Christopher W. Brown

QEPCAD B: a program for computing with semi-algebraic sets using CADs

This report introduces QEPCAD, B, a program for computing with real algebraic sets using cylindrical algebraic decomposition (CAD). QEPCAD B both extends and improves upon the QEPCAD system for quantifier elimination by partial cylindrical algebraic decomposition written by Hoon Hong in the early 1990s. This paper briefly discusses some of the improvements in the implementation of CAD and quantifier elimination via CAD, and provides somewhat more detail on extensions to the system that go beyond quantifier elimination. The author is responsible for most of the extended features of QEPCAD B, but improvements to the basic CAD implementation and to the SACLIB library on which QEPCAD is based are the results of many peoples work, including: George E. Collins, Mark J. Encarnación, Hoon Hong, Jeremy Johnson, Werner Krandick, Richard Liska, Scott McCallum, Nicolas Robidoux, and Stanly Steinberg. Source code, documentation and installation instructions for QEPCAD B are all available at www.cs.usna.edu/~qepcad.

Improved Projection for Cylindrical Algebraic Decomposition

McCallum?s projection operator for cylindrical algebraic decomposition (CAD) represented a huge step forward for the practical utility of the CAD algorithm. This paper presents a simple theorem showing that the mathematics in McCallum?s paper actually point to a better projection operator than he proposes?a reduced McCallum projection. The reduced projection has the potential to not simply speed up CAD computation for problems that are currently solvable in practice, but actually increase the scope of problems that can realistically be attacked via CADs. Additionally, the same methods are used to show that McCallum?s projection can be reduced still further when CAD is applied to certain types of commonly occurring quantifier elimination problems.

The complexity of quantifier elimination and cylindrical algebraic decomposition

This paper has two parts. In the first part we give a simple and constructive proof that quantifier elimination in real algebra is doubly exponential, even when there is only one free variable and all polynomials in the quantified input are linear. The general result is not new, but we hope the simple and explicit nature of the proof makes it interesting. The second part of the paper uses the construction of the first part to prove some results on the effects of projection order on CAD construction -- roughly that there are CAD construction problems for which one order produces a constant number of cells and another produces a doubly exponential number of cells, and that there are problems for which all orders produce a doubly exponential number of cells. The second of these results implies that there is a true singly vs. doubly exponential gap between the worst-case running times of several modern quantifier elimination algorithms and CAD-based quantifier elimination when the number of quantifier alternations is constant.

Simple CAD Construction and its Applications

This paper presents a method for the simplification of truth-invariant cylindrical algebraic decompositions (CADs). Examples are given that demonstrate the usefulness of the method in speeding up the solution formula construction phase of the CAD-based quantifier elimination algorithm. Applications of the method to the construction of truth-invariant CADs for very large quantifier-free formulas and quantifier elimination of non-prenex formulas are also discussed.

Supporting Computer Science Curriculum: Exploring and Learning Linked Lists with iList

We developed two versions of a system, called iList, that helps students learn linked lists, an important topic in computer science curricula. The two versions of iList differ on the level of feedback they can provide to the students, specifically in the explanation of syntax and execution errors. The system has been fielded in multiple classrooms in two institutions. Our results indicate that iList is effective, is considered interesting and useful by the students, and its performance is getting closer to the performance of human tutors. Moreover, the system is being developed in the context of a study of human tutoring, which is guiding the evolution of iList with empirical evidence of effective tutoring.

Algorithmic methods for investigating equilibria in epidemic modeling

The calculation of threshold conditions for models of infectious diseases is of central importance for developing vaccination policies. These models are often coupled systems of ordinary dierential equations, in which case the computation of threshold conditions can be reduced to the question of stability of the disease-free equilibrium. This paper shows how computing threshold conditions for such models can be done fully algorithmically using quantifier elimination for real closed fields and related simplification methods for quantifier-free formulas. Using ecient quantifier

On using bi-equational constraints in CAD construction

This paper introduces an improved method for constructing cylindrical algebraic decompositions (CADs) for formulas with two polynomial equations as implied constraints. The fundamental idea is that neither of the varieties of the two polynomials is actually represented by the CAD the method produces, only the variety defined by their common zeros is represented. This allows for a substantially smaller projection factor set, and for a CAD with many fewer cells.In the current theory of CADs, the fundamental object is to decompose n-space into regions in which a polynomial equation is either identically true or identically false. With many polynomials, one seeks a decomposition into regions in which each polynomial equation is identically true or false independently. The results presented here are intended to be the first step in establishing a theory of CADs in which systems of equations are fundamental objects, so that given a system we seek a decomposition into regions in which the system is identically true or false --- which means each equation is no longer considered independently. Quantifier elimination problems of this form (systems of equations with side conditions) are quite common, and this approach has the potential to bring large problems of this type into the scope of what can be solved in practice. The special case of formulas containing two polynomial equations as constraints is an important one, but this work is also intended to be extended in the future to the more general case.

Learning Linked Lists: Experiments with the iList System

This paper presents the first experiments with an Intelligent Tutoring System in the domain of linked lists, a fundamental topic in Computer Science. The system has been deployed in an introductory college-level Computer Science class, and engendered significant learning gains. A constraint-based approach has been adopted in the design and implementation of the system. We describe the system architecture, its current functionalities, and the future directions of its development.

Guaranteed solution formula construction

Quantifier elimination transfornls i1 description of it semialgebraic set as a formula with quantified variables into a dcscrilAion of t,llc set as a qant,ificr-free forniula. Quant,ifier clirnination by cylindrical algebraic di:c.omI)ositioIl (C-AD) does this via an intermediate representation of the seniialgebraic set as a CAD. A quantifier-free formula representation of t.lic set is then construct,4 frown t.liis CAD. One dcsirablc property of the CAD-l-xwtl quant,ificr clirnination algorithm is its abilit,y to prodwr .simplc solution fornnll;Ls via methods such >Ls that. of [7]. These nlet.hods require that the CAD produced as an int,ernlediate representation bc yi,ojcction-flefinclble, which is not. the cast for all cluant,ificr-elimina.t,i~n problems. This paper presents an efficient method for transforrning an xrbitrar? C!-%D int.o a l)rojection-defirlable CAD: thereb?; rendering smple solution fornlula construction nwthods applicable lo all quantifier cliniination prol)lcIns.

Generating proactive feedback to help students stay on track

In a tutoring system based on an exploratory environment, it is also important to provide direct guidance to students We endowed iList, our linked list tutor, with the ability to generate proactive feedback using a procedural knowledge model automatically constructed from the interaction of previous students with the system We compared the new version of iList with its predecessors and human tutors Our evaluation shows that iList is effective in helping students learn.