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### Optimising Problem Formulation for Cylindrical Algebraic Decomposition [chapter]

Russell Bradford, James H. Davenport, Matthew England, David Wilson
2013 Lecture Notes in Computer Science
This leads to new heuristics for choosing: the variable ordering for a CAD problem, a designated equational constraint, and formulations for truth-table invariant CADs (TTICADs).  ...  Cylindrical algebraic decomposition (CAD) is an important tool for the study of real algebraic geometry with many applications both within mathematics and elsewhere.  ...  The authors would like to thank Scott McCallum for many useful conversations on TTICAD and Chris Brown for sharing his work on the Joukowsky transformation.  ...

### Real Quantifier Elimination in the RegularChains Library [chapter]

Changbo Chen, Marc Moreno Maza
2014 Lecture Notes in Computer Science
In this paper, we report on the implementation of a QE procedure, called QuantifierElimination, based on the CAD implementations in the Regu-larChains library.  ...  Cylindrical algebraic decomposition (CAD) is one of the main tools for handling quantifier elimination of nonlinear input formulas.  ...  Moreover, in the same paper, a systematic way for making use of equational constraints is presented. In [4] , an RC-CAD based quantifier elimination algorithm was proposed.  ...

### A Preprocessor Based on Clause Normal Forms and Virtual Substitutions to Parallelize Cylindrical Algebraic Decomposition [article]

2013 arXiv   pre-print
The Cylindrical Algebraic Decomposition (CAD) algorithm is a comprehensive tool to perform quantifier elimination over real closed fields.  ...  Since parallelizability of CAD depends on the structure of given prenex formula, we introduce some structural notions to study the performance of CAD with the proposed preprocessor.  ...  Equational constraints in a prenex formula eliminate the need for many constructions, thus speeding up CAD.  ...

### Formulating problems for real algebraic geometry [article]

Matthew England
2014 arXiv   pre-print
We discuss issues of problem formulation for algorithms in real algebraic geometry, focussing on quantifier elimination by cylindrical algebraic decomposition.  ...  and indeed choosing the correct algorithm variant for a problem, is key to improving the practical use of both quantifier elimination and symbolic real algebraic geometry in general.  ...  Variable ordering When using CAD for QE we must project quantified variables first, but we are free to project the other variables in any order (and to change the order within quantifier blocks).  ...

### Quantifier elimination by cylindrical algebraic decomposition based on regular chains

Changbo Chen, Marc Moreno Maza
2014 Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation - ISSAC '14
A quantifier elimination algorithm by cylindrical algebraic decomposition based on regular chains is presented.  ...  We report on an implementation of our algorithm in the RegularChains library in Maple and illustrate its effectiveness by examples.  ...  ALGORITHM In this section, we demonstrate how to do quantifier elimination via RC-CAD. Algorithm 1 presents the main steps of QE based on RC-CAD.  ...

### Quantifier elimination by cylindrical algebraic decomposition based on regular chains

Changbo Chen, Marc Moreno Maza
2016 Journal of symbolic computation
A quantifier elimination algorithm by cylindrical algebraic decomposition based on regular chains is presented.  ...  We report on an implementation of our algorithm in the RegularChains library in Maple and illustrate its effectiveness by examples.  ...  ALGORITHM In this section, we demonstrate how to do quantifier elimination via RC-CAD. Algorithm 1 presents the main steps of QE based on RC-CAD.  ...

### Recent Advances in Real Geometric Reasoning [chapter]

James H. Davenport, Matthew England
2015 Lecture Notes in Computer Science
In the 1930s Tarski showed that real quantifier elimination was possible, and in 1975 Collins gave a remotely practicable method, albeit with doubly-exponential complexity, which was later shown to be  ...  We discuss some of the recent major advances in Collins method: such as an alternative approach based on passing via the complexes, and advances which come closer to "solving the question asked" rather  ...  -Redlog [SS03] ; this Reduce package implements CAD along with other quantifier elimination methods such as virtual substitution.  ...

### A "Piano Movers" Problem Reformulated

David Wilson, James H. Davenport, Matthew England, Russell Bradford
2013 2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
We provide a new formulation for which both a CAD can be constructed and from which an actual path could be determined if one exists, and analyse the CADs produced using this approach for variations of  ...  Producing a CAD for the original formulation of this problem is still infeasible after 25 years of improvements in both CAD theory and computer hardware.  ...  We start by constructing the projection set with respect to the equational constraint, producing 11 polynomials in y, w, z.  ...

