A. De Luca

Bio: A. De Luca is an academic researcher from ARCO. The author has contributed to research in topics: Fuzzy mathematics & Type-2 fuzzy sets and systems. The author has an hindex of 9, co-authored 14 publications receiving 2212 citations. Previous affiliations of A. De Luca include Sapienza University of Rome.

A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory

Abstract: A functional defined on the class of generalized characteristic functions (fuzzy sets), called “entropy≓, is introduced using no probabilistic concepts in order to obtain a global measure of the indefiniteness connected with the situations described by fuzzy sets. This “entropy≓ may be regarded as a measure of a quantity of information which is not necessarily related to random experiments. Some mathematical properties of this functional are analyzed and some considerations on its applicability to pattern analysis are made.

Reverberations and control of neural networks.

Abstract: The simulation of neural networks, such as the brain cortex, which have a diffuse and rather uniform structure quite unlike the simple block-structure of extant computers, leads naturally to the study of functions and principles which only in part fall within the scope of Automata Theory. Systems of decision equations must be studied with a view especially to obtaining practical means for the prevision and computation of diffuse reverberations of wanted general characteristics, with the exclusion of all others. This amounts to deriving constraints on the allowed variability of the couplings among elements during learning processes, failing which the behavior of the simulator would become uncontrollable for practical purposes. A simple mathematical treatment is presented, which essentially linearizes these problems by an appropriate use of matrix algebra and permits a straightforward study of the wanted conditions, as well as of the controlling elements which may have to be added to the network.

Fuzzy sets and decision theory

Abstract: The problem of making decisions to classify the objects of a certain universe into two or more suitable classes has been considered in the setting of fuzzy sets theory. A measure of the total amount of uncertainty that arises in making decisions has been proposed in the general case. This quantity reduces to the “entropy” of a fuzzy set in the case of two classes. Other quantities which play a relevant role in this theory are the “energy” and the “effective power” of a fuzzy set, defined as ∑ i = 1 N w i f i a n d φ ∑ i = 1 N f i , respectively, where w is a nonnegative weight function and φ a nonnegative constant. If w = constant and φ ≠ 0, the energy is proportional to the effective power and, therefore, to the “power” of the fuzzy set. The maximum of the uncertainty has been calculated in some cases of interest, keeping constant the total energy and effective power. In particular the Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein distributions are derived. Some applications to decision theory are considered in the case of both deterministic and probabilistic decisions. Finally, the analogies that exist between the previous concepts and the thermodynamic ones are discussed.

Decision equation for binary systems. Application to neuronal behavior.

Abstract: The exact mathematical treatment is given for a non linear equation describing the delayed yes-or-no response to a binary system to external stimulations, in some typical cases of interest. Comparison is made with neurophysiological data on the frequency rate of firings of stimulated neurons; the same equation, however, can be conceivably applied to a vast variety of phenomena. The procedures followed to solve the problems that arise in connection with this equation could be extended to more general types of non linear equations.

Pattern Recognition with Fuzzy Objective Function Algorithms

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Outline of a New Approach to the Analysis of Complex Systems and Decision Processes

Abstract: The approach described in this paper represents a substantive departure from the conventional quantitative techniques of system analysis. It has three main distinguishing features: 1) use of so-called ``linguistic'' variables in place of or in addition to numerical variables; 2) characterization of simple relations between variables by fuzzy conditional statements; and 3) characterization of complex relations by fuzzy algorithms. A linguistic variable is defined as a variable whose values are sentences in a natural or artificial language. Thus, if tall, not tall, very tall, very very tall, etc. are values of height, then height is a linguistic variable. Fuzzy conditional statements are expressions of the form IF A THEN B, where A and B have fuzzy meaning, e.g., IF x is small THEN y is large, where small and large are viewed as labels of fuzzy sets. A fuzzy algorithm is an ordered sequence of instructions which may contain fuzzy assignment and conditional statements, e.g., x = very small, IF x is small THEN Y is large. The execution of such instructions is governed by the compositional rule of inference and the rule of the preponderant alternative. By relying on the use of linguistic variables and fuzzy algorithms, the approach provides an approximate and yet effective means of describing the behavior of systems which are too complex or too ill-defined to admit of precise mathematical analysis.

Fuzzy Set Theory - and Its Applications

Abstract: Fuzzy Set Theory - And Its Applications, Third Edition is a textbook for courses in fuzzy set theory. It can also be used as an introduction to the subject. The character of a textbook is balanced with the dynamic nature of the research in the field by including many useful references to develop a deeper understanding among interested readers. The book updates the research agenda (which has witnessed profound and startling advances since its inception some 30 years ago) with chapters on possibility theory, fuzzy logic and approximate reasoning, expert systems, fuzzy control, fuzzy data analysis, decision making and fuzzy set models in operations research. All chapters have been updated. Exercises are included.

A computational approach to fuzzy quantifiers in natural languages

Abstract: The generic term fuzzy quantifier is employed in this paper to denote the collection of quantifiers in natural languages whose representative elements are: several, most, much, not many, very many, not very many, few, quite a few, large number, small number, close to five, approximately ten, frequently, etc. In our approach, such quantifiers are treated as fuzzy numbers which may be manipulated through the use of fuzzy arithmetic and, more generally, fuzzy logic. A concept which plays an essential role in the treatment of fuzzy quantifiers is that of the cardinality of a fuzzy set. Through the use of this concept, the meaning of a proposition containing one or more fuzzy quantifiers may be represented as a system of elastic constraints whose domain is a collection of fuzzy relations in a relational database. This representation, then, provides a basis for inference from premises which contain fuzzy quantifiers. For example, from the propositions “Most U's are A's” and “Most A's are B's,” it follows that “Most2 U's are B's,” where most2 is the fuzzy product of the fuzzy proportion most with itself. The computational approach to fuzzy quantifiers which is described in this paper may be viewed as a derivative of fuzzy logic and test-score semantics. In this semantics, the meaning of a semantic entity is represented as a procedure which tests, scores and aggregates the elastic constraints which are induced by the entity in question.