Eduardo Mizraji

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• Professor at Universidad de la República de Uruguay

In 1929 Jan Lukasiewicz used, apparently for the first time, his Polish notation to represent the operations of formal logic. This is a parenthesis-free notation, which also implies that logical functions are operators preceding the variables on which they act. In the 1980s, within the framework of research into mathematical models on the parallel...

We investigate here the connectives of the three-valued logical systems of Lukasiewicz, Kleene and Heyting using matrix operators and vector truth values. Several novel results follow from this formalism. The modalities possibility and necessity are represented by monadic matrices. These modalities, and other monadic connectives, are represented by...

Life is a natural phenomenon ineluctably subject to the laws and principles of physics. In this framework, thermodynamics has a crucial role, since living beings are structured on a molecular and cellular basis that can only be maintained with extensive energy consumption. This imposes that living beings are necessarily open systems. But the surviv...

The Self Organizing Map (SOM) published by Teuvo Koho-nen in 1982 is a neural unsupervised model capable of organizing data on a topographic neural surface. Furthermore, this model shows the remarkable ability to preserve topological connections during the interaction between two neuronal layers (for example, the retina and the surface of a model c...

Explaining the foundation of cognitive abilities in the processing of information by neural systems has been in the beginnings of biophysics since McCulloch and Pitts pioneered work within the biophysics school of Chicago in the 1940s and the interdisciplinary cyberneticist meetings in the 1950s, inseparable from the birth of computing and artifici...

In this essay we propose, based on the ideas of L. Rapkine and J. Monod on the physical reasons for aesthetic appreciation, that the interest of creative scientists in Jorge Luis Borges' works is produced by an apparently contradictory effect: on the one hand, the serenity that these texts induce in the mind of an innovative person and, on the othe...

Mechanisms that ensure the stability of dynamical systems are of vital importance, in particular in our globalized and increasingly interconnected world. The so-called connectivity-stability dilemma denotes the theoretical finding that increased connectivity between the components of a large dynamical system drastically reduces its stability. This...

Los humanos habitamos dos mundos paralelos. Por un lado, residimos en la biosfera, región del planeta Tierra de la que obtenemos lo necesario para sobrevivir, como maquinarias biológicas que somos. Pero a la vez, nuestras vidas transcurren en un mundo de información. En la biosfera nacemos, nos alimentamos y morimos, sometidos a las rigurosas exige...

Esta presentación fue la base de una conferencia realizada en Buenos Aires en 2016 en el Ciclo de Charlas y Conferencias "Borges: Ficciones de un tiempo infinito", evento que durante varias semanas evocó los 30 años del fallecimiento del escritor.

Se expone un panorama de los diversos itinerarios seguidos por generaciones de investigadores para acercarse a comprender al pensamiento como un fenómeno natura surgido en los cerebros con cierta escala de complejidad

This work is based on ideas supported by some of the biologists who discovered foundational facts of twentieth-century biology and who argued that Maxwell's demons are physically implemented by biological devices. In particular, JBS Haldane first, and later J. Monod, A, Lwoff and F. Jacob argued that enzymes and molecular receptors implemented Maxw...

In this article we analyze the way in which logic gates become operators of linear algebra, and we study in particular the interesting properties of a logical operator, the square root of negation. This root is a square complex matrix. We show that through this operation it is possible to reveal the full computational potential of some fundamental...

In this work we investigate the representation of counterfactual conditionals using the vector logic, a matrix-vectors formalism for logical functions and truth values. Inside this formalism, the counterfactuals can be transformed in complex matrices preprocessing an implication matrix with one of the square roots of NOT, a complex matrix. This mat...

This book exposes the representation, developed by the author, of the logical operators using matrix algebra and describes some applications. After an elementary introduction to logical calculus and to matrix algebra, the book describes how neurocomputational models inspired this matrix-vector formalism. Then, the book presents the basic matrix log...

Human brains seem to represent categories of objects and actions as locations in a continuous semantic space across the cortical surface that reflects the similarity among categories. This vision of the semantic organization of information in the brain, suggested by recent experimental findings, is in harmony with the well-known topographically org...

There exists a dynamic interaction between the world of information and the world of concepts, which is seen as a quintessential byproduct of the cultural evolution of individuals as well as of human communities. The feeling of understanding (FU) is that subjective experience that encompasses all the emotional and intellectual processes we undergo...

