Automated Theorem Proving in GeoGebra: Current Achievements

GeoGebra is an open-source educational mathematics software tool, with millions of users worldwide. It has a number of features (integration of computer algebra, dynamic geometry, spreadsheet, etc.), primarily focused on facilitating student experiments, and not on formal reasoning. Since including automated deduction tools in GeoGebra could bring a whole new range of teaching and learning scenarios, and since automated theorem proving and discovery in geometry has reached a rather mature stage, we embarked on a project of incorporating and testing a number of different automated provers for geometry in GeoGebra. In this paper, we present the current achievements and status of this project, and discuss various relevant challenges that this project raises in the educational, mathematical and software contexts. We will describe, first, the recent and forthcoming changes demanded by our project, regarding the implementation and the user interface of GeoGebra. Then we present our vision of the educational scenarios that could be supported by automated reasoning features, and how teachers and students could benefit from the present work. In fact, current performance of GeoGebra, extended with automated deduction tools, is already very promising—many complex theorems can be proved in less than 1 second. Thus, we believe that many new and exciting ways of using GeoGebra in the classroom are on their way

An axiomatic approach for solving geometric problems symbolically

technical reportThis paper describes a new approach for solving geometric constraint problems and problems in geometry theorem proving. We developed a rewrite-rule mechanism operating on geometric predicates. Termination and completeness of the problem solving algorithm can be obtained through well foundedness and confluence of the set of rewrite rules. To guarantee these properties we adapted the Knuth-Bendix completion algorithm to the specific requirements of the geometric problem. A symbolic, geometric solution has the advantage over the usual algebraic approach that it speaks the language of geometry. Therefore, it has the potential to be used in many practical applications in interactive Computer Aided Design

Some new results on majority-logic codes for correction of random errors

The main advantages of random error-correcting majority-logic codes and majority-logic decoding in general are well known and two-fold. Firstly, they offer a partial solution to a classical coding theory problem, that of decoder complexity. Secondly, a majority-logic decoder inherently corrects many more random error patterns than the minimum distance of the code implies is possible. The solution to the decoder complexity is only a partial one because there are circumstances under which a majority-logic decoder is too complex and expensive to implement. [Continues.

Mathematical warrants, objects and actions in higher school mathematics

'Higher school mathematics' connotes typical upper secondary school and early college mathematics. The mathematics at this level is characterised by moves to (1) rigour in justification,(2) abstraction in content and (3) fluency in symbolic manipulation. This thesis investigates these three transitions - towards rigour, abstraction, and tluencyusing philosophical method: for each of the three transitions a proposition is presented and arguments are given in favour of that proposition. These arguments employ concepts and results from contemporary English language-medium philosophy and also rely crucially on classroom issues or accounts of mathematical experience both to elucidate meaning and for the domain of application. These three propositions, with their arguments, are the three sub-theses at the centre of the thesis as a whole. The first of these sub-theses (1) argues that logical deduction, quasi-empiricism and visualisation are mathematical warrants, while authoritatively based justification is essentially non-mathematical. The second sub-thesis (2) argues that the reality of mathematical entities of the sort encountered in the higher school mathematics curriculum is actual not metaphoric. The third sub-thesis (3) claims that certain 'mathematical action' can be construed as non-propositional mathematical knowledge. The application of these general propositions to mathematics in education yields the following: 'coming to know mathematics' involves:(1) using mathematical warrants for justification and self conviction; (2) ontological commitment to mathematical objects; and (3)developing a capability to execute some mathematical procedures automatically

Ultrasonic propagation in cancellous bone

The thesis covers two main areas of work. The first is detailed experimental work and the second is the evaluation of existing ultrasonic theories in attempt to apply them to the propagation in cancellous bone. The work is related to a new technique which uses ultrasonic attenuation to measure and predict osteoporosis, especially in the elderly population.The ultrasonic attenuation, the longitudinal ultrasonic velocity, the scattering effect and the attenuation as a function of frequency were measured on a range of cancellous bone samples, from healthy to severely osteoporotic, and also a few cortical samples. The cancellous bone was human os calces and vertebrae. The relationships between the ultrasonic propagation and the structural parameters and density of the bone were investigated, and were considered both for whole bones and separate purely cancellous samples. Image analysis of photomicrographs taken under low magnification was carried out to find the architectural parameters of the bone structure. The ultrasonic measurements were also compared with quantitative computed tomography assessment and compressive strength testing.Many theories which are currently used to evaluate ultrasonic propagation in a porous material are reviewed, and three particular ones are developed in detail and applied to models of cancellous bone. The self consistent theory (SCT), Biot's theory and the multiple scattering theory based on the work of Waterman and Truell were all assessed for their limits with repect to this particular application, and each had its own deficiencies. The Blot theory, however, proved the most successful at predicting the experimental attenuation results observed, but still only in a limited way

Reports to the President

A compilation of annual reports for the 1985-1986 academic year, including a report from the President of the Massachusetts Institute of Technology, as well as reports from the academic and administrative units of the Institute. The reports outline the year's goals, accomplishments, honors and awards, and future plans