Realizability and recursive mathematics

Section 1: Philosophy, logic and constructivityPhilosophy, formal logic and the theory of computation all bear on problems in the foundations of constructive mathematics. There are few places where these, often competing, disciplines converge more neatly than in the theory of realizability structures. Uealizability applies recursion-theoretic concepts to give interpretations of constructivism along lines suggested originally by Heyting and Kleene. The research reported in the dissertation revives the original insights of Kleene—by which realizability structures are viewed as models rather than proof-theoretic interpretations—to solve a major problem of classification and to draw mathematical consequences from its solution.Section 2: Intuitionism and recursion: the problem of classificationThe internal structure of constructivism presents an interesting problem. Mathematically, it is a problem of classification; for philosophy, it is one of conceptual organization. Within the past seventy years, constructive mathematics has grown into a jungle of fullydeveloped "constructivities," approaches to the mathematics of the calculable which range from strict finitism through hyperarithmetic model theory. The problem we address is taxonomic: to sort through the jungle, set standards for classification and determine those features which run through everything that is properly "constructive."There are two notable approaches to constructivity; these must appear prominently in any proposed classification. The most famous is Brouwer's intuitioniam. Intuitionism relies on a complete constructivization of the basic mathematical objects and logical operations. The other is classical recursive mathematics, as represented by the work of Dekker, Myhill, and Nerode. Classical constructivists use standard logic in a mathematical universe restricted to coded objects and recursive operations.The theorems of the dissertation give a precise answer to the classification problem for intuitionism and classical constructivism. Between these realms arc connected semantically through a model of intuitionistic set theory. The intuitionistic set theory IZF encompasses all of the intuitionistic mathematics that does not involve choice sequences. (This includes all the work of the Bishop school.) IZF has as a model a recursion-theoretic structure, V(A7), based on Kleene realizability. Since realizability takes set variables to range over "effective" objects, large parts of classical constructivism appear over the model as inter¬ preted subsystems of intuitionistic set theory. For example, the entire first-order classical theory of recursive cardinals and ordinals comes out as an intuitionistic theory of cardinals and ordinals under realizability. In brief, we prove that a satisfactory partial solution to the classification problem exists; theories in classical recursive constructivism are identical, under a natural interpretation, to intuitionistic theories. The interpretation is especially satisfactory because it is not a Godel-style translation; the interpretation can be developed so that it leaves the classical logical forms unchanged.Section 3: Mathematical applications of the translation:The solution to the classification problem is a bridge capable of carrying two-way mathematical traffic. In one direction, an identification of classical constructivism with intuitionism yields a certain elimination of recursion theory from the standard mathematical theory of effective structures, leaving pure set theory and a bit of model theory. Not only are the theorems of classical effective mathematics faithfully represented in intuitionistic set theory, but also the arguments that provide proofs of those theorems. Via realizability, one can find set-theoretic proofs of many effective results, and the set-theoretic proofs are often more straightforward than their recursion-theoretic counterparts. The new proofs are also more transparent, because they involve, rather than recursion theory plus set theory, at most the set-theoretic "axioms" of effective mathematics.Working the other way, many of the negative ("cannot be obtained recursively") results of classical constructivism carry over immediately into strong independence results from intuitionism. The theorems of Kalantari and Retzlaff on effective topology, for instance, turn into independence proofs concerning the structure of the usual topology on the intuitionistic reals.The realizability methods that shed so much light over recursive set theory can be applied to "recursive theories" generally. We devote a chapter to verifying that the realizability techniques can be used to good effect in the semantical foundations of computer science. The classical theory of effectively given computational domains a la Scott can be subsumed into the Kleene realizability universe as a species of countable noneffective domains. In this way, the theory of effective domains becomes a chapter (under interpre¬ tation) in an intuitionistic study of denotational semantics. We then show how the "extra information" captured in the logical signs under realizability can be used to give proofs of classical theorems about effective domains.Section 4: Solutions to metamathematical problems:The realizability model for set theory is very tractible; in many ways, it resembles a Boolean-valued universe. The tractibility is apparent in the solutions it offers to a number of open problems in the metamathematics of constructivity. First, there is the perennial problem of finding and delimiting in the wide constructive universe those features that correspond to structures familiar from classical mathematics. In the realizability model, it is easy to locate the collection of classical ordinals and to show that they form, intuitionistically, a set rather than a proper class. Also, one interprets an argument of Dekker and Myhill to prove that the classical powerset of the natural numbers contains at least continuum-many distinct cardinals.Second, a major tenet of Bishop's program for constructivity has been that constructive mathematics is "numerical:" all the properties of constructive objects, including the real numbers, can be represented as properties of the natural numbers. The realizability model shows that Bishop's numericalization of mathematics can, in principle, be accomplished. Every set over the model with decidable equality and every metric space is enumerated by a collection of natural numbers

Reducibilities in recursive function theory.

