Building and Refining Abstract Planning Cases by Change of Representation Language

ion is one of the most promising approaches to improve the performance of problem solvers. In several domains abstraction by dropping sentences of a domain description -- as used in most hierarchical planners -- has proven useful. In this paper we present examples which illustrate significant drawbacks of abstraction by dropping sentences. To overcome these drawbacks, we propose a more general view of abstraction involving the change of representation language. We have developed a new abstraction methodology and a related sound and complete learning algorithm that allows the complete change of representation language of planning cases from concrete to abstract. However, to achieve a powerful change of the representation language, the abstract language itself as well as rules which describe admissible ways of abstracting states must be provided in the domain model. This new abstraction approach is the core of Paris (Plan Abstraction and Refinement in an Integrated System), a system in which abstract planning cases are automatically learned from given concrete cases. An empirical study in the domain of process planning in mechanical engineering shows significant advantages of the proposed reasoning from abstract cases over classical hierarchical planning.Comment: See http://www.jair.org/ for an online appendix and other files accompanying this articl

Extensible Automated Constraint Modelling

In constraint solving, a critical bottleneck is the formulationof an effective constraint model of a given problem. The CONJURE system described in this paper, a substantial step forward over prototype versions of CONJURE previously reported, makes a valuable contribution to the automation of constraint modelling by automatically producing constraint models from their specifications in the abstract constraint specification language ESSENCE. A set of rules is used to refine an abstract specification into a concrete constraint model. We demonstrate that this set of rules is readily extensible to increase the space of possible constraint models CONJURE can produce. Our empirical results confirm that CONJURE can reproduce successfully the kernels of the constraint models of 32 benchmark problems found in the literature

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

Estimating Sparse Signals Using Integrated Wideband Dictionaries

In this paper, we introduce a wideband dictionary framework for estimating sparse signals. By formulating integrated dictionary elements spanning bands of the considered parameter space, one may efficiently find and discard large parts of the parameter space not active in the signal. After each iteration, the zero-valued parts of the dictionary may be discarded to allow a refined dictionary to be formed around the active elements, resulting in a zoomed dictionary to be used in the following iterations. Implementing this scheme allows for more accurate estimates, at a much lower computational cost, as compared to directly forming a larger dictionary spanning the whole parameter space or performing a zooming procedure using standard dictionary elements. Different from traditional dictionaries, the wideband dictionary allows for the use of dictionaries with fewer elements than the number of available samples without loss of resolution. The technique may be used on both one- and multi-dimensional signals, and may be exploited to refine several traditional sparse estimators, here illustrated with the LASSO and the SPICE estimators. Numerical examples illustrate the improved performance