Boolean Algebra Research Papers - Academia.edu (original) (raw)

This paper describes work carried out to explore the role of a learning companion as a teachable student of the human student. A LCS for Binary Boolean Algebra has been developed to explore the hypothesis that a learning companion with... more

This paper describes work carried out to explore the role of a learning companion as a teachable student of the human student. A LCS for Binary Boolean Algebra has been developed to explore the hypothesis that a learning companion with less expertise than ...

The irregular shape packing problem is a combinatorial optimization problem that consists of arranging items on a container in such way that no item overlaps. In this paper we adopt a solution that places the items sequentially, touching... more

The irregular shape packing problem is a combinatorial optimization problem that consists of arranging items on a container in such way that no item overlaps. In this paper we adopt a solution that places the items sequentially, touching the already placed items or the container. To place a new item without overlaps, the collision free region for the new item is robustly computed using non manifold Boolean operations. A simulated annealing algorithm controls the items sequence of placement, the item's placement and orientation. In this work, the placement occurs at collision free region's vertices. Several results with benchmark datasets obtained from the literature are reported. Some of them are the best already reported in the literature. To improve the computational cost performance of the algorithm, a parallelization method to determine the collision free region is proposed. We demonstrated two possible algorithms to compute the collision free region, and only one of them can be parallelized. The results showed that the parallelized version is better than the sequential approach only for datasets with very large number of items. The computational cost of the non manifold Boolean operation algorithm is strongly dependent on the number of vertices of the original polygons.

Is a whole something more than the sum of its parts? Are there things composed of the same parts? If you divide an object into parts, and divide those parts into smaller parts, will this process ever come to an end? Can something lose... more

Is a whole something more than the sum of its parts? Are there things composed of the same parts? If you divide an object into parts, and divide those parts into smaller parts, will this process ever come to an end? Can something lose parts or gain new ones without ceasing to be the thing it is? Does any multitude of things (including disparate things such as you, this book, and the tail of a cat) compose a whole of some sort? Questions such as these have occupied us for at least as long as philosophy has existed. They define the field that has come to be known as mereology-the study of all relations of part to whole and of part to part within a whole-and have deep and far-reaching ramifications in metaphysics as well as in logic, the foundations of mathematics, the philosophy of language, the philosophy of science, and beyond. In Mereology, A. J. Cotnoir and Achille C. Varzi have compiled decades of advanced research into a comprehensive, up-to-date, and formally rigorous picture. The early chapters cover the more classical aspects of mereology; the rest of the book deals with variants and extensions. Whether you are an established professional philosopher, an interested student, or a newcomer, inside you will find all the tools you need to join this ever-evolving field of inquiry and theorize about all things mereological.

If I should pick three thinkers that influenced me the most, I would name Nietzsche, Robert Greene (self-help book writer), and Lao Tzu. Although most portions of the text written by my third picked thinker are shrouded in ambiguity, I... more

If I should pick three thinkers that influenced me the most, I would name Nietzsche, Robert Greene (self-help book writer), and Lao Tzu. Although most portions of the text written by my third picked thinker are shrouded in ambiguity, I personally believe that his Tao philosophy is so grand and deep that it even embraces the other two respectable writers. I personally concede, though, that I do not have high regards for China as a nation. Nevertheless, the profundity of the Tao Te Ching was sufficient to grab my attention, which led to creation of this blog. I was born and raised in a country that has a high regard for Confucianism. Although I was not familiar with the name of Confucius when I was young, I can see now that part of my value system was greatly influenced by him. However, I personally conclude that there is a reason why his philosophy is eclipsed by Lao Tzu. I wish to discuss the content of the Tao Te Ching in detail chapter by chapter. Thanks.