### Cylindrical algebraic decompositions for boolean combinations

Russell Bradford, James H. Davenport, Matthew England, Scott McCallum, David Wilson
2013 Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation - ISSAC '13
We generalise the theory of equational constraints to design an algorithm which will efficiently construct a TTICAD for a wide class of problems, producing stronger results than when using equational constraints  ...  This article makes the key observation that when using cylindrical algebraic decomposition (CAD) to solve a problem with respect to a set of polynomials, it is not always the signs of those polynomials  ...  Strzeboński for assistance in perform ing the Mathematica tests and to the anonymous referees for useful comments. We also thank the rest of the Trian gular Sets seminar at Bath (A. Locatelli, G.  ...

### An implementation of CAD in Maple utilising problem formulation, equational constraints and truth-table invariance [article]

Matthew England
2013 arXiv   pre-print
Our implementation was in contrast to Maple's in-built CAD command, based on a quite separate theory.  ...  We describe how the CADs produced using equational constraints are able to take advantage of not just improved projection but also improvements in the lifting phase.  ...  respects these blocks (as required when using CAD for quantifier elimination).  ...

### Cylindrical Algebraic Sub-Decompositions

D. J. Wilson, R. J. Bradford, J. H. Davenport, M. England
2014 Mathematics in Computer Science
Cylindrical algebraic decompositions (CADs) are a key tool in real algebraic geometry, used primarily for eliminating quantifiers over the reals and studying semi-algebraic sets.  ...  We define two new types of sub-CAD: variety sub-CADs which are those cells in a CAD lying on a designated variety; and layered sub-CADs which have only those cells of dimension higher than a specified  ...  The authors would also like to thank Professor Gregory Sankaran for his thoughts and feedback on the topic, and Professor Scott McCallum for many stimulating conversations on TTICAD.  ...

### Applying Machine Learning to the Problem of Choosing a Heuristic to Select the Variable Ordering for Cylindrical Algebraic Decomposition [chapter]

Zongyan Huang, Matthew England, David Wilson, James H. Davenport, Lawrence C. Paulson, James Bridge
2014 Lecture Notes in Computer Science
Cylindrical algebraic decomposition(CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields.  ...  When using CAD, there is often a choice for the ordering placed on the variables. This can be important, with some problems infeasible with one variable ordering but easy with another.  ...  It can also makes use of an equational constraint automatically (via the projection operator [38] ) when one is explicit in the formula, (where explicit means the formula is a conjunction of the equational  ...

### Comparing machine learning models to choose the variable ordering for cylindrical algebraic decomposition [article]

Matthew England, Dorian Florescu
2019 arXiv   pre-print
We experimented with the NLSAT dataset and the Regular Chains Library CAD function for Maple 2018.  ...  We address the problem of selecting the variable ordering for cylindrical algebraic decomposition (CAD), an important algorithm in Symbolic Computation.  ...  Acknowledgements The authors are supported by EPSRC Project EP/R019622/1: Embedding Machine Learning within Quantifier Elimination Procedures.  ...

### Verification and synthesis using real quantifier elimination

Thomas Sturm, Ashish Tiwari
2011 Proceedings of the 36th international symposium on Symbolic and algebraic computation - ISSAC '11
We therefore automatically combine three established software components: virtual subtitution based quantifier elimination in Reduce/Redlog, cylindrical algebraic decomposition implemented in Qepcad, and  ...  Existing off-the-shelf quantifier elimination procedures are not successful in eliminating quantifiers from many of our benchmarks.  ...  Acknowledgments We would like to thank Andreas Weber for encouraging and supporting the first author in visiting SRI.  ...

### Need Polynomial Systems Be Doubly-Exponential? [chapter]

James H. Davenport, Matthew England
2016 Lecture Notes in Computer Science
of equational constraints.  ...  We first note that [Mayr and Ritscher, 2013] shows that the doubly exponential nature of Gröbner bases is with respect to the dimension of the ideal, not the number of variables.  ...  We are also grateful to Professor Buchberger for reminding JHD that Gröbner Bases were applicable to CAD complexity.  ...
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