Martin-Lof's intuitionistic type theory (Type Theory) is a formal system that serves not only as a foundation of constructive mathematics but also as a dependently typed programming language. Dependent types are types that depend on values of other types. Type Theory is based on the Curry-Howard isomorphism which relates computer programs with mathematical proofs so that we can do computer-aided formal reasoning and write certified programs in programming languages like Agda, Epigram etc. Martin Lof proposed two variants of Type Theory which are differentiated by the treatment of equality. In Intensional Type Theory, propositional equality defined by identity types does not imply definitional equality, and type checking is decidable. In Extensional Type Theory, propositional equality is identified with definitional equality which makes type checking undecidable. Because of the good computational properties, Intensional Type Theory is more popular, however it lacks some important extensional concepts such as functional extensionality and quotient types. This thesis is about quotient types. A quotient type is a new type whose equality is redefined by a given equivalence relation. However, in the usual formulation of Intensional Type Theory, there is no type former to create a quotient. We also lose canonicity if we add quotient types into Intensional Type Theory as axioms. In this thesis, we first investigate the expected syntax of quotient types and explain it with categorical notions. For quotients which can be represented as a setoid as well as defined as a set without a quotient type former, we propose to define an algebraic structure of quotients called definable quotients. It relates the setoid interpretation and the set definition via a normalisation function which returns a normal form (canonical choice) for each equivalence class. It can be seen as a simulation of quotient types and it helps theorem proving because we can benefit from both representations. However this approach cannot be used for all quotients. It seems that we cannot define a normalisation function for some quotients in Type Theory, e.g. Cauchy reals and finite multisets. Quotient types are indeed essential for formalisation of mathematics and reasoning of programs. Then we consider some models of Type Theory where types are interpreted as structured objects such as setoids, groupoids or weak omega-groupoids. In these models equalities are internalised into types which means that it is possible to redefine equalities. We present an implementation of Altenkirch's setoid model and show that quotient types can be defined within this model. We also describe a new extension of Martin-Lof type theory called Homotopy Type Theory where types are interpreted as weak omega-groupoids. It can be seen as a generalisation of the groupoid model which makes extensional concepts including quotient types available. We also introduce a syntactic encoding of weak omega-groupoids which can be seen as a first step towards building a weak omega-groupoids model in Intensional Type Theory. All of these implementations were performed in the dependently typed programming language Agda which is based on intensional Martin-Lof type theory.

Homotopy type theory (HoTT) is a branch of mathematics that combines and benefits from a variety of fields, most importantly homotopy theory, higher dimensional category theory, and, of course, type theory. We present several original results in homotopy type theory which are related to the truncation level of types, a concept due to Voevodsky. To begin, we give a few simple criteria for determining whether a type is 0-truncated (a set), inspired by a well-known theorem by Hedberg, and these criteria are then generalised to arbitrary n. This naturally leads to a discussion of functions that are weakly constant, i.e. map any two inputs to equal outputs. A weakly constant function does in general not factor through the propositional truncation of its domain, something that one could expect if the function really did not depend on its input. However, the factorisation is always possible for weakly constant endofunctions, which makes it possible to define a propositional notion of anonymous existence. We additionally find a few other non-trivial special cases in which the factorisation works. Further, we present a couple of constructions which are only possible with the judgmental computation rule for the truncation. Among these is an invertibility puzzle that seemingly inverts the canonical map from Nat to the truncation of Nat, which is perhaps surprising as the latter type is equivalent to the unit type. A further result is the construction of strict n-types in Martin-Lof type theory with a hierarchy of univalent universes (and without higher inductive types), and a proof that the universe U(n) is not n-truncated. This solves a hitherto open problem of the 2012/13 special year program on Univalent Foundations at the Institute for Advanced Study (Princeton). The main result of this thesis is a generalised universal property of the propositional truncation, using a construction of coherently constant functions. We show that the type of such coherently constant functions between types A and B, which can be seen as the type of natural transformations between two diagrams over the simplex category without degeneracies (i.e. finite non-empty sets and strictly increasing functions), is equivalent to the type of functions with the truncation of A as domain and B as codomain. In the general case, the definition of natural transformations between such diagrams requires an infinite tower of conditions, which exists if the type theory has Reedy limits of diagrams over the ordinal omega. If B is an n-type for some given finite n, (non-trivial) Reedy limits are unnecessary, allowing us to construct functions from the truncation of A to B in homotopy type theory without further assumptions. To obtain these results, we develop some theory on equality diagrams, especially equality semi-simplicial types. In particular, we show that the semi-simplicial equality type over any type satisfies the Kan condition, which can be seen as the simplicial version of the fundamental result by Lumsdaine, and by van den Berg and Garner, that types are weak omega-groupoids. Finally, we present some results related to formalisations of infinite structures that seem to be impossible to express internally. To give an example, we show how the simplex category can be implemented so that the categorical laws hold strictly. In the presence of very dependent types, we speculate that this makes the Reedy approach for the famous open problem of defining semi-simplicial types work.

