Validating plans with continuous effects

A critical element in the use of PDDL2.1, the modelling language developed for the International Planning Competition series, has been the common understanding of the semantics of the language. The fact that this has been implemented in plan validation software was vital to the progress of the competition. However, the validation of plans using actions with continuous effects presents new challenges (that precede the challenges presented by planning with those effects). In this paper we review the need for continuous effects, their semantics and the problems that arise in validation of plans that include them. We report our progress in implementing the semantics in an extended version of the plan validation software

Counting polynomial roots in Isabelle/HOL: A formal proof of the Budan-Fourier theorem

Many problems in computer algebra and numerical analysis can be reduced to counting or approximating the real roots of a polynomial within an interval. Existing verified root-counting procedures in major proof assistants are mainly based on the classical Sturm theorem, which only counts distinct roots. In this paper, we have strengthened the root-counting ability in Isabelle/HOL by first formally proving the Budan-Fourier theorem. Subsequently, based on Descartes' rule of signs and Taylor shift, we have provided a verified procedure to efficiently over-approximate the number of real roots within an interval, counting multiplicity. For counting multiple roots exactly, we have extended our previous formalisation of Sturm's theorem. Finally, we combine verified components in the developments above to improve our previous certified complex-root-counting procedures based on Cauchy indices. We believe those verified routines will be crucial for certifying programs and building tactics.ERC Advanced Grant ALEXANDRIA (Project 742178

Completeness of zero curve tracing for analytic functions

AbstractEmpirical data for the polynomial confirm the efficiency of exploring function structure as a means of isolating the zeros of a scalar analytic function defined on a disk (Klip, 1985). In exceptional cases the tracing is not complete which means that certain arcs may not have been traced. A mathematical basis for the algorithm which investigates and restores completeness, the so-called completeness algorithm, is presented. The main theorem is that completeness of the tracing, as verified at the isolated zeros, is sufficient for completeness of the tracing. Its proof evolves along various lemmas, one of which provides a new condition for the location of the critical points. The paper concludes with a brief description of the completeness algorithm

Towards justifying computer algebra algorithms in Isabelle/HOL

As verification efforts using interactive theorem proving grow, we are in need of certified algorithms in computer algebra to tackle problems over the real numbers. This is important because uncertified procedures can drastically increase the size of the trust base and under- mine the overall confidence established by interactive theorem provers, which usually rely on a small kernel to ensure the soundness of derived results. This thesis describes an ongoing effort using the Isabelle theorem prover to certify the cylindrical algebraic decomposition (CAD) algorithm, which has been widely implemented to solve non-linear problems in various engineering and mathematical fields. Because of the sophistication of this algorithm, people are in doubt of the correctness of its implementation when deploying it to safety-critical verification projects, and such doubts motivate this thesis. In particular, this thesis proposes a library of real algebraic numbers, whose distinguishing features include a modular architecture and a sign determination algorithm requiring only rational arithmetic. With this library, an Isabelle tactic based on univariate CAD has been built in a certificate-based way: external, untrusted code delivers solutions in the form of certificates that are checked within Isabelle. To lay the foundation for the multivariate case, I have formalised various analytical results including Cauchy’s residue theorem and the bivariate case of the projection theorem of CAD. During this process, I have also built a tactic to evaluate winding numbers through Cauchy indices and verified procedures to count complex roots in some domains. The formalisation effort in this thesis can be considered as the first step towards a certified computer algebra system inside a theorem prover, so that various engineering projections and mathematical calculations can be carried out in a high-confidence framework

Evaluating Winding Numbers and Counting Complex Roots Through Cauchy Indices in Isabelle/HOL

Abstract: In complex analysis, the winding number measures the number of times a path (counter-clockwise) winds around a point, while the Cauchy index can approximate how the path winds. We formalise this approximation in the Isabelle theorem prover, and provide a tactic to evaluate winding numbers through Cauchy indices. By further combining this approximation with the argument principle, we are able to make use of remainder sequences to effectively count the number of complex roots of a polynomial within some domains, such as a rectangular box and a half-plane