Massachusetts Institute of Technology. Dept. of Mathematics. Thesis. 1966. Ph.D.Bibliography: leaves 102-103.Ph.D

A 2-D Numerical Simulation and Analysis of a Simple Band Model for the Priz Spatial Light Modulator

This dissertation discusses the development of and analyzes the first complete, 2-D numerical simulation of the PRIZ. The simulation is based upon a simple band model of the PRIZ: a single donor, a single trap, and free electron carriers. Modeled mechanisms include photogeneration, energy level transitions, injection, drift currents, diffusion currents, photorefraction and diffraction. The model goes beyond the previous charge and field dynamics of 1-D numerical models to include optical effects, and it eliminates the oversimplifications and assumptions used in earlier mathematical models with closed solutions. Sensitivity analyses and selected simulations provide a better understanding of the dynamic imaging phenomena. The device output depends on the relative dominance or strength of the fields in the positive or negative space charge region. Transverse drift is as important as charge mirror imaging and injection current in determining peak output and self-erasure. The simulations show a broad range of unreported behavior both before and after writ-beam turnoff, including sharp transients; reintensification with and without phase reversals; and even strong intensification after a turnoff. Finally, the 2-D model is shown to be a reasonable representation of the PRIZ by comparing simulated output with experimental data from the literature

Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022

The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing

Browning's knowledge and criticism of music

Thesis (M.A.)--Boston University, 1948. This item was digitized by the Internet Archive