Sanborn C. Brown and John J. McCarthy. / "July 29, 1948." / Includes bibliographical references. / Army Signal Corps No. W-36-039 sc-32037.

by Gabriel R. Bitran and Thomas L. Magnanti. / "3-97-76."--handwritten on t.p. Cover title. / Bibliography: p. 38-42. / Supported in part by the U.S. Army Research Office (Durham) under contract no. DAHC04-732-0032 Supported in part by a Grant-in-Aid from Coca-Cola, U.S.A, administered at M.I.T. as OSP 27857

La tesis aborda la aparición y desarrollo de la tercera dimensión en la pintura, es decir, la progresiva incorporación del volumen como recurso expresivo de este arte, a priori bidimensional. Analiza la evolución histórica del fenómeno desde las primeras pinceladas densas, hasta las manifestaciones más abiertamente tridimensionales, donde la propia materia y elementos añadidos se convierten en la razón de ser de la obra. También estudia la cuestión desde una perspectiva filosófica y estética, analizando la propia naturaleza de la pintura en busca de las motivaciones que han empujado a los pintores al empleo de estrategias tridimensionales, así como de las aportaciones que estas hayan podido suponer. La parte experimental está constituida por un conjunto de obras pictóricas, fruto del proceso de documentación, estudio y reflexión, complementándolo y formando un todo homogéneo. / Material painting and three-dimensional. Evolution and current situation. Theoretical and practical considerations. The thesis addresses the question of the emergence and development of the third dimension in painting, i.e. the progressive incorporation of volume like an expressive resource of this art, initially two-dimensional. It analyze the historical evolution of the phenomenon from the first strokes dense, until more openly three-dimensional demonstrations, where the material itself and added elements become the raison d'être of the work. It also examines the question from a philosophical and aesthetic perspective, analyzing the nature of painting in search of the motivations that drove painters to the use of three-dimensional strategies, as well as the contributions that these have been assumed. The experimental part is composed of a set of paintings, the result of the process of documentation, study and reflection, supplementing and forming a homogeneous whole.

Ramón Stolz Viciano (Valencia 1903-1958) es el autor de una obra extraordinaria tanto en calidad como en cantidad, muy poco conocida para el público en general y para la crítica de arte en particular debido a la práctica ausencia de bibliografía sobre el tema y también a las escasas exposiciones realizadas sobre su obra. Especialista en la técnica del fresco, una técnica en progresivo desuso desde el siglo XVIII, es autor también de una ingente producción al óleo. Stolz Viciano destaca también como dibujante excepcional, y en opinión de su amigo el profesor Enrique Lafuente Ferrari, sus dibujos tienen pocos rivales entre los de cualquier artista español del presente o del pasado. En esta tesis se presenta un catálogo razonado de su obra mural, de caballete y dibujos.Stolz Viciano destaca también como "pintor de oficio". Sus conocimientos técnicos sobresalen sin duda ya que eran muy pocos los que podían por entonces pintar al fresco, una técnica en progresivo y práctico desuso. Estos conocimientos, adquiridos de forma autodidacta con la práctica constante que favorecía su talante disciplinado y su afán experimentador, se vieron reforzados con sus trabajos como restaurador de todas las cúpulas al fresco del Pilar de Zaragoza, además de la reconstrucción integral de una de Francisco Bayeu y de la parte arruinada de la de Palomino, en la Basílica de la Virgen de los Desamparados de Valencia, entre 1940 y 1941. Desde su puesto como profesor en la Escuela de Bellas Artes de San Fernando de "Prácticas de ornamentación" primero -desde 1932- y después desde la cátedra de "Procedimientos artísticos" ganada con el número uno en 1943, intentó inculcar a varias generaciones de pintores la necesidad de aprender