On Flows, Paths, Roots, and Zeros

This thesis has two parts; in the first of which we give new results for various network flow problems. (1) We present a novel dual ascent algorithm for min-cost flow and show that an implementation of it is very efficient on certain instance classes. (2) We approach the problem of numerical stability of interior point network flow algorithms by giving a path following method that works with integer arithmetic solely and is thus guaranteed to be free of any nu-merical instabilities. (3) We present a gradient descent approach for the undirected transship-ment problem and its special case, the single source shortest path problem (SSSP). For distrib-uted computation models this yields the first SSSP-algorithm with near-optimal number of communication rounds. The second part deals with fundamental topics from algebraic computation. (1) We give an algorithm for computing the complex roots of a complex polynomial. While achieving a com-parable bit complexity as previous best results, our algorithm is simple and promising to be of practical impact. It uses a test for counting the roots of a polynomial in a region that is based on Pellet's theorem. (2) We extend this test to polynomial systems, i.e., we develop an algorithm that can certify the existence of a k-fold zero of a zero-dimensional polynomial system within a given region. For bivariate systems, we show experimentally that this approach yields signifi-cant improvements when used as inclusion predicate in an elimination method.Im ersten Teil dieser Dissertation präsentieren wir neue Resultate für verschiedene Netzwerkflussprobleme. (1)Wir geben eine neue Duale-Aufstiegsmethode für das Min-Cost-Flow- Problem an und zeigen, dass eine Implementierung dieser Methode sehr effizient auf gewissen Instanzklassen ist. (2)Wir behandeln numerische Stabilität von Innere-Punkte-Methoden fürNetwerkflüsse, indem wir eine solche Methode angeben die mit ganzzahliger Arithmetik arbeitet und daher garantiert frei von numerischen Instabilitäten ist. (3) Wir präsentieren ein Gradienten-Abstiegsverfahren für das ungerichtete Transshipment-Problem, und seinen Spezialfall, das Single-Source-Shortest-Problem (SSSP), die für SSSP in verteilten Rechenmodellen die erste mit nahe-optimaler Anzahl von Kommunikationsrunden ist. Der zweite Teil handelt von fundamentalen Problemen der Computeralgebra. (1) Wir geben einen Algorithmus zum Berechnen der komplexen Nullstellen eines komplexen Polynoms an, der eine vergleichbare Bitkomplexität zu vorherigen besten Resultaten hat, aber vergleichsweise einfach und daher vielversprechend für die Praxis ist. (2)Wir erweitern den darin verwendeten Pellet-Test zum Zählen der Nullstellen eines Polynoms auf Polynomsysteme, sodass wir die Existenz einer k-fachen Nullstelle eines Systems in einer gegebenen Region zertifizieren können. Für bivariate Systeme zeigen wir experimentell, dass eine Integration dieses Ansatzes in eine Eliminationsmethode zu einer signifikanten Verbesserung führt

Trusting Computations: a Mechanized Proof from Partial Differential Equations to Actual Program

Computer programs may go wrong due to exceptional behaviors, out-of-bound array accesses, or simply coding errors. Thus, they cannot be blindly trusted. Scientific computing programs make no exception in that respect, and even bring specific accuracy issues due to their massive use of floating-point computations. Yet, it is uncommon to guarantee their correctness. Indeed, we had to extend existing methods and tools for proving the correct behavior of programs to verify an existing numerical analysis program. This C program implements the second-order centered finite difference explicit scheme for solving the 1D wave equation. In fact, we have gone much further as we have mechanically verified the convergence of the numerical scheme in order to get a complete formal proof covering all aspects from partial differential equations to actual numerical results. To the best of our knowledge, this is the first time such a comprehensive proof is achieved.Comment: N° RR-8197 (2012). arXiv admin note: text overlap with arXiv:1112.179

Data driven methods for updating fault detection and diagnosis system in chemical processes

