Publication: Estudio de la endogamia de los circuitos booleanos
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2020
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Abstract
El objetivo de este trabajo consistió en desarrollar un experimento que permita analizar múltiples tipos de circuitos booleanos y las funciones que estos computan, de manera que podamos analizar las relaciones entre ambos conceptos.
En el primer capítulo, se expone una definición formal del concepto archiconocido de circuito booleano, se analiza el crecimiento doblemente exponencial del conjunto de circuitos con m bits de entrada y se dota de un orden total a dicho conjunto. En el segundo capítulo, se demuestran una serie de resultados sobre vectores con determinadas propiedades que nos permiten recorrer de manera eficiente el conjunto de circuitos booleanos que tienen determinada profundidad y anchura. Este algoritmo se ha implementado en C++ utilizando técnicas de programación concurrente y estructuras de datos adecuadas para garantizar la eficiencia del mismo.
A lo largo del tercer capítulo se exponen distintas métricas para medir la endogamia de los circuitos booleanos, así como intuiciones que justifican estas definiciones. Además, se incorpora un breve cuarto capítulo que versa sobre un tipo de gramática libre de contexto particular que genera funciones booleanas y su correspondencia con los circuitos booleanos.
Finalmente, este trabajo concluye con un quinto y último capítulo en el que se analizan los resultados obtenidos en este experimento y se exponen las conclusiones consecuentes.
The aim of this project was to develop an experiment that allow us to analyze different kinds of boolean circuits and the functions they compute to study the relationship between both concepts. In the first chapter, a formal definition of boolean circuits is introduced, the double exponential growth of the set of circuits with m bits of input is analyzed and a total order in this set is defined. During the second chapter, some results about vectors of some kind are proved, which allow us to efficiently sweep the set of boolean circuits with fixed depth and width. This algorithm has been implemented in C++ using concurrent programming and adequate data structures to ensure its efficiency. During the third chapter, some metrics to measure the endogamy of boolean circuits are defined and some ideas beyond these definitions are added. In addition, there is a fourth brief chapter where the relationship between a special type of context-free grammar and its equivalence with our boolean circuits are discussed. To conclude, a final fifth chapter is added to analyse the results obtained in the experiment and the conclusions these results led us to.
The aim of this project was to develop an experiment that allow us to analyze different kinds of boolean circuits and the functions they compute to study the relationship between both concepts. In the first chapter, a formal definition of boolean circuits is introduced, the double exponential growth of the set of circuits with m bits of input is analyzed and a total order in this set is defined. During the second chapter, some results about vectors of some kind are proved, which allow us to efficiently sweep the set of boolean circuits with fixed depth and width. This algorithm has been implemented in C++ using concurrent programming and adequate data structures to ensure its efficiency. During the third chapter, some metrics to measure the endogamy of boolean circuits are defined and some ideas beyond these definitions are added. In addition, there is a fourth brief chapter where the relationship between a special type of context-free grammar and its equivalence with our boolean circuits are discussed. To conclude, a final fifth chapter is added to analyse the results obtained in the experiment and the conclusions these results led us to.
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Trabajo de Fin de Grado en Ingeniería Informática y Matemáticas, Facultad de Informática UCM, Departamento de Sistemas Informáticos y Computación, Curso 2019/2020