1,638 research outputs found
Applications of the Galois Model LFSR in Cryptography
The linear feedback shift-register is a widely used tool for generating cryptographic sequences. The properties of the Galois model discussed here offer many opportunities to improve the implementations that already exist. We explore the overall properties of the phases of the Galois model and conjecture a relation with modular Golomb rulers. This conjecture points to an efficient method for constructing non-linear filtering generators which fulfil Golic s design criteria in order to maximise protection against his inversion attack. We also produce a number of methods which can improve the rate of output of sequences by combining particular distinct phases of smaller elementary sequences
Gated recurrent neural networks discover attention
Recent architectural developments have enabled recurrent neural networks
(RNNs) to reach and even surpass the performance of Transformers on certain
sequence modeling tasks. These modern RNNs feature a prominent design pattern:
linear recurrent layers interconnected by feedforward paths with multiplicative
gating. Here, we show how RNNs equipped with these two design elements can
exactly implement (linear) self-attention, the main building block of
Transformers. By reverse-engineering a set of trained RNNs, we find that
gradient descent in practice discovers our construction. In particular, we
examine RNNs trained to solve simple in-context learning tasks on which
Transformers are known to excel and find that gradient descent instills in our
RNNs the same attention-based in-context learning algorithm used by
Transformers. Our findings highlight the importance of multiplicative
interactions in neural networks and suggest that certain RNNs might be
unexpectedly implementing attention under the hood
Facets and Levels of Mathematical Abstraction
International audienceMathematical abstraction is the process of considering and manipulating operations, rules, methods and concepts divested from their reference to real world phenomena and circumstances, and also deprived from the content connected to particular applications. There is no one single way of performing mathematical abstraction. The term "abstraction" does not name a unique procedure but a general process, which goes many ways that are mostly simultaneous and intertwined ; in particular, the process does not amount only to logical subsumption. I will consider comparatively how philosophers consider abstraction and how mathematicians perform it, with the aim to bring to light the fundamental thinking processes at play, and to illustrate by significant examples how much intricate and multi-leveled may be the combination of typical mathematical techniques which include axiomatic method, invarianceprinciples, equivalence relations and functional correspondences.L'abstraction mathématique consiste en la considération et la manipulation d'opérations, règles et concepts indépendamment du contenu dont les nantissent des applications particulières et du rapport qu'ils peuvent avoir avec les phénomènes et les circonstances du monde réel. L'abstraction mathématique emprunte diverses voies. Le terme " abstraction " ne désigne pasune procédure unique, mais un processus général où s'entrecroisent divers procédés employés successivement ou simultanément. En particulier, l'abstraction mathématique ne se réduit pas à la subsomption logique. Je vais étudier comparativement en quels termes les philosophes expliquent l'abstraction et par quels moyens les mathématiciens la mettent en oeuvre. Je voudrais parlà mettre en lumière les principaux processus de pensée en jeu et illustrer par des exemples divers niveaux d'intrication de techniques mathématiques récurrentes, qui incluent notamment la méthode axiomatique, les principes d'invariance, les relations d'équivalence et les correspondances fonctionnelles
Predicting the approximate functional behaviour of physical systems
This dissertation addresses the problem of the computer prediction of the approximate
behaviour of physical systems describable by ordinary differential equations.Previous approaches to behavioural prediction have either focused on an exact
mathematical description or on a qualitative account. We advocate a middle ground: a
representation more coarse than an exact mathematical solution yet more specific than a
qualitative one. What is required is a mathematical expression, simpler than the exact
solution, whose qualitative features mirror those of the actual solution and whose
functional form captures the principal parameter relationships underlying the behaviour of
the real system. We term such a representation an approximate functional solution.Approximate functional solutions are superior to qualitative descriptions because they
reveal specific functional relationships, restore a quantitative time scale to a process and
support more sophisticated comparative analysis queries. Moreover, they can be superior to
exact mathematical solutions by emphasizing comprehensibility, adequacy and practical
utility over precision.Two strategies for constructing approximate functional solutions are proposed. The first
abstracts the original equation, predicts behaviour in the abstraction space and maps this
back to the approximate functional level. Specifically, analytic abduction exploits
qualitative simulation to predict the qualitative properties of the solution and uses this
knowledge to guide the selection of a parameterized trial function which is then tuned with
respect to the differential equation. In order to limit the complexity of a proposed
approximate functional solution, and hence maintain its comprehensibility,
back-of-the-envelope reasoning is used to simplify overly complex expressions in a
magnitude extreme. If no function is recognised which matches the predicted behaviour,
segment calculus is called upon to find a composite function built from known primitives
and a set of operators. At the very least, segment calculus identifies a plausible structure
for the form of the solution (e.g. that it is a composition of two unknown functions).
