzy mathematics in a uniform way by Zadeh Extension Principle (see Zadeh [1975]b). So, it is natural to ask the following question: Given a crisp logic, does there exist a canonical way to extend it in a fuzzy logic?
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In this paper we develop a point-free system of geometry based on the notions of region, parthood, and ovality, the last one being a region-based counterpart of the notion of convex set. In order to show that the system we propose is... more
In this paper we develop a point-free system of geometry based on the notions of region, parthood, and ovality, the last one being a region-based counterpart of the notion of convex set. In order to show that the system we propose is sufficient to reconstruct an affine geometry we make use of a theory of a Polish mathematician Aleksander Śniatycki from [15], in which the concept of half-plane is assumed as basic.
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Effective domain theory is applied to fuzzy logic. The aim is to give suitable notions of semi-decidable and decidableL-subset and to investigate about the effectiveness of the fuzzy deduction apparatus.