Approximation problems in linear and non-linear analysis

En esta tesis estudiamos problemas relacionados con aplicaciones de varios tipos que alcanzan su norma u operadores que alcanzan su radio numérico. Tras un capítulo introductorio donde se comentan las notaciones, los principales conceptos, y un resumen histórico del estado del arte, hay 4 capítulos de contenido matemático donde se estudian diversos tipos de problemas. En el capítulo 2, se estudian clases de operadores entre espacios de Banach tales que cuando casi alcanzan su norma (respectivamente, su radio numérico) en un punto (respectivamente, un estado), necesariamente la alcanzan en un punto cercano (respectivamente, en un estado cercano). Se obtienen resultados positivos para dominios finito dimensionales, funcionales, operadores compactos, y operadores adjuntos, y se estudia si las condiciones obtenidas para dichos resultados son las óptimas o no. También se caracterizan por completo los operadores diagonales que pertenecen a dichas clases cuando el dominio es un espacio de Banach clásico de sucesiones, obteniendo como consecuencia que las proyecciones canónicas pertenecen a dichas clases. Se obtienen varios ejemplos positivos y negativos relativos a estas clases en varios espacios, y se estudia la relación entre ambas clases. En el capítulo 3, se introduce y estudia la propiedad de Bishop-Phelps-Bollobás para el radio numérico restringida al contexto de los operadores compactos. Adaptando los resultados existentes, se obtiene una primera gran colección de espacios de Banach clásicos que satisfacen la propiedad para operadores compactos como consecuencia de que la satisfacen en general. Se obtiene un resultado técnico que afirma que si ciertas proyecciones de un espacio de Banach satisfacen nuestra propiedad para operadores compactos de una forma uniforme, entonces el propio espacio de Banach también cumple dicha propiedad, y este resultado se usa para probar que los preduales isométricos de l_1 cumplen la propiedad. El resultado principal del capítulo afirma que si L es cualquier espacio topológico localmente compacto y Hausdorff, entonces el espacio C_0(L) de funciones continuas en L que se anulan en el infinito siempre tiene la propiedad para operadores compactos (algo que para operadores en general no se sabe por ahora), y para probarlo se demuestra primero un tipo de propiedad de aproximación simultánea para estos espacios y sus duales (basada en una construcción topológica) en conjunto con el lema técnico de las proyecciones. En el capítulo 4 se introducen nociones naturales de alcanzamiento de norma para productos tensoriales proyectivos de espacios de Banach y operadores nucleares entre espacios de Banach. Tras caracterizar los elementos que alcanzan su norma en términos de operadores clásicos, se demuestra que existen tensores y operadores nucleares que alcanzan sus respectivas normas, pero que no todos ellos lo hacen. Se encuentran múltiples resultados positivos sobre la densidad de los elementos de estos espacios que alcanzan su norma, y se prueba definitivamente que dicha densidad no siempre es cierta para tensores, aunque la pregunta análoga para operadores nucleares sigue abierta. Finalmente, en el capítulo 5 se estudian problemas sobre la espaciabilidad del conjunto de funciones Lipschitz que alcanzan su norma fuertemente sobre un espacio métrico infinito M, SNA(M). Primero demostramos que si el métrico es infinito, entonces SNA(M) es n-lineable para todo n. Vemos que todo espacio de Banach puede ser formado con funciones de SNA(M) para un métrico apropiado, y estudiamos posibles restricciones sobre qué espacios se pueden formar si el métrico es pequeño (separable, o incluso sigma-precompacto). Finalmente, se hace un estudio exhaustivo sobre la posibilidad de encontrar c_0 en SNA(M). Se demostró recientemente que eso se puede conseguir siempre isomorfamente, y nosotros demostramos que no siempre se puede hacer isométricamente, aunque vemos que en muchas ocasiones sí se puede. El capítulo concluye con un pequeño estudio para el caso no separable. Finalmente, hay un capítulo de conclusiones y preguntas abiertas, y una exhaustiva y extensa lista de referencias. // En aquesta tesi estudiem problemes relacionats amb diversos tipus de funcions que assoleixen la seua norma o operadors que assoleixen el seu radi numèric. Després d'un capítol introductori on es comenten les notacions, els conceptes principals, i un resum històric de l'estat de l'art, hi ha 4 capítols de contingut matemàtic on s'estudien diversos tipus de problemes. Al capítol 2, s'estudien classes d'operadors entre espais de Banach tals que quan gairebé assoleixen la seua norma (respectivament, el radi numèric) en un punt (respectivament, un estat), necessàriament l'assoleixen en un punt proper (respectivament, en un estat proper). S'obtenen resultats positius per a dominis de dimensió finita, funcionals, operadors compactes i operadors adjunts, i s'estudia si les condicions obtingudes per a aquests resultats són les òptimes o no. També es caracteritzen completament els operadors diagonals que pertanyen a aquestes classes quan el domini és un espai de Banach clàssic de successions, obtenint com a conseqüència que les projeccions canòniques pertanyen a aquestes classes. S'obtenen diversos exemples positius i negatius relatius a aquestes classes a diversos espais, i s'estudia la relació entre ambdues classes. Al capítol 3, s'introdueix i s'estudia la propietat de Bishop-Phelps-Bollobás per al radi numèric restringida al context dels operadors compactes. Adaptant-hi els resultats existents, s'obté una primera gran col·lecció d'espais de Banach clàssics que satisfan la propietat per a operadors compactes com a conseqüència que la satisfan en general. S'obté un resultat tècnic que afirma que si certes projeccions d'un espai de Banach satisfan la nostra propietat per a operadors compactes d'una forma uniforme, aleshores el mateix espai de Banach també compleix aquesta propietat, i aquest resultat s'empra per tal de provar que els