Equation parsing capitalizes on this partial information to look for a set of termwise
interactions which, when interpreted, expose a particular solution of the equation.The second, and more direct, strategy for constructing an approximate functional solution is
embodied in the closed form approximation technique. This extends approximation
methods to equations which lack a closed form solution. This involves solving the
differential equation exactly, as an infinite series, and obtaining an approximate functional
solution by constructing a closed form function whose Taylor series is close to that of the
exact solutionThe above techniques dovetail together to achieve a style of reasoning closer to that of an
engineer or physicist rather than a mathematician. The key difference being to sacrifice the
goal of finding the correct solution of the differential equation in favour of finding an
approximation which is adequate for the purpose to which the knowledge will be put.
Applications to Intelligent Tutoring and Design Support Systems are suggested
Formal methods applied to the analysis of phylogenies: Phylogenetic model checking
Los árboles filogenéticos son abstracciones útiles para modelar y caracterizar la evolución de un conjunto de especies o poblaciones respecto del tiempo. La proposición, verificación y generalización de hipótesis sobre un árbol filogenético inferido juegan un papel importante en el estudio y comprensión de las relaciones evolutivas. Actualmente, uno de los principales objetivos cientÃficos es extraer o descubrir los mensajes biológicos implÃcitos y las propiedades estructurales subyacentes en la filogenia. Por ejemplo, la integración de información genética en una filogenia ayuda al descubrimiento de genes conservados en todo o parte del árbol, la identificación de posiciones covariantes en el ADN o la estimación de las fechas de divergencia entre especies. Consecuentemente, los árboles ayudan a comprender el mecanismo que gobierna la deriva evolutiva. Hoy en dÃa, el amplio espectro de métodos y herramientas heterogéneas para el análisis de filogenias enturbia y dificulta su utilización, además del fuerte acoplamiento entre la especificación de propiedades y los algoritmos utilizados para su evaluación (principalmente scripts ad hoc). Este problema es el punto de arranque de esta tesis, donde se analiza como solución la posibilidad de introducir un entorno formal de verificación de hipótesis que, de manera automática y modular, estudie la veracidad de dichas propiedades definidas en un lenguaje genérico e independiente (en una lógica formal asociada) sobre uno de los múltiples softwares preparados para ello. La contribución principal de la tesis es la propuesta de un marco formal para la descripción, verificación y manipulación de relaciones causales entre especies de forma independiente del código utilizado para su valoración. Para ello, exploramos las caracterÃsticas de las técnicas de model checking, un paradigma en el que una especificación expresada en lógica temporal se verifica con respecto a un modelo del sistema que representa una implementación a un cierto nivel de detalle. Se ha aplicado satisfactoriamente en la industria para el modelado de sistemas y su verificación, emergiendo del ámbito de las ciencias de la computación. Las contribuciones concretas de la tesis han sido: A) La identificación e interpretación de los árboles filogeneticos como modelos de la evolución, adaptados al entorno de las técnicas de model checking. B) La definición de una lógica temporal que captura las propiedades filogenéticas habituales junto con un método de construcción de propiedades. C) La clasificación de propiedades filogenéticas, identificando categorÃas de propiedades según estén centradas en la estructura del árbol, en las secuencias o sean hÃbridas. D) La extensión de las lógicas y modelos para contemplar propiedades cuantitativas de tiempo, probabilidad y de distancias. E) El desarrollo de un entorno para la verificación de propiedades booleanas, cuantitativas y paramétricas. F) El establecimiento de los principios para la manipulación simbolica de objetos filogenéticos, p. ej., clados. G) La explotación de las herramientas de model checking existentes, detectando sus problemas y carencias en el campo