la técnica, de manejar el oficio, una herramienta imprescindible que, lejos de coartar la espontaneidad o comprometer la personalidad, como no se cansaba de repetir, es el único medio que permite al pintor expresar su pensamiento con formas y colores. De la Academia de la Escuela fue elegido también Académico en febrero de 1958, unos meses antes de su muerte, ocasión en la que se publicó el volumen de los Discursos leídos con motivo de la recepción como académico de Don Ramón Stolz Viciano el 23 de febrero de 1958.En este libro se describe a un artista especialista en la técnica del fresco, dedicado a grandes trabajos de solemne inspiración histórica y religiosa. Ramón Stolz era indudablemente especialista en la técnica del fresco y, en opinión una vez más de Lafuente Ferrari, el mayor fresquista español desde Lucas Jordán. La envergadura de los encargos que recibió a lo largo de su vida (Capilla de la Comunión en la Basílica de los Desamparados de Valencia, Iglesia del Consejo Superior de Investigaciones Científicas en Madrid, Gobierno Civil de Santander, la cúpula sobre la Via Sacra del Pilar o la gran cúpula del Monumento a los Caídos en Pamplona entre otras) revelan un reconocimiento de sus clientes que no se ajusta al desconocimiento generalizado de su producción. Pero Stolz no se dedicó en exclusiva a la pintura mural. También pintó mucho (y muy bien) al óleo. Las visitas a familiares y amigos permitieron el descubrimiento de un ingente número de óleos y dibujos de los que no existía absolutamente ninguna relación y que desvelan su faceta más mundana y sensual o su preferencia por la mujer, manifiesta en los bellos desnudos o cuadros de tema mitológico, y también en sus cuadros de género, en los que destacan sus elegantes valencianas; o su habilidad como retratista o como pintor de bodegones. En esta obra resulta llamativo el contraste no sólo en los temas sino en la concepción e incluso estilo frente a los grandes ciclos murales, un contraste que se ha pretendido entender y explicar en el presente trabajo, mediante la puesta en relación de su obra con la de otras corrientes coetáneas o del pasado y el análisis de sus escritos sobre el oficio y la técnica pictórica. / AABSTRACTRamón Stolz Viciano (Valencia 1903-1958) was a painter almost unknown both to the general public and to the specialised critics, due to the scarcely existant bibliography as well as to the few exhibitions of his works. His life work is astonishing in terms of amount and quality. He had an extraordinary knowledge of pictorial techniques, the subject of his lectures at the School of Fine Arts of San Fernando in Madrid, where he taught since 1932 until his death. He specialised himself in the fresco technique, one in progressively abandoned (desuso) since XVIII century. He first restored all the fresco paintings at the Basílica del Pilar in Zaragoza, ten domes by Goya, González Velázquez and the Bayeu brothers. In 1941, after this great work, he was comissioned by the officials?? (Board of directors) of the Basílica to paint al fresco?? (in the fresco technique) a vault over the main chorus. From here on he would be forever devoted to huge tasks comissioned by the Church as well as by public and governmental institutions: the great dome of the Monument to the Dead in War in Pamplona, the San Isidro ermita in Aranjuez, the whole pictorial decoration of the Holy Spirit Church in Madrid, and the Ateneo Reception Hall in Valencia are among the many walls decorated by Stolz. He was an outstanding drawer, a condition which helped him to sign more than 5000 square meters of fresco painting, along with hundreds of oil canvases until now desperdigados: portraits, costume painting, historical and mithological canvases, still lives and nudes, being women, always elegant and beautiful, his favourite subject. In this thesis all his fresco paintings, and almost four hundred of his canvases and drawings have been catalogued, analysing his work in relation to different influential contemporaneous and past art movements.

