On the Formalisation of the Metatheory of the Lambda Calculus and Languages with Binders

Este trabajo trata sobre el razonamiento formal veri cado por computadora involucrando lenguajes con operadores de ligadura. Comenzamos presentando el Cálculo Lambda, para el cual utilizamos la sintaxis histórica, esto es, sintaxis de primer orden con sólo un tipo de nombres para las variables ligadas y libres. Primeramente trabajamos con términos concretos, utilizando la operación de sustitución múltiple de nida por Stoughton como la operación fundamental sobre la cual se de nen las conversiones alfa y beta. Utilizando esta sintaxis desarrollamos los principales resultados metateóricos del cálculo: los lemas de sustitución, el teorema de Church-Rosser y el teorema de preservación de tipo (Subject Reduction) para el sistema de asignación de tipos simples. En una segunda formalización reproducimos los mismos resultados, esta vez basando la conversion alfa sobre una operación más sencilla, que es la de permutación de nombres. Utilizando este mecanismo, derivamos principios de inducción y recursión que permiten trabajar identificando términos alfa equivalentes, de modo tal de reproducir la llamada convención de variables de Barendregt. De este modo, podemos imitar las demostraciones al estilo lápiz y papel dentro del riguroso entorno formal de un asistente de demostración. Como una generalización de este último enfoque, concluimos utilizando técnicas de programación genérica para definir una base para razonar sobre estructuras genéricas con operadores de ligadura. Definimos un universo de tipos de datos regulares con información de variables y operadores de ligadura, y sobre éstos definimos operadores genéricos de formación, eliminación e inducción. También introducimos una relación de alfa equivalencia basada en la operación de permutación y derivamos un principio de iteración/inducción que captura la convención de variables anteriormente mencionada. A modo de ejemplo, mostramos cómo definir el Cálculo Lambda y el sistema F en nuestro universo, ilustrando no sólo la reutilización de las pruebas genéricas, sino también cuán sencillo es el desarrollo de nuevas pruebas en estos casos. Todas las formalizaciones de esta tesis fueron realizadas en Teoría Constructiva de Tipos y verificadas utilizando el asistente de pruebas AgdaThis work is about formal, machine-checked reasoning on languages with name binders. We start by considering the ʎ-calculus using the historical ( rst order) syntax with only one sort of names for both bound and free variables. We rst work on the concrete terms taking Stoughton's multiple substitution operation as the fundamental operation upon which the ά and ß-conversion are de ned. Using this syntax we reach well-known meta-theoretical results, namely the Substitution lemmas, the Church-Rosser theorem and the Subject Reduction theorem for the system of assignment of simple types. In a second formalisation we reproduce the same results, this time using an approach in which -conversion is de ned using the simpler operation of name permutation. Using this we derive induction and recursion principles that allow us to work by identifying terms up to -conversion and to reproduce the so-called Barendregt's variable convention [4]. Thus, we are able to mimic pencil and paper proofs inside the rigorous formal setting of a proof assistant. As a generalisation of the latter, we conclude by using generic programming techniques to de ne a framework for reasoning over generic structures with binders. We de ne a universe of regular datatypes with variables and binders information, and over these we de ne generic formation, elimination, and induction operations. We also introduce an ά equivalence relation based on the swapping operation, and are able to derive an -iteration/induction principle that captures Barendregt's variable convention. As an example, we show how to de ne the ʎ calculus and System F in our universe, and thereby we are able to illustrate not only the reuse of the generic proofs but also how simple the development of new proofs becomes in these instances. All formalisations in this thesis have been made in Constructive Type Theory and completely checked using the Agda proof assistan

Reference and Reinterpretation

Reference is the relation held to obtain between an expression and what a speaker or thinker intends the expression to represent. Reference is a component of interpretation, the process of giving terms, sentences, and thoughts semantic content. An example of reference in a formal context involves the natural numbers, where each one can be taken to have a corresponding set-theoretic counterpart as its referent. In an informal context reference is exemplified by the relation between a name and the specific name-bearer when a speaker or thinker utters or has the name in mind. Recent debates over reference have concerned the mechanism of reference: How is it that we can refer? In informal contexts, externalists see the reference relation as explicable in terms of the salient causal relations involved in the naming of a thing, or a class of things, and the ensuing causal chains leading to a term’s use. Opponents of this view—internalists—see the reference relation as being conceptually direct, and they take the external approach to rely on untenable metaphysical assumptions about the world’s structure. Moreover, some internalists take the permutability—i.e. the consistent reinterpretation—of certain referential schemes to confound the externalist picture of reference. In this thesis I focus on the reference of theoretical terms in science, and I argue for an externalist treatment of natural kinds and other theoretical elements. Along the way I offer a defense of the externalist’s pre-theoretic metaphysical assumptions and emphasize their central role in the interpretation of scientific languages. The externalist approach acknowledges the necessary constraints on reference-fixing that account for the schemes we employ, and this, I argue, confounds the permutation strategy

L'atomisme, le holisme et la quête d'une tierce alternative viable

Selon John McDowell, l'atomisme et le holisme sont chacun incapables de porter fruit. Plutôt que d'osciller futilement entre ces deux pôles, il croit que nous devrions repenser notre façon de concevoir la relation liant l'esprit et le monde. Inspiré par certains passages de Kant, il nous invite donc à reconsidérer l'expérience de telle sorte qu'on y admette d'entrée de jeu l'exercice d'une liberté distinctement humaine-l'étendue de l'esprit devenant ainsi dénuée de toute contrainte externe. À notre avis, McDowell a plus de succès lorsqu'il dépeint le va-et-vient entre l'atomisme et le holisme que lorsqu'il propose une façon d'échapper à ce mouvement. Nous croyons que la fusion qu'il cherche à développer ne tient pas la route dans la mesure où, d'un point de vue naturaliste, il y a bel et bien lieu de distinguer la réceptivité empirique et la spontaneité conceptuelle. À l'encontre de McDowell, nous soutenons qu'il n'y a oscillation entre ces facultés que si l'on endosse une inférence allant du statut non-atomique des représentations au holisme, saut inductif qui repose sur une approche spéculative que nous rejetons. Le premier chapitre cherche à démontrer comment les théories holistes de filière quinéenne se fondent sur des présupposés spéculatifs et comment les éléments plus louables de la philosophie de McDowell à cet égard sont rendus impuissants par son assentiment à la critique que fait W. Sellars du "mythe du Donné". Le second chapitre reconstruit méticuleusement l'argument fort complexe qu'étale McDowell dans Mind and World, pour ensuite critiquer sa suggestion que la culture et l'éducation induisent chez l'être humain une attitude critique pouvant remplacer la friction produite par l'expérience. Le troisième chapitre soutient que la thèse de Sellars voulant que l'expérience peut causer mais non justifier nos représentations détruirait non seulement la connaissance empirique mais aussi la capacité de tirer des inférences. Enfin, le quatrième chapitre présente une nouvelle vision "constrictive" qui, par l'entremise des notions de coercition et de complexité, reconnait que la représentation du monde met en jeu une échelle plus large que l'atome mais plus petite que le tout. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Atomisme, Holisme, Représentation, John McDowell

Formal methods and digital systems validation for airborne systems

This report has been prepared to supplement a forthcoming chapter on formal methods in the FAA Digital Systems Validation Handbook. Its purpose is as follows: to outline the technical basis for formal methods in computer science; to explain the use of formal methods in the specification and verification of software and hardware requirements, designs, and implementations; to identify the benefits, weaknesses, and difficulties in applying these methods to digital systems used on board aircraft; and to suggest factors for consideration when formal methods are offered in support of certification. These latter factors assume the context for software development and assurance described in RTCA document DO-178B, 'Software Considerations in Airborne Systems and Equipment Certification,' Dec. 1992

Art and Peripheral Digital Activity

As digital technologies have become more pervasive, this thesis argues, there has been an accompanying expansion of a phenomenon here called ‘peripheral digital activity.’ This activity includes unplanned and unexpected events that arise in conjunction with digital technologies and that are poorly classified using conventional notions of ‘interaction,’ ‘user experience,’ or purposeful ‘use.’ To ground this idea, the thesis looks to artistic strategies that might critically investigate the concept of peripheral digital activity, in this case arguing that Alfred North Whitehead’s philosophy of organism, with its emphasis on whole-part relations, holds special relevance. The thesis proposes an original Whitehead-centred analysis of art as a manner or ‘mode’ of decision procedure. Developing this analysis via Whitehead’s notion of actual entities and through a discussion of digital function, the thesis examines the practice of contemporary artist Tino Sehgal. Reading through theories of social and participatory art, the thesis arrives at Sehgal’s proposition of ‘cleaner conceptualism.’ Outlining a systems-based interpretation of cleaner conceptualism, Sehgal’s constructed situations are contrasted with the idea of art as a decision procedure proposed by Sol LeWitt. Whereas LeWitt organizes his idea of decision procedures as a dualist critique of instrumental rationalism, Sehgal creates a new mode of monist decision procedure. Using this monist strategy, Sehgal mobilises participants, collectors, and curators in a way that is entangled with and presupposes digital function even as his practice foregrounds non-technological body-to-body human engagement. The thesis claims that Sehgal’s practice is one strategy for critically investigating the effects of peripheral digital activity. Proposing directions for future research, the thesis ends with a Coda that provides a preliminary analysis of the paintings Laura Owens as a diagnostic tool for investigating digital functional augmentation