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Lazard-style CAD and Equational Constraints

Authors:

James Harold Davenport,

Akshar Sajive Nair,

Gregory Kumar Sankaran,

Ali Kemal UncuAuthors Info & Claims

ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation

Pages 218 - 226

https://doi.org/10.1145/3597066.3597090

Published: 24 July 2023 Publication History

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Abstract

McCallum-style Cylindrical Algebra Decomposition (CAD) is a major improvement on the original Collins version, and has had many subsequent advances, notably for total or partial equational constraints. But it suffers from a problem with nullification. The recently-justified Lazard-style CAD does not have this problem. However, transporting the equational constraints work to Lazard-style does reintroduce nullification issues. This paper explains the problem, and the solutions to it, based on the second author’s Ph.D. thesis and the Brown–McCallum improvement to Lazard.

With a single equational constraint, we can gain the same improvements in Lazard-style as in McCallum-style CAD. Moreover, our approach does not fail where McCallum would due to nullification. Unsurprisingly, it does not achieve the same level of improvement as it does in the non-nullified cases. We also consider the case of multiple equational constraints.

References

[1]

R.J. Bradford, J.H. Davenport, M. England, S. McCallum, and D.J. Wilson. 2013. Cylindrical Algebraic Decompositions for Boolean Combinations. In Proceedings ISSAC 2013. 125–132. https://doi.org/10.1145/2465506.2465516

Digital Library

[2]

R.J. Bradford, J.H. Davenport, M. England, S. McCallum, and D.J. Wilson. 2016. Truth table invariant cylindrical algebraic decomposition. J. Symbolic Comp. 76 (2016), 1–35. https://doi.org/10.1016/j.jsc.2015.11.002

Digital Library

[3]

C.W. Brown. 2001. Improved Projection for Cylindrical Algebraic Decomposition. J. Symbolic Comp. 32 (2001), 447–465. https://doi.org/10.1006/jsco.2001.0463

Digital Library

[4]

C.W. Brown and S. McCallum. 2020. Enhancements to Lazard’s Method for Cylindrical Algebraic Decomposition. In Computer Algebra in Scientific Computing CASC 2020(Springer Lecture Notes in Computer Science, Vol. 12291), F. Boulier, M. England, T.M. Sadykov, and E.V. Vorozhtsov (Eds.). 129–149. https://doi.org/10.1007/978-3-030-60026-6_8

Digital Library

[5]

C. Chen, M. Moreno Maza, B. Xia, and L. Yang. 2009. Computing Cylindrical Algebraic Decomposition via Triangular Decomposition. In Proceedings ISSAC 2009, J. May (Ed.). 95–102. https://doi.org/10.1145/1576702.1576718

Digital Library

[6]

G.E. Collins. 1975. Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition. In Proceedings 2nd. GI Conference Automata Theory & Formal Languages(Springer Lecture Notes in Computer Science, Vol. 33). 134–183. https://doi.org/10.1007/3-540-07407-4_17

[7]

G.E. Collins. 1998. Quantifier elimination by cylindrical algebraic decomposition — twenty years of progress. In Quantifier Elimination and Cylindrical Algebraic Decomposition, B.F. Caviness and J.R. Johnson (Eds.). Springer Verlag, Wien, 8–23. https://doi.org/10.1007/978-3-7091-9459-1_2

[8]

J.H. Davenport and M. England. 2016. Need Polynomial Systems be Doubly-exponential?. In International Congress on Mathematical Software ICMS 2016(Springer Lecture Notes in Computer Science, Vol. 9725), G-M. Greuel, T. Koch, P. Paule, and A. Sommese (Eds.). 157–164. https://doi.org/10.1007/978-3-319-42432-3_20

[9]

J.H. Davenport and M. England. 2017. The Potential and Challenges of CAD with Equational Constraints for SC-Square. In Proceedings MACIS 2017: Mathematical Aspects of Computer and Information Sciences. 280–285. https://doi.org/10.1007/978-3-319-72453-9_22

[10]

J.H. Davenport, Z.P. Tonks, and A.K. Uncu. 2023. A Poly-algorithmic Approach to Quantifier Elimination. https://arxiv.org/abs/2302.06814.

[11]

A. Dolzmann, A. Seidl, and Th. Sturm. 2004. Efficient Projection Orders for CAD. In Proceedings ISSAC 2004, J. Gutierrez (Ed.). 111–118. https://doi.org/10.1145/1005285.1005303

Digital Library

[12]

M. England, R.J. Bradford, and J.H. Davenport. 2020. Cylindrical Algebraic Decomposition with Equational Constraints. In Symbolic Computation and Satisfiability Checking: special issue of Journal of Symbolic Computation, J.H. Davenport, M. England, A. Griggio, T. Sturm, and C. Tinelli (Eds.). Vol. 100. 38–71. https://doi.org/10.1016/j.jsc.2019.07.019

Digital Library

[13]