This thesis explores applications of the theory of containers within automated theorem proving. Container theory provides a foundational analysis of data types as containers, specified by a type $S$ of shapes and a function P assigning to each shape its set of positions for data.More importantly, a representation theorem guarantees that polymorphic functions between container data types are given by container morphisms, which are characterised by mappings between shapes and positions. Container theory is interesting, in this context, for the following reasons. A mechanism for representing and reasoning with ellipsis (the dots in x_1, x_2, ... , x_n) in lists, existing in the literature, has proved to be very useful for formalisations involving abstractions. Success with this mechanism came by means of a meta-level representation through which many functions that normally require recursive definitions can be given explicit ones. As a result, not only can induction and generalisation be eliminated from proofs but, by means of an associated portrayal system, the resulting proofs are also intuitive and much closer to informal mathematical proofs. This ellipsis mechanism, however, is not based on any formal theory, making it rather exiguous in comparison with rival techniques. There also remains questions about its scope and applications. Our aim is to improve this ellipsis mechanism. In this connection, we hypothesize that the theory of containers provides a formal underpinning for such representations. In order to test our hypothesis, we identify limitations of the ellipsis mechanism and show how they can be addressed within the theory of containers. We subsequently develop a new reasoning system based on containers, which does not suffer from these limitations. This judicious container-based system endorses representations of polymorphic rewrite rules using arithmetic, which naturally lends itself to applications of arithmetic decision procedures. We exploit this facet to develop a new technique for deciding properties of lists. Our technique is developed within a quasi-container setting: shape maps are given as piecewise-linear functions, while a new representation is derived for re-indexing functions that obviates the need for dependent types, which are fundamental in a judicious container approach. We show that this new setting enables us to represent and reason about a large class of properties.

El presente trabajo tiene como objetivo fundamental analizar la obra del pintor José Lucas desde un enfoque semiótico, como un acto de “lectura” que actualiza su significado y su sentido desde una perspectiva intertextual. Se han desarrollado además, otros objetivos como concebir la pintura como una obra abierta, teniendo en cuenta conceptos como el lector implícito y el intertexto lector; realizar un enfoque de valores heterosémicos de la obra por su encuentro intertextual con el mundo artístico del autor y con el mundo estético del lector; proporcionar enlaces intertextuales con la obra artística de José Lucas desde el reconocimiento y comprensión del alcance poliédrico de su producción en diversos aspectos y textos como pretextos poéticos que los une; e interpretar la pintura ecfrástica de José Lucas como elemento mediador entre el cuadro y la realidad. Esta tesis doctoral ha seguido una metodología cualitativa, fundamentada principalmente en la interpretación de datos a través de un análisis triangular: investigación del tesinando, revisión del director de la Tesis, confrontación con las ideas del autor. La recogida de datos realizada para este estudio se puede dividir en tres grandes categorías: la observación directa, las entrevistas en profundidad y el empleo de documentos. Es importante destacar que también se han conseguido numerosos textos inéditos que nos han acercado a la obra de José Lucas, firmas como las de Jaime Siles, Antonio Colinas, Luis Antonio de Villena, Antonio Lucas, Xavier Ribalta, Juan Antonio Molina, José Antonio Lozano,… Las conclusiones alcanzadas por este trabajo parten de un enfoque de los valores heterosémicos. Sobre la discutida indeterminación semántica del signo pictórico es necesario considerar como esencial su carácter polisémico o polivalente al explicar las consecuencias efectivas de dicha indeterminación, ya que se trata de un significado indeterminado, pero no por ello menos real. La imposición del arte moderno ha supuesto la abdicación del orden canónico de la referencialidad realista lo que ha contribuido a “abrir” relativamente las posibilidades de sugerencia del texto plástico; potenciando a su vez la colaboración fantástica de los receptores. Es lo que sucede con las obras de José Lucas. Para poder dar sentido y establecer determinados significados referenciales, el intertexto lector de la persona que contemple sus obras debe ser amplio y debe tener presente los códigos y raíces de los que bebe el artista. Las relaciones entre texto e imagen, en la obra de José Lucas, establecen múltiples correspondencias que unas veces tienen su origen en el texto y otra en la imagen, o al revés, generando una dialéctica que germina en una síntesis de ambas a partir de una semiotización (en el sentido de Peirce) de una de ellas o de ambas juntas. Los mecanismos estilísticos de los que José Lucas se sirve para construir su obra ecfrástica son múltiples y muy variados. Pero lo más interesante, sin duda, es que nacen de un profundo conocimiento de las relaciones entre pintura y literatura, que constituye, a su vez, un modo concreto de acercarse a la realidad e interpretarla. Un modo en que es fundamental, como en todo ejercicio ecfrástico, la presencia del poema ausente como elemento de mediación entre la subjetividad del yo que pinta y la objetividad primera, ya perdida, de la que el poema se ha valido. En esto José Lucas se destaca, desde la cualidad mediadora de su pintura, como un autor profundamente original, a la par que enraizado en una nueva tradición, la del ut poesis pictura, inagotable en sus ramificaciones y derivaciones. / Intertextuality, Literature and Fine Arts in José Lucas. An investigation from a semiotic-analytical perspective. Mª Encarnación Navarro Martínez. Universidad de Murcia. 