Using Machine Learning to Decide When to Precondition Cylindrical Algebraic Decomposition with Groebner Bases

Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. However, it can be expensive, with worst case complexity doubly exponential in the size of the input. Hence it is important to formulate the problem in the best manner for the CAD algorithm. One possibility is to precondition the input polynomials using Groebner Basis (GB) theory. Previous experiments have shown that while this can often be very beneficial to the CAD algorithm, for some problems it can significantly worsen the CAD performance. In the present paper we investigate whether machine learning, specifically a support vector machine (SVM), may be used to identify those CAD problems which benefit from GB preconditioning. We run experiments with over 1000 problems (many times larger than previous studies) and find that the machine learned choice does better than the human-made heuristic.

[1]  Matthew England,et al.  Need Polynomial Systems Be Doubly-Exponential? , 2016, ICMS.

[2]  Matthew England,et al.  The Complexity of Cylindrical Algebraic Decomposition with Respect to Polynomial Degree , 2016, CASC.

[3]  D Aspinall,et al.  Optimising Problem Formulation for Cylindrical Algebraic Decomposition , 2013 .

[4]  Mark A. Hall,et al.  Correlation-based Feature Selection for Discrete and Numeric Class Machine Learning , 1999, ICML.

[5]  J. Ross Quinlan,et al.  Induction of Decision Trees , 1986, Machine Learning.

[6]  Thorsten Joachims,et al.  Making large-scale support vector machine learning practical , 1999 .

[7]  William H. Press,et al.  Numerical recipes in C (2nd ed.): the art of scientific computing , 1992 .

[8]  Christopher W. Brown.  Constructing a single open cell in a cylindrical algebraic decomposition , 2013, ISSAC '13.

[9]  Adam W. Strzebonski.  Cylindrical algebraic decomposition using local projections , 2016, J. Symb. Comput..

[10]  Matthew England,et al.  Cylindrical algebraic decompositions for boolean combinations , 2013, ISSAC '13.

[11]  Changbo Chen,et al.  Truth Table Invariant Cylindrical Algebraic Decomposition by Regular Chains , 2014, CASC.

[12]  Hoon Hong,et al.  Synthesis of optimal numerical algorithms using real quantifier elimination (case study: square root computation) , 2014, ISSAC.

[13]  Claude E. Shannon,et al.  The mathematical theory of communication , 1950 .

[14]  Hirokazu Anai,et al.  Efficient Subformula Orders for Real Quantifier Elimination of Non-prenex Formulas , 2015, MACIS.

[15]  Pierre Baldi,et al.  Assessing the accuracy of prediction algorithms for classification: an overview , 2000, Bioinform..

[16]  Matthew England,et al.  Applying machine learning to the problem of choosing a heuristic to select the variable ordering for cylindrical algebraic decomposition , 2014, CICM.

[17]  Bruno Buchberger,et al.  Speeding-up Quantifier Elimination by Gr?bner Bases , 1991 .

[18]  Usama M. Fayyad,et al.  Multi-Interval Discretization of Continuous-Valued Attributes for Classification Learning , 1993, IJCAI.

[19]  George E. Collins,et al.  The SAC-2 Computer Algebra System , 1985, European Conference on Computer Algebra.

[20]  Scott McCallum,et al.  An Improved Projection Operation for Cylindrical Algebraic Decomposition , 1985, European Conference on Computer Algebra.

[21]  Christopher W. Brown.  Improved Projection for Cylindrical Algebraic Decomposition , 2001, J. Symb. Comput..

[22]  Christopher W. Brown,et al.  Algorithmic methods for investigating equilibria in epidemic modeling , 2006, J. Symb. Comput..

[23]  Changbo Chen,et al.  Real Quantifier Elimination in the RegularChains Library , 2014, ICMS.

[24]  Bruno Buchberger,et al.  Bruno Buchberger's PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal , 2006, J. Symb. Comput..

[25]  George E. Collins,et al.  Partial Cylindrical Algebraic Decomposition for Quantifier Elimination , 1991, J. Symb. Comput..

[26]  Changbo Chen,et al.  Problem Formulation for Truth-Table Invariant Cylindrical Algebraic Decomposition by Incremental Triangular Decomposition , 2014, CICM.