D. Lazard. 1994. An Improved Projection Operator for Cylindrical Algebraic Decomposition. In Proceedings Algebraic Geometry and its Applications: Collections of Papers from Shreeram S. Abhyankar’s 60th Birthday Conference, C.L. Bajaj (Ed.). 467–476. https://doi.org/10.1007/978-1-4612-2628-4_29

[14]

S. McCallum. 1985. An Improved Projection Operation for Cylindrical Algebraic Decomposition. Technical Report 578. Computer Science University Wisconsin at Madison. https://minds.wisconsin.edu/bitstream/handle/1793/58594/TR578.pdf?sequence=1

[15]

S. McCallum. 1998. An Improved Projection Operation for Cylindrical Algebraic Decomposition. In Quantifier Elimination and Cylindrical Algebraic Decomposition, B.F. Caviness and J.R. Johnson (Eds.). Springer-Verlag, Wien, 242–268. https://doi.org/10.1007/978-3-7091-9459-1_12

[16]

S. McCallum. 1999. On Projection in CAD-Based Quantifier Elimination with Equational Constraints. In Proceedings ISSAC ’99, S. Dooley (Ed.). 145–149. https://doi.org/10.1145/309831.309892

Digital Library

[17]

S. McCallum. 2001. On Propagation of Equational Constraints in CAD-Based Quantifier Elimination. In Proceedings ISSAC 2001, B. Mourrain (Ed.). 223–230. https://doi.org/10.1145/384101.384132

Digital Library

[18]

S. McCallum, A. Parusiński, and L. Paunescu. 2019. Validity proof of Lazard’s method for CAD construction. J. Symbolic Comp. 92 (2019), 52–69. https://doi.org/10.1016/j.jsc.2017.12.002

[19]

A.S. Nair. 2021. Curtains in Cylindrical Algebraic Decomposition. Ph. D. Dissertation. University of Bath. https://researchportal.bath.ac.uk/en/studentTheses/curtains-in-cylindrical-algebraic-decomposition

[20]

A.S. Nair, J.H. Davenport, and G.K. Sankaran. 2020. Curtains in CAD: Why Are They a Problem and How Do We Fix Them? In Mathematical Software — ICMS 2020, (Springer Lecture Notes in Computer Science, Vol. 12097)A.M. Bigatti, J. Carette, J.H. Davenport, M. Joswig, and T. de Wolff (Eds.). 17–26. https://doi.org/10.1007/978-3-030-52200-1_2

Digital Library

[21]

A. Nair, J. Davenport, G. Sankaran, and S. McCallum. 2019. Lazard’s CAD exploiting equality constraints. ACM Comm. Computer Algebra 53 (2019), 138–141. https://doi.org/10.1145/3377006.3377020

Digital Library

[22]

Z. Tonks. 2021. Poly-algorithmic Techniques in Real Quantifier Elimination. Ph. D. Dissertation. University of Bath. https://researchportal.bath.ac.uk/en/studentTheses/poly-algorithmic-techniques-in-real-quantifier-elimination

[23]

D.J. Wilson, R.J. Bradford, and J.H. Davenport. 2012. Speeding up Cylindrical Algebraic Decomposition by Gröbner Bases. In Proceedings CICM 2012, J. Jeuring et al. (Ed.). 280–294. https://doi.org/10.1007/978-3-642-31374-5_19

Digital Library

Cited By

Michel LMathonet PZenaïdi N(2024)On Minimal and Minimum Cylindrical Algebraic DecompositionsProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3669704(316-323)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1145/3666000.3669704
Davenport JTonks ZUncu A(2023)A Poly-algorithmic Approach to Quantifier Elimination2023 25th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC)10.1109/SYNASC61333.2023.00013(44-51)Online publication date: 11-Sep-2023
https://doi.org/10.1109/SYNASC61333.2023.00013

Index Terms

Lazard-style CAD and Equational Constraints
1. Computing methodologies
  1. Symbolic and algebraic manipulation
2. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics

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cover image ACM Other conferences

ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation

July 2023

567 pages

ISBN:9798400700392

DOI:10.1145/3597066

Editors:
Alicia Dickenstein
University of Buenos Aires, Argentina
,
Elias Tsigaridas
INRIA Paris and Sorbonne University, France
,
Gabriela Jeronimo
University of Buenos Aires, Argentina

Copyright © 2023 Owner/Author.

This work is licensed under a Creative Commons Attribution-NoDerivatives International 4.0 License.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 24 July 2023

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ISSAC 2023

ISSAC 2023: International Symposium on Symbolic and Algebraic Computation 2023

July 24 - 27, 2023

Tromsø, Norway

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Cited By

Michel LMathonet PZenaïdi N(2024)On Minimal and Minimum Cylindrical Algebraic DecompositionsProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3669704(316-323)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1145/3666000.3669704
Davenport JTonks ZUncu A(2023)A Poly-algorithmic Approach to Quantifier Elimination2023 25th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC)10.1109/SYNASC61333.2023.00013(44-51)Online publication date: 11-Sep-2023
https://doi.org/10.1109/SYNASC61333.2023.00013

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