Claudio Moraga: A Passion for Multi-Valued Logic and Soft Computing

When, in February 2008, I visited the European Centre for Soft Computing (ECSC) in Mieres Asturias, Spain, for the first time to give a talk, I stayed for about a week. It was Enric Trillas who extended the invitation, and it was Claudio Moraga who made my first weekend in Asturias—this then unknown and foreign landscape—enjoyable. One of the first places of interest he showed me was the old church of San Julián de los Prados, or Santullano (built between the years 812 and 842 AD) in Oviedo’s suburb Pumarí, close to the A-6 motorway.

What follows is a virtual discussion between two imaginary characters, Carla and Karl, concerning the widespread idea that, with fuzzy sets, the Aristotle’s principle of contradiction fails. Most thinkers see this principle as a guarantee for reasoning on solid grounds, and look at its failure with a suspicion of heterodoxy.

This chapter is devoted to do some calculations from the tables of the operations \((+, \cdot , ')\) in the three valued logic of Łukasiewicz, Gödel, Kleene, Bochvar and Post. The chapter is divided in two parts: the first one dealing with the Sheffer stroke, and the second one deals with models of Conjectures from a Basic Flexible Algebra.

In this contribution we collect a few considerations and remarks on such apparently unrelated topics as: an early paper by Norbert Wiener on the Nature of Mathematics; mathematical logic’s heritage on the formalization of reasoning; cognitive aspects on the modalities of drawing conclusions. We hope that reading the present paper will show that they are, neverthless, related in some way at least for what regards the problem of reasoning in the presence of uncertainty, showing a network of concepts that can help considering again the innovating aspects of fuzziness—in our opinion a more than fit homage to Claudio Moraga’s interdisciplinary approach to fuzziness.

In this homage to Prof. Moraga, firstly a short introduction and definition of Fuzzy Deformable Prototypes, introduced by the author in 2000, referring some of the most interesting applications of this concept, mainly concerning prediction systems is presented. Then, there is a short introduction to interval fuzzy sets with the aim of showing some reflections on why it could be interesting to use interval fuzzy sets in-stead of standard ones for dealing with Fuzzy Deformable Prototypes and some guidelines for the representation and inference mechanisms required for applications.

In the following we give a survey on important epochs in the development of the electrical means for transmission and processing of information between humans and machines. The goal is to contribute to a kind of holistic knowledge of the function of the different complex systems and devices of modern information technology. We are convinced that a historical projection on the different stages of development can be of great help. However, by the limited space of this paper we can do this only by a sketch. The list of relevant literature which we give at the end may help to get the desirable deeper knowledge.

The purpose of this work is design an automatic starter for the synchronizing equipment (SE) in power systems. Such a starter leads to a faster and secure decision concerning the introduction (i.e., a parallel switch) of a ready reserve generator to the power system as a type of ancillary service. The applied method is based on a hybrid neural model (HNM). The HNM consists of a feedforward, three-layer neural network using neurons with a sigmoid activation, and a perceptron with a biased hard limiter. The adopted HNM is excited by signals of the generator’s operating status, current load regime, and an available power of the wind power plant park (WPPP). The logical decision-making is used to find out the actual load regime and the available power from WPPP relevant to building of HNM’s input. The automatic starter of the SE enables a reduction of the time spent in seeing whether or not the rescue action will imply resorting to the ready reserve power. Such a reduction is certainly a contribution to the efforts of preserving a power system’s integrity during the critical situations (e.g. generating unit/area outages). HNM has the ability to recognize the crisis symptoms immediately, and to consequently suggest an introduction of the ready-reserve (RR) generator (a supplemental reserve) through SE.

Reed–Muller (RM) expressions are an important class of functional expressions for binary valued (Boolean) functions which have a double interpretation, as analogues to both Taylor series or Fourier series in classical mathematical analysis. In matrix notation, the set of basic functions in terms of which they are defined can be represented by a binary triangular matrix. Reed-Muller-Fourier (RMF) expressions are a generalisation of RM expressions to multiple valued functions preserving properties of RM expressions including the triangular structure of the transform matrix. In this paper, we discuss different methods for computing RMF coefficients over different data structure efficiently in terms of space and time. In particular, we consider algorithms. corresponding to Cooley-Tukey and constant geometry algorithms for Fast Fourier transform. We also consider algorithms based on various decompositions borrowed from the decomposition of the Pascal matrix and related computing algorithms.

In classical multiple-valued logic its values are encoded by integers. This complicates the use of multiple-valued logic as a basic model, which can be utilized in an artificial neuron, because the values of k-valued logic encoded by integers 0, 1, 2, ..., k are not normalized. To overcome this obstacle, it was suggested to encode the values of k-valued logic by complex numbers located on the unit circle, namely by the kth roots of unity. It is described in the paper how this model of multiple-valued logic over the field of complex numbers was suggested and how it was used to develop a multi-valued neuron (MVN). Then it is considered how a feedforward neural network based on MVN—a multilayer neural network with multi-valued neurons (MLMVN) was designed and its derivative-free learning algorithm based on the error-correction learning rule was presented. Different applications of MLMVN, which outperforms many other machine learning tools in terms of learning speed and generalization capability are also observed.

This is short overview of the authors’ research in the area of the sequential or recursive Bayesian estimation of recurrent neural networks. Our approach is founded on the joint estimation of synaptic weights, neuron outputs and structure of the recurrent neural networks. Joint estimation enables generalization of the training heuristic known as teacher forcing, which improves the training speed, to the sequential training on noisy data. By applying Gaussian mixture approximation of relevant probability density functions, we have derived training algorithms capable to deal with non-Gaussian (multi modal or heavy tailed) noise on training samples. Finally, we have used statistics, recursively updated during sequential Bayesian estimation, to derive criteria for growing and pruning of synaptic connections and hidden neurons in recurrent neural networks.

Data words are an extension of traditional strings that have at each position, besides a symbol from some finite alphabet, a data value from some infinite domain. Class-memory automata constitute an automata model for data words with a decidable emptiness problem. The paper improves the previous result that class-memory automata are strictly more expressive than register automata, another automata model for data words. More specifically, it shows that weak class-memory automata, a restriction of class-memory automata introduced by Cotton-Barratt, Murawski, and Ong are strictly more powerful than the extension of register automata by a data-guessing facility. While weak deterministic class-memory automata yield a restriction of class-memory automata which is closed under Boolean operations, the paper also proposes an extension of deterministic class-memory automata with this property.

Improvement of time series forecasting accuracy is an active research area that has significant importance in many practical domains. Ensemble methods have gained considerable attention from machine learning and soft computing communities in recent years. There are several practical and theoretical reasons, mainly statistical reasons, why an ensemble may be preferred. Ensembles are recognized as one of the most successful approaches to prediction tasks. Previous theoretical studies of ensembles have shown that one of the key reasons for this performance is diversity among ensemble members. Several methods exist to generate diversity. Extensive works in literature suggest that substantial improvements in accuracy can be achieved by combining forecasts from different models. The focus of this chapter will be on ensemble for time series prediction. We describe the use of ensemble methods to compare different models for time series prediction and extensions to the classical ensemble methods for neural networks for classification and regression prediction by using different model architectures. Design, implementation and application will be the main topics of the chapter, and more specifically: conditions under which ensemble based systems may be more beneficial than their single machine; algorithms for generating individual components of ensemble systems; and various procedures through which they can be combined. Various ensemble based algorithms will be analyzed: Bagging, Adaboost and Negative Correlation; as well as combination rules and decision templates. Finally, future directions will be time series forecasting, machine fusion and others areas in which ensemble of machines have shown great promise.

Even limited to 4-bits reversible functions, the synthesis of optimal reversible circuits becomes an arduous task owing to the extremely large problem space. The current paper tries to answer the following question: is it possible to implement optimal 4-bit reversible circuits without relying on existing partial solutions libraries? A distributed linear genetic programming based-approach (DRIMEP2) is presented. It consists of a hierarchical topology with a new communication policy to allow the evolutionary algorithm to explore and exploit the search space in an efficient way. To test the effectivity and the efficiency of the proposed system, the design of 69 benchmarks (4-bits reversible functions) was performed. With respect to good results available in the literature, a gate count reduction up to 60 % was achieved with an average of 16.82 % (for the two first benchmark groups where the gate count of the circuit was considered by the reference authors) and a quantum cost reduction up to 62.71 % was reached with an average of 10.79 % (for the two remaining benchmark groups where the quantum cost of the circuit was considered by the reference authors).

Bent functions are functions that have the largest distance to all linear functions. Due to this property bent functions hedge statistical attacks against cryptosystems. This contribution reflects some steps on the way of the specification of Boolean bent functions by O. S. Rothaus, over the description of such bent functions using Boolean differential equations by Bernd Steinbach, the enumeration of bent functions by Natalia Tokareva, the evaluation of classes of bent functions using the Special Normal Form (SNF) by Bernd Steinbach and Christian Posthoff, the embedding of bent functions into other properties needed in cryptosystems by Jon T. Buttler and Tsutomu Sasao, the extension of such properties by Chunhui Wu and Bernd Steinbach, to the generalization to multi-valued bent function by Claudio Moraga et al.

In this chapter we summarize the fruitful relationship between Prof. Claudio Moraga and the University of Santiago de Compostela. For almost two decades Prof. Moraga was a regular visitor at our departments as well as a lecturer at our doctoral and summer courses. A milestone in this long term relationship was his key role in the hosting and organization of one of the most important international scientific meetings the Intelligent Systems Group at the University of Santiago de Compostela ever organized.

By using the concept of possible worlds as system states, it is possible to express a system’s internal state with the configuration of the system’s variables. In the same way, the (usually incomplete and not necessarily correct) belief of an intelligent agent about the system’s state can be expressed by a set of possible worlds. If this belief is to be changed due to more accurate information about the system’s true state, it is reasonable to incorporate the new information while at the same time abandon as little information as possible, that is, to minimally change the belief of the agent. In this paper we define semantical distances between possible worlds based on the background beliefs of an agent which are represented as a conditional knowledge base, by defining distances on the syntax of the (semantical) conditional structure. With these distances, we instantiate AGM belief change operators that incorporate new information into the belief state and implement the principle of minimal change by selecting a set of worlds that are closest to the actual beliefs. We demonstrate that using the background knowledge to calculate distances allows us to change the belief state of the agent in a way that is semantically more correct than using, e.g., Dalal’s distance. We finally discuss that defining the distances on a syntactical distance on conditional structures allows us to implement the resulting belief change operators more efficiently.

In this contribution we deal with a global monotonicity condition for the class of associative extended aggregation functions. We insist on the idea that global monotonicity can be taken as a minimum requirement for an extended aggregation function to be considered consistent.

The amount of information available nowadays is almost incalculable, presenting new opportunities to gain insight from this data. In this chapter we present some of the work done in field of Distributed Machine Learning and discuss a problem not often mentioned in the literature. The problem is related when the distributed information comes from different contexts. Different contexts can be defined as the different underlying laws of probability governing the data. This is a problem not always addressed, where the majority of the contributions assume that between distributed sources, there is no difference in the underlying law of probability. In this chapter a distributed regression model is presented that addresses this problem.

Clustering has become an important tool for every data scientist as it allows to perform exploratory data analysis and summarize large amounts of data. Specifically for text data, clustering faces other challenges derived from the high-dimensional space into which the data is represented. Furthermore and in spite of the fact that important contributions have already been made, scalability presents an important challenge when the whole-data-in-memory approach is no longer valid for real scenarios where data is collected in massive volumes. This chapter reviews the recent contributions on high-dimensional text data clustering with particular emphasis on scalability issues and also on the impact of the curse of dimensionality over the distance-based clustering methods.

I met Claudio Moraga at the European Centre for Soft Computing back in 2008. My friend and colleague Enric Trillas introduced him to me and soon I understood Claudio was not only a scientist in the truest and highest meaning of the word but also a very special human being: friendly, empathic, cordial and friend of his friends. Among his many educational achievements, he got a master degree in engineering in his youth from the Massachusetts Institute of Technology, MIT.