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Concept# Boolean algebra

Summary

In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values true and false, usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (and) denoted as ∧, disjunction (or) denoted as ∨, and the negation (not) denoted as ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction, and division. Boolean algebra is therefore a formal way of describing logical operations, in the same way that elementary algebra describes numerical operations.
Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847), and set forth more fully in his An Investigation of the Laws of Thought (1854).
According to Huntington, the term "Boo

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Ce cours couvre les fondements des systèmes numériques. Sur la base d'algèbre Booléenne et de circuitscombinatoires et séquentiels incluant les machines d'états finis, les methodes d'analyse et de synthèse de systèmelogiques sont étudiées et appliquée

Study the basics of representation theory of groups and associative algebras.

1ère année: bases nécessaires à la représentation informatique 2D (3D).
Passage d'un à plusieurs logiciels: compétence de choisir les outils adéquats en 2D et en 3D.
Mise en relation des outils de CAO et DIM: regard critique et aptitude à choisir les méthodes nécessaires au résultat recherché.

Many natural optimization problems are NP-hard, which implies that they are probably hard to solve exactly in the worst-case. However, it suffices to get reasonably good solutions for all (or even most) instances in practice. This paper presents a new algorithm for computing approximate solutions in Theta(N) for the maximum exact 3-satisfiability (MAX-E-3-SAT) problem by using supervised learning methodology. This methodology allows us to create a learning algorithm able to fix Boolean variables by using local information obtained by the Survey Propagation algorithm. By performing an accurate analysis, on random conjunctive normal form instances of the MAX-E-3-SAT with several Boolean variables, we show that this new algorithm, avoiding any decimation strategy, can build assignments better than a random one, even if the convergence of the messages is not found. Although this algorithm is not competitive with state-of-the-art maximum satisfiability solvers, it can solve substantially larger and more complicated problems than it ever saw during training.

A language is said to be homogeneous when all its words have the same length. Homogeneous languages thus form a monoid under concatenation. It becomes freely commutative under the simultaneous actions of every permutation group G(n) on the collection of homogeneous languages of length n is an element of N. One recovers the isothetic regions from (Haucourt 2017, to appear (online since October 2017)) by considering the alphabet of connected subsets of the space vertical bar G vertical bar, viz the geometric realization of a finite graph G. Factoring the geometric model of a conservative program amounts to parallelize it, and there exists an efficient factoring algorithm for isothetic regions. Yet, from the theoretical point of view, one wishes to go beyond the class of conservative programs, which implies relaxing the finiteness hypothesis on the graph G. Provided that the collections of n-dimensional isothetic regions over G (denoted by R-n vertical bar G vertical bar) are co -unital distributive lattices, the prime decomposition of isothetic regions is given by an algorithm which is, unfortunately, very inefficient. Nevertheless, if the collections R-n vertical bar G vertical bar satisfy the stronger property of being Boolean algebras, then the efficient factoring algorithm is available again. We relate the algebraic properties of the collections R-n vertical bar G vertical bar to the geometric properties of the space I GI. On the way, the algebraic structure R-n vertical bar G vertical bar is proven to be the universal tensor product, in the category of semilattices with zero, of n copies of the algebraic structure R-1 vertical bar G vertical bar.

2019Data classification, in the present context, concerns the use of computers in order to create systems that learn how to automatically decide to which of a predefined set of classes a given object belongs. Boolean concepts have long been present in data classification, and still are today, at various levels. We consider a kind of Boolean elements, called patterns, which consist of specific conjunctions of Boolean facts. Assuming the existence of two classes of objects, the positive and the negative, patterns can be expressed as conditionals of the type "If A and B and not C, then positive", where A, B, and C are Boolean values, each associated with a specific attribute describing the objects to be classified. From the classification system point of view, patterns are basic Boolean expressions that can be processed in a purely mathematical manner. However, if the values of the conjunction have an intuitive meaning, then this type of representation provides an intuitive perspective of the classification system, which can be interpreted by humans. The use of sets of patterns for building classification systems is justified by the fact that certain conjunctions of the type described above are matched by objects of one class but not by those of the other class. In this sense, patterns possess distinguishing properties that can be used to determine whether an object belongs to a class (if it matches certain patterns) or not. In this thesis, we consider the use of classification systems based on patterns, which we have seen to be inherently Boolean, in the context of classification tasks where nothing is Boolean, that is to say, where the objects are described by multi-valued or continuous attributes and instead of a positive/negative decision, suitable to situations where only two classes are involved, the output is rather the selection of a class among several. The work presented here extends along different directions. It starts by the study of a suitable way of transforming the attributes describing the objects to be classified into equivalent Boolean attributes. We describe the constraints that must be satisfied by this transformation and present a procedure allowing it to be done in a short amount of time. Another research direction explores the possibility of generating multi-class divisions with systems whose output is Boolean. This study has the benefit of also being applicable to other types of classification systems adapted to two-class situations, but whose internal language is not necessarily Boolean. This is the case of some current state-of-the-art systems. To realize this adaptation, the common approach consists in decomposing the original problem into several two-class sub-problems and generating an independent classification system to solve each one of them. The final class decision is obtained from the combination of the partial Boolean decisions. We analyze and compare several existing procedures for this purpose, and in addition propose an interesting new procedure. Finally, we consider the possibility of sharing patterns among several classes and develop a Boolean-based classification system which is directly adapted to multi-class problems, but maintaining the property of being understandable by humans. We show that the performance of this type of system is not in all cases comparable to the best available systems, but they have the advantage of providing, in certain circumstances, a better understanding of the problem at hand.