A variety of problems emerged investigating electronic circuits, computer devices and cellular automata motivated a number of attempts to create a differential and integral calculus for Boolean functions. In the present article, we extend this kind of calculus in order to include the semantic of classical logical operations. We show that this exten...

Natural languages can express some logical propositions that humans are able to understand. We illustrate this fact with a famous text that Conan Doyle attributed to Holmes: “It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth”. This is a subtle logical statement usually fel...

Natural languages can express some logical propositions that humans are able to understand. We illustrate this fact with a famous text that Conan Doyle attributed to Holmes: “It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth”. This is a subtle logical statement usually fel...

We present a neural network model that can execute some of the procedures used in the information sciences literature. In particular we offer a simplified notion of topic and how to implement it using neural networks that use the Kronecker tensor product. We show that the topic detecting mechanism is related to Naive Bayes statistical classifiers,...

We organize our behavior and store structured information with many procedures that require the coding of spatial and temporal order in specific neural modules. In the simplest cases, spatial and temporal relations are condensed in prepositions like “below” and “above”, “behind” and “in front of”, or “before” and “after”, etc. Neural operators lie...

A variety of problems emerged investigating electronic circuits, computer devices and cellular automata motivated a number of attempts to create a differential and integral calculus for Boolean functions. In the present article, we extend this kind of calculus in order to include the semantic of classical logical operations. We show that this exten...

Numerous cortical disorders affect language. We explore the connection between the observed language behavior and the underlying substrates by adopting a neurocomputational approach. To represent the observed trajectories of the discourse in patients with disorganized speech and in healthy participants, we design a graphical representation for the...

Numerous cortical disorders affect language. We explore the connection between the observed language behavior and the underlying substrates by adopting a neurocomputational approach. To represent the observed trajectories of the discourse in patients with disorganized speech and in healthy participants, we design a graphical representation for the...

Natural languages can express some logical propositions that humans, with similar cultural traditions, are able to immediately and correctly compute. We illustrate this fact with a famous text that Arthur Conan Doyle attributed to Sherlock Holmes: the 'old maxim' mentioned in 'The Adventure of the Beryl Coronet', where Holmes said: 'It is an old ma...

Ensayo sobre la relación áurea, la serie de Fibonacci y algunas morfologías naturales

Several psychiatric and neurological conditions affect the semantic organization and content of a patient's speech. Specifically, the discourse of patients with schizophrenia is frequently characterized as lacking coherence. The evaluation of disturbances in discourse is often used in diagnosis and in assessing treatment efficacy, and is an importa...

The ability of the human brain to carry out logical reasoning can be interpreted, in general, as a by-product of adaptive capacities of complex neural networks. Thus, we seek to base abstract logical operations in the general properties of neural networks designed as learning modules. We show that logical operations executable by McCulloch-Pitts bi...

Este libro trata sobre la conexión entre la búsqueda de las leyes del razonamiento formal, la neurociencia, y la teoría de redes.

Revista Uruguay Ciencia N°10

Cognitive functions rely on the extensive use of information stored in the brain, and the searching for the relevant information for solving some problem is a very complex task. Human cognition largely uses biological search engines, and we assume that to study cognitive function we need to understand the way these brain search engines work. The ap...

In this article, we show the existence of a formal convergence between the matrix models of biological memories and the vector space models designed to extract information from large collections of documents. We first show that, formally, the term-by-document matrix (a mathematical representation of a set of codified documents) can be interpreted a...

Vector logic is a matrix–vector representation of the logical calculus inspired in neural network models. In this algebraic formalism, the truth values map on orthonormal Q-dimensional vectors, the monadic operations are represented by square matrices, and the dyadic operations produce rectangular matrices that act on the Kronecker product of the v...

Graph-theoretical methods have recently been used to analyze certain properties of natural and social networks. In this work, we have investigated the early stages in the growth of a Uruguayan academic network, the Biology Area of the Programme for the Development of Basic Science (PEDECIBA). This transparent social network is a territory for the e...

The study of complex macromolecular binding systems reveals that a high number of states and processes are involved in their mechanism of action, as has become more apparent with the sophistication of the experimental techniques used. The resulting information is often difficult to interpret because of the complexity of the scheme (large size and p...

Latent Semantic Analysis (LSA) is a well-known method for information retrieval. It has also been applied as a model of cognitive processing and word-meaning acquisition. This dual importance of LSA derives from its capacity to modulate the meaning of words by contexts, dealing successfully with polysemy and synonymy. The underlying reasons that ma...

