Asymptotically almost all \lambda-terms are strongly normalizing

We present quantitative analysis of various (syntactic and behavioral) properties of random \lambda-terms. Our main results are that asymptotically all the terms are strongly normalizing and that any fixed closed term almost never appears in a random term. Surprisingly, in combinatory logic (the translation of the \lambda-calculus into combinators), the result is exactly opposite. We show that almost all terms are not strongly normalizing. This is due to the fact that any fixed combinator almost always appears in a random combinator

Encoding the Factorisation Calculus

Jay and Given-Wilson have recently introduced the Factorisation (or SF-) calculus as a minimal fundamental model of intensional computation. It is a combinatory calculus containing a special combinator, F, which is able to examine the internal structure of its first argument. The calculus is significant in that as well as being combinatorially complete it also exhibits the property of structural completeness, i.e. it is able to represent any function on terms definable using pattern matching on arbitrary normal forms. In particular, it admits a term that can decide the structural equality of any two arbitrary normal forms. Since SF-calculus is combinatorially complete, it is clearly at least as powerful as the more familiar and paradigmatic Turing-powerful computational models of Lambda Calculus and Combinatory Logic. Its relationship to these models in the converse direction is less obvious, however. Jay and Given-Wilson have suggested that SF-calculus is strictly more powerful than the aforementioned models, but a detailed study of the connections between these models is yet to be undertaken. This paper begins to bridge that gap by presenting a faithful encoding of the Factorisation Calculus into the Lambda Calculus preserving both reduction and strong normalisation. The existence of such an encoding is a new result. It also suggests that there is, in some sense, an equivalence between the former model and the latter. We discuss to what extent our result constitutes an equivalence by considering it in the context of some previously defined frameworks for comparing computational power and expressiveness

Weak orthogonality implies confluence : the higher-order case

In this paper we prove confluence for weakly orthogonal Higher-Order Rewriting Systems. This generalises all the known `confluence by orthogonality' results

Reescritura de términos y sustituciones explícitas

La operación de sustitución constituye un engranaje básico en los fundamentos de la teoría de lenguajes de programación. Juega un rol central en el lambda cálculo (por ende, en lenguajes de programación funcional), en unificación de primer orden y de orden superior (por ende, en lenguajes de programación basados en el paradigma lógico), en modalidades de pasaje de parámetros (por ende, en lenguajes de programación imperativos), etc. Recientemente, investigadores en informática se han interesado en el pasaje de la noción usual de la sustitución, atómica, y de gruesa granularidad, hacia una noción más refinada, de más fina granularidad. La noción de sustitución es transportada del metalenguaje (nuestro lenguaje de discurso) al lenguaje objeto (nuestro lenguaje de estudio). Como consecuencia de ello se obtienen los llamados cálculos de sustituciones explícitas. Estos son de sumo interés a la hora de estudiar la interpretación operacional de los formalismos en cuestión y constituyen los objetos de interés de esta tesis. Se desarrollan los siguientes tres ejes de estudio: Primero, se consideran estrategias de reescritura perpetuas en lambda cálculos con sustituciones explícitas. Estas son estrategias de reescritura que preservan la posibilidad de reducciones infinitas. Se propone una caracterización inductiva del conjunto de términos que no poseen reducciones infinitas (los llamados fuertemente normalizantes). Un lambda cálculo polimórfico con sustituciones explícitas también es analizado, incluyendo propiedades tales como subject reduction y normalización fuerte. Segundo, colocamos el ς-cálculo de M. Abadi and L. Cardelli enriquecido con sustituciones explícitas bajo el microscopio. Este cálculo se encuentra en un nivel semejante de abstracción al lambda cálculo pero se basa en objetos en lugar de funciones. Propiedades tales como simulación del lambda cálculo, confluencia y preservación de la normalización fuerte (aquellos términos que son fuertemente normalizantes en ς también lo son en ς con sustituciones explícitas) son consideradas. Finalmente, dirigimos nuestra atención a la tarea de relacionar la reescritura de orden superior con aquella de primer orden. Fijamos una variante de los ERS (apodados SERSdb) de Z. Khasidashvili como nuestro formalismo de orden superior de partida y definimos un proceso de conversión que permite codificar cualquier SERSdb como un sistema de reescritura de primer orden. En este último, cada paso de reescritura se lleva a cabo módulo una teoría ecuacional determinada por un cálculo de sustituciones explícitas. La misma se formula de manera genérica a través de una presentación de cálculos de sustituciones explícitas basada en macros y axiomas sobre estas macros, parametrizando de esta manera al procedimiento de conversión sobre cualquier cálculo de sustituciones explícitas que obedece la presentación basada en macros. El procedimiento de conversión se encarga de codificar pattern matching de orden superior y sustitución en el entorno de reescritura de primer orden. Asimismo, propiedades que relacionan la noción de reescritura en el orden superior con aquella de primer orden son analizadas en detalle. Se identifica una clase de SERSdb para los cuales el sistema de primer orden resultante de su conversión no requiere una teoría ecuacional para implementar pattern matching de orden superior, bastando para ello matching sintáctico. También se argumenta que esta clase de sistemas de orden superior es apropiada para transferir resultados del entorno de reescritura de orden superior a aquella de primer orden. A modo de ejemplo no-trivial de ello, estudiamos la transferencia del teorema de standarización (fuerte).Substitution spans many areas in programming language theory. It plays a central role in the lambda calculus (hence functional programming), in first and higher-order unikation (hence logic programming), parameter passing methods (hence imperative programming), etc. Recently researchers became interested in shifting from the usual atomic, coarse grained view of substitution to a more refined, fine grained one. Substitution is promoted from the metalevel (our language of discourse) to the object-level (our language of study). This is interesting when studying the operational interpretation of the formalisms in question. Calculi of object-level or explicit substitution is the concern of this thesis. The following three study axes are developed. First we consider perpetual rewrite strategies in lambda calculi of explicit substitutions. These are rewrite strategies that preserve the possibility of inhite derivations. Also, we study how to characterize inductively the set of terms that do not possess infinite derivations (the strongly normalizing terms). Polymorphic lambda calculus with explicit substitutions shall receive our attention too, including properties such as subject reduction and strong normalization. Secondly, we put the ς-calculus of M.Abadi and L.Cardelli augmented with explicit substitutions under the microscope. This calculus is at the level of the lambda calculus but is based on objects instead of functions. Properties such as simulation of the lambda calculus, confluence and preservation of strong normalization (terms which are strongly normalizing in ς are also strongly normalizii in ς with explicit substitutions) are considered. Finally, we address the task of reducing higher-order rewriting to first-order rewriting. We fix a variant of Z-Khasidashvili's ERS (dubbed SERSdb) as our departing formalism and provide a conversion procedure to encode any ERS as a first-order rewrite system in which a rewrite step takes place modulo an equational theory determined by a calculus of explicit substitutions. The latter is achieved with the aid of a macro-based presentation of calculi of explicit substitutions, thus parametrizing the conversion procedure over any calculus of explicit substitutions in compliance with the aforementioned presentation. The conversion procedure is in charge of encoding higherorder pattern matching and substitution in the first-order framework. Properties relating the rewrite relation in the higher-order framework and that of the resulting first-order system are studied in detail. We then identify a class of SERSdb for which the resulting first-order system does not require the equational theory to implement higher-order pattern matching, thus contenting itself with syntactic matching. It is argued that this class of systems is appropriate for transferring results from the first-order framework to the higher-order one. As a non-trivial example we study the transfer of the (strong) standardization theorem.Fil:Bonelli, Eduardo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina

Zero-padding Network Coding and Compressed Sensing for Optimized Packets Transmission

Ubiquitous Internet of Things (IoT) is destined to connect everybody and everything on a never-before-seen scale. Such networks, however, have to tackle the inherent issues created by the presence of very heterogeneous data transmissions over the same shared network. This very diverse communication, in turn, produces network packets of various sizes ranging from very small sensory readings to comparatively humongous video frames. Such a massive amount of data itself, as in the case of sensory networks, is also continuously captured at varying rates and contributes to increasing the load on the network itself, which could hinder transmission efficiency. However, they also open up possibilities to exploit various correlations in the transmitted data due to their sheer number. Reductions based on this also enable the networks to keep up with the new wave of big data-driven communications by simply investing in the promotion of select techniques that efficiently utilize the resources of the communication systems. One of the solutions to tackle the erroneous transmission of data employs linear coding techniques, which are ill-equipped to handle the processing of packets with differing sizes. Random Linear Network Coding (RLNC), for instance, generates unreasonable amounts of padding overhead to compensate for the different message lengths, thereby suppressing the pervasive benefits of the coding itself. We propose a set of approaches that overcome such issues, while also reducing the decoding delays at the same time. Specifically, we introduce and elaborate on the concept of macro-symbols and the design of different coding schemes. Due to the heterogeneity of the packet sizes, our progressive shortening scheme is the first RLNC-based approach that generates and recodes unequal-sized coded packets. Another of our solutions is deterministic shifting that reduces the overall number of transmitted packets. Moreover, the RaSOR scheme employs coding using XORing operations on shifted packets, without the need for coding coefficients, thus favoring linear encoding and decoding complexities. Another facet of IoT applications can be found in sensory data known to be highly correlated, where compressed sensing is a potential approach to reduce the overall transmissions. In such scenarios, network coding can also help. Our proposed joint compressed sensing and real network coding design fully exploit the correlations in cluster-based wireless sensor networks, such as the ones advocated by Industry 4.0. This design focused on performing one-step decoding to reduce the computational complexities and delays of the reconstruction process at the receiver and investigates the effectiveness of combined compressed sensing and network coding

