Exploiting the Hierarchical Structure of Rule-Based Specifications for Decision Planning

Rule-based specifications have been very successful as a declarative approach in many domains, due to the handy yet solid foundations offered by rule-based machineries like term and graph rewriting. Realistic problems, however, call for suitable techniques to guarantee scalability. For instance, many domains exhibit a hierarchical structure that can be exploited conveniently. This is particularly evident for composition associations of models. We propose an explicit representation of such structured models and a methodology that exploits it for the description and analysis of model- and rule-based systems. The approach is presented in the framework of rewriting logic and its efficient implementation in the rewrite engine Maude and is illustrated with a case study.

Implementation of standard testbeds for numerical relativity

We discuss results that have been obtained from the implementation of the initial round of testbeds for numerical relativity which was proposed in the first paper of the Apples with Apples Alliance. We present benchmark results for various codes which provide templates for analyzing the testbeds and to draw conclusions about various features of the codes. This allows us to sharpen the initial test specifications, design a new test and add theoretical insight.Comment: Corrected versio

A multi-level preconditioned Krylov method for the efficient solution of algebraic tomographic reconstruction problems

Classical iterative methods for tomographic reconstruction include the class of Algebraic Reconstruction Techniques (ART). Convergence of these stationary linear iterative methods is however notably slow. In this paper we propose the use of Krylov solvers for tomographic linear inversion problems. These advanced iterative methods feature fast convergence at the expense of a higher computational cost per iteration, causing them to be generally uncompetitive without the inclusion of a suitable preconditioner. Combining elements from standard multigrid (MG) solvers and the theory of wavelets, a novel wavelet-based multi-level (WMG) preconditioner is introduced, which is shown to significantly speed-up Krylov convergence. The performance of the WMG-preconditioned Krylov method is analyzed through a spectral analysis, and the approach is compared to existing methods like the classical Simultaneous Iterative Reconstruction Technique (SIRT) and unpreconditioned Krylov methods on a 2D tomographic benchmark problem. Numerical experiments are promising, showing the method to be competitive with the classical Algebraic Reconstruction Techniques in terms of convergence speed and overall performance (CPU time) as well as precision of the reconstruction.Comment: Journal of Computational and Applied Mathematics (2014), 26 pages, 13 figures, 3 table

Thomas Decomposition of Algebraic and Differential Systems

In this paper we consider disjoint decomposition of algebraic and non-linear partial differential systems of equations and inequations into so-called simple subsystems. We exploit Thomas decomposition ideas and develop them into a new algorithm. For algebraic systems simplicity means triangularity, squarefreeness and non-vanishing initials. For differential systems the algorithm provides not only algebraic simplicity but also involutivity. The algorithm has been implemented in Maple

Teaching and learning algebra word problems : a thesis presented in partial fulfilment of the requirements for the degree of Master of Educational Studies in Mathematics, Massey University, Palmerston North, New Zealand

This study reports on a classroom design experiment into the teaching and learning of algebra word problems. The study was set in the mathematics department of a coeducational secondary school, and involved two teachers and 30 Year 12 students. The teachers and the researcher worked collaboratively to design and implement an intervention that focused explicitly on translation between word problems and algebra. Two issues were considered: the impact of the intervention on students, and the impact of the study on teachers. Students' responses to classroom activities, supported by individual student interviews, were used to examine their approaches to solving algebra word problems. Video-stimulated focus group interviews explored students' responses to classroom activities, and informed the ongoing planning and implementation of classroom activities. Data about the impact on teachers' understandings, beliefs and practices was gathered through individual interviews and classroom observations as well as the ongoing dialogue of the research team. The most significant impact on students related to their understandings of algebra as a tool. Some students were able to combine their new-found translation skills with algebraic manipulation skills to solve word problems algebraically. However, other students had difficulties at various stages of the translation process. Factors identified as supporting student learning included explicit objectives and clarity around what was to be learnt, the opportunity for students to engage in conversations about their thinking and to practise translating between verbal and symbolic forms, structured progression of learning tasks, time to consolidate understandings, and, a heuristic for problem solving. Participation in the project impacted on teachers in two ways: firstly, with regards to the immediate intervention of teaching algebra; and secondly, with regards to teaching strategies for mathematics in general. Translation activities provided a tool for teachers to engage students in mathematical discussion, enabling them to elicit and build on student thinking. As teachers developed new understandings about how their students approached word problems they gained insight into the importance of selecting problems for which students needed to use algebra. However, teachers experienced difficulty designing quality instructional activities, including algebra word problems, that pressed for algebraic thinking. The focus on translation within the study encouraged a shift in teacher practice away from a skills-focus toward a problem-focus. Whilst it was apparent that instructional focus on translation shifted teachers and students away from an emphasis on procedure, it was equally clear that translation alone is insufficient as an intervention. Students need both procedural and relational understandings to develop an understanding of the use of algebra as a tool to solve word problems. Students also need to develop fluency with a range of strategies, including algebra, in order to be able to select appropriate strategies to solve particular problems. This study affirmed for teachers that teaching with a focus on understanding can provide an effective and efficient method for increasing students' motivation, interest and success

Automated Generation of User Guidance by Combining Computation and Deduction

Herewith, a fairly old concept is published for the first time and named "Lucas Interpretation". This has been implemented in a prototype, which has been proved useful in educational practice and has gained academic relevance with an emerging generation of educational mathematics assistants (EMA) based on Computer Theorem Proving (CTP). Automated Theorem Proving (ATP), i.e. deduction, is the most reliable technology used to check user input. However ATP is inherently weak in automatically generating solutions for arbitrary problems in applied mathematics. This weakness is crucial for EMAs: when ATP checks user input as incorrect and the learner gets stuck then the system should be able to suggest possible next steps. The key idea of Lucas Interpretation is to compute the steps of a calculation following a program written in a novel CTP-based programming language, i.e. computation provides the next steps. User guidance is generated by combining deduction and computation: the latter is performed by a specific language interpreter, which works like a debugger and hands over control to the learner at breakpoints, i.e. tactics generating the steps of calculation. The interpreter also builds up logical contexts providing ATP with the data required for checking user input, thus combining computation and deduction. The paper describes the concepts underlying Lucas Interpretation so that open questions can adequately be addressed, and prerequisites for further work are provided.Comment: In Proceedings THedu'11, arXiv:1202.453

On the determination of cusp points of 3-R\underline{P}R parallel manipulators

This paper investigates the cuspidal configurations of 3-RPR parallel manipulators that may appear on their singular surfaces in the joint space. Cusp points play an important role in the kinematic behavior of parallel manipulators since they make possible a non-singular change of assembly mode. In previous works, the cusp points were calculated in sections of the joint space by solving a 24th-degree polynomial without any proof that this polynomial was the only one that gives all solutions. The purpose of this study is to propose a rigorous methodology to determine the cusp points of 3-R\underline{P}R manipulators and to certify that all cusp points are found. This methodology uses the notion of discriminant varieties and resorts to Gr\"obner bases for the solutions of systems of equations