On the Relation Between Representations and Computability

Computability and decidability are intimately linked problems which have interested computer scientists and mathematicians for a long time, especially during the last century. Work performed by Turing, Church, Godel, Post, Kleene and other authors considered the questions "What is computable?" and "What is an algorithm?". Very important results with plenty of implica- tions were obtained, such as the halting theorem [12], the several solutions to the Entscheidungsproblem [12, 5], the Church-Turing thesis [12] or Godel's incompleteness theorem. Further work was performed on topics which as of today have remained purely theoretical but which have o ered us a great understanding of computability and related questions. Some of this work in- cludes the one related to degrees of recursive unsolvability [1] [7] and Rice's theorem [11]. Several formalisms were described and compared, some of the most im- portant ones being Turing machines and -calculus. These formalisms were mathematical constructions which allowed the study of the concept of com- putation or calculation and all of its related questions. We have found that an often ignored detail and, as we show, important aspect of computability is related to representation. In particular, we show that the computability of an abstract problem can only be considered once a choice of representation has been made. We inquire to what extent this is essential and what e ects it may have and in what manner. We o er a wide discussion on its implications, a formalisation of these considerations and some important results deriving from these formalisations. In particular, the main result of the work is a proof that computably enumerable repre- sentations cannot be strictly stronger or weaker than other representations. We also discuss the Church-Turing thesis with particular interest, inquiring about its deep meaning and the actual facts and false assumptions related to it. Furthermore, we consider the relationship between representation and the so-called representation degrees and the degrees of recursive unsolvability de- rived from the concept of oracle machine. We show that these two concepts o er parallel hierarchies which are very similar in their construction but quite di erent in their essential meaning and properties.La computabilidad y la decidibilidad son problemas estrechamente relacionados que han interesado ampliamente a informáticos y matemáticos, especialmente a lo largo del ultimo siglo. Los trabajos realizados por Turing, Church, Godel, Post, Kleene y otros autores se planteaban las preguntas "Qué es computable?" y "Qué es un algoritmo?". Se lograron muchos resultados importantes con multitud de implicaciones, como el teorema de la parada [12], la solución al Entscheidungsproblem [12, 5], la hipótesis de Church-Turing [12] o el teorema de incompletidud de Godel. Gran cantidad del trabajo posterior se realizó en relación a otros temas que han permanecido hasta hoy en el campo de la teoría pero que nos han permitido entender en mayor medida la computabilidad y problemas relacionados. Por ejemplo, el relacionado con los grados de indecibilidad [1] [7] y el teorema de Rice [11]. Varios formalismos fueron descritos y comparados, algunos de los más importantes son las máquinas de Turing y el cálculo lambda. Estos formalismos constituían construcciones matemáticas que permitían el estudio del concepto de computación o cálculo y todas las preguntas relacionadas. Un aspecto comúnmente ignorado y relevante de la computabilidad está relacionado con la representación. En particular, percatamos que la com- putabilidad de un problema abstracto sólo puede ser considerada una vez se ha producido una elección de representación. Nos preguntamos hasta qué punto esto es esencial y qué efectos puede tener y de qué manera. Ofrecemos una amplia discusión sobre sus implicaciones, una formalización de estas consideraciones y algunos resultados importantes derivados de las mismas. En particular, el resultado principal del trabajo es una demostración de que las representaciones computacionalmente enumerables no pueden ser más fuertes o más débiles que otras. Realizamos una discusión especialmente enfrascada en relación a la tesis de Church-Turing, su significado más profundo y los hechos y falacias que giran en torno a ella. Además, consideramos la relación existente entre la representación y los llamados grados de representación, y los grados de indecibilidad derivados del concepto de máquina oráculo. Demostramos que estos dos conceptos ofrecen jerarquías paralelas con una construcción muy similar pero notablemente distintas en su significado esencial y sus propiedades

La apariencia en Gracián

[Resumen] El artículo estudia el concepto de “apariencia”, en el ámbito de la Filosofía de la naturaleza, como una segunda naturaleza del ser, tras un repaso al marco general de la narrativa graciana. Se analiza la fábula del pavo real desde el ámbito moral y se muestra que la apariencia es una exigencia ontológica en el campo normativo y, en el campo táctico, un punto de partida del triunfo en la vida cortesana.[Sommaire] L’article étudie le concept d’“apparence” dans le domaine de la Philosophie de la nature, comme étant une deuxième nature de l’être, après avoir procédé à une révision du cadre général de la narration graciane. On analyse la fable du paon royal du point de vue moral et on montre que l’apparence est une exigence ontologique dans le champ normatif et, dans le champ tactique, un point de départ du triomphe dans la vie courtoise.[Abstract] The article focuses on the concept of “appearance”, in the area of the Philosophy of nature. After reviewing Gracián’s narrative, the author interprets the concept of appearance as the “second nature” of being. The fable of the royal turkey is analyzed from a moral perspective, and it is the author’s contention that “appearance” is an ontological exigency in the normative field and, in the tactical field, a point of departure in order to attain success in courtier life

El Hegelianismo de la inédita Filosofía Lógica de Unamuno.

Sin resume

A geometric description of the extreme Khovanov cohomology

We prove that the hypothetical extreme Khovanov cohomology of a link is the cohomology of the independence simplicial complex of its Lando graph. We also provide a family of knots having as many non-trivial extreme Khovanov cohomology modules as desired, that is, examples of H-thick knots which are as far of being H-thin as desired.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo Regiona

Raman spectroscopy as a versatile tool to study organic biradicals

Since -conjugated organic molecules were probed as potential semiconducting materials, suitable for replacing the widely used silicon technologies, their structural, optical and conductive properties have been under study to improve their application in organic electronics and to make possible their ad hoc synthesis. In this sense, the modification of the -electron delocalization path is one of the available tools to tune the properties of the molecules to obtain the desired characteristics for the fabrication of these devices. One of the parameters employed to tailor -conjugated organic molecules for organic electronics is the diradical character. A progressive change in the diradical contribution to the ground electronic state structure can tune some of the main system features, highlighting the HOMO-LUMO energy gap and the aggregation mode. The main drawback of this approach is the loss of chemical stability when increasing the diradical character of these molecules. On the other hand, the -electron delocalization can be interrupted introducing a perpendicularly conjugated path. The competition of these two cross-conjugated patterns leads to a new 2-dimensional delocalization scenario that changes the electronic properties of the studied materials. In this project, we present a stable quinoidal quaterthiophene diradical that possess outstanding stability and conductivity properties. [1] The combination of the diradical character together with the possibility to delocalize the electron density through two different perpendicular paths explain its exceptional behavior in comparison with the other members of the series, or with its linearly conjugated analogues. The balance between these two properties has been evaluated through UV-Vis-NIR electronic spectroscopy and Raman and IR vibrational spectroscopy in the neutral and charged forms of the target molecule and similar non-cross-conjugated samples.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

El uso del N‐Vivo como apoyo al análisis de datos. Aplicación a la investigación sobre el Aula de Mayores de La Línea

Este artículo trata de sintetizar las características de un instrumento de análisis de datos cualitativos como es el N‐Vivo y las posibilidades de uso; con el fin de comentar entre los asistentes a las Jornadas si pudiera ser la herramienta que utilizáramos para la parte cualittiva de la investigación que estamos llevando a cabo dentro del JUTE.This article tries to synthesize the characteristics of an instrument of analysis of qualitative data as it is N‐Vivo and the possibilities of use, with the purpose of commenting with the assistants to the Days if it could be the tool that we used for the qualitative part of the investigation which we are carrying out within the JUTE