Corina Reischer

Learning with permutably homogeneous multiple-valued multiple-threshold perceptrons

The (n,k,s)-perceptrons partition the input space V ⊂ Rn into s+1 regions using s parallel hyperplanes. Their learning abilities are examined in this research paper. The previously studied homogeneous (n,k,k−1)-perceptron learning algorithm is generalized to the permutably homogeneous (n,k,s)-perceptron learning algorithm with guaranteed convergence property. We also introduce a high capacity learning method that learns any permutably homogeneously separable k-valued function given as input.

On set-valued functions and Boolean collections

The notion of a Boolean collection of set is introduced, and several combinatorial aspects of these collections are exploited. These collections of set appear to play a role in the approximation of non-Boolean set-valued functions by Boolean functions and, therefore, are relevant in the study of biocircuits and in the study of circuits based on frequency multiplexing, where set-valued functions are used.<<ETX>>

Local similarity manifolds

SummaryA local similarity manifold is defined as a locally affine manifold for which the transition functions of an affine atlas are similarity transformations inRn. The main result of this paper is that, for n≧3, the compact local similarity manifolds (which are not locally Euclidean) are given by the formula M=(Rn{0} G, where G is a group of covering transformations such that G={ht0k¦h ε H, k εZ, H being a finite orthogonal group without fixed points inRn{0},and t0 being some conformal linear transformation ofRn which commutes with H.

On functional entropy

A numerical characteristic of functions between finite sets that satisfies certain properties related to common operations applied to functions is defined. Unlike the notion of entropy of a probability distribution, the entropy of a function has an algebraic rather than a probabilistic character, although the two notions are clearly related. Applications of this notion to the study of finite functions are indicated.<<ETX>>

The relative information generating function

Abstract The aim of the paper is to introduce the relative information generating function whose derivatives give well-known statistical indices such as the Kullback-Leibler divergence between two probability distributions and Watanabes measure of interdependence. It contains Golombs generating function as a particular case, and takes simple forms for the binomial and the Poisson distributions, neither of which fits into Golombs model. Besides its unifying role, the relative information generating function suggests new information indices as, for instance, the standard deviation of the variation of information.

On The Principle of Minimum interdependence

Applying the Principle of Minimum Interdependence (abbreviated as PIM) we determine the largest product probability distribution compatible with some data on the interdependence between two or several random variables (some mixed moments of random variables). The general solution both in the discrete case and in the continuous one is given. A special analysis is done for the normal random variables. Particularly, the minimum interdependence between normal random variables compatible with the given covariance matrix is computed.

Some remarks on entropic distance, entropic measure of connexion and Hamming distance

— The relationship between entropie distance, entropie measure of connexion or interdependence and Hamming distance is investigated. Some applications to the classification ofthe families of curves are also given. Résumé. — On étudie la relation entre la distance entropique, la mesure entropique de connexion ou interdépendance et la distance de Hamming, avec quelques applications à la classification de familles de courbes.

Associative Algebraic Structures in the Set of Boolean Functions and Some Applications in Automata Theory

A class of linear algebraic structures over the set of Boolean functions is presented. Afterwards, from these structures a new one is studied—the right-weak algebra. By using the properties of this algebra, some new results are established about the commutativity of the superposition, the ideals, the annulators, and the properties of the iteration.

Classification of functions and enumeration of bases of set logic under Boolean compositions

This paper discusses some classification and enumeration problems in r-valued set logic, which is the logic of functions mapping n-tuples of subsets into subsets over r values. Boolean functions are convenient choice as building blocks in the design of set logic. B-maximal sets are maximal sets containing all Boolean functions, where Boolean functions are those obtained from /spl cup/, /spl cap/ and /sup -/ by composition (constants are not involved in the compositions). We give the number of n-place functions in each B-maximal set and find some properties of intersection of B-maximal sets in r-valued set logic. These properties are used to classify all 2-valued and 3-valued set logic functions according to the B-maximal sets to which they belong to. We prove that there are 8 and 200 such classes of functions respectively in 2-valued and 3-valued set logic. For each class of functions we give a one-place example function and its total number of one-place set logic functions. Finally, we study the B-Sheffer functions, i.e. functions which are complete under compositions with Boolean functions. We find the number of n-place B-Sheffer functions of 2-valued set logic and give a lower bound and an upper bound on the number of n-place B-Sheffer functions of 3-valued set logic. Also we enumerate all the classes of B-bases of 2-valued and 3-valued set logic.

Completeness criteria in set-valued logic under compositions with union and intersection

This paper discusses the Boolean completeness problems in r-valued set logic, which is the logic of functions mapping n-tuples of subsets into subsets over r values. Boolean functions are convenient choice as building blocks in the design of set logic circuits. Given a set S of Boolean functions, a set of functions F is S-complete if any set logic function can be composed from F once all Boolean functions from S are added to F. For the special case U=[/spl cup/, /spl cap/], we characterize all U-maximal sets in r-valued set logic. A set F is then U-complete if it is not a subset of any of these U-maximal sets, which is a completeness criterion in r-valued set logic under compositions with U functions.