[27]  M. Morari,et al.  Nonlinear parametric optimization using cylindrical algebraic decomposition , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[28]  Hirokazu Anai,et al.  An effective implementation of a symbolic-numeric cylindrical algebraic decomposition for quantifier elimination , 2009, SNC '09.

[29]  Changbo Chen,et al.  Quantifier elimination by cylindrical algebraic decomposition based on regular chains , 2014, J. Symb. Comput..

[30]  B. Matthews.  Comparison of the predicted and observed secondary structure of T4 phage lysozyme. , 1975, Biochimica et biophysica acta.

[31]  Jean Charles Faugère,et al.  A new efficient algorithm for computing Gröbner bases without reduction to zero (F5) , 2002, ISSAC '02.

[32]  Matthew England,et al.  Program Verification in the Presence of Complex Numbers, Functions with Branch Cuts etc , 2012, 2012 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing.

[33]  A. Meyer,et al.  The complexity of the word problems for commutative semigroups and polynomial ideals , 1982 .

[34]  Andreas Seidl,et al.  Efficient projection orders for CAD , 2004, ISSAC '04.

[35]  George E. Collins,et al.  Cylindrical Algebraic Decomposition I: The Basic Algorithm , 1984, SIAM J. Comput..

[36]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[37]  Matthew England,et al.  Improving the Use of Equational Constraints in Cylindrical Algebraic Decomposition , 2015, ISSAC.

[38]  William H. Press,et al.  Numerical recipes in C , 2002 .

[39]  Changbo Chen,et al.  Computing cylindrical algebraic decomposition via triangular decomposition , 2009, ISSAC '09.

[40]  James H. Davenport,et al.  Real Quantifier Elimination is Doubly Exponential , 1988, J. Symb. Comput..

[41]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[42]  Bernhard Schölkopf,et al.  Kernel Methods in Computational Biology , 2005 .

[43]  Matthew England,et al.  Choosing a Variable Ordering for Truth-Table Invariant Cylindrical Algebraic Decomposition by Incremental Triangular Decomposition , 2014, ICMS.

[44]  Heinz Kredel,et al.  Gröbner Bases Using SAC-2 , 1985, European Conference on Computer Algebra.

[45]  Chih-Jen Lin,et al.  A Practical Guide to Support Vector Classication , 2008 .

[46]  Ian H. Witten,et al.  The WEKA data mining software: an update , 2009, SKDD.

[47]  Geoff Holmes,et al.  Benchmarking Attribute Selection Techniques for Discrete Class Data Mining , 2003, IEEE Trans. Knowl. Data Eng..

[48]  James H. Davenport,et al.  A repository for CAD examples , 2013, ACCA.

[49]  H. Hong.  An improvement of the projection operator in cylindrical algebraic decomposition , 1990, ISSAC '90.

[50]  Liyun Dai,et al.  Constructing fewer open cells by GCD computation in CAD projection , 2014, ISSAC.

[51]  James H. Davenport,et al.  Speeding Up Cylindrical Algebraic Decomposition by Gröbner Bases , 2012, AISC/MKM/Calculemus.

[52]  Nello Cristianini,et al.  Kernel Methods for Pattern Analysis , 2003, ICTAI.

[53]  James P. Bridge,et al.  Machine Learning for First-Order Theorem Proving , 2014, J. Autom. Reason..

[54]  Isabelle Guyon,et al.  An Introduction to Variable and Feature Selection , 2003, J. Mach. Learn. Res..

[55]  Lawrence C. Paulson,et al.  MetiTarski: Past and Future , 2012, ITP.

[56]  Adam W. Strzebonski,et al.  Cylindrical Algebraic Decomposition using validated numerics , 2006, J. Symb. Comput..

[57]  Matthew England,et al.  Truth table invariant cylindrical algebraic decomposition , 2014, J. Symb. Comput..

[58]  James H. Davenport,et al.  The complexity of quantifier elimination and cylindrical algebraic decomposition , 2007, ISSAC '07.

[59]  D. S. Arnon,et al.  Algorithms in real algebraic geometry , 1988 .