A suite of quantum algorithms for the shortestvector problem

Crytography has come to be an essential part of the cybersecurity infrastructure that provides a safe environment for communications in an increasingly connected world. The advent of quantum computing poses a threat to the foundations of the current widely-used cryptographic model, due to the breaking of most of the cryptographic algorithms used to provide confidentiality, authenticity, and more. Consequently a new set of cryptographic protocols have been designed to be secure against quantum computers, and are collectively known as post-quantum cryptography (PQC). A forerunner among PQC is lattice-based cryptography, whose security relies upon the hardness of a number of closely related mathematical problems, one of which is known as the shortest vector problem (SVP). In this thesis I describe a suite of quantum algorithms that utilize the energy minimization principle to attack the shortest vector problem. The algorithms outlined span the gate-model and continuous time quantum computing, and explore methods of parameter optimization via variational methods, which are thought to be effective on near-term quantum computers. The performance of the algorithms are analyzed numerically, analytically, and on quantum hardware where possible. I explain how the results obtained in the pursuit of solving SVP apply more broadly to quantum algorithms seeking to solve general real-world problems; minimize the effect of noise on imperfect hardware; and improve efficiency of parameter optimization.Open Acces

CLASS: A Logical Foundation for Typeful Programming with Shared State

Software construction depends on imperative state sharing and concurrency, which are naturally present in several application domains and are also exploited to improve the structure and efficiency of computer programs. However, reasoning about concurrency and shared mutable state is hard, error-prone and the source of many programming bugs, such as memory leaks, data corruption, deadlocks and non-termination. In this thesis, we develop CLASS: a core session-based language with a lightweight substructural type system, that results from a principled extension of the propositions-astypes correspondence with second-order classical linear logic. More concretely, CLASS offers support for session-based communication, mutex-protected first-class reference cells, dynamic state sharing, generic polymorphic algorithms, data abstraction and primitive recursion. CLASS expresses and types significant realistic programs, that manipulate memoryefficient linked data structures (linked lists, binary search trees) with support for updates in-place, shareable concurrent ADTs (counters, stacks, functional and imperative queues), resource synchronisation methods (fork-joins, barriers, dining philosophers, generic corecursive protocols). All of these examples are guaranteed to be safe, a result that follows by the logical approach. The linear logical foundations guarantee that well-typed CLASS programs do not go wrong: they never deadlock on communication or reference cell acquisition, do not leak memory and always terminate, even if they share complex data structures protected by synchronisation primitives. Furthermore, since we follow a propositions-as-types approach, we can reason about the behaviour of concurrent stateful processes by algebraic program manipulation. The feasibility of our approach is witnessed by the implementation of a type checker and interpreter for CLASS, which validates and guides the development of many realistic programs. The implementation is available with an open-source license, together with several examples.A construção de software depende de estado partilhado imperativo e concorrência, que estão naturalmente presentes em vários domínios de aplicação e que também são explorados para melhorar o a estrutura e o desempenho dos programas. No entanto, raciocinar sobre concorrência e estado mutável partilhado é difícil e propenso à introdução de erros e muitos bugs de programação, tais como fugas de memória, corrupção de dados, programas bloqueados e programas que não terminam a sua execução. Nesta tese, desenvolvemos CLASS: uma linguagem baseada em sessões, com um sistema de tipos leve e subestrutural, que resulta de uma extensão metodológica da correspondência proposições-como-tipos com a lógica linear clássica de segunda ordem. Mais concretamente, a linguagem CLASS oferece suporte para comunicação baseada em sessões, células de memória protegidas com mutexes de primeira classe, partilha dinâmica de estado, algoritmos polimórficos genéricos, abstração de dados e recursão primitiva. A linguagem CLASS expressa e tipifica programas realistas significativos, que manipulam estruturas de dados ligadas eficientes (listas ligadas, árvores de pesquisa binária) suportando actualização imperativa local, TDAs partilhados e concorrentes (contadores, pilhas, filas funcionais e imperativas), métodos de sincronização e partilha de recursos (bifurcar-juntar, barreiras, jantar de filósofos, protocolos genéricos corecursivos). Todos estes exemplos são seguros, uma garantia que resulta da nossa abordagem lógica. Os fundamentos, baseados na lógica linear, garantem que programas em CLASS bem tipificados não incorrem em erros: nunca bloqueiam, quer na comunicação, quer na aquisição de células de memória, nunca causam fugas de memória e terminam sempre, mesmo que compartilhem estruturas de dados complexas protegidas por primitivas de sincronização. Além disso, uma vez que seguimos uma abordagem de proposições-comotipos, podemos raciocinar sobre o comportamento de processos concorrentes, que usam estado, através de manipulação algébrica. A viabilidade da nossa abordagem é evidenciada pela implementação de um verificador de tipos e interpretador para a linguagem CLASS, que valida e orienta o desenvolvimento de vários programs realistas. A implementação está disponível com uma licença de acesso livre, juntamente com inúmeros exemplos

