A Certified Polynomial-Based Decision Procedure for Propositional Logic

In this paper we present the formalization of a decision procedure for Propositional Logic based on polynomial normalization. This formalization is suitable for its automatic verification in an applicative logic like Acl2. This application of polynomials has been developed by reusing a previous work on polynomial rings [19], showing that a proper formalization leads to a high level of reusability. Two checkers are defined: the first for contradiction formulas and the second for tautology formulas. The main theorems state that both checkers are sound and complete. Moreover, functions for generating models and counterexamples of formulas are provided. This facility plays also an important role in the main proofs. Finally, it is shown that this allows for a highly automated proof development

Enhanced cryptographic approaches for SCADA network security.

Due to the overwhelming increase in open source code, off-the-shelf software packages, third party and vendor codes, along with the ease of getting information about hacking network security systems and attacking the well known holes in security systems, the problem of having a secure network system is much more difficult than before this boom in technology and information broadcast. What makes the problem even worse is trying to secure a network for real time control, such as a network using supervisory control and data acquisition (SCADA) systems, because now the problem has two faces: securing the real time control system and at the same time keeping the response time of the system in the acceptable range for the transactions\u27 level of service. There is a strong trend to chose security frameworks that have been popular in the e-commerce sites of the web, particularly because they proven to be very mature and secure for more than one and half decades. Examples include the transport level security (TLS) and its predecessor secured socket layer (SSL) framework that is based on the very popular public key cryptography and key distribution algorithms, such as Rivest, Shamir and Adleman (RSA), elliptic curve cryptography (ECC), and Diffie-Hellman. Despite the fact that these algorithms proved to be very powerful against most types of attacks, they are not tailored to secure SCADA networks, and consequently cause a significant degradation in the performance time of real time transactions. This dissertation offers two novel encryption algorithms for securing a SCADA network, the N-Secrecy and the Security Spectrum algorithms. N-Secrecy gave very good results when compared with the SSL; with N-Secrecy performance time in the range of one thousandth of the SSL. The Security Spectrum approach moved the encryption methodology from using numerical representations into using a physical representation based on modeling the conditions of the two communicating parties with a system of non-linear polynomials and then using computer algebra techniques. Both approaches have the potential to significantly enhance the security of commercial SCADA installations

Minimum Distance Estimation in Categorical Conditional Independence Models

One of the oldest and most fundamental problems in statistics is the analysis of cross-classified data called contingency tables. Analyzing contingency tables is typically a question of association - do the variables represented in the table exhibit special dependencies or lack thereof? The statistical models which best capture these experimental notions of dependence are the categorical conditional independence models; however, until recent discoveries concerning the strongly algebraic nature of the conditional independence models surfaced, the models were widely overlooked due to their unwieldy implicit description. Apart from the inferential question above, this thesis asks the more basic question - suppose such an experimental model of association is known, how can one incorporate this information into the estimation of the joint distribution of the table? In the traditional parametric setting several estimation paradigms have been developed over the past century; however, traditional results are not applicable to arbitrary categorical conditional independence models due to their implicit nature. After laying out the framework for conditional independence and algebraic statistical models, we consider three aspects of estimation in the models using the minimum Euclidean (L2E), minimum Pearson chi-squared, and minimum Neyman modified chi-squared distance paradigms as well as the more ubiquitous maximum likelihood approach (MLE). First, we consider the theoretical properties of the estimators and demonstrate that under general conditions the estimators exist and are asymptotically normal. For small samples, we present the results of large scale simulations to address the estimators' bias and mean squared error (in the Euclidean and Frobenius norms, respectively). Second, we identify the computation of such estimators as an optimization problem and, for the case of the L2E, propose two different methods by which the problem can be solved, one algebraic and one numerical. Finally, we present an R implementation via two novel packages, mpoly for symbolic computing with multivariate polynomials and catcim for fitting categorical conditional independence models. It is found that in general minimum distance estimators in categorical conditional independence models behave as they do in the more traditional parametric setting and can be computed in many practical situations with the implementation provided