Polynomial systems are fundamental tools in the solution of hard problems in science and engineering such as robotics, automated reasoning, artificial intelligence and signal processing. Similarly, from the early days of the digital era,... more

Polynomial systems are fundamental tools in the solution of hard problems in science and engineering such as robotics, automated reasoning, artificial intelligence and signal processing. Similarly, from the early days of the digital era, Boolean variables have been the foundations of the computer operations. Hence, the application of common algebraic techniques to Boolean algebra is used now as a method to solve complex Boolean equation systems that before were only intended to solve using Boolean logic techniques. The aim of this project is to demonstrate that Zhegalkin polynomials (also known as Algebraic Normal Form - ANF) are an alternative way to represent Boolean functions. In order to test the hypothesis, a Zhegalkin SAT Solver (ZPSAT) was developed. The results conducted after the testing concluded that ZPSAT can solve a conjunction of XOR equations efficiently in terms of reliability and computing time.
The heuristic used to build ZPSAT was based mainly on the concepts used by the Horn Formulae and a Fast-Multiplication method of two ANF polynomials known as Mobius transform. The proposed SAT Solver use a frequency table which records all the variables and the number of times it appears in the total equation system. Similarly, a goal table where the target values for each polynomial is generated during the SAT Solver pre-processing stage. These two elements constitute the “core” of the Heuristic and will help to control the backtracking process, restoring the previous variables values when the satisfiable number of polynomials obtained by the current iteration is below than the ones in the previous loop. Hence, applying this heuristic it is possible to achieve the maximum factorization and reduction of the whole equations system. The SAT Solver succeed when it finds the first solution to the equations system. Successive intents are executed until all the monomials inside the unsatisfiable polynomials are solve. Otherwise, the SAT Solver will conclude that the whole equation system is unsatisfiable.
This paper is organized as follows: Section 2 introduces some essential concepts to understand the scope and the solution proposed. In Section 3 the methodology used to design a solution to the Zhegalkin equation system is explained. Section 4 presents the ZPSAT algorithm and describes its implementation and testing. Finally, Section 4 summarizes the results and conclusions.

The boolean satisfiability problem was the first example of a NP-complete problem: a boolean formula can only be considered satisfiable if there is a set of variable bindings that evaluates said formula to true. The task of finding such... more

The boolean satisfiability problem was the first example of a NP-complete problem: a boolean formula can only be considered satisfiable if there is a set of variable bindings that evaluates said formula to true. The task of finding such solution is done by so-called SAT solvers. This project nevertheless focuses on constructing a solver for propositional formulae based on the DPLL algorithm, realizing the Watched Literals technique. Exploration will also be done regarding the performance impact of the implementation decisions, followed by further discussion about possible optimizations.

Mediante el análisis de Boole, el uso de automatismos eléctricos y la programación en PLC se quiere resolver una problemática que involucra variables de control ON – OFF. Con efectos de aplicación a la vida real se describe el... more

Mediante el análisis de Boole, el uso de automatismos eléctricos y la programación en PLC se quiere resolver una problemática que involucra variables de control ON – OFF. Con efectos de aplicación a la vida real se describe el funcionamiento, construcción y materiales del dispositivo diseñado.

In this paper, we study three representations of lattices by means of a set with a binary relation of compatibility in the tradition of Ploscica. The standard representations of complete ortholattices and complete perfect Heyting... more

In this paper, we study three representations of lattices by means of a set with a binary
relation of compatibility in the tradition of Ploscica. The standard representations of
complete ortholattices and complete perfect Heyting algebras drop out as special cases
of the first representation, while the second covers arbitrary complete lattices, as well
as complete lattices equipped with a negation we call a protocomplementation. The
third topological representation is a variant of that of Craig, Haviar, and Priestley. We
then extend each of the three representations to lattices with a multiplicative unary
modality; the representing structures, like so-called graph-based frames, add a second
relation of accessibility interacting with compatibility. The three representations
generalize possibility semantics for classical modal logics to non-classical modal logics,
motivated by a recent application of modal orthologic to natural language semantics.

