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Iterated Resultants and Rational Functions in Real Quantifier Elimination (2312.16210v2)

Published 23 Dec 2023 in cs.SC and math.AG

Abstract: This paper builds and extends on the authors' previous work related to the algorithmic tool, Cylindrical Algebraic Decomposition (CAD), and one of its core applications, Real Quantifier Elimination (QE). These topics are at the heart of symbolic computation and were first implemented in computer algebra systems decades ago, but have recently received renewed interest as part of the ongoing development of SMT solvers for non-linear real arithmetic. First, we consider the use of iterated univariate resultants in traditional CAD, and how this leads to inefficiencies, especially in the case of an input with multiple equational constraints. We reproduce the workshop paper [Davenport and England, 2023], adding important clarifications to our suggestions first made there to make use of multivariate resultants in the projection phase of CAD. We then consider an alternative approach to this problem first documented in [McCallum and Brown, 2009] which redefines the actual object under construction, albeit only in the case of two equational constraints. We correct an unhelpful typo and provide a proof missing from that paper. We finish by revising the topic of how to deal with SMT or Real QE problems expressed using rational functions (as opposed to the usual polynomial ones) noting that these are often found in industrial applications. We revisit a proposal made in [Uncu, Davenport and England, 2023] for doing this in the case of satisfiability, explaining why such an approach does not trivially extend to more complicated quantification structure and giving a suitable alternative.