Parametric reduced-order aeroelastic modelling for analysis, dynamic system interpolation and control of flexible aircraft

This work presents an integral framework to derive aeroelastic models for very flexible aircraft that can be used in design routines, operational envelope analysis and control applications. Aircraft are modelled using a nonlinear geometrically-exact beam model coupled with an Unsteady Vortex-Lattice Method aerodynamic solver, capable of capturing important nonlinear couplings and effects that significantly impact the flight characteristics of very flexible aircraft. Then, complete linearised expressions of the aircraft system about trim reference conditions at possibly large deformations are presented. The nature of the aerodynamic models results in a high-dimensional system that requires of model reduction methods for efficient analysis and manipulation. Krylov-subspace model reduction methods are implemented to reduce the dimensionality of the multi-input multi-output linearised aerodynamic model and achieve a very significant reduction in the size of the size of the system. The reduced aerodynamic model is then coupled with a modal expression of the linearised beam model, resulting in a compact aeroelastic state-space that can be efficiently used on desktop hardware for linear analysis or as part of internal control models. These have been used to explore the design space of a very flexible wing with complex aeroelastic properties to determine the flutter boundaries, for which experimental data has become available that validates the methods presented herein. Additionally, they have been integrated in a model predictive control framework, where the reduced linear aerodynamic model is part of the control model, and the simulation plant is the nonlinear flight dynamic/aeroelastic model connected as a hardware-in-the-loop platform. Finally, in order to accelerate the design space exploration of very flexible structures, state-space interpolation methods are sought to obtain, with a few linearised models sampled across the domain, interpolated state-spaces anywhere in the parameter-space in a fast and accurate manner. The performance of the interpolation schemes is heavily dependent on the location of the sampling points on the design space, therefore, a novel adaptive Bayesian sampling scheme is presented to choose these points in an optimal approach that minimises the interpolation error function.Open Acces

Automatic goal distribution strategies for the execution of committed choice logic languages on distributed memory parallel computers

There has been much research interest in efficient implementations of the Committed Choice Non-Deterministic (CCND) logic languages on parallel computers. To take full advantage of the speed gains of parallel computers, methods need to be found to automatically distribute goals over the machine processors, ideally with as little involvement from the user as possible.In this thesis we explore some automatic goal distribution strategies for the execu¬ tion of the CCND languages on commercially available distributed memory parallel computers.There are two facets to the goal distribution strategies we have chosen to explore:DEMAND DRIVEN: An idle processor requests work from other processors. We describe two strategies in this class: one in which an idle processor asks only neighbouring processors for spare work, the nearest-neighbour strategy; and one where an idle processor may ask any other processor in the machine for spare work, the allprocessors strategy.WEIGHTS: Using a program analysis technique devised by Tick, weights are attached to goals; the weights can be used to order the goals so that they can be executed and distributed out in weighted order, possibly increasing performance.We describe a framework in which to implement and analyse goal distribution strategies, and then go on to describe experiments with demand driven strategies, both with and without weights. The experiments were made using two of our own implementations of Flat Guarded Horn Clauses — an interpreter and a WAM-like system — executing on a MEIKO T800 Transputer Array configured in a 2-D mesh topology.Analysis of the results show that the all-processors strategies are promising (AP-NW), adding weights had little positive effect on performance, and that nearest-neighbours strategies can reduce performance due to bad load balancing.We also describe some preliminary experiments for a variant of the AP-NW strategy: goals which suspend on one variable are sent to the processor that controls that variable, the processes-to-data strategy. And we briefly look at some preliminary results of executing programs on large numbers of processors (> 30)

Twisting Lattice and Graph Techniques to Compress Transactional Ledgers

International audienceKeeping track of financial transactions (e.g., in banks and blockchains) means keeping track of an ever-increasing list of exchanges between accounts. In fact, many of these transactions can be safely " forgotten " , in the sense that purging a set of them that compensate each other does not impact the network's semantic meaning (e.g., the accounts' balances). We call nilcatenation a collection of transactions having no effect on a network's semantics. Such exchanges may be archived and removed, yielding a smaller, but equivalent ledger. Motivated by the computational and analytic benefits obtained from more compact representations of numerical data, we formalize the problem of finding nilcatenations, and propose detection methods based on graph and lattice-reduction techniques. Atop interesting applications of this work (e.g., decoupling of centralized and distributed databases), we also discuss the original idea of a " community-serving proof of work " : finding nilcatenations constitutes a proof of useful work, as the periodic removal of nilcatenations reduces the transactional graph's size