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Lazard's CAD exploiting equality constraints

Published: 17 December 2019 Publication History

Abstract

McCallum improved the original Collins CAD projection operator (assuming well-orientation) and reduced the projection set even further for quantifier elimination problems which have equality constraints [6, 7]. Lazard provided a projection operator (and corresponding lifting process) that reduces the projection set as compared to McCallum's and is unconditional like Collins' original algorithm [2]. Our research extends Lazard's work by providing a modification that reduces the projection set even further when there is a single equality constraint in the quantifier elimination problem (as in [6]). We also report a slight error in [7].

References

[1]
G.E. Collins. Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition. In Proceedings 2nd. GI Conference Automata Theory & Formal Languages, pages 134--183, 1975.
[2]
G.E. Collins. Quantifier elimination by cylindrical algebraic decomposition --- twenty years of progess. In B.F. Caviness and J.R. Johnson, editors, Quantifier Elimination and Cylindrical Algebraic Decomposition, pages 8--23. Springer Verlag, Wien, 1998.
[3]
M. England, R.J. Bradford, and J.H. Davenport. Cylindrical Algebraic Decomposition with Equational Constraints. In J.H. Davenport, M. England, A. Griggio, T. Sturm, and C. Tinelli, editors, Symbolic Computation and Satisfiability Checking. Journal of Symbolic Computation (to appear), 2019.
[4]
D. Lazard. An Improved Projection Operator for Cylindrical Algebraic Decomposition. In C.L. Bajaj, editor, Proceedings Algebraic Geometry and its Applications: Collections of Papers from Shreeram S. Abhyankar's 60th Birthday Conference, pages 467--476, 1994.
[5]
S. McCallum. An Improved Projection Operation for Cylindrical Algebraic Decomposition. PhD thesis, University of Wisconsin-Madison Computer Science, 1984.
[6]
S. McCallum. On Projection in CAD-Based Quantifier Elimination with Equational Constraints. In S. Dooley, editor, Proceedings ISSAC '99, pages 145--149, 1999.
[7]
S. McCallum. On Propagation of Equational Constraints in CAD-Based Quantifier Elimination. In B. Mourrain, editor, Proceedings ISSAC 2001, pages 223--230, 2001.
[8]
S. McCallum. Error in [7]. E-mail 2019 January 5th, 2019.
[9]
S. McCallum and H. Hong. On Lazard's Valuation and CAD Construction. http://www.arxiv.org/abs/1501.06563, 2015.
[10]
S. McCallum, A. Parusiński, and L. Paunescu. Validity proof of Lazard's method for CAD construction. J. Symbolic Comp., 92:52--69, 2019.
[11]
A.S. Nair, J.H. Davenport, and G.K. Sankaran. On Benefits of Equality Constraints in Lex-Least Invariant CAD (Extended Abstract). In SC-Square 2019: Satisfiability Checking and Symbolic Computation, 2019. URL: https://researchportal.bath.ac.uk/en/publica/on-benefits-of-equality-constraints-in-lex-least-invariant-cad-ex.

Cited By

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  • (2023)Lazard-style CAD and Equational ConstraintsProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597090(218-226)Online publication date: 24-Jul-2023

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Published In

cover image ACM Communications in Computer Algebra
ACM Communications in Computer Algebra  Volume 53, Issue 3
September 2019
72 pages
ISSN:1932-2232
EISSN:1932-2240
DOI:10.1145/3377006
Issue’s Table of Contents
Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the Owner/Author.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 17 December 2019
Published in SIGSAM-CCA Volume 53, Issue 3

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Cited By

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  • (2023)Lazard-style CAD and Equational ConstraintsProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597090(218-226)Online publication date: 24-Jul-2023

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