Relative Entailment Among Probabilistic Implications

We study a natural variant of the implicational fragment of propositional logic. Its formulas are pairs of conjunctions of positive literals, related together by an implicational-like connective; the semantics of this sort of implication is defined in terms of a threshold on a conditional probability of the consequent, given the antecedent: we are dealing with what the data analysis community calls confidence of partial implications or association rules. Existing studies of redundancy among these partial implications have characterized so far only entailment from one premise and entailment from two premises, both in the stand-alone case and in the case of presence of additional classical implications (this is what we call "relative entailment"). By exploiting a previously noted alternative view of the entailment in terms of linear programming duality, we characterize exactly the cases of entailment from arbitrary numbers of premises, again both in the stand-alone case and in the case of presence of additional classical implications. As a result, we obtain decision algorithms of better complexity; additionally, for each potential case of entailment, we identify a critical confidence threshold and show that it is, actually, intrinsic to each set of premises and antecedent of the conclusion

Strong size properties

We prove that countable aposyndesis, finite-aposyndesis, continuum chainability, acyclicity (for n≥ 3), and acyclicity for locally connected continua are strong size properties. As a consequence of our results we obtain that arcwise connectedness is a strong size property which is originally proved by Hosokawa

Data analysis through graph decomposition

This work is developed within the field of data mining and data visualization. Under the premise that many of the algorithms give as result huge amounts of data impossible to handle for the users, we work with the decomposition of Gaifman graphs and its variations as an option for data visualization. In fact, we apply the decomposition method based on the so-called 2-structures. This decomposition method has been theoretical developed but has not any practical application yet in this field, being this part of our contribution. Thus, from a dataset we construct the Gaifman graph (and possible variations of it) that represents information about co-occurrence patterns. More precisely, the construction of the Gaifman graphs from the dataset is based on co-occurrence, or lack of it, of items in the dataset. That is, those pair of items that appear together in any transaction are connected and those items that never appear together are disconnected. We may do the natural completion of the graph adding the absent edges with a different kind of edges, in this way we get a complete graph with two equivalence classes on its edges. Now, think of the graph where the kind of edges are determined by the multiplicity of the items that they connect, that is, by the number of transactions that contains the pair of items that the edge connects. In this case we have as many equivalence relations as different multiplicities, and we may apply some discretization methods on them to get different variations of the graphs. All these variations can be seen as 2-structures. The application of the 2-structure decomposition method produces as result a hierarchical visualization of the co-occurrences on data. In fact, the decomposition method is based on clan decomposition. Given a 2-structure defined on U, a set of vertices C, C subset of U, is a clan if, for each z not in C, z may not distinguish among the elements of C. We connect this decomposition with an associated closure space, developing this intuition by introducing a construction of implication sets, named clan implications. Based on the definition of a clan, let x, y be elements of any clan C, if there is z such that sees in a different way x and y, that is the edges (x,z) and (x,y) are in different equivalence classes, so z in C; this is equivalent to C logically entails the implication xy then z. Throughout the thesis, in order to explain our work in a constructive way, we first work with the case of having only two equivalence classes and its corresponding nomenclature (modules), and then we extend the theory to work with more equivalence classes. Our main contributions are: an algorithm (with its full implementation) for the clan decomposition method; the theorems that support our approach, and examples of its application to demonstrate its usability.Este trabajo está desarrollado dentro del área de Mineria de Datos y Visualización de Datos. Bajo la premisa de que muchos algoritmos dan como resultados un gran número de datos imposibles de manejar por los usuarios, proponemos trabajar con la descomposición de grafos de Gaifman y sus variantes como una opción para visualizar datos. De hecho, aplicamos un método de descomposición basado en las llamadas 2-structures. Este método de descomposición ha sido teóricamente desarrollado pero hasta ahora no había tenido una aplicación práctica en esta área, siendo ésta parte de nuestra contribución. Así, partiendo de la base de datos contruimos un grafo de Gaifman (y posiblemente variantes de él) que representa información sobre los patrones de co-ocurrencias. Esto es, aquellos pares de items que aparecen juntos en cualquier transacción son conectados, mientras que aquellos que nunca aparecen juntos están desconectados. Podemos completar naturalmente el grafo añadiendo las aristas ausentes como un diferente tipo de arista, en este sentido, obtenemos un grafo completo con dos clases de equivalencia sobre sus aristas. Ahora, piense en el grafo donde el tipo de aristas está determinado por la multiplicidad de los items que las aristas conectan, esto es, el número de transacciones que contienen el par de items que la arista conecta. En este caso tenemos tantas relaciones de equivalencia como diferentes multiplicidades, podemos aplicar algunos métodos de discretización sobre ellos para así obtener diferentes variantes de grafos, todas estas variaciones pueden ser vistas como 2-structures. La aplicación del método de descomposición de 2-structures produce como resultados una visualización jerárquica de las co-ocurrencias de los datos. De hecho, el método de descomposición está basado en la descomposición de clanes. Dada una 2-structure definida sobre U, un conjunto de vertices C, C subconjunto de U, es un clan si para cada z que no está en C, z no distingue los elementos de C, esto es, z está conectado a los elementos de C con el mismo tipo de aristas. En nuestro trabajo, conectamos esta descomposición con un espacio de cerrados asociado, desarrollamos esta parte del trabajo introduciendo una construcción de un conjunto de implicaciones, llamado clan implications. Basándonos en la definición de clan, sea x, y elemento de cualquier clan C, si existe z tal que las aristas (x,z) y (y,z) están en diferentes clases de equivalencia, z deber estar en C; esto es equivalente a que C lógicamente ocasiona la implicación xy entonces z. A lo largo de esta tesis, con el fin de explicar nuestro trabajo de una manera constructiva, primero trabajamos con sólo dos clases de equivalencia y su nomenclatura correspondiente (modules, en lugar de clanes), para después extender la teoría a más clases de equivalencia. Nuestras contribuciones principales son: un algoritmo (que implementamos) para le método de descomposición de clanes; los teoremas que respaldan nuestro trabajo, y ejemplos de sus aplicaciones con el fin de ilustrar su usabilidad.Postprint (published version

