Mechanised metamathematics : an investigation of first-order logic and set theory in constructive type theory

In this thesis, we investigate several key results in the canon of metamathematics, applying the contemporary perspective of formalisation in constructive type theory and mechanisation in the Coq proof assistant. Concretely, we consider the central completeness, undecidability, and incompleteness theorems of first-order logic as well as properties of the axiom of choice and the continuum hypothesis in axiomatic set theory. Due to their fundamental role in the foundations of mathematics and their technical intricacies, these results have a long tradition in the codification as standard literature and, in more recent investigations, increasingly serve as a benchmark for computer mechanisation. With the present thesis, we continue this tradition by uniformly analysing the aforementioned cornerstones of metamathematics in the formal framework of constructive type theory. This programme offers novel insights into the constructive content of completeness, a synthetic approach to undecidability and incompleteness that largely eliminates the notorious tedium obscuring the essence of their proofs, as well as natural representations of set theory in the form of a second-order axiomatisation and of a fully type-theoretic account. The mechanisation concerning first-order logic is organised as a comprehensive Coq library open to usage and contribution by external users.In dieser Doktorarbeit werden einige Schlüsselergebnisse aus dem Kanon der Metamathematik untersucht, unter Verwendung der zeitgenössischen Perspektive von Formalisierung in konstruktiver Typtheorie und Mechanisierung mit Hilfe des Beweisassistenten Coq. Konkret werden die zentralen Vollständigkeits-, Unentscheidbarkeits- und Unvollständigkeitsergebnisse der Logik erster Ordnung sowie Eigenschaften des Auswahlaxioms und der Kontinuumshypothese in axiomatischer Mengenlehre betrachtet. Aufgrund ihrer fundamentalen Rolle in der Fundierung der Mathematik und ihrer technischen Schwierigkeiten, besitzen diese Ergebnisse eine lange Tradition der Kodifizierung als Standardliteratur und, besonders in jüngeren Untersuchungen, eine zunehmende Bedeutung als Maßstab für Mechanisierung mit Computern. Mit der vorliegenden Doktorarbeit wird diese Tradition fortgeführt, indem die zuvorgenannten Grundpfeiler der Methamatematik uniform im formalen Rahmen der konstruktiven Typtheorie analysiert werden. Dieses Programm ermöglicht neue Einsichten in den konstruktiven Gehalt von Vollständigkeit, einen synthetischen Ansatz für Unentscheidbarkeit und Unvollständigkeit, der großteils den berüchtigten, die Essenz der Beweise verdeckenden, technischen Aufwand eliminiert, sowie natürliche Repräsentationen von Mengentheorie in Form einer Axiomatisierung zweiter Ordnung und einer vollkommen typtheoretischen Darstellung. Die Mechanisierung zur Logik erster Ordnung ist als eine umfassende Coq-Bibliothek organisiert, die offen für Nutzung und Beiträge externer Anwender ist

Some results on embeddings of algebras, after de Bruijn and McKenzie

In 1957, N. G. de Bruijn showed that the symmetric group Sym(\Omega) on an infinite set \Omega contains a free subgroup on 2^{card(\Omega)} generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality \leq card(\Omega), Sym(\Omega) contains a coproduct of 2^{card(\Omega)} copies of A, not only in the variety of all groups, but in any variety of groups to which A belongs. His key lemma is here generalized to an arbitrary variety of algebras \bf{V}, and formulated as a statement about functors Set --> \bf{V}. From this one easily obtains analogs of the results stated above with "group" and Sym(\Omega) replaced by "monoid" and the monoid Self(\Omega) of endomaps of \Omega, by "associative K-algebra" and the K-algebra End_K(V) of endomorphisms of a K-vector-space V with basis \Omega, and by "lattice" and the lattice Equiv(\Omega) of equivalence relations on \Omega. It is also shown, extending another result from de Bruijn's 1957 paper, that each of Sym(\Omega), Self(\Omega) and End_K (V) contains a coproduct of 2^{card(\Omega)} copies of itself. That paper also gave an example of a group of cardinality 2^{card(\Omega)} that was {\em not} embeddable in Sym(\Omega), and R. McKenzie subsequently established a large class of such examples. Those results are shown to be instances of a general property of the lattice of solution sets in Sym(\Omega) of sets of equations with constants in Sym(\Omega). Again, similar results -- this time of varying strengths -- are obtained for Self(\Omega), End_K (V), and Equiv(\Omega), and also for the monoid \Rel of binary relations on \Omega. Many open questions and areas for further investigation are noted.Comment: 37 pages. Copy at http://math.berkeley.edu/~gbergman/papers is likely to be updated more often than arXiv copy Revised version includes answers to some questions left open in first version, references to results of Wehrung answering some other questions, and some additional new result