Modern industrial processes are becoming more complex, and consequently monitoring them has become a challenging task. Fault Detection and Diagnosis (FDD) as a key element of process monitoring, needs to be investigated because of its essential role in decision making processes. Among available FDD methods, data driven approaches are currently receiving increasing attention because of their relative simplicity in implementation. Regardless of FDD types, one of the main traits of reliable FDD systems is their ability of being updated while new conditions that were not considered at their initial training appear in the process. These new conditions would emerge either gradually or abruptly, but they have the same level of importance as in both cases they lead to FDD poor performance. For addressing updating tasks, some methods have been proposed, but mainly not in research area of chemical engineering. They could be categorized to those that are dedicated to managing Concept Drift (CD) (that appear gradually), and those that deal with novel classes (that appear abruptly). The available methods, mainly, in addition to the lack of clear strategies for updating, suffer from performance weaknesses and inefficient required time of training, as reported. Accordingly, this thesis is mainly dedicated to data driven FDD updating in chemical processes. The proposed schemes for handling novel classes of faults are based on unsupervised methods, while for coping with CD both supervised and unsupervised updating frameworks have been investigated. Furthermore, for enhancing the functionality of FDD systems, some major methods of data processing, including imputation of missing values, feature selection, and feature extension have been investigated. The suggested algorithms and frameworks for FDD updating have been evaluated through different benchmarks and scenarios. As a part of the results, the suggested algorithms for supervised handling CD surpass the performance of the traditional incremental learning in regard to MGM score (defined dimensionless score based on weighted F1 score and training time) even up to 50% improvement. This improvement is achieved by proposed algorithms that detect and forget redundant information as well as properly adjusting the data window for timely updating and retraining the fault detection system. Moreover, the proposed unsupervised FDD updating framework for dealing with novel faults in static and dynamic process conditions achieves up to 90% in terms of the NPP score (defined dimensionless score based on number of the correct predicted class of samples). This result relies on an innovative framework that is able to assign samples either to new classes or to available classes by exploiting one class classification techniques and clustering approaches.Los procesos industriales modernos son cada vez más complejos y, en consecuencia, su control se ha convertido en una tarea desafiante. La detección y el diagnóstico de fallos (FDD), como un elemento clave de la supervisión del proceso, deben ser investigados debido a su papel esencial en los procesos de toma de decisiones. Entre los métodos disponibles de FDD, los enfoques basados en datos están recibiendo una atención creciente debido a su relativa simplicidad en la implementación. Independientemente de los tipos de FDD, una de las principales características de los sistemas FDD confiables es su capacidad de actualización, mientras que las nuevas condiciones que no fueron consideradas en su entrenamiento inicial, ahora aparecen en el proceso. Estas nuevas condiciones pueden surgir de forma gradual o abrupta, pero tienen el mismo nivel de importancia ya que en ambos casos conducen al bajo rendimiento de FDD. Para abordar las tareas de actualización, se han propuesto algunos métodos, pero no mayoritariamente en el área de investigación de la ingeniería química. Podrían ser categorizados en los que están dedicados a manejar Concept Drift (CD) (que aparecen gradualmente), y a los que tratan con clases nuevas (que aparecen abruptamente). Los métodos disponibles, además de la falta de estrategias claras para la actualización, sufren debilidades en su funcionamiento y de un tiempo de capacitación ineficiente, como se ha referenciado. En consecuencia, esta tesis está dedicada principalmente a la actualización de FDD impulsada por datos en procesos químicos. Los esquemas propuestos para manejar nuevas clases de fallos se basan en métodos no supervisados, mientras que para hacer frente a la CD se han investigado los marcos de actualización supervisados y no supervisados. Además, para mejorar la funcionalidad de los sistemas FDD, se han investigado algunos de los principales métodos de procesamiento de datos, incluida la imputación de valores perdidos, la selección de características y la extensión de características. Los algoritmos y marcos sugeridos para la actualización de FDD han sido evaluados a través de diferentes puntos de referencia y escenarios. Como parte de los resultados, los algoritmos sugeridos para el CD de manejo supervisado superan el rendimiento del aprendizaje incremental tradicional con respecto al puntaje MGM (puntuación adimensional definida basada en el puntaje F1 ponderado y el tiempo de entrenamiento) hasta en un 50% de mejora. Esta mejora se logra mediante los algoritmos propuestos que detectan y olvidan la información redundante, así como ajustan correctamente la ventana de datos para la actualización oportuna y el reciclaje del sistema de detección de fallas. Además, el marco de actualización FDD no supervisado propuesto para tratar fallas nuevas en condiciones de proceso estáticas y dinámicas logra hasta 90% en términos de la puntuación de NPP (puntuación adimensional definida basada en el número de la clase de muestras correcta predicha). Este resultado se basa en un marco innovador que puede asignar muestras a clases nuevas o a clases disponibles explotando una clase de técnicas de clasificación y enfoques de agrupamientoPostprint (published version

Performance of Machine Learning and Big Data Analytics paradigms in Cybersecurity and Cloud Computing Platforms

The purpose of the research is to evaluate Machine Learning and Big Data Analytics paradigms for use in Cybersecurity. Cybersecurity refers to a combination of technologies, processes and operations that are framed to protect information systems, computers, devices, programs, data and networks from internal or external threats, harm, damage, attacks or unauthorized access. The main characteristic of Machine Learning (ML) is the automatic data analysis of large data sets and production of models for the general relationships found among data. ML algorithms, as part of Artificial Intelligence, can be clustered into supervised, unsupervised, semi-supervised, and reinforcement learning algorithms