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References (44)
  1. S.S. Abhyankar. Algebraic Geometry for Scientists and Engineers, volume 35 of Mathematical Surveys and Monographs. American Mathematical Society, 1990.
  2. SC22{}^{2}start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT: Satisfiability Checking meets Symbolic Computation (Project Paper). In Proceedings CICM 2016, pages 28–43, 2016. doi:10.1007/978-3-319-42547-4_3.
  3. Deciding the Consistency of Non-Linear Real Arithmetic Constraints with a Conflict Driven Search Using Cylindrical Algebraic Coverings. Journal of Logical and Algebraic Methods in Programming, 119, 2021. doi:10.1016/j.jlamp.2020.100633.
  4. New Perspectives in Symbolic Computation and Satisfiability Checking (Dagstuhl Seminar 22072). Dagstuhl Reports, 12(2):67–86, 2022. URL: https://drops.dagstuhl.de/opus/volltexte/2022/16931, doi:10.4230/DagRep.12.2.67.
  5. The SMT-LIB Standard: Version 2.6. http://smtlib.cs.uiowa.edu/papers/smt-lib-reference-v2.6-r2021-05-12.pdf, 2021.
  6. C.W. Brown. Open Non-uniform Cylindrical Algebraic Decompositions. In Proceedings ISSAC 2015, pages 85–92, 2015. doi:10.1145/2755996.2756654.
  7. C.W. Brown. Why adding non-vanishing conditions on all denominators is problematic. Communication to Davenport 13 October, 2023.
  8. The Complexity of Quantifier Elimination and Cylindrical Algebraic Decomposition. In C.W. Brown, editor, Proceedings ISSAC 2007, pages 54–60, 2007. doi:10.1145/1277548.1277557.
  9. Supporting Proving and Discovering Geometric Inequalities in GeoGebra by using Tarski. Automated Deduction in Geometry ADG 2021 EPTCS, 352:156–166, 2021. URL: https://arxiv.org/abs/2201.00544v1.
  10. C.W. Brown and S. McCallum. On using bi-equational constraints in CAD construction. In Proceedings ISSAC 2005, pages 76–83, 2005. doi:10.1145/1073884.1073897.
  11. L. Busé and B. Mourrain. Explicit Factors of some Iterated Resultants and Discriminants. Math. Comp., 78:345–386, 2009. doi:10.1090/S0025-5718-08-02111-X.
  12. B. Caviness and J. Johnson. Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts & Monographs in Symbolic Computation. Springer-Verlag, 1998. doi:10.1007/978-3-7091-9459-1.
  13. G.E. Collins. Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition. In H. Brakhage, editor, Proceedings 2nd. GI Conference Automata Theory & Formal Languages, volume 33 of Springer Lecture Notes in Computer Science, pages 134–183, 1975. doi:10.1007/3-540-07407-4_17.
  14. Iterated resultants in CAD. In Erika Ábrahám and Thomas Sturm, editors, Proceedings of the 8th SC-Square Workshop, CEUR-WS Proceedings, volume 3455, pages 54–60, 2023. URL: http://ceur-ws.org/Vol-3455/.
  15. Symbolic Computation and Satisfiability Checking: special issue of Journal of Symbolic Computation, volume 100. Elsevier, 2020.
  16. Lazard-style CAD and Equational Constraints. In G. Jeronimo, editor, Proceedings ISSAC 2023, pages 218–226, 2023. doi:10.1145/3597066.3597090.
  17. Tereso del Río and Matthew England. New Heuristic to Choose a Cylindrical Algebraic Decomposition Variable Ordering Motivated by Complexity Analysis. In François Boulier, Matthew England, Timur M. Sadykov, and Evgenii V. Vorozhtsov, editors, Computer Algebra in Scientific Computing CASC 2022, volume 13366 of Lecture Notes in Computer Science, pages 300–317, 2022. doi:10.1007/978-3-031-14788-3_17.
  18. Improving the Use of Equational Constraints in Cylindrical Algebraic Decomposition. In D. Robertz, editor, Proceedings ISSAC 2015, pages 165–172, 2015. doi:10.1145/2755996.2756678.
  19. Cylindrical Algebraic Decomposition with Equational Constraints. In Davenport et al. [15], pages 38–71. doi:10.1016/j.jsc.2019.07.019.
  20. J.P. Jouanolou. Le Formalisme du Résultant. Advances in Mathematics, 90:117–263, 1991. doi:10.1016/0001-8708(91)90031-2.
  21. D. Jovanović and L. de Moura. Solving Non-Linear Arithmetic. In Proceedings IJCAR 2012, pages 339–354, 2012. doi:10.1007/978-3-642-31365-3_27.
  22. D. Kapur and T. Saxena. Comparison of various multivariate resultant formulations. In Proceedings ISSAC 1915, pages 187–194, 1995. doi:https://doi.org/10.1145/220346.220370.
  23. G. Kremer and E. Ábrahám. Fully Incremental CAD. In Davenport et al. [15], pages 11–37. doi:https://doi.org/10.1016/j.jsc.2019.07.018.
  24. Cylindrical algebraic coverings for quantifiers. In Uncu and Barbosa [38], pages 1–9. URL: https://ceur-ws.org/Vol-3458/.
  25. D. Lazard and S. McCallum. Iterated Discriminants. J. Symbolic Comp., 44:1176–1193, 2009. doi:10.1016/j.jsc.2008.05.006.
  26. M. Marden. Geometry of polynomials, 2nd edition. American Mathematical Soc., 1966.
  27. S. McCallum. An Improved Projection Operation for Cylindrical Algebraic Decomposition. PhD thesis, University of Wisconsin-Madison Computer Science, 1984.
  28. S. McCallum. An Improved Projection Operation for Cylindrical Algebraic Decomposition of Three-dimensional Space. J. Symbolic Comp., 5:141–161, 1988. doi:10.1016/S0747-7171(88)80013-0.
  29. S. McCallum. An Improved Projection Operation for Cylindrical Algebraic Decomposition. [12], pages 242–268. doi:10.1016/S0747-7171(88)80010-5.
  30. S. McCallum. Factors of Iterated Resultants and Discriminants. J. Symbolic Comp., 27:367–385, 1999. doi:doi:10.1006/jsco.1998.0257.
  31. S. McCallum. On Projection in CAD-Based Quantifier Elimination with Equational Constraints. In S. Dooley, editor, Proceedings ISSAC ’99, pages 145–149, 1999. doi:10.1145/309831.309892.
  32. S. McCallum. On Propagation of Equational Constraints in CAD-Based Quantifier Elimination. In B. Mourrain, editor, Proceedings ISSAC 2001, pages 223–230, 2001. doi:10.1145/384101.384132.
  33. S. McCallum and C.W. Brown. On delineability of varieties in CAD-based quantifier elimination with two equational constraints. In Proceedings ISSAC 2009, pages 71–78, 2009. doi:10.1145/1576702.1576715.
  34. Validity proof of Lazard’s method for CAD construction. J. Symbolic Comp., 92:52–69, 2019. doi:10.1016/j.jsc.2017.12.002.
  35. S. McCallum and F. Winkler. Differential resultants. ITM Web of Conferences Article 01005, 20, 2018. doi:10.1051/itmconf/20182001005.
  36. S. McCallum and F. Winkler. Resultants: Algebraic and Differential. Technical Report RISC18-08, Johannes Kepler University, 2018.
  37. A.S. Nair. Curtains in Cylindrical Algebraic Decomposition. PhD thesis, University of Bath, 2021. URL: https://researchportal.bath.ac.uk/en/studentTheses/curtains-in-cylindrical-algebraic-decomposition.
  38. Ali Kemal Uncu and Haniel Barbosa, editors. Proceedings of the 7th SC-Square Workshop, volume 3458, 2023. URL: https://ceur-ws.org/Vol-3458/.
  39. Smt-solving induction proofs of inequalities. In Uncu and Barbosa [38], pages 10–24. URL: https://ceur-ws.org/Vol-3458/paper2.pdf.
  40. F. Vale-Enriquez and C.W. Brown. Polynomial Constraints and Unsat Cores in TARSKI. In J.H. Davenport, M. Kauers, G. Labahn, and J. Urban, editors, Proceedings Mathematical Software — ICMS 2018, pages 466–474, 2018. doi:10.1007/978-3-319-96418-8_55.
  41. B.L. van der Waerden. Modern Algebra Vol. II (trans. Theodore J. Benac). Frederick Ungar, 1950.
  42. B.L. van der Waerden. Modern Algebra Vol. II (trans. F. Blum and J.R. Schulenberger). Frederick Ungar, 1970.
  43. B.L. van der Waerden. About [13]. Private communication to G.E. Collins, 1975.
  44. R.J. Walker. Algebraic Curves. Dover, 1962.

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