de filogenia y proponiendo mejoras. H) El desarrollo de técnicas "ad hoc" para obtener ganancia de complejidad alrededor de dos frentes: distribución de los cálculos y datos, y el uso de sistemas de información. Los puntos A-F se centran en las aportaciones conceptuales de nuestra aproximación, mientras que los puntos G-H enfatizan la parte de herramientas e implementación. Los contenidos de la tesis están contrastados por la comunidad cientÃfica mediante las siguientes publicaciones en conferencias y revistas internacionales. La introducción de model checking como entorno formal para analizar propiedades biológicas (puntos A-C) ha llevado a la publicación de nuestro primer artÃculo de congreso [1]. En [2], desarrollamos la verificación de hipótesis filogenéticas sobre un árbol de ejemplo construido a partir de las relaciones impuestas por un conjunto de proteÃnas codificadas por el ADN mitocondrial humano (ADNmt). En ese ejemplo, usamos una herramienta automática y genérica de model checking (punto G). El artÃculo de revista [7] resume lo básico de los artÃculos de congreso previos y extiende la aplicación de lógicas temporales a propiedades filogenéticas no consideradas hasta ahora. Los artÃculos citados aquà engloban los contenidos presentados en las Parte I--II de la tesis. El enorme tamaño de los árboles y la considerable cantidad de información asociada a los estados (p.ej., la cadena de ADN) obligan a la introducción de adaptaciones especiales en las herramientas de model checking para mantener un rendimiento razonable en la verificación de propiedades y aliviar también el problema de la explosión de estados (puntos G-H). El artÃculo de congreso [3] presenta las ventajas de rebanar el ADN asociado a los estados, la partición de la filogenia en pequeños subárboles y su distribución entre varias máquinas. Además, la idea original del model checking rebanado se complementa con la inclusión de una base de datos externa para el almacenamiento de secuencias. El artÃculo de revista [4] reúne las nociones introducidas en [3] junto con la implementación y resultados preliminares presentados [5]. Este tema se corresponde con lo presentado en la Parte III de la tesis. Para terminar, la tesis reaprovecha las extensiones de las lógicas temporales con tiempo explÃcito y probabilidades a fin de manipular e interrogar al árbol sobre información cuantitativa. El artÃculo de congreso [6] ejemplifica la necesidad de introducir probabilidades y tiempo discreto para el análisis filogenético de un fenotipo real, en este caso, el ratio de distribución de la intolerancia a la lactosa entre diversas poblaciones arraigadas en las hojas de la filogenia. Esto se corresponde con el CapÃtulo 13, que queda englobado dentro de las Partes IV--V. Las Partes IV--V completan los conceptos presentados en ese artÃculo de conferencia hacia otros dominios de aplicación, como la puntuación de árboles, y tiempo continuo (puntos E-F). La introducción de parámetros en las hipótesis filogenéticas se plantea como trabajo futuro. Referencias [1] Roberto Blanco, Gregorio de Miguel Casado, José Ignacio Requeno, and José Manuel Colom. Temporal logics for phylogenetic analysis via model checking. In Proceedings IEEE International Workshop on Mining and Management of Biological and Health Data, pages 152-157. IEEE, 2010. [2] José Ignacio Requeno, Roberto Blanco, Gregorio de Miguel Casado, and José Manuel Colom. Phylogenetic analysis using an SMV tool. In Miguel P. Rocha, Juan M. Corchado RodrÃguez, Florentino Fdez-Riverola, and Alfonso Valencia, editors, Proceedings 5th International Conference on Practical Applications of Computational Biology and Bioinformatics, volume 93 of Advances in Intelligent and Soft Computing, pages 167-174. Springer, Berlin, 2011. [3] José Ignacio Requeno, Roberto Blanco, Gregorio de Miguel Casado, and José Manuel Colom. Sliced model checking for phylogenetic analysis. In Miguel P. Rocha, Nicholas Luscombe, Florentino Fdez-Riverola, and Juan M. Corchado RodrÃguez, editors, Proocedings 6th International Conference on Practical Applications of Computational Biology and Bioinformatics, volume 154 of Advances in Intelligent and Soft Computing, pages 95-103. Springer, Berlin, 2012. [4] José Ignacio Requeno and José Manuel Colom. Model checking software