preduals isomètrics de l_1 compleixen la propietat. El resultat principal del capítol afirma que si L és qualsevol espai topològic localment compacte i Hausdorff, aleshores l'espai C_0(L) de funcions contínues a L que s'anul·len a l'infinit sempre té la propietat per a operadors compactes (cosa que per a operadors en general no se sap ara com ara), i per provar-ho es demostra primer un tipus de propietat d'aproximació simultània per a aquests espais i els seus duals (basada en una construcció topològica) conjuntament amb el lema tècnic de les projeccions. Al capítol 4 s'introdueixen nocions naturals d'assoliment de norma per a productes tensorials projectius d'espais de Banach i operadors nuclears entre espais de Banach. Després de caracteritzar els elements que assoleixen la seua norma en termes d'operadors clàssics, es demostra que hi ha tensors i operadors nuclears que assoleixen les seues normes respectives, però que no tots ells ho fan. Es troben múltiples resultats positius sobre la densitat dels elements d'aquests espais que assoleixen la seva norma, i es prova definitivament que aquesta densitat no sempre és certa per a tensors, mentre que la pregunta anàloga per a operadors nuclears roman oberta. Finalment, al capítol 5 s'estudien problemes sobre l'espaiabilitat del conjunt de funcions Lipschitz que assoleixen la seva norma fortament sobre un espai mètric infinit M, SNA(M). Primer es demostra que si el mètric és infinit, aleshores SNA(M) és n-lineable per a tot n. Es prova que tot espai de Banach pot ser format amb funcions de SNA(M) per a un mètric apropiat, i estudiem possibles restriccions sobre quins espais es poden formar si el mètric és xicotet (separable, o fins i tot sigma-precompacte). Finalment, es fa un estudi exhaustiu sobre la possibilitat de trobar c_0 a SNA(M). Es va demostrar recentment que això es pot aconseguir sempre isomorfament, i nosaltres demostrem que no sempre es pot fer isomètricament, encara que veiem que moltes vegades sí que es pot. El capítol conclou amb un xicotet estudi per al cas no separable. Finalment, hi ha un capítol de conclusions i preguntes obertes, i una llista exhaustiva i extensa de referències.In this thesis we study problems related to several types of mappings that attain their norm or operators that attain their numerical radius. After an introductory chapter where the notations, the main concepts, and a historical summary of the state of the art are discussed, there are 4 chapters of mathematical content where various types of problems are studied. In Chapter 2, we study classes of operators between Banach spaces such that when they almost attain their norm (respectively, their numerical radius) at one point (respectively, a state), they necessarily attain it at a nearby point (respectively, at a nearby state). Positive results are obtained for finite dimensional domains, functiona, compact operators, and adjoint operators, and it is studied whether the conditions obtained for said results are optimal or not. The diagonal operators that belong to these classes are also completely characterized when the domain is a classical Banach sequence space, obtaining as a consequence that the canonical projections belong to those classes. Several positive and negative examples related to these classes are obtained for several spaces, and the relationship between both classes is studied. In chapter 3, the Bishop-Phelps-Bollobás property for the numerical radius restricted to the context of compact operators is introduced and studied. Adapting the existing results, a first large collection of classical Banach spaces is obtained which satisfy the property for compact operators as a consequence of their satisfying it in general. A technical result is obtained stating that if certain projections of a Banach space satisfy the property for compact operators in a uniform way, then the Banach space itself also satisfies this property, and this result is used to prove that the isometric preduals of l_1 satisfy the property. The main result of the chapter asserts that if L is any locally compact and Hausdorff topological space, then the space C_0(L) of continuous functions on L that vanish at infinity always has the property for compact operators (something that for operators in general is not known for now), and to prove it we first prove a kind of simultaneous approximation property for these spaces and their duals (based on a topological construction) together with the technical lemma about projections. In Chapter 4, natural notions of norm attainment for projective tensor products of Banach spaces and nuclear operators between Banach spaces are introduced. After characterizing the elements that attain their norm in terms of classical operators, it is shown that there are tensors and nuclear operators that attain their respective norms, but that not all of them do. Several positive results are found about the density of the norm-attaining elements in these spaces, but it is proven that such density does not always hold for tensors, although the analogous question for nuclear operators remains open. Finally, in chapter 5 we study problems about the spaceability of the set of Lipschitz functions that strongly attain their norm over an infinite metric space M, SNA(M). We first show that if the metric is infinite, then SNA(M) is n-linear for all n. We see that every Banach space can be formed with functions of SNA(M) for an appropriately chosen metric space, and we study possible restrictions on which spaces can be formed if the metric is small (separable, or even sigma-precompact). Finally, an exhaustive study is carried about the possibility of finding c_0 in SNA(M). It was recently shown that this can always be achieved isomorphically, and we show that it cannot always be done isometrically, although we also see that this can be done on many occasions. The chapter concludes with a small study for the non-separable case. Finally, there is a chapter of conclusions and open questions, and an exhaustive and extensive list of references