Interval Linear Constraint Solving Using the Preconditioned Interval Gauss-Seidel Method

We propose the use of the preconditioned interval Gauss-Seidel method as the backbone of an efficient linear equality solver in a CLP(Interval) language. The method, as originally designed, works only on linear systems with square coefficient matrices. Even imposing such a restriction, a naive incorporation of the traditional preconditioning algorithm in a CLP language incurs a high worst-case time complexity of O(n^4), where n is the number of variables in the linear system. In this paper, we generalize the algorithm for general linear systems with m constraints and n variables, and give a novel incremental adaptation of preconditioning of O(n 2 (n + m)) complexity. The efficiency of the incremental preconditioned interval Gauss-Seidel method is demonstrated using large-scale linear systems

Interval linear constraint solving in constraint logic programming.

by Chong-kan Chiu.Thesis (M.Phil.)--Chinese University of Hong Kong, 1994.Includes bibliographical references (leaves 97-103).Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Related Work --- p.2Chapter 1.2 --- Organizations of the Dissertation --- p.4Chapter 1.3 --- Notations --- p.4Chapter 2 --- Overview of ICLP(R) --- p.6Chapter 2.1 --- Basics of Interval Arithmetic --- p.6Chapter 2.2 --- Relational Interval Arithmetic --- p.8Chapter 2.2.1 --- Interval Reduction --- p.8Chapter 2.2.2 --- Arithmetic Primitives --- p.10Chapter 2.2.3 --- Interval Narrowing and Interval Splitting --- p.13Chapter 2.3 --- Syntax and Semantics --- p.16Chapter 3 --- Limitations of Interval Narrowing --- p.18Chapter 3.1 --- Computation Inefficiency --- p.18Chapter 3.2 --- Inability to Detect Inconsistency --- p.23Chapter 3.3 --- The Newton Language --- p.27Chapter 4 --- Design of CIAL --- p.30Chapter 4.1 --- The CIAL Architecture --- p.30Chapter 4.2 --- The Inference Engine --- p.31Chapter 4.2.1 --- Interval Variables --- p.31Chapter 4.2.2 --- Extended Unification Algorithm --- p.33Chapter 4.3 --- The Solver Interface and Constraint Decomposition --- p.34Chapter 4.4 --- The Linear and the Non-linear Solvers --- p.37Chapter 5 --- The Linear Solver --- p.40Chapter 5.1 --- An Interval Gaussian Elimination Solver --- p.41Chapter 5.1.1 --- Naive Interval Gaussian Elimination --- p.41Chapter 5.1.2 --- Generalized Interval Gaussian Elimination --- p.43Chapter 5.1.3 --- Incrementality of Generalized Gaussian Elimination --- p.47Chapter 5.1.4 --- Solvers Interaction --- p.50Chapter 5.2 --- An Interval Gauss-Seidel Solver --- p.52Chapter 5.2.1 --- Interval Gauss-Seidel Method --- p.52Chapter 5.2.2 --- Preconditioning --- p.55Chapter 5.2.3 --- Increment ality of Preconditioned Gauss-Seidel Method --- p.58Chapter 5.2.4 --- Solver Interaction --- p.71Chapter 5.3 --- Comparisons --- p.72Chapter 5.3.1 --- Time Complexity --- p.72Chapter 5.3.2 --- Storage Complexity --- p.73Chapter 5.3.3 --- Others --- p.74Chapter 6 --- Benchmarkings --- p.76Chapter 6.1 --- Mortgage --- p.78Chapter 6.2 --- Simple Linear Simultaneous Equations --- p.79Chapter 6.3 --- Analysis of DC Circuit --- p.80Chapter 6.4 --- Inconsistent Simultaneous Equations --- p.82Chapter 6.5 --- Collision Problem --- p.82Chapter 6.6 --- Wilkinson Polynomial --- p.85Chapter 6.7 --- Summary and Discussion --- p.86Chapter 6.8 --- Large System of Simultaneous Equations --- p.87Chapter 6.9 --- Comparisons Between the Incremental and the Non-Incremental Preconditioning --- p.89Chapter 7 --- Concluding Remarks --- p.93Chapter 7.1 --- Summary and Contributions --- p.93Chapter 7.2 --- Future Work --- p.95Bibliography --- p.9

Static analysis of an actor-based process calculus by abstract interpretation

Le modèle des Acteurs, introduit par HEWITT et AGHA à la fin des années 80, décrit un système concurrent comme un ensemble d'agents autonomes au comportement non uniforme et communiquant de façon point-à-point par l'envoi de messages étiquetés. Le calcul CAP, proposé par COLAÇO, est un calcul de processus basé sur ce modèle qui permet de décrire sans encodage complexe des systèmes réalistes non triviaux. Ce calcul permet, entre autre, la communication de comportements via les messages et est, en ce sens, un calcul d'ordre supérieur. L'analyse de propriétés sur ce calcul a déjà fait l'objet de plusieurs travaux, essentiellement par inférence de type en utilisant des types comportementaux et du sous-typage. Par ailleurs, des travaux plus récents, effectués par VENET puis FERET, proposent une utilisation de l'interprétation abstraite pour l'analyse de calculs de processus. Ces approches permettent de calculer des propriétés non uniformes : elles permettent, par exemple, de différencier les instances récursives d'un même processus. Cette thèse s'inscrit donc dans la suite de ces deux approches, en appliquant l'interprétation abstraite à l'analyse de CAP. Suivant le cadre proposé par FERET, CAP est, tout d'abord, exprimé dans une forme non standard facilitant les analyses. L'ensemble des configurations atteignables est ensuite sur-approximé via une représentation, correcte par construction, dans des domaines abstraits. Des domaines abstraits généraux sont ensuite introduits afin d'améliorer les analyses existantes ou de représenter des propriétés locales à un sous-terme. Des propriétés spécifiques à CAP, la linéarité des termes et l'absence de messages orphelins, sont alors étudiées dans ce cadre. Des domaines spécifiques sont définis et utilisés pour vérifier ces propriétés. Le cadre présenté permet de lever toutes les restrictions existantes des analyses précédentes quant à la forme des termes ou l'utilisation du passage de comportement. L'intégralité des analyses présentées a été implantée dans un prototype. ABSTRACT : The Actor model, introduced by HEWITT and AGHA in the late 80s, describes a concurrent communicating system as a set of autonomous agents, with non uniform interfaces and communicating by the use of labeled messages. The CAP process calculus, proposed by COLAÇO, is based on this model and allows to describe non trivial realistic systems, without the need of complex encodings. CAP is a higher-order calculus: messages can carry actor behaviors. Multiple works address the analysis of CAP properties, mainly by the use of inferencebased type systems using behavioral types and sub-typing. Otherwise, ore recent works, by VENET and later FERET, propose the use of abstract interpretation to analyze process calculi. These approaches allow to compute non-uniform properties. For example, they are able to differentiate recursive instances of the same thread. This thesis is at the crossroad of these two approaches, applying abstract interpretation to the analysis of CAP. Following the framework of FERET, CAP is firstly expressed in a non standard form, easing its analysis. The set of reachable states is then over-approximated via a sound by construction representation within existing abstract domains. New general abstract domains are then introduced in order to improve the accuracy of existing analyses or to represent local properties. CAP specific properties such as the linearity of terms or the absence of orphan messages, are then considered in this framework. Specific abstract domains are defined and used to check these properties. The proposed framework is able to relax any existing restriction of previous analyses such as constraints on the shape of terms or limitation in the use of CAP behavior passing. The whole analyses have been implemented in a prototyp