In traditional semantics for classical logic and its extensions, such as modal logic, propositions are interpreted as subsets of a set, as in discrete duality, or as clopen sets of a Stone space, as in topological duality. A point in such... more

In traditional semantics for classical logic and its extensions, such as modal logic, propositions are interpreted as subsets of a set, as in discrete duality, or as clopen sets of a Stone space, as in topological duality. A point in such a set can be viewed as a "possible world," with the key property of a world being primeness—a world makes a disjunction true only if it makes one of the disjuncts true—which classically implies totality—for each proposition, a world either makes the proposition true or makes its negation true. This chapter surveys a more general approach to logical semantics, known as possibility semantics, which replaces possible worlds with possibly partial "possibilities." In classical possibility semantics, propositions are interpreted as regular open sets of a poset, as in set-theoretic forcing, or as compact regular open sets of an upper Vietoris space, as in the recent theory of "choice-free Stone duality." The elements of these sets, viewed as possibilities, may be partial in the sense of making a disjunction true without settling which disjunct is true. We explain how possibilities may be used in semantics for classical logic and modal logics and generalized to semantics for intuitionistic logics. The goals are to overcome or deepen incompleteness results for traditional semantics, to avoid the nonconstructivity of traditional semantics, and to provide richer structures for the interpretation of new languages.

From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of... more

From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models can be topologized in such a way that the theory can be recovered from its space of models. The situation can be cast as a formal duality relating two categories of syntax and semantics, mediated by homming into a common dualizing object, in this case 2. In the present work, we generalize the entire arrangement from propositional to first-order logic. Boolean algebras are replaced by Boolean categories presented by theories in first-order logic, and spaces of models are replaced by topological groupoids of models and their isomorphisms. A duality between the resulting categories of syntax and semantics, expressed first in the form of a contravariant adjunction, is established by homming into a common dualizing object, now Sets\SetsSets, regarded once as a boolean category, and once as a groupoid equipped with an intrinsic topology. The overall framework of our investigation is provided by topos theory. Direct proofs of the main results are given, but the specialist will recognize toposophical ideas in the background. Indeed, the duality between syntax and semantics is really a manifestation of that between algebra and geometry in the two directions of the geometric morphisms that lurk behind our formal theory. Along the way, we construct the classifying topos of a decidable coherent theory out of its groupoid of models via a simplified covering theorem for coherent toposes.

Resumen Lo que distingue a los diálogos formales de los materiales es que en estos últimos la formulación de la Regla Socrática prescribe una forma de interacción que permite al Proponente basar la afirmación de una proposición elemental... more

Resumen Lo que distingue a los diálogos formales de los materiales es que en estos últimos la formulación de la Regla Socrática prescribe una forma de interacción que permite al Proponente basar la afirmación de una proposición elemental en identidades específicas a la proposición en cuestión. El objetivo del presente artículo es describir sucintamente la forma de producir diálogos materiales y al mismo tiempo discutir sus vínculos con el desarrollo de lenguajes totalmente interpretados en la Teoría Constructiva de Tipos (TCT). A modo de ilustración discutiremos brevemente la formulación de la Regla Socrática para el conjunto de números naturales, para la noción de identidad predicativa y para el conjunto de Booleanos.

Karnaugh map (K-map) and Quine-McCluskey algorithm (QM algorithm) are two methods to simplify Boolean algebra functions. They cannot always optimize their own results completely. Then we have to use optimization laws to reduce the number... more

Karnaugh map (K-map) and Quine-McCluskey algorithm (QM algorithm) are two methods to simplify Boolean algebra functions. They cannot always optimize their own results completely. Then we have to use optimization laws to reduce the number of operators but the human error might lead us to a false answer. Taking out the common factor works on the results of K-map and QM algorithm. Usage of the common factor in both solutions is explained in the paper as an added step and with three examples to make challenges for it.

Rules and laws of Boolean algebra are very essential for the simplification of a long and complex logic equation. Applying the Boolean algebra basic concept, such a kind of logic equation could be simplified in a more simple and efficient... more

Rules and laws of Boolean algebra are very essential for the simplification of a long and complex logic equation. Applying the Boolean algebra basic concept, such a kind of logic equation could be simplified in a more simple and efficient form.Mainly, the standard rules of Boolean algebra are given in operator ‘+’ and ‘x’, based on the AND and OR logic gates equations. For some logic designs, it is commonly that logic problems are writtenin terms of XOR format.This paper tries to conduct something different. It will analyze, describe, and derive Boolean algebra rules related to logic equations using exclusive-or (XOR) logic gate.