Mathematics in Computer Science

, Volume 11, Issue 3–4, pp 483–502 | Cite as

A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

Open Access
Article

Abstract

Effective quantifier elimination procedures for first-order theories provide a powerful tool for generically solving a wide range of problems based on logical specifications. In contrast to general first-order provers, quantifier elimination procedures are based on a fixed set of admissible logical symbols with an implicitly fixed semantics. This admits the use of sub-algorithms from symbolic computation. We are going to focus on quantifier elimination for the reals and its applications giving examples from geometry, verification, and the life sciences. Beyond quantifier elimination we are going to discuss recent results with a subtropical procedure for an existential fragment of the reals. This incomplete decision procedure has been successfully applied to the analysis of reaction systems in chemistry and in the life sciences.

Keywords

Real quantifier elimination and decision Satisfiability Virtual substitution Subtropical methods Real geometry Verification Reaction systems Stability analysis 

Mathematics Subject Classification

68U99 

References

  1. 1.
    Arnon, D.S.: Algorithms for the geometry of semi-algebraic sets. Technical Report 436, Computer Science Department, University of Wisconsin-Madison, Ph.D. Thesis (1981)Google Scholar
  2. 2.
    Basu, S., Pollack, R., Roy, M.-F.: On the combinatorial and algebraic complexity of quantifier elimination. J. ACM 43(6), 1002–1045 (1996)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Boulier, F., Lefranc, M., Lemaire, F., Morant, P.-E., Ürgüplü, A.: On proving the absence of oscillations in models of genetic circuits. In: Proceedings of the AB 2007, volume 4545 of LNCS, pp. 66–80. Springer (2007)Google Scholar
  4. 4.
    Boulier, F., Lefranc, M., Lemaire, F., Morant, P.-E.: Applying a rigorous quasi-steady state approximation method for proving the absence of oscillations in models of genetic circuits. In: Proceedings of the AB 2008, volume 5147 of LNCS, pp. 56–64. Springer (2008)Google Scholar
  5. 5.
    Brown, C.W., Gross C.: Efficient preprocessing methods for quantifier elimination. In: Proceedings of the CASC 2006, volume 4194 of LNCS, pp. 89–100. Springer (2006)Google Scholar
  6. 6.
    Brown, C.W.: QEPCAD B: a program for computing with semi-algebraic sets using CADs. ACM SIGSAM Bull. 37(4), 97–108 (2003)CrossRefMATHGoogle Scholar
  7. 7.
    Brown, C.W., Košta, M.: Constructing a single cell in cylindrical algebraic decomposition. J. Symb. Comput. 70, 14–48 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Doctoral dissertation, Mathematical Institute, University of Innsbruck, Innsbruck, Austria (1965)Google Scholar
  9. 9.
    Chou, S.-C.: Mechanical Geometry Theorem Proving. Mathematics and Its Applications. D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, Tokyo (1988)Google Scholar
  10. 10.
    Clarke, B.L.: Stability of complex reaction networks. In: Prigogine, I., Rice, Stuart A. (eds.) Advances in Chemical Physics, vol. 43. Wiley, Hoboken (1980)CrossRefGoogle Scholar
  11. 11.
    Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition—preliminary report. ACM SIGSAM Bull. 8(3), 80–90 (1974). Proc. EUROSAM ’74Google Scholar
  12. 12.
    Collins, G.E.: Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. In: Automata Theory and Formal Languages. 2nd GI Conference, volume 33 of LNCS, pp. 134–183. Springer (1975)Google Scholar
  13. 13.
    Collins, G.E.: Quantifier elimination by cylindrical algebraic decomposition—twenty years of progress. In: Caviness, B.F., Johnson, J.R. (eds.) Quantifier Elimination and Cylindrical Algebraic Decomposition, pp. 8–23. Springer, Berlin (1998)CrossRefGoogle Scholar
  14. 14.
    Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comput. 12(3), 299–328 (1991)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Davenport, J.H., Heintz, J.: Real quantifier elimination is doubly exponential. J. Symb. Comput. 5(1–2), 29–35 (1988)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Davis, M.: Mathematical Procedures for Decision Problems. Final Report on Ordnance Research and Development Project No. TB2-0001 (1954)Google Scholar
  17. 17.
    Dolzmann A., Sturm T. Redlog User Manual, 2nd edn. Technical Report MIP-9905, FMI, Universität Passau, Germany (1999)Google Scholar
  18. 18.
    Dolzmann, A., Sturm, T., Weispfenning, V.: A new approach for automatic theorem proving in real geometry. J. Autom. Reason. 21(3), 357–380 (1998)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Dolzmann, A., Sturm, T.: Redlog: computer algebra meets computer logic. ACM SIGSAM Bull. 31(2), 2–9 (1997)CrossRefGoogle Scholar
  20. 20.
    Dolzmann, A., Sturm, T.: Simplification of quantifier-free formulae over ordered fields. J. Symb. Comput. 24(2), 209–231 (1997)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Errami, H., Eiswirth, M., Grigoriev, D., Seiler, W.M., Sturm, T., Weber, A.: Efficient methods to compute Hopf bifurcations in chemical reaction networks using reaction coordinates. In: Proceedings of the CASC 2013, volume 8136 of LNCS, pp. 88–99. Springer (2013)Google Scholar
  22. 22.
    Errami, H., Seiler, W.M., Eiswirth, M., Weber, A.: Computing Hopf bifurcations in chemical reaction networks using reaction coordinates. In: Proceedings of the CASC 2012, volume 7442 of LNCS. Springer (2012)Google Scholar
  23. 23.
    Errami, H., Eiswirth, M., Grigoriev, D., Seiler, W.M., Sturm, T., Weber, A.: Detection of Hopf bifurcations in chemical reaction networks using convex coordinates. J. Comput. Phys. 291, 279–302 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Fussmann, G.F., Ellner, S.P., Shertzer, K.W., Hairston Jr., N.G.: Crossing the Hopf bifurcation in a live predator–prey system. Science 290(5495), 1358–1360 (2000)CrossRefGoogle Scholar
  25. 25.
    Gatermann, K., Eiswirth, M., Sensse, A.: Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems. J. Symb. Comput. 40(6), 1361–1382 (2005)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Godbole, D.N., Lygeros, J.: Longitudinal control of the lead car of a platoon. IEEE Trans. Veh. Technol. 43(4), 1125–1135 (1994)CrossRefGoogle Scholar
  27. 27.
    Grigoriev, D.: Complexity of deciding Tarski algebra. J. Symb. Comput. 5(1–2), 65–108 (1988)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Gulwani, S., Tiwari, A.: Constraint-based approach for analysis of hybrid systems. In: Proceedings of the CAV 2008, volume 5123 of LNCS, pp. 190–203. Springer (2008)Google Scholar
  29. 29.
    Hilbert, D.: Grundlagen der Geometrie, 13th edn. Teubner Studienbücher Mathematik. Teubner, Stuttgart (1987)MATHGoogle Scholar
  30. 30.
    Hong, H.: Comparison of several decision algorithms for the existential theory of the reals. Technical Report 91-41.0, RISC, Johannes Kepler University, A-4040 Linz, Austria (1991)Google Scholar
  31. 31.
    Hong, H., Liska, R., Steinberg, S.: Testing stability by quantifier elimination. J. Symb. Comput. 24(2), 161–187 (1997)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Jirstrand, M.: Cylindrical algebraic decomposition—an introduction. Technical Report 1995-10-18, Department of Electrical Engineering, Linköping University, Linköping, Sweden (1995)Google Scholar
  33. 33.
    Kahoui, M.El, Weber, A.: Deciding Hopf bifurcations by quantifier elimination in a software-component architecture. J. Symb. Comput. 30(2), 161–179 (2000)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Kapur, D.: Using Gröbner bases to reason about geometry problems. J. Symb. Comput. 2(4), 399–408 (1986)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Košta, M.: New concepts for real quantifier elimination by virtual substitution. Doctoral dissertation, Saarland University, Germany (2016)Google Scholar
