Programming language Formian.

Formex algebra is a powerful tool for the generation of data used in the design and analysis of space structures. However, for the algebra to be of practical use, it is necessary to have a means of employing the concepts on a computer. This is the particular problem which this thesis addresses. The solution proposed here is Formian, an interactive programming language, which being modelled on formex algebra allows complex configurations to be generated from a few concise and yet readily understood statements. Formian is designed to allow problems of data generation to be tackled in a single programming environment. The thesis describes the raison d'etre for the Formian programming language and the steps taken to create the language and to provide a practical and reliable implementation in the form of a computer program. A complete description of the language structure is given. This includes an overview of formex algebra. The use of Formian from a designer's viewpoint is provided by interspersing the description with practical examples

Behavioural hybrid process calculus

Process algebra is a theoretical framework for the modelling and analysis of the behaviour of concurrent discrete event systems that has been developed within computer science in past quarter century. It has generated a deeper nderstanding of the nature of concepts such as observable behaviour in the presence of nondeterminism, system composition by interconnection of concurrent component systems, and notions of behavioural equivalence of such systems. It has contributed fundamental concepts such as bisimulation, and has been successfully used in a wide range of problems and practical applications in concurrent systems. We believe that the basic tenets of process algebra are highly compatible with the behavioural approach to dynamical systems. In our contribution we present an extension of classical process algebra that is suitable for the modelling and analysis of continuous and hybrid dynamical systems. It provides a natural framework for the concurrent composition of such systems, and can deal with nondeterministic behaviour that may arise from the occurrence of internal switching events. Standard process algebraic techniques lead to the characterisation of the observable behaviour of such systems as equivalence classes under some suitably adapted notion of bisimulation

Robust Computer Algebra, Theorem Proving, and Oracle AI

In the context of superintelligent AI systems, the term "oracle" has two meanings. One refers to modular systems queried for domain-specific tasks. Another usage, referring to a class of systems which may be useful for addressing the value alignment and AI control problems, is a superintelligent AI system that only answers questions. The aim of this manuscript is to survey contemporary research problems related to oracles which align with long-term research goals of AI safety. We examine existing question answering systems and argue that their high degree of architectural heterogeneity makes them poor candidates for rigorous analysis as oracles. On the other hand, we identify computer algebra systems (CASs) as being primitive examples of domain-specific oracles for mathematics and argue that efforts to integrate computer algebra systems with theorem provers, systems which have largely been developed independent of one another, provide a concrete set of problems related to the notion of provable safety that has emerged in the AI safety community. We review approaches to interfacing CASs with theorem provers, describe well-defined architectural deficiencies that have been identified with CASs, and suggest possible lines of research and practical software projects for scientists interested in AI safety.Comment: 15 pages, 3 figure

Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gr{\"o}bner bases

This paper is a sequel to "Computing diagonal form and Jacobson normal form of a matrix using Groebner bases", J. of Symb. Computation, 46 (5), 2011. We present a new fraction-free algorithm for the computation of a diagonal form of a matrix over a certain non-commutative Euclidean domain over a computable field with the help of Gr\"obner bases. This algorithm is formulated in a general constructive framework of non-commutative Ore localizations of $G$-algebras (OLGAs). We split the computation of a normal form of a matrix into the diagonalization and the normalization processes. Both of them can be made fraction-free. For a matrix $M$ over an OLGA we provide a diagonalization algorithm to compute $U,V$ and $D$ with fraction-free entries such that $UMV=D$ holds and $D$ is diagonal. The fraction-free approach gives us more information on the system of linear functional equations and its solutions, than the classical setup of an operator algebra with rational functions coefficients. In particular, one can handle distributional solutions together with, say, meromorphic ones. We investigate Ore localizations of common operator algebras over $K[x]$ and use them in the unimodularity analysis of transformation matrices $U,V$. In turn, this allows to lift the isomorphism of modules over an OLGA Euclidean domain to a polynomial subring of it. We discuss the relation of this lifting with the solutions of the original system of equations. Moreover, we prove some new results concerning normal forms of matrices over non-simple domains. Our implementation in the computer algebra system {\sc Singular:Plural} follows the fraction-free strategy and shows impressive performance, compared with methods which directly use fractions. Since we experience moderate swell of coefficients and obtain simple transformation matrices, the method we propose is well suited for solving nontrivial practical problems.Comment: 25 pages, to appear in Journal of Symbolic Computatio

Social Mathworking: The Effects of Online Reflection on Algebra I Students\u27 Sense of Community and Perceived Learning

The purpose of this study was to determine if online reflections through social networking affect students\u27 sense of community and levels of perceived conceptual learning in Algebra I courses. Social constructivism, connectivism, and computer-mediated communication in relation to reflective practices form the theoretical and practical framework for the use of Web 2.0 technologies in this investigation. A quasi-experimental nonequivalent control group design was used to examine Algebra I students\u27 sense of community as measured by the Sense of Classroom Community Index, and perceived learning as measured by Perceived Learning Instrument. The sample consisted of 27 Algebra I students at a Central Florida middle school. There were 14 participants in the experimental group and 13 students in the control group. Both groups completed pre and posttest survey instruments for the independent variables of sense of community and perceived learning. The tests were separated by four weeks of instruction on Algebra 1 course content and participation in discourse through face-to-face and discussion board formats. Independent t-tests were employed in data analysis. The results of the study revealed no significant differences between experimental and control groups in relation to students\u27 sense of community and perceived learning. However, the findings support curriculum design targeted to those concepts Algebra I students have the most difficulty with, and advance the understanding of students\u27 cognitive development and feelings regarding comfort when communicating their mathematical thinking through Web 2.0 technologies