A Quick Overview on the Quantum Control Approach to the Lambda Calculus

In this short overview, we start with the basics of quantum computing, explaining the difference between the quantum and the classical control paradigms. We give an overview of the quantum control line of research within the lambda calculus, ranging from untyped calculi up to categorical and realisability models. This is a summary of the last 10+ years of research in this area, starting from Arrighi and Dowek's seminal work until today.Comment: In Proceedings LSFA 2021, arXiv:2204.0341

The Vectorial λ\lambda-Calculus

We describe a type system for the linear-algebraic $\lambda$-calculus. The type system accounts for the linear-algebraic aspects of this extension of $\lambda$-calculus: it is able to statically describe the linear combinations of terms that will be obtained when reducing the programs. This gives rise to an original type theory where types, in the same way as terms, can be superposed into linear combinations. We prove that the resulting typed $\lambda$-calculus is strongly normalising and features weak subject reduction. Finally, we show how to naturally encode matrices and vectors in this typed calculus.Comment: Long and corrected version of arXiv:1012.4032 (EPTCS 88:1-15), to appear in Information and Computatio

Language Support for Programming High-Performance Code

Nowadays, the computing landscape is becoming increasingly heterogeneous and this trend is currently showing no signs of turning around. In particular, hardware becomes more and more specialized and exhibits different forms of parallelism. For performance-critical codes it is indispensable to address hardware-specific peculiarities. Because of the halting problem, however, it is unrealistic to assume that a program implemented in a general-purpose programming language can be fully automatically compiled to such specialized hardware while still delivering peak performance. One form of parallelism is single instruction, multiple data (SIMD). Part I of this thesis presents Sierra: an extension for C ++ that facilitates portable and effective SIMD programming. Part II discusses AnyDSL. This framework allows to embed a so-called domain-specific language (DSL) into a host language. On the one hand, a DSL offers the application developer a convenient interface; on the other hand, a DSL can perform domain-specific optimizations and effectively map DSL constructs to various architectures. In order to implement a DSL, one usually has to write or modify a compiler. With AnyDSL though, the DSL constructs are directly implemented in the host language while a partial evaluator removes any abstractions that are required in the implementation of the DSL.Die Rechnerlandschaft wird heutzutage immer heterogener und derzeit ist keine Trendwende in Sicht. Insbesondere wird die Hardware immer spezialisierter und weist verschiedene Formen der Parallelität auf. Für performante Programme ist es unabdingbar, hardwarespezifische Eigenheiten zu adressieren. Wegen des Halteproblems ist es allerdings unrealistisch anzunehmen, dass ein Programm, das in einer universell einsetzbaren Programmiersprache implementiert ist, vollautomatisch auf solche spezialisierte Hardware übersetzt werden kann und dabei noch Spitzenleistung erzielt. Eine Form der Parallelität ist „single instruction, multiple data (SIMD)“. Teil I dieser Arbeit stellt Sierra vor: eine Erweiterung für C++, die portable und effektive SIMD-Programmierung unterstützt. Teil II behandelt AnyDSL. Dieses Rahmenwerk ermöglicht es, eine sogenannte domänenspezifische Sprache (DSL) in eine Gastsprache einzubetten. Auf der einen Seite bietet eine DSL dem Anwendungsentwickler eine komfortable Schnittstelle; auf der anderen Seiten kann eine DSL domänenspezifische Optimierungen durchführen und DSL-Konstrukte effektiv auf verschiedene Architekturen abbilden. Um eine DSL zu implementieren, muss man gewöhnlich einen Compiler schreiben oder modifizieren. In AnyDSL werden die DSL-Konstrukte jedoch direkt in der Gastsprache implementiert und ein partieller Auswerter entfernt jegliche Abstraktionen, die in der Implementierung der DSL benötigt werden

A linear proof language for second-order intuitionistic linear logic

We present a polymorphic linear lambda-calculus as a proof language for second-order intuitionistic linear logic. The calculus includes addition and scalar multiplication, enabling the proof of a linearity result at the syntactic level.Comment: 18 pages + appendix. arXiv admin note: text overlap with arXiv:2201.1122

Beyond Logic. Proceedings of the Conference held in Cerisy-la-Salle, 22-27 May 2017

The project "Beyond Logic" is devoted to what hypothetical reasoning is all about when we go beyond the realm of "pure" logic into the world where logic is applied. As such extralogical areas we have chosen philosophy of science as an application within philosophy, informatics as an application within the formal sciences, and law as an application within the field of social interaction. The aim of the conference was to allow philosophers, logicians and computer scientists to present their work in connection with these three areas. The conference took place 22-27 May, 2017 in Cerisy-la-Salle at the Centre Culturel International de Cerisy. The proceedings collect abstracts, slides and papers of the presentations given, as well as a contribution from a speaker who was unable to attend

Parsing/theorem-proving for logical grammar CatLog3

CatLog3 is a 7000 line Prolog parser/theorem-prover for logical categorial grammar. In such logical categorial grammar syntax is universal and grammar is reduced to logic: an expression is grammatical if and only if an associated logical statement is a theorem of a fixed calculus. Since the syntactic component is invariant, being the logic of the calculus, logical categorial grammar is purely lexicalist and a particular language model is defined by just a lexical dictionary. The foundational logic of continuity was established by Lambek (Am Math Mon 65:154–170, 1958) (the Lambek calculus) while a corresponding extension including also logic of discontinuity was established by Morrill and Valentín (Linguist Anal 36(1–4):167–192, 2010) (the displacement calculus). CatLog3 implements a logic including as primitive connectives the continuous (concatenation) and discontinuous (intercalation) connectives of the displacement calculus, additives, 1st order quantifiers, normal modalities, bracket modalities, and universal and existential subexponentials. In this paper we review the rules of inference for these primitive connectives and their linguistic applications, and we survey the principles of Andreoli’s focusing, and of a generalisation of van Benthem’s count-invariance, on the basis of which CatLog3 is implemented.Peer ReviewedPostprint (author's final draft