Parallel Manipulators

In recent years, parallel kinematics mechanisms have attracted a lot of attention from the academic and industrial communities due to potential applications not only as robot manipulators but also as machine tools. Generally, the criteria used to compare the performance of traditional serial robots and parallel robots are the workspace, the ratio between the payload and the robot mass, accuracy, and dynamic behaviour. In addition to the reduced coupling effect between joints, parallel robots bring the benefits of much higher payload-robot mass ratios, superior accuracy and greater stiffness; qualities which lead to better dynamic performance. The main drawback with parallel robots is the relatively small workspace. A great deal of research on parallel robots has been carried out worldwide, and a large number of parallel mechanism systems have been built for various applications, such as remote handling, machine tools, medical robots, simulators, micro-robots, and humanoid robots. This book opens a window to exceptional research and development work on parallel mechanisms contributed by authors from around the world. Through this window the reader can get a good view of current parallel robot research and applications

Berechnung und Anwendungen Approximativer Randbasen

This thesis addresses some of the algorithmic and numerical challenges associated with the computation of approximate border bases, a generalisation of border bases, in the context of the oil and gas industry. The concept of approximate border bases was introduced by D. Heldt, M. Kreuzer, S. Pokutta and H. Poulisse in "Approximate computation of zero-dimensional polynomial ideals" as an effective mean to derive physically relevant polynomial models from measured data. The main advantages of this approach compared to alternative techniques currently in use in the (hydrocarbon) industry are its power to derive polynomial models without additional a priori knowledge about the underlying physical system and its robustness with respect to noise in the measured input data. The so-called Approximate Vanishing Ideal (AVI) algorithm which can be used to compute approximate border bases and which was also introduced by D. Heldt et al. in the paper mentioned above served as a starting point for the research which is conducted in this thesis. A central aim of this work is to broaden the applicability of the AVI algorithm to additional areas in the oil and gas industry, like seismic imaging and the compact representation of unconventional geological structures. For this purpose several new algorithms are developed, among others the so-called Approximate Buchberger Möller (ABM) algorithm and the Extended-ABM algorithm. The numerical aspects and the runtime of the methods are analysed in detail - based on a solid foundation of the underlying mathematical and algorithmic concepts that are also provided in this thesis. It is shown that the worst case runtime of the ABM algorithm is cubic in the number of input points, which is a significant improvement over the biquadratic worst case runtime of the AVI algorithm. Furthermore, we show that the ABM algorithm allows us to exercise more direct control over the essential properties of the computed approximate border basis than the AVI algorithm. The improved runtime and the additional control turn out to be the key enablers for the new industrial applications that are proposed here. As a conclusion to the work on the computation of approximate border bases, a detailed comparison between the approach in this thesis and some other state of the art algorithms is given. Furthermore, this work also addresses one important shortcoming of approximate border bases, namely that central concepts from exact algebra such as syzygies could so far not be translated to the setting of approximate border bases. One way to mitigate this problem is to construct a "close by" exact border bases for a given approximate one. Here we present and discuss two new algorithmic approaches that allow us to compute such close by exact border bases. In the first one, we establish a link between this task, referred to as the rational recovery problem, and the problem of simultaneously quasi-diagonalising a set of complex matrices. As simultaneous quasi-diagonalisation is not a standard topic in numerical linear algebra there are hardly any off-the-shelf algorithms and implementations available that are both fast and numerically adequate for our purposes. To bridge this gap we introduce and study a new algorithm that is based on a variant of the classical Jacobi eigenvalue algorithm, which also works for non-symmetric matrices. As a second solution of the rational recovery problem, we motivate and discuss how to compute a close by exact border basis via the minimisation of a sum of squares expression, that is formed from the polynomials in the given approximate border basis. Finally, several applications of the newly developed algorithms are presented. Those include production modelling of oil and gas fields, reconstruction of the subsurface velocities for simple subsurface geometries, the compact representation of unconventional oil and gas bodies via algebraic surfaces and the stable numerical approximation of the roots of zero-dimensional polynomial ideals