  36. 36.
    Košta, M., Sturm, T., Dolzmann, A.: Better answers to real questions. J. Symb. Comput. 74, 255–275 (2016)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Kutzler, B.A., Stifter, S.: On the application of Buchberger’s algorithm to automated geometry theorem proving. J. Symb. Comput. 2(4), 389–397 (1986)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Liu, W.-M.: Criterion of Hopf bifurcations without using eigenvalues. J. Math. Anal. Appl. 182(1), 250–256 (1994)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Loos, R., Weispfenning, V.: Applying linear quantifier elimination. Comput. J. 36(5), 450–462 (1993)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    McCallum, S.: An improved projection operation for cylindrical algebraic decomposition of three-dimensional space. J. Symb. Comput. 5(1–2), 141–161 (1988)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    McPhee, N.F., Chou, S.-C., Gao, X.-S.: Mechanically proving geometry theorems using a combination of Wu’s method and Collins’ method. In: Proceedings of CADE-12, volume 814 of LNAI, pp. 401–415. Springer (1994)Google Scholar
  42. 42.
    Mincheva, M., Roussel, M.R.: Graph-theoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models. J. Math. Biol. 55(1), 61–86 (2007)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Niu, W., Wang, D.: Algebraic approaches to stability analysis of biological systems. Math. Comput. Sci. 1(3), 507–539 (2008)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Novak, B., Pataki, Z., Ciliberto, A., Tyson, J.J.: Mathematical model of the cell division cycle of fission yeast. Chaos 11(1), 277–286 (2001)CrossRefMATHGoogle Scholar
  45. 45.
    Prajna, S., Jadbabaie, A., Pappas, G.J.: A framework for worst-case and stochastic safety verification using barrier certificates. IEEE Trans. Autom. Control 52(8), 1415–1428 (2007)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Prestel, A.: Lectures on formally real fields, volume 1093 of Lecture Notes in Mathematics. Springer (1984)Google Scholar
  47. 47.
    Puri, A., Varaiya, P.: Driving safely in smart cars. In: Proceedings of the 1995 American Control Conference. IEEE (1995)Google Scholar
  48. 48.
    Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals. Part II: the general decision problem. Preliminaries for quantifier elimination. J. Symb. Comput. 13(3), 301–328 (1992)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Ritt, J.F.: Differential Equations from the Algebraic Standpoint, volume 14 of Colloquium Publications. American Mathematical Society, New York (1932)Google Scholar
  50. 50.
    Ritt, J.F.: Differential Algebra, volume 33 of Colloquium Publications. American Mathematical Society, Providence (1950)Google Scholar
  51. 51.
    Seidenberg, A.: An elimination theory for differential algebra. Univ. Calif. Publ. Math. New Ser. 3(2), 31–66 (1956)MathSciNetGoogle Scholar
  52. 52.
    Seidenberg, A.: Some remarks on Hilbert’s Nullstellensatz. Arch. Math. 7(4), 235–240 (1956)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Seidenberg, A.: On \(k\)-constructable sets, \(k\)-elementary formulae, and elimination theory. J. für die reine und angewandte Math. 239–240, 256–267 (1969)MathSciNetMATHGoogle Scholar
  54. 54.
    Seidl, A., Sturm, T.: A generic projection operator for partial cylindrical algebraic decomposition. In: Proceedings of the ISSAC 2003, pp. 240–247. ACM (2003)Google Scholar
  55. 55.
    Sensse, A., Hauser, M.J.B., Eiswirth, M.: Feedback loops for Shilnikov chaos the peroxidase–oxidase reaction. J. Chem. Phys. 125(1), 014901-1–014901-12 (2006)CrossRefGoogle Scholar
  56. 56.
