Reasoning in combinations of theories

Verification problems are often expressed in a language which mixes several theories. A natural question to ask is whether one can use decision procedures for individual theories to construct a decision procedure for the union theory. In the cases where this is possible one has a powerful method at hand to handle complex theories effectively. The setup considered in this thesis is that of one base theory which is extended by one or more theories. The question is if and when a given ground satisfiability problem in the extended setting can be effectively reduced to an equi-satisfiable problem over the base theory. A case where this reductive approach is always possible is that of so-called local theory extensions. The theory of local extensions is developed and some applications concerning monotone functions are given. Then the theory of local theory extensions is generalized in order to deal with data structures that exhibit local behavior. It will be shown that a suitable fragment of both the theory of arrays and the theory of pointers is local in this broader sense. Finally, the case of more than one theory extension is discussed. In particular, a modularity result is given that under certain circumstances the locality of each of the extensions lifts to locality of the entire extension. The reductive approach outlined above has become particularly relevant in recent years due to the rise of powerful solvers for background theories common in verification tasks. These so-called SMT-solvers effectively handle theories such as real linear or integer arithmetic. As part of this thesis, a program called H-PILoT was implemented which carries out reductive reasoning for local theory extensions. H-PILoT found applications in mathematics, multiple-valued logics, data-structures and reasoning in complex systems.Verifikationsaufgaben kombinieren oft verschiedene Theorien. Eine naheliegende Frage ist, ob man Entscheidungsverfahren für die Einzeltheorien auf die gesamte Theorie übertragen kann. In den Fällen, wo das möglich ist, hat man eine mächtige Technik zur Hand, um mit komplexen Theorien effizient umgehen zu können. Der Ansatz, der in dieser Arbeit betrachtet wird, ist stets der einer Hintergrundtheorie, die durch eine oder mehrere Theorien erweitert wird. Die Frage ist dann, ob und wann sich eine gegebene Beweisanfrage bezüglich der Theorieerweiterung effektiv auf eine äquivalente Beweisanfrage bezüglich der Hintergrundtheorie reduzieren lässt. Ein Fall, in dem diese Reduzierung immer möglich ist, ist derjenige der lokalen Theorieerweiterungen. Die Theorie der lokalen Erweiterungen wird entwickelt, und es werden einige Anwendungen für monotone Funktionen gegeben. Danach wird die Theorie der lokalen Erweiterungen verallgemeinert, um mit Datenstrukturen umgehen zu können, die Lokalitätseigenschaften aufweisen. Es wird gezeigt, dass sowohl ein geeignetes Fragment der Theorie der Arrays wie auch der Theorie der Zeiger lokal im erweiterten Sinne sind. Schließlich wird der Fall mehrerer Theorieerweiterungen betrachtet. Insbesondere wird ein Modularitätsresultat gezeigt, das besagt, dass unter gewissen Umständen die Lokalität der einzelnen Erweiterungen hinreichend ist, um die Lokalität der gesamten Erweiterung zu gewährleisten. Die oben erwähnte Reduzierung von Beweisaufgaben ist in jüngster Zeit besonders relevant geworden, weil leistungsfähige Beweiser für gängige Hintergrundtheorien entwickelt worden sind. Diese sogenannten SMT-Beweiser behandeln Theorien wie z.B. lineare Arithmetik der ganzen oder der reellen Zahlen effektiv. Als Teil der vorgelegten Arbeit wurde ein Programm namens H-PILoT entwickelt, welches die Reduzierung von Beweisaufgaben für lokale Theorien durchführt. H-PILoT hat Anwendungen in der Mathematik, bei mehrwertigen Logiken, bei der Verifikation von Datenstrukturen und in der Verifikation komplexer Systeme gefunden

Lectures on Loop Quantum Gravity

Quantum General Relativity (QGR), sometimes called Loop Quantum Gravity, has matured over the past fifteen years to a mathematically rigorous candidate quantum field theory of the gravitational field. The features that distinguish it from other quantum gravity theories are 1) background independence and 2) minimality of structures. Background independence means that this is a non-perturbative approach in which one does not perturb around a given, distinguished, classical background metric, rather arbitrary fluctuations are allowed, thus precisely encoding the quantum version of Einstein's radical perception that gravity is geometry. Minimality here means that one explores the logical consequences of bringing together the two fundamental principles of modern physics, namely general covariance and quantum theory, without adding any experimentally unverified additional structures. The approach is purposely conservative in order to systematically derive which basic principles of physics have to be given up and must be replaced by more fundamental ones. QGR unifies all presently known interactions in a new sense by quantum mechanically implementing their common symmetry group, the four-dimensional diffeomorphism group, which is almost completely broken in perturbative approaches. These lectures offer a problem -- supported introduction to the subject.Comment: 90 pages, Latex, 18 figures, uses graphicx and pstricks for coloured text and graphics, based on lectures given at the 271st WE Heraeus Seminar ``Aspects of Quantum Gravity: From Theory to Experimental Search'', Bad Honnef, Germany, February 25th -- March 1st, to appear in Lecture Notes in Physic

Bipolar cascade lasers

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2000.Includes bibliographical references.This thesis addresses issues of the design and modeling of the Bipolar Cascade Laser (BCL), a new type of quantum well laser. BCLs consist of multiple single stage lasers electrically coupled via tunnel junctions. The BCL ideally operates by having each injected electron participate in a recombination event in the topmost active region, then tunnel from the valence band of the first active region into the conduction band of the next active region, participate in another recombination event, and so on through each stage of the cascade. As each electron may produce more than one photon the quantum efficiency of the device can, in theory, exceed 100%. This work resulted in the first room temperature, continuous-wave operation of a BCL, with a record 99.3% differential slope efficiency. The device was fully characterized and modeled to include light output and voltage versus current bias, modulation response and thermal properties. A new singlemode bipolar cascade laser, the bipolar cascade antiresonant reflecting optical waveguide laser, was proposed and modeled.by Steven G. Patterson.Ph.D