New theoretical instruments, as goal-directed neural networks models and geometric representations based on semantic graphs, open new approaches for our understanding of the schizophrenic speech. The neuropathologic disorders of the schizophrenia can be simulated using neural models, and these models can eventually explain the origin of goal confus...

We approach the general problem of the emergence of complexity in natural, artificial or formal systems, by studying the particular problems that arise during the investigation of elementary cellular automata (ECA). Our objective is to investigate the origin of complex behaviours in the universe of ECA by using a formalism, named “vector logic”, th...

The search for “complexity signatures” in natural laws is a main concern for researchers working in many different fields, going from physics to biology. Very simple laws are able to produce unforeseen behaviors, and from their “simplicity” sometimes it is not possible to predict anything about the potential complexities they are able to produce wh...

Latent Semantic Analysis (LSA) is a well known method for information retrieval. It has also been applied as a model of cognitive processing and word-meaning acquisition. This dual importance of LSA derives from its capacity to modulate the meaning of words by contexts, dealing successfully with polysemy and synonymy. The underlying reasons that ma...

The development of neural network models has greatly enhanced the comprehension of cognitive phenomena. Here, we show that models using multiplicative processing of inputs are both powerful and simple to train and understand. We believe they are valuable tools for cognitive explorations. Our model can be viewed as a subclass of networks built on si...

Graphs have been increasingly utilized in the characterization of complex networks from diverse origins, including different kinds of semantic networks. Human memories are associative and are known to support complex semantic nets; these nets are represented by graphs. However, it is not known how the brain can sustain these semantic graphs. The vi...

This work concerns the interaction between two classical problems: the forecasting of the dynamical behaviors of elementary cellular automata (ECA) from its intrinsic mathematical laws and the conditions that determine the emergence of complex dynamics. To approach these problems, and inspired by the theory of reversible logical gates, we decompose...

In the context of modular neural networks we show that it is possible to define logical gates as matrix associative memories. This matrix formalism is able to represent logical decisions as dynamical dialogues among neural modules. If the interactions among these modules are mapped onto a subspace of positive identifications they give rise to dynam...

We named "Minsky's problem" the challenge of building up a cognitive architecture able to perform a good diagnosis based on multiple criteria that arrive one by one as successive clues. This is a remarkable human information processing capability, and a desirable ability for an artificial expert system. We present a general cognitive design that so...

Modular neural networks structured as associative memories are capable of processing inputs built from tensorial products of vectors. In this context, the operators of propositional and modal logic can be represented as modular distributed memories that can process not only classical Boolean but also fuzzy evaluations of truth-values of sentences....

Este libro trata sobre la evolución biológica, la cibernética y las moléculas

Context-dependent associative memories are models that allow the retrieval of different vectorial responses given a same vectorial stimulus, depending on the context presented to the memory. The contextualization is obtained by doing the Kronecker product between two vectorial entries to the associative memory: the key stimulus and the context. The...

Cooperativity, the departure from hyperbolic behaviour of the fractional saturation of a receptor at equilibrium (Y) for different values of ligand concentration (L), is an essential property of many physiological mechanisms and a first clue to the existence of conformational transitions and allosteric interactions. Here we investigate the properti...

We propose a first approximation to the modeling of cognitive decisions based on the theory of associative memories. The basic structure are the matrix memories of Anderson-Kohonen adapted to include the modulation of associations via tensorial preprocessing of inputs. This view admits an easy implementation of logical operations as matrix operator...

Vector logic is a mathematical model of the propositional calculus in which the logical variables are represented by vectors and the logical operations by matrices. In this framework, many tautologies of classical logic are intrinsic identities between operators and, consequently, they are valid beyond the bivalued domain. The operators can be expr...

Computer calculations on sequences provide specific answers to specific questions. On the other hand, graphical representations of sequences are perhaps better suited for drawing one's attention towards unsuspected features. Three types of graphical coding of nucleic acid sequences, in which the four nucleotides G, C, A, T (or U) are replaced with...

Vector logic is a mathematical model of logic in which the truth values are mapped on elements of a vector space. The binary logical functions are performed by rectangular matrices operating on the Kronecker product of their vectorial arguments. The binary operators acting on vectors representing ambiguous (fuzzy) truth values, generate many-valued...

A system of networks, consisting of a first net that constructs the Kronecker product between two vectors and then sends it to a second net that sustains a correlation memory, defines a context-dependent associative memory. In the real nervous system of higher mammals, the anatomy of the neural connections surely exhibits a considerable amount of l...