for phylogenetic trees using distribution and database methods. Journal of Integrative Bioinformatics, 10(3):229-233, 2013. [5] José Ignacio Requeno and José Manuel Colom. Speeding up phylogenetic model checking. In Mohd Saberi Mohamad, Loris Nanni, Miguel P. Rocha, and Florentino Fdez-Riverola, editors, Proceedings 7th International Conference on Practical Applications of Computational Biology and Bioinformatics, volume 222 of Advances in Intelligent Systems and Computing, pages 119-126. Springer, Berlin, 2013. [6] José Ignacio Requeno and José Manuel Colom. Timed and probabilistic model checking over phylogenetic trees. In Miguel P. Rocha et al., editors, Proceedings 8th International Conference on Practical Applications of Computational Biology and Bioinformatics, Advances in Intelligent and Soft Computing. Springer, Berlin, 2014. [7] José Ignacio Requeno, Gregorio de Miguel Casado, Roberto Blanco, and José Manuel Colom. Temporal logics for phylogenetic analysis via model checking. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 10(4):1058-1070, 2013
Complexity measures for classes of sequences and cryptographic apllications
Pseudo-random sequences are a crucial component of cryptography, particularly
in stream cipher design. In this thesis we will investigate several measures of
randomness for certain classes of finitely generated sequences.
We will present a heuristic algorithm for calculating the k-error linear complexity
of a general sequence, of either finite or infinite length, and results on the
closeness of the approximation generated.
We will present an linear time algorithm for determining the linear complexity
of a sequence whose characteristic polynomial is a power of an irreducible element,
again presenting variations for both finite and infinite sequences. This algorithm
allows the linear complexity of such sequences to be determined faster than was
previously possible.
Finally we investigate the stability of m-sequences, in terms of both k-error
linear complexity and k-error period. We show that such sequences are inherently
stable, but show that some are more stable than others
Random Neural Networks and Optimisation
In this thesis we introduce new models and learning algorithms for the Random
Neural Network (RNN), and we develop RNN-based and other approaches for the
solution of emergency management optimisation problems.
With respect to RNN developments, two novel supervised learning algorithms are
proposed. The first, is a gradient descent algorithm for an RNN extension model
that we have introduced, the RNN with synchronised interactions (RNNSI), which
was inspired from the synchronised firing activity observed in brain neural circuits.
The second algorithm is based on modelling the signal-flow equations in RNN as a
nonnegative least squares (NNLS) problem. NNLS is solved using a limited-memory
quasi-Newton algorithm specifically designed for the RNN case.
Regarding the investigation of emergency management optimisation problems,
we examine combinatorial assignment problems that require fast, distributed and
close to optimal solution, under information uncertainty. We consider three different
problems with the above characteristics associated with the assignment of
emergency units to incidents with injured civilians (AEUI), the assignment of assets
to tasks under execution uncertainty (ATAU), and the deployment of a robotic
network to establish communication with trapped civilians (DRNCTC).
AEUI is solved by training an RNN tool with instances of the optimisation problem
and then using the trained RNN for decision making; training is achieved using
the developed learning algorithms. For the solution of ATAU problem, we introduce
two different approaches. The first is based on mapping parameters of the
optimisation problem to RNN parameters, and the second on solving a sequence of
minimum cost flow problems on appropriately constructed networks with estimated
arc costs. For the exact solution of DRNCTC problem, we develop a mixed-integer
linear programming formulation, which is based on network flows. Finally, we design
and implement distributed heuristic algorithms for the deployment of robots
when the civilian locations are known or uncertain
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