    Sturm, T., Tiwari, A.: Verification and synthesis using real quantifier elimination. In: Proceedings of the ISSAC 2011, pp. 329–336. ACM (2011)Google Scholar
  57. 57.
    Sturm, T., Weber, A.: Investigating generic methods to solve Hopf bifurcation problems in algebraic biology. In: Proceedings of the AB 2008, volume 5147 of LNCS, pp. 200–215. Springer (2008)Google Scholar
  58. 58.
    Sturm, T., Weispfenning, V.: Computational geometry problems in Redlog. In: Automated Deduction in Geometry, volume 1360 of LNAI, pp. 58–86. Springer (1998)Google Scholar
  59. 59.
    Sturm, T., Weispfenning, V.: Rounding and blending of solids by a real elimination method. In: Proceedings of the IMACS World Congress 1997, volume 2, pp. 727–732. Wissenschaft & Technik Verlag, Berlin (1997)Google Scholar
  60. 60.
    Sturm, T.: An algebraic approach to offsetting and blending of solids. In: Proceedings of the CASC 2000, pp. 367–382. Springer (2000)Google Scholar
  61. 61.
    Sturm, T.: New domains for applied quantifier elimination. In: Proceedings of the CASC 2006, volume 4194 of LNCS. Springer (2006)Google Scholar
  62. 62.
    Sturm, T.: Real Quantifier Elimination in Geometry. Doctoral dissertation, Universität Passau, Germany (1999)Google Scholar
  63. 63.
    Sturm, T.: Subtropical real root finding. In: Proceedings of the ISSAC 2015, pp. 347–354. ACM (2015)Google Scholar
  64. 64.
    Sturm, T., Weber, A., Abdel-Rahman, E.O., El Kahoui, M.: Investigating algebraic and logical algorithms to solve Hopf bifurcation problems in algebraic biology. Math. Comput. Sci. 2(3), 493–515 (2009)MathSciNetCrossRefMATHGoogle Scholar
  65. 65.
    Tarski, A.: A decision method for elementary algebra and geometry. Prepared for publication by J. C. C. McKinsey. In: RAND Report R109, August 1948, Revised May 1951, 2nd Edition, RAND (1957)Google Scholar
  66. 66.
    Tiwari, A.: Approximate reachability for linear systems. In: Proceedings of the HSCC 2003, volume 2623 of LNCS, pp. 514–525. Springer (2003)Google Scholar
  67. 67.
    Tyson, J.J., Chen, K., Novak, B.: Network dynamics and cell physiology. Nat. Rev. Mol. Cell Biol. 2(12), 908–916 (2001)CrossRefGoogle Scholar
  68. 68.
    Wagner, C., Urbanczik, R.: The geometry of the flux cone of a metabolic network. Biophys. J. 89(6), 3837–3845 (2005)CrossRefGoogle Scholar
  69. 69.
    Wang, D.: Reasoning about geometric problems using an elimination method. In: Automated Practical Reasoning, Texts and Monographs in Symbolic Computation, pp. 147–185. Springer (1995)Google Scholar
  70. 70.
    Wang, D.: An elimination method for polynomial systems. J. Symb. Comput. 16(2), 83–114 (1993)MathSciNetCrossRefMATHGoogle Scholar
  71. 71.
    Weber, A., Sturm, T., Abdel-Rahman, E.O.: Algorithmic global criteria for excluding oscillations. Bull. Math. Biol. 73(4), 899–916 (2011)MathSciNetCrossRefMATHGoogle Scholar
  72. 72.
    Weispfenning, V.: The complexity of linear problems in fields. J. Symb. Comput. 5(1–2), 3–27 (1988)MathSciNetCrossRefMATHGoogle Scholar
  73. 73.
    Weispfenning, V.: Quantifier elimination for real algebra—the quadratic case and beyond. Appl. Algebra Eng. Commun. Comput. 8(2), 85–101 (1997)MathSciNetCrossRefMATHGoogle Scholar
  74. 74.
    Wu, W.-T.: Basic principles of mechanical theorem proving in elementary geometries. J. Syst. Sci. Math. Sci. 4(3), 207–235 (1984)MathSciNetGoogle Scholar
  75. 75.
    Wu, W.-T.: Basic principles of mechanical theorem proving in elementary geometries. J. Autom. Reason. 2(3), 219–252 (1986)CrossRefMATHGoogle Scholar

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Authors and Affiliations

  1. 1.University of Lorraine, CNRS, Inria, and LORIANancyFrance
  2. 2.Max Planck Institute for Informatics and Saarland UniversitySaarbrückenGermany

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