La noción de principalidad en especialización de tipos

Cuando se consideran las maneras de producir programas, la técnica principal queviene a la mente de cada uno de nosotros es la de escribir el programa a mano. Aunqueexisten técnicas de derivación y herramientas para producir programas automáticamente,su aplicación está usualmente restringida a cierta clase de problemas, o ciertos dominios (como los generadores de parsers, o las interfaces visuales). En los casos donde talestécnicas se pueden aplicar, la productividad y la confiabilidad se ven ampliamente incrementadas. Por esa razón, nos interesamos en la producción automática de programasen un marco general. La especialización de programas es una manera particular de producir programasautomáticamente. En ella se utiliza un programa fuente general dado para generardiversas versiones particulares, especializadas, del mismo, cada una resolviendo una instanciaparticular del problema original. La técnica más conocida y más ampliamenteestudiada de especialización de programas es llamada evaluación parcial; se la ha utilizadocon éxito en varias áreas de aplicación diferentes. Sin embargo, la evaluaciónparcial tiene problemas cuando se considera la producción automática de programas contipos. La especialización de tipos es una forma de especialización de programas que puedeproducir automáticamente programas con tipos a partir de uno fuente. Comprendediversas técnicas muy poderosas, tales como especialización polivariante, especializaciónde constructores, conversión de clausuras; es la primera de las variantes de especializaciónde programas que puede generar tipos arbitrarios a partir de un único programa fuente. Creemos que la especialización de tipos puede ser la base sobre la que desarrollar unmarco de producción automática de programas. En esta tesis consideramos la especialización de programas, extendiéndola para producirprogramas polimórficos. Ilustramos eso considerando un intérprete para un lambdacálculo con tipos a la Hindley-Milner, y especializándolo con cualquier programa objetopara producir un programa residual que sea esencialmente igual que el original. En labúsqueda de la generación de polimorfismo, extendemos la especialización de tipos paraque pueda expresar la especialización de programas con información estática incompleta,y probamos que para cada término podemos inferir una especialización particular quepuede ser usada para reconstruir cada uno de las otras especializaciones de tal término. Llamamos especialización de tipos principal a tal técnica, debido a la analogía de estapropiedad con la noción de tipos principales. Nuestra presentación clarifica muchos delos problemas existentes en la especialización de tipos, lo cual puede ser usado como unaguía en la búsqueda de soluciones para ellos. La presentación se divide en cuatro partes. En la primera Parte, presentamos la Especialización de Tipos en su forma original, junto con material complementario. Enla Parte II desarrollamos la presentación de la Especialización de Tipos Principal, explicandotodos los detalles técnicos, dando varios ejemplos, y presentando nuestra implementación del prototipo. En la Parte III describimos las posibilidades de la formulaciónnueva, proveyendo una extensión de la Especialización de Tipos para generar programaspolimórficos. Finalmente, la última parte describe trabajos relacionados, trabajosfuturos, y concluye. Este trabajo es el resultado de siete a˜nos de investigación, realizados durante misestudios de doctorado.When considering the ways in which programs are produced, the main techniquethat comes to everybody’s mind is writing by hand. Although derivation techniquesand tools for producing programs exist, their application is usually restricted to certainkind of problems, or certain domains (such as parsing generators, or visual interfaces). In those cases where such techniques can be applied, productivity and reliability arehighly boosted. For that reason, we are concerned with the automatic production ofprograms in a general setting. Program specialisation is a particular way to produce programs automatically. Agiven, general source program is used to generate several particular, specialised versionsof it, each one solving a particular instance of the original problem. The best-known andthoroughly studied technique for program specialisation is called partial evaluation; ithas been successfully used in several different application areas. But partial evaluationfalls short when automatic production of typed programs is considered. Type specialisation is a form of program specialisation that can automatically producetyped programs from some general source one. It comprises several powerful techniques,such as polyvariant specialisation, constructor specialisation, and closure conversion,and it is the first variant of program specialisation that can generate arbitrarytypes from a single source program. We believe that type specialisation can be the basisin which to develop a framework for automatic program production. In this thesis we consider type specialisation, extending it to produce polymorphicprograms. We illustrate that by considering an interpreter for Hindley-Milner typedlambda-calculus, and specialising it to any given object program, producing a residualprogram that is essentially the same as the original one. In achieving the generationof polymorphism, we extend type specialisation to be able to express specialisation ofprograms with incomplete static information, and prove that for each term we can infera particular specialisation that can be used to reconstruct every other for that term. Wecall that principal type specialisation because of the analogy this property has with thenotion of principal types. Our presentation clarifies some of the problems existing intype specialisation, clarification that can be used as a guide in the search for solutionsto them. The presentation is divided in four parts. In the first Part we present Type Specialisationin its original form, together with some background material. In Part II wedevelop the presentation of Principal Type Specialisation, explaining all the technicaldetails, offering several examples, and presenting our prototype implementation. In Part III we describe the posibilities of the new formulation, by providing an extensionof Type Specialisation to generate polymorphic programs. And finally, in the last Partwe describe related and future work and conclude. This work is the result of seven years of research, performed during my PhD studies.Fil: Martínez López, Pablo E.. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina