Papers by Giangiacomo Gerla
Riprendendo il discorso iniziato in un nostro articolo già apparso su questo periodico, si consid... more Riprendendo il discorso iniziato in un nostro articolo già apparso su questo periodico, si considera un argomento fondamentale della teoria ingenua degli insiemi, il teorema detto di Cantor-Bernstein. Lo scopo è fornire materiale per possibili percorsi didattici legati alla nozione di equipotenza.
Due nuove dimostrazioni del Teorema di Cantor-Bernstein, 2011
Si espongono ed analizzano due dimostrazioni del teorema di Cantor Bernstein
Studia Logica, Jun 27, 2007
We define the notion of "potential existence" by starting from the fact that in multi-valued logi... more We define the notion of "potential existence" by starting from the fact that in multi-valued logic the existential quantifier is interpreted by the least upper bound operator. Besides, we try to define in a general way how to pass from potential into actual existence.
Logic and Logical Philosophy, Oct 9, 2013
This is an exploratory paper whose aim is to investigate the potentialities of bilattice theory f... more This is an exploratory paper whose aim is to investigate the potentialities of bilattice theory for an adequate definition of the deduction apparatus for multi-valued logic. We argue that bilattice theory enables us to obtain a nice extension of the graded approach to fuzzy logic. To give an example, a completeness theorem for a logic based on Boolean algebras is proved.
Information Processing and Management of Uncertainty, 2004
IEEE Transactions on Automatic Control, Oct 1, 1982
ABSTRACT
Journal of Mathematical Analysis and Applications, Dec 1, 1997
The aim of this paper is to show that fuzzy logic is a suitable tool to manage several types of p... more The aim of this paper is to show that fuzzy logic is a suitable tool to manage several types of probability-like functionals. Namely, we show that the superadditive functions, the necessities, the upper and lower probabilities, and the envelopes can be considered theories of suitable fuzzy logics. Some general results about the compactness in fuzzy logic are also obtained.
Studia Logica, Mar 1, 2005
The paper concerns fuzzy logic programming. As an example, we show that is not restrictive to con... more The paper concerns fuzzy logic programming. As an example, we show that is not restrictive to confine ourselves to fuzzy Herbrand interpretations in giving a semantics for fuzzy programs. Also, we show that the resulting apparatus gives a unifying theoretical framework for fuzzy control.
Springer eBooks, 2001
Let U be any complete lattice. Then, as observed in Chapter 4, in fuzzy logic it would be mislead... 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)”.
Springer eBooks, 2001
The concepts of a decidable subset and a recursively enumerable subset are crucial for first orde... 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.
Soft Computing, Feb 27, 2009
In order to give a suitable framework for a synonymy-based logic programming, we argue about the ... more In order to give a suitable framework for a synonymy-based logic programming, we argue about the possibility of reducing fuzzy logic programming to classical logical programming. More precisely, we show that given a fuzzy program in a language L, we can translate it into an equivalent classical program in a (meta-)language L m in which every predicate name in L becomes a constant in L m : This enables us to admit in L m meta-relations among predicates and therefore, in particular, the synonymy.
Studia Logica, Jun 1, 1989
We propose the notion of partial recursiveness and strong partial recursiveness for fuzzy maps. W... more We propose the notion of partial recursiveness and strong partial recursiveness for fuzzy maps. We prove that a fuzzy map f is partial recursive if and only if it is computable by a Turing fuzzy machine and that f is strongly partial recursive and deterministic if and only if it is computable via a deterministic Turing fuzzy machine. This gives a simple and manageable tool to investigate about the properties of the fuzzy machines.
De Gruyter eBooks, Dec 31, 2008
This paper is devoted to some mathematical considerations on the geometrical ideas contained in P... more This paper is devoted to some mathematical considerations on the geometrical ideas contained in PNK, CN and, successively, in PR. Mainly, we will emphasize that these ideas give very promising suggestions for a modern point-free foundation of geometry.
Fuzzy Sets and Systems, Mar 1, 1986
ABSTRACT Given a set X, we take into consideration the lattice F(X, ∗[0, 1]) of the nonstandard f... 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.
Notre Dame Journal of Formal Logic, Apr 1, 2008
De Cock and Kerre, in considering Poincaré paradox, observed that the intuitive notion of "approx... more De Cock and Kerre, in considering Poincaré paradox, observed that the intuitive notion of "approximate similarity" cannot be adequately represented by the fuzzy equivalence relations. In this note we argue that the deduction apparatus of fuzzy logic gives adequate tools with which to face the question. Indeed, a first-order theory is proposed whose fuzzy models are plausible candidates for the notion of approximate similarity. A connection between these structures and the point-free metric spaces is also established.
Stochastica: revista de matemática pura y aplicada, 1987
ABSTRACT In this paper we propose a general approach to the theory of fuzzy algebras, while the e... more 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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Papers by Giangiacomo Gerla