The ability to reason spatially is an important skill required for engineers, particularly in engineering design and construction. One aspect of spatial reasoning is visualizing and constructing three-dimensional (3D) solid objects from two-dimensional (2D) projections. To assist in teaching this to engineering students, an instructional software system is being developed at the University of Illinois. This instructional software system is comprised of the Visual Sweeper and the Visual Teacher. The Visual Sweeper is a geometric framework for solving missing view problems. In missing view problems, students create 3D solid objects from two 2D projections by applying operations inverse to orthographic projection. The Visual Teacher, which is the focus of this article, is an intelligent critiquing and tutoring module that gives feedback to the student regarding partial solutions to missing view problems. The Visual Teacher is comprised of a Recognizer and a Critiquer. The Recognizer identifies which solution solid the student's partial solution is closest to. Based on the solution solid and a student's partial solution, the Criti-quer gives critique and advice to the student. The Recognizer is based on an algorithm for bipartite graph matching, while the Critiquer uses a rule-based approach. This paper describes the Visual Teacher, gives examples of how it can be used, presents preliminary evaluation results, and discusses the system's assumptions and limitations.

We study here schedulers for a class of rules that naturally arise in the context of rule-based constraint programming. We systematically derive a scheduler for them from a generic iteration algorithm of Apt (2000). We apply this study to so-called membership rules of Apt and Monfroy (2001). This leads to an implementation that yields a considerably better performance for these rules than their execution as standard CHR rules. Finally, we show how redundant rules can be identified and how appropriately reduced sets of rules can be computed.

We study here a natural situation when constraint programming can be entirely reduced to rule-based programming. To this end we explain first how one can compute on constraint satisfaction problems using rules represented by simple first-order formulas. Then we consider constraint satisfaction problems that are based on predefined, explicitly given constraints. To solve them we first derive rules from these explicitly given constraints and limit the computation process to a repeated application of these rules, combined with labeling. We consider two types of rule here. The first type, that we call equality rules, leads to a new notion of local consistency, called rule consistency that turns out to be weaker than arc consistency for constraints of arbitrary arity (called hyper-arc consistency in Marriott & Stuckey (1998)). For Boolean constraints rule consistency coincides with the closure under the well-known propagation rules for Boolean constraints. The second type of rules, that we call membership rules, yields a rule-based characterization of arc consistency. To show feasibility of this rule-based approach to constraint programming, we show how both types of rules can be automatically generated, as CHR rules of Frühwirth (1995). This yields an implementation of this approach to programming by means of constraint logic programming. We illustrate the usefulness of this approach to constraint programming by discussing various examples, including Boolean constraints, two typical examples of many valued logics, constraints dealing with Waltz's language for describing polyhedral scenes, and Allen's qualitative approach to temporal logic.