Behind Misheard Lyrics

Phonetics deals with the production of different sounds of speech made by people. It explains how the vocal tract produces the sounds that people make as they talk. A lot of words can be misheard because of the complexity that goes into creating certain sounds. This is especially common with song lyrics, with thousands of song lyrics being misheard or misinterpreted all the time. There are several reasons behind why this happens with so many songs and this can be seen when lyrics are transcribed. When one word or sound is misheard and sounds like another, this usually means that the words share one or more phonetic feature. Sometimes words will have a similar manner or place of articulation. Articulators are what comes together to make a sound. Other times, it will have to do with the voicing of word (voiced or voiceless), vowel height or frontness of the tongue, rounding of the lips or the tongue being lax or tense. There are so many elements that go into saying and understanding words and there is so much happening in the mouth with each sound being made that it is very easy to hear words incorrectly. Looking into just one set of lyrics and the way people hear them can give others an understanding of how language really works and how complex it really is. Everyone has their own unique way of speaking and hearing words, whether it is English or another language, and examining lyrics is one way to come to this understanding

Data analysis through graph decomposition

This work is developed within the field of data mining and data visualization. Under the premise that many of the algorithms give as result huge amounts of data impossible to handle for the users, we work with the decomposition of Gaifman graphs and its variations as an option for data visualization. In fact, we apply the decomposition method based on the so-called 2-structures. This decomposition method has been theoretical developed but has not any practical application yet in this field, being this part of our contribution. Thus, from a dataset we construct the Gaifman graph (and possible variations of it) that represents information about co-occurrence patterns. More precisely, the construction of the Gaifman graphs from the dataset is based on co-occurrence, or lack of it, of items in the dataset. That is, those pair of items that appear together in any transaction are connected and those items that never appear together are disconnected. We may do the natural completion of the graph adding the absent edges with a different kind of edges, in this way we get a complete graph with two equivalence classes on its edges. Now, think of the graph where the kind of edges are determined by the multiplicity of the items that they connect, that is, by the number of transactions that contains the pair of items that the edge connects. In this case we have as many equivalence relations as different multiplicities, and we may apply some discretization methods on them to get different variations of the graphs. All these variations can be seen as 2-structures. The application of the 2-structure decomposition method produces as result a hierarchical visualization of the co-occurrences on data. In fact, the decomposition method is based on clan decomposition. Given a 2-structure defined on U, a set of vertices C, C subset of U, is a clan if, for each z not in C, z may not distinguish among the elements of C. We connect this decomposition with an associated closure space, developing this intuition by introducing a construction of implication sets, named clan implications. Based on the definition of a clan, let x, y be elements of any clan C, if there is z such that sees in a different way x and y, that is the edges (x,z) and (x,y) are in different equivalence classes, so z in C; this is equivalent to C logically entails the implication xy then z. Throughout the thesis, in order to explain our work in a constructive way, we first work with the case of having only two equivalence classes and its corresponding nomenclature (modules), and then we extend the theory to work with more equivalence classes. Our main contributions are: an algorithm (with its full implementation) for the clan decomposition method; the theorems that support our approach, and examples of its application to demonstrate its usability.Este trabajo está desarrollado dentro del área de Mineria de Datos y Visualización de Datos. Bajo la premisa de que muchos algoritmos dan como resultados un gran número de datos imposibles de manejar por los usuarios, proponemos trabajar con la descomposición de grafos de Gaifman y sus variantes como una opción para visualizar datos. De hecho, aplicamos un método de descomposición basado en las llamadas 2-structures. Este método de descomposición ha sido teóricamente desarrollado