The Second NASA Formal Methods Workshop 1992

The primary goal of the workshop was to bring together formal methods researchers and aerospace industry engineers to investigate new opportunities for applying formal methods to aerospace problems. The first part of the workshop was tutorial in nature. The second part of the workshop explored the potential of formal methods to address current aerospace design and verification problems. The third part of the workshop involved on-line demonstrations of state-of-the-art formal verification tools. Also, a detailed survey was filled in by the attendees; the results of the survey are compiled

A brief overview of NASA Langley's research program in formal methods

An overview of NASA Langley's research program in formal methods is presented. The major goal of this work is to bring formal methods technology to a sufficiently mature level for use by the United States aerospace industry. Towards this goal, work is underway to design and formally verify a fault-tolerant computing platform suitable for advanced flight control applications. Also, several direct technology transfer efforts have been initiated that apply formal methods to critical subsystems of real aerospace computer systems. The research team consists of six NASA civil servants and contractors from Boeing Military Aircraft Company, Computational Logic Inc., Odyssey Research Associates, SRI International, University of California at Davis, and Vigyan Inc

Unprovability and phase transitions in Ramsey theory

The first mathematically interesting, first-order arithmetical example of incompleteness was given in the late seventies and is know as the Paris-Harrington principle. It is a strengthened form of the finite Ramsey theorem which can not be proved, nor refuted in Peano Arithmetic. In this dissertation we investigate several other unprovable statements of Ramseyan nature and determine the threshold functions for the related phase transitions. Chapter 1 sketches out the historical development of unprovability and phase transitions, and offers a little information on Ramsey theory. In addition, it introduces the necessary mathematical background by giving definitions and some useful lemmas. Chapter 2 deals with the pigeonhole principle, presumably the most well-known, finite instance of the Ramsey theorem. Although straightforward in itself, the principle gives rise to unprovable statements. We investigate the related phase transitions and determine the threshold functions. Chapter 3 explores a phase transition related to the so-called infinite subsequence principle, which is another instance of Ramsey’s theorem. Chapter 4 considers the Ramsey theorem without restrictions on the dimensions and colours. First, generalisations of results on partitioning α-large sets are proved, as they are needed later. Second, we show that an iteration of a finite version of the Ramsey theorem leads to unprovability. Chapter 5 investigates the template “thin implies Ramsey”, of which one of the theorems of Nash-Williams is an example. After proving a more universal instance, we study the strength of the original Nash-Williams theorem. We conclude this chapter by presenting an unprovable statement related to Schreier families. Chapter 6 is intended as a vast introduction to the Atlas of prefixed polynomial equations. We begin with the necessary definitions, present some specific members of the Atlas, discuss several issues and give technical details

Formal methods and digital systems validation for airborne systems

This report has been prepared to supplement a forthcoming chapter on formal methods in the FAA Digital Systems Validation Handbook. Its purpose is as follows: to outline the technical basis for formal methods in computer science; to explain the use of formal methods in the specification and verification of software and hardware requirements, designs, and implementations; to identify the benefits, weaknesses, and difficulties in applying these methods to digital systems used on board aircraft; and to suggest factors for consideration when formal methods are offered in support of certification. These latter factors assume the context for software development and assurance described in RTCA document DO-178B, 'Software Considerations in Airborne Systems and Equipment Certification,' Dec. 1992

Discounting Behavior and Environmental Decisions

Discounting plays a major role in the life cycle of environmental and natural resource policies. Evaluating centuries-scale problems like climate change with standard discount rates yields results that many find ethically unacceptable. Paradoxes abound. Low discount rates are urged for determining the net benefits of climate change, while households fail to undertake energy conservation actions that have payback periods of only a few years. Efforts to uncover discount rates from revealed and stated preferences suggest that a variety of confounding factors may be simultaneously in play. Common property resources provide an example of how market failures can lead to behavior consistent with extreme discounting that can be addressed through effective policy. Finally, politicians who make ultimate policy decisions may have incentives to act in accordance with discount rates not socially optimal