We describe a kind of logical calculus based on matrix operators. This representation is inspired on a neural network model for a context-dependent associative memory. The binary operations of classical propositional calculus (e.g. conjunction, disjunction, implication, Sheffer's connective) are represented by rectangular matrices that operate over...

Two approaches to the understanding of biological sequences are confronted. While the recognition of particular signals in sequences relies on complex physical interactions, the problem is often analysed in terms of the presence or absence of literal motifs (strings) in the sequence. We present here a test-case for evaluating the potential of this...

In this article we present a method that allows conditioning of the response of a linear distributed memory to a variable context. This method requires a system of two neural networks. The first net constructs the Kronecker product between the vector input and the vector context, and the second net supports a linear associative memory. This system...

En el curso de nuestro siglo, los científicos construyeron una biología admirablemente sólida, hoy en plena expansión y con límites no visibles. Esta consolidación de la ciencia biológica es, en buena medida, la consecuencia de un postulado, establecido vagamente en la segunda mitad del siglo XIX, y enunciado precisamente por Jacques Loeb en la pri...

In this work we use mathematical models with discrete and distributed time delays to analyse the stability of metabolic pathways controlled by end product. We assume the kinetics of the intermediates of the path to be unknown, and we cover the lack of information by using a time delay. We find that above a definite substrate value, there is a criti...

In this article we represent an enzyme capable of exhibiting more than one conformational state as a viscoelastic unit embedded in a fluid medium. We show how this viscoelastic unit is thermally activated to make transitions between equilibrium states, and propose this model as a mesoscopic representation for transitions between conformational stat...

In this work, we develop a minimal two-cycle model for the action of DNA girase. One of the cycles describes the ATP dependent chemicomechanical transduction performed by the enzyme. The other cycle describes the relaxing activity exhibited by girase on supercoiled DNA in the absence of substrates. Supercoiling of DNA is described as a random walk...

When, in a nucleic acid sequence, the four letters C, G, A, T (or U) are replaced by suitable graphical symbols, some patterns become immediately apparent. Two sets of symbols, constructed for the analysis of either purine/pyrimidine alternations, or of regions of complementarity within a sequence are shown. In addition, another mode of coding is p...

Stereograms containing two similar or dissimilar linear textures, either on the same surface or at two different depths, were tested on seventy subjects. Whereas random textures usually produced correct percepts, regular textures consistently led to errors of stereoscopic interpretations, including a reversal of hollows into bumps, dissociation of...

In this paper we analyze the organization imposed by the energy input during the migration of enzymes on DNA. We attempt to measure that organization by means of a concept proposed by A.A. Jarkievich in 1961. We found relationships among a Jarkievich measure, the energy dissipation, and the fluctuations in the kinematic velocity of the enzyme on th...

Kinetic models for the mode of action of processive and non-processive DNA-helicases are detailed. Fluxes at the steady state are analyzed, and the random walk of the enzymes on the DNA is studied in connection with the rate constants of the chemical reactions involved in the transformation of substrate to products. Finally, the constants of the ki...

We study the behavior of homogeneous fluctuations near an instability point of the Brusselator as the concentration of substrates changes slowly in time. We compute the transient and asymptotic behavior of the covariance matrix in the extensive regime. If the system is quenched to a distance ∼(system size)−1/2 near the instability point an enhancem...

In this paper we communicate some of the results of the statistical study of 15,000 patients that consulted the National Institute of Rheumatology of the Ministry of Public Health of Uruguay, between 1972 and 1981. We specially analyse here the frequencies of different rheumatic diseases and the sex and age distribution of some diseases. It is also...

We present a simple model based on the kinetics of DNA-dependent ATPases where the probability for enzyme migration on the linear activating polymer depends on the rate equations at the steady state. It is shown how the chemical velocity of the reaction is correlated to the average kinematic velocity along the polymer. The implications of this resu...

Se estudia una cadena de reacciones cuyo primer paso está catalizado por una enzima alostérica inhibida por el producto final. La enzima se comporta como el modelo de Monod-Wyman-Changeux de unión exclusiva. Bajo condiciones simples se reduce la descripción matemática de toda Ia cadena a una única ecuación diferencial con retardo que describe la sí...

Martini's model makes it possible to study various epidemiological situations arising in the study of infectious diseases: endemicity, recurrence, presence of a reservoir of pathogenic agent and active immunization. Numerical applications to epidemiological data concerning measles in Uruguay is presented.