pero hasta ahora no había tenido una aplicación práctica en esta área, siendo ésta parte de nuestra contribución. Así, partiendo de la base de datos contruimos un grafo de Gaifman (y posiblemente variantes de él) que representa información sobre los patrones de co-ocurrencias. Esto es, aquellos pares de items que aparecen juntos en cualquier transacción son conectados, mientras que aquellos que nunca aparecen juntos están desconectados. Podemos completar naturalmente el grafo añadiendo las aristas ausentes como un diferente tipo de arista, en este sentido, obtenemos un grafo completo con dos clases de equivalencia sobre sus aristas. Ahora, piense en el grafo donde el tipo de aristas está determinado por la multiplicidad de los items que las aristas conectan, esto es, el número de transacciones que contienen el par de items que la arista conecta. En este caso tenemos tantas relaciones de equivalencia como diferentes multiplicidades, podemos aplicar algunos métodos de discretización sobre ellos para así obtener diferentes variantes de grafos, todas estas variaciones pueden ser vistas como 2-structures. La aplicación del método de descomposición de 2-structures produce como resultados una visualización jerárquica de las co-ocurrencias de los datos. De hecho, el método de descomposición está basado en la descomposición de clanes. Dada una 2-structure definida sobre U, un conjunto de vertices C, C subconjunto de U, es un clan si para cada z que no está en C, z no distingue los elementos de C, esto es, z está conectado a los elementos de C con el mismo tipo de aristas. En nuestro trabajo, conectamos esta descomposición con un espacio de cerrados asociado, desarrollamos esta parte del trabajo introduciendo una construcción de un conjunto de implicaciones, llamado clan implications. Basándonos en la definición de clan, sea x, y elemento de cualquier clan C, si existe z tal que las aristas (x,z) y (y,z) están en diferentes clases de equivalencia, z deber estar en C; esto es equivalente a que C lógicamente ocasiona la implicación xy entonces z. A lo largo de esta tesis, con el fin de explicar nuestro trabajo de una manera constructiva, primero trabajamos con sólo dos clases de equivalencia y su nomenclatura correspondiente (modules, en lugar de clanes), para después extender la teoría a más clases de equivalencia. Nuestras contribuciones principales son: un algoritmo (que implementamos) para le método de descomposición de clanes; los teoremas que respaldan nuestro trabajo, y ejemplos de sus aplicaciones con el fin de ilustrar su usabilidad

MORE ON STRONG SIZE PROPERTIES

We continue our study of strong size maps. We show that strong size levels for the n-fold hyperspace of a continuum contain (n-1)-cells. We give two constructions of strong size maps. We introduce reversible strong size properties. We prove that each of the following properties: being a continuum chainable continuum, being a locally connected continuum, and being a continuum with the property of Kelley, is a reversible strong size property. Following Professors Goodykoontz and Nadler, we define admissible strong size maps and show that the levels of admissible strong size maps for the n-fold hyperspace of a locally connected continuum are homeomorphic to the Hilbert cube. Professor Benjamín Espinoza defined Whitney preserving maps for the hyperspace of subcontinua of a continuum. We define strong size preserving maps and show that this class of maps coincides with the class of homeomorphisms

Minima Nonblockers and Blocked Sets of a Continuum

Given a continuum $X$ and an element $x \in X$, $\pi(x)$ is the smallest set that contains $x$ and does not block singletons, and $B(x)$ is the set of all elements blocked by ${x}$. We prove that for each $x \in X$, $B(x)$ is connected, $B(x) \subset \pi(x)$, and that if $B(x)$ is closed, then $B(x)=\pi(x)$. Among other results, we prove that if $X$ is a Kelley continuum and $\pi(x)$ is proper, then $B(x)=\pi(x)$. Finally, we prove that for a certain class of dendroids, the family of minima non-blockers coincides with the family of connected non-blockers.Comment: 13 pages, 3 figures, submitted to Topology and its Application

La práctica de la simulación en la solución de problemas de probabilidad: el caso de los estudiantes del nivel medio superior

En este trabajo en proceso presentamos los resultados de la primera fase de nuestra investigación (análisis preliminar), que pretende reconocer a la práctica o la estrategia de la simulación que realizan los estudiantes al momento de resolver problemas de probabilidad y con ello las cuestiones en probabilidad será de gran sencillez teniendo a la herramienta de la simulación. En ello sostenemos que la práctica de la simulación enriquece al conocimiento matemático del ser humano y en particular a la probabilidad