2015 Abstract: The main objective of this work is to analyse Jose Lucas’ working life from a semiotic viewpoint, as an act of interpretation, this document is trying to present his meaning and his sense from an intertextual perspective. Other objectives that have been developed are “to conceive of” the painting as open work, having in mind concepts like “implicit reader” or “intertextual reader”; focussing on “heterosemic values” of the work for the intertextual encounter between the artistic world of the autor and the reader’s aesthetic world. This document tries to give “intertextual” links with the artistic work of José Lucas from the recognition and understanding of the “poliedric range” of his output in different aspects and texts as a poetic pretext which links both. and interpreting the ecfrastica painting of José Lucas with an element of mediation between the painting and reality. This PhD thesis follows a qualitative methodology, based mainly on the interpretation of information through a triangular analysis: the PHD candidate’s research, review by the supervisor of the thesis and engagement with the author’s ideas. The information discovered by these studies can be divided into three main categories: direct observation, in-depth interviews and the use of different documents. It’s important to emphasise the large amount of unpublished documents which helped in getting closer to an understanding of Jose Lucas’ work; names like Jaime Siles, Antonio Colinas, Luis Antonio de Villena, Antonio Lucas, Xavier Ribalta, Juan Antonio Molina, José Antonio Lozano,… The conclusions reached by this work started from a ‘heterosemic value’ standpoint. On the uncertain meaning of the pictorial signs, it is necessary to consider how essentially polemic and versatile his character was for explaining the effective consequences of that uncertainty – that even though the behaviour and meaning of the sign is uncertain, it is no less real for him. The imposition of modern art has created the abdication of the canonic order of referencing realism which has contributed to opening up the possibilities of plasticity of suggestion in text; strengthening the collaboration of the receiver. It is this that has occured with José Lucas´ work. To be able to give sense and establish some meanings, the person who contemplates his life´s work needs a lot of context about where and from whom is the artist’s inspiration. The relations between between text and image in José Lucas work, establish numerous links which sometimes have their origin in the text and sometimes in the image or vice versa. This generates a dialectic which germinates in a synthesis of both from a semioticisation (in Peirce´s meaning) of one of them or both together. The stylistic mechanisms used by José Lucas to build his “ecfrásica” work are knowledge of the relations between painting and literature which are at the same time a way to be close with reality and interpret it. This is essential in every “ecfrástico” exercise, the presence of the absent poem like an element of mediation between the subjectivity of the “I” which is painted and the first objectivity already lost. In this camp José Lucas is noted from his mediator painting as a deeply original author rooted in a new tradition “the ut poesis picture” tradition, unquenchable of ramifications and derivations

Les vint-i-sis taules policromades –frontals, laterals d’altar i retaules primitius- que s’analitzen en aquesta tesi doctoral han estat assenyalades repetidament com a obres clau de la transició artística del romànic al gòtic. Representen l’evolució d’un llenguatge pictòric que correspon al període artístic denominat gòtic lineal català. Aquestes taules, han estat estudiades a nivell iconogràfic i estilístic des de les últimes dècades del segle XIX. Els estudis artístics, dotats de certa subjectivitat i focalitzats en aspectes documentals, històrics i estilístics, encara no han desvelat tots els elements claus per a la completa comprensió de l’obra. Fins al moment, no existien estudis exhaustius individualitzats i sistemàtics que permetessin relacionar les tipologies, els materials i les estructures del suport del llegat de taules policromades d’estil gòtic lineal català, moment clau en l’evolució estructural de les taules, en el què conviuen mobles d’altar -frontals i laterals- i els primers retaules. Per tant, la taula policromada continuava requerint investigacions que exploressin la seva tecnologia, els procediments i els recursos tècnics emprats per a determinar la seva elaboració i així establir la metodologia de treball de confecció del suport de cada taller, per