The Germanic Development of the Pre-Modern Notion of Number From c. 1750 to Bolzano’s Rein analytischer Beweis

[ES]La historiografía y la filosofía de la matemática usualmente describen las reformas educativas llevadas a cabo en Francia y los territorios germánicos durante la última década del siglo XVIII y la primera del siglo XIX de la siguiente manera: mientras que en Francia tales reformas se realizaron tras la Revolución Francesa, particularmente con la creación de la École polytechnique y la École normale en París en 1794 y 1795, respectivamente, las reformas germánicas comenzaron alrededor de 1810 con la fundación de la Universität zu Berlin.1 Esta descripción, si bien acaso sea útil para resumir el periodo antes mencionado cuando el interés radica en lo que ocurrió antes –por ejemplo, el desarrollo del análisis matemático en el siglo XVIII– o después – por ejemplo, el desarrollo del análisis real y la teoría de conjuntos en el siglo XIX–, ha contribuido a una falta de comprensión, cuando no a una mala compresión, de lo que ocurrió en la matemática no sólo entre tales desarrollos sino además durante los propios siglos XVIII y XIX

Formalising Mathematics in Simple Type Theory

Despite the considerable interest in new dependent type theories, simple type theory (which dates from 1940) is sufficient to formalise serious topics in mathematics. This point is seen by examining formal proofs of a theorem about stereographic projections. A formalisation using the HOL Light proof assistant is contrasted with one using Isabelle/HOL. Harrison's technique for formalising Euclidean spaces is contrasted with an approach using Isabelle/HOL's axiomatic type classes. However, every formal system can be outgrown, and mathematics should be formalised with a view that it will eventually migrate to a new formalism

Reconceptualizing Mathematics Education

This dissertation is to explore theoretically mathematics education in the United States and the need for reconcepualizing mathematics education. Mathematics education needs reconceptualizing because students know very little mathematics by the time they graduate from high school. Mathematics has become a subject to be feared and dreaded for centuries. High school teachers blame middle school teachers, middle school teachers blame elementary teachers, and elementary teachers blame parents for their students\u27 lack of preparedness in mathematics. Elementary teachers express frustration in teaching mathematics because of their own lack of content knowledge and lack of preparation for the mathematics component of their profession. Regardless of who is to blame, most students entering high school are not prepared to problem solve nor are they interested in mathematics except as the dreaded requirement needed to graduate. Because I have been involved in mathematics education for more than three decades, I have seen many programs come and go. I have seen different types of pedagogy be the in way to teach mathematics. Naturally, technology has influenced mathematics education tremendously in the last decade. Unfortunately, many mathematics educators use technology as a crutch instead of using it to enhance mathematics education. Mathematics education in the United States has been debated for over three centuries. The debate is ongoing. Standardized testing has become a way of life in schools today. Teachers are expected to tell the students exactly what they are supposed to know in mathematics. Standardized tests do not allow students to be creative or struggle in their quest for knowledge because teachers must make sure they have covered the material for the test. The No Child Left Behind Act of 2001 (NCLBA) adds to the problem of mathematics education. The shortage of mathematics teachers throughout the nation is acute. Compliance with the NCLBA requires more mathematics teachers than can possibly be found. My purpose in writing this dissertation is to convey my thoughts and ideas about how the study of mathematics developed, how mathematics education progressed throughout history how mathematics education is progressing today, and how mathematics education will progress in the future. In my opinion, teacher preparation of elementary and middle school teachers will be a very strong component in the reconceptualization of mathematics education. Mathematics teachers at all levels should be grounded in a history of mathematics and be cognizant of the development of mathematics education throughout the relatively short history of America. Furthermore, a dialogue must be implemented and maintained between mathematics educators at all levels. With the implementation of this dialogue, mathematics education will become a subject of intrigue and beauty and will no longer remain the subject to be feared and dreaded