Relaxation and its Role in Vision

It is argued that a visual system, especially one which handles imperfect data, needs a way of selecting the best consistent combination from among the many interrelated, locally plausible hypotheses about how parts or aspects of the visual input may be interpreted. A method is presented in which each hypothesis is given a supposition value between 0 and 1. A parallel relaxation I operator, based on the plausibilities of hypotheses and the logical relations between them, is then used to modify the supposition values, and the process is repeated until the best consistent set of hypotheses have supposition values of approximately 1, and the rest have values of approximately 0. The method is incorporated in a program which can interpret configurations of overlapping rectangles as puppets. For this task it is possible to formulate all the potentially relevant hypotheses before using relaxation to select the best consistent set. For more complex tasks, it is necessary to use relaxation on the locally plausible interpretations to guide the search for locally less obvious ones. Ways of doing this are discussed. Finally, an implemented system is presented which allows the user to specify schemas and inference rules, and uses relaxation to control the building of a network of instances of the schemas, when presented with data about some instances and relations between the

Volume I: Acquisition Research: The Foundation for Innovation

Proceedings Paper (for Acquisition Research Program)Accordingly, the year 2006 was especially significant for the NPS Acquisition Research Program in taking major strides toward expanding the program''s reach in important ways to other institutions.'' The number of research institutions participating as collaborators grew to 35 with the formation of a Virtual University Consortium.'' Most noteworthy was, as mentioned above, our securing sponsorship from USD(AT&L) to fund research proposals selected from a nationwide call, or Broad Agency Announcement (BAA) (copy available at www.acquisitionresearch.org).'' We''re truly excited at the prospects of receiving innovative and cutting edge proposals from the top minds around the country.'' We trust that this new sponsorship will act like good seeds sown in fertile soil, yielding rich fruits of profitable acquisition research for many years to come.Naval Postgraduate School Acquisition Research ProgramApproved for public release; distribution is unlimited

Environmental Impact of Aviation and Sustainable Solutions

Environmental Impact of Aviation and Sustainable Solutions is a compilation of review and research articles in the broad field of aviation and the environment. Over three sections and thirteen chapters, this book covers topics such as aircraft design and materials, combustor modeling, atomization, airport pollution, sonic boom and street noise pollution, emission mitigation strategies, and environmentally friendly contributions from a Russian aviation pioneer. This volume is a useful reference for both researchers and students interested in learning about various aspects of aviation and the environmen

Assessment of thermal and visual micro-climate of a traditional commercial street in a hot arid climate

PhD ThesisIn the hot arid contexts, the impact of urban climate is often associated with negative effects on outdoor thermal comfort and an increase in the urban heat island (UHI) effect. The primary aim of this research is to investigate the outdoor thermal performance of traditional commercial urban streets located in the hot arid context of Cairo in Egypt. A number of methods were used including field measurements and social surveys. Consequently, urban air flows, temperature and daylight simulations to assess existing and possible improvement scenarios to extend pedestrian thermal and visual comfort were tested. The field measurements were conducted in order to first assess the UHI intensity in the urban street, and to investigate the effectiveness of the traditional design solutions in ensuring comfortable outdoor conditions based on human-biometeorological assessment methods. Validation of results was carried out by comparing measured and simulated results of thermal conditions in the commercial spine ENVI-met is a three dimensional microclimatic model based on computational fluid dynamics (CFD) models and is designed to simulate surface-air interactions in urban environments. It was used to calculate the mean radiant temperature and obtaining the microclimatic maps with problematic areas concerning the pedestrian's thermal comfort for the existing urban configurations. Outdoor thermal comfort was assessed based on a thermal sensation survey and the physiological equivalent temperature (PET), with a comfort range of (24oC - 32oC). To improve outdoor thermal conditions at pedestrian level seven different shading scenarios addressing the form and the opening of shading devices were simulated using CFD Fluent, based on two dependant variables including air temperature distribution and wind velocity. The daylight analysis software (DIVA) was used to evaluate the solar access for the tested cases. The findings show that typology and the opening locations are one of the paramount factors in providing a temperature reduction in the urban scale. As the air temperature was reduced by (2.3oC) for the best case compared to the base leading to a lower PET for the best case recording 32.9oC against 35oC for the base case

Recent Advances in Small-Angle Neutron Scattering

Over the decades, small-angle neutron scattering has became a definitive method for structural investigation on the mesoscale between a few Angstrom up to a few 100 nm. This makes it an indispensable tool for non-destructive material investigations in fields ranging from chemistry and biology, over material sciences to solid state physics, especially taking into account the fundamental nature of neutrons, which makes it possible to probe different isotopes and, therefore, enhance contrast by choosing an appropriate isotope distribution or to probe the spin state of the investigated materials. This Special Issue is dedicated to elucidate the advances made with SANS over the last few years, which includes new instrumentation, sample environment and experimental control, as well as novel approaches and experimental techniques. The ideas and approaches collected here will serve both the experienced experimenter as well as the novice to appraise whether their specific experimental setup is feasible with new ideas