contribuir amb noves dades a la determinació de l’autoria i de la procedència. La investigació científico-tècnica presentada en aquesta tesi permet relacionar les solucions tecnològiques del suport i agrupar les estructures segons la seva tipologia, evidenciat que les taules d’estil gòtic lineal català són obres claus per comprendre l’evolució estructural de les taules, tant transcendental en aquest període caracteritzat per la convivència d’ambdues formes, sent el retaule l’hereu directe del frontal. Els resultats objectius dels estudis tècnics i científics del suport aporten informació inèdita per a la comprensió de les taules policromades, demostrant que el suport, fins ara una font d’informació infravalorada, és un element més a tenir en compte per reconstruir la història de l’obra, essent un complement imprescindible dels estudis historicoartístics. / The twenty-six polychrome tables –altarpieces, laterals and primary retables- analyzed in this research have been repeatedly identified as a key work of the transition from Romanesque to Gothic art. They represent the evolution of a pictorial language that corresponds to the artistic period known as Catalan Linear Gothic.These tables have been ichnographically and stylistically studied since the last decades of the nineteenth century. Artistic studies, which can be somehow subjective and focused on documental, historical and stylistic aspects, have not yet revealed all the key elements for the complete understanding of the work.So far, there are no systematic and individualized exhaustive studies to relate support typologies, materials and structures of the polychromatic Catalan Linear Gothic tables, period of significance in the structural evolution of the tables, in which coexist the altarpieces and the earliest retables.Therefore, the tables still require research that explores the technology, procedures and technical resources in order to determine their development and thus, to establish the methodology of each workshop, contributing with new data to determine the authorship and origin of the tables.The scientific and technical research presented in this thesis provide relationships among the different technology solutions of the support and group them by type, evidencing that Catalan Linear Gothic tables are essential to understand the structural evolution of the tables, which is transcendental in this period characterized by the coexistence of both forms, being the retable a direct successor of the altarpiece.The objective results of technical and scientific studies provide new information to support the understanding of polychrome tables, demonstrating that the support, hitherto an undervalued source of information, is another element to be considered in order to reconstruct the history of the fine work, being essential for historical and artistic studies.

The rapidly increasing reach of computation into our everyday public settings presents new and significant challenges for the design of interfaces. One key feature of these settings is the increased presence of third parties to interaction, watching or passing-by as conduct with an interface takes place. This thesis assumes a performative perspective on interaction in public, presenting a framework derived from four empirical studies of interaction in a diverse series of public places---museums and galleries, city streets and funfairs---as well as observations on a variety of computer science, art and sociological literatures. As these settings are explored, a number of basic framework concepts are built up: * The first study chapter presents a deployment of an interactive exhibit within an artistic installation, introducing a basic division of roles and the ways in which visitors may be seen as `audience to manipulations of interactive devices by `participants . It also examines how visitors in an audience role may transition to active participant and vice versa. * The second study chapter describes a storytelling event that employed a torch-based interface. This chapter makes a distinction between non-professional and professional members of settings, contrasting the role of `actor with that of participants. * The third study chapter examines a series of scientific and artistic performance events that broadcast live telemetry data from a fairground ride to a watching audience. The study expands the roles introduced in previous chapters through making a further distinction between `behind-the-scenes ---in which `orchestrators operate---and `centre-stage settings---in which actors present the rider s experience to the audience. * The final study chapter presents a performance art game conducted on city streets, in which participants follow a series of often ambiguous clues in order to lead them to their goal. This chapter introduces a further `front-of-house setting, the notion of a circumscribing performance `frame in which the various roles are situated, and the additional role of the `bystander as part of this. These observations are brought together into a design framework which analyses other literature to complement the earlier studies. This framework seeks to provide a new perspective on and language for human-computer interaction (HCI), introducing a series of sensitising concepts, constraints and strategies for design that may be employed in order to approach the various challenges presented by interaction in public settings.