New Domains for Applied Quantifier Elimination

  • Thomas Sturm
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4194)

Abstract

We address various aspects of our computer algebra-based computer logic system redlog. There are numerous examples in the literature for successful applications of redlog to practical problems. This includes work by the group around the redlog developers as well as by many others. redlog is, however, not at all restricted to the real numbers but comprises a variety of other domains. We particularly point at the immense potential of quantifier elimination techniques for the integers. We also address another new redlog domain, which is queues over arbitrary basic domains. Both have most promising applications in practical computer science, viz. automatic loop parallelization and software security.

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References

  1. 1.
    Dolzmann, A., Sturm, T.: Redlog: Computer algebra meets computer logic. ACM SIGSAM Bulletin 31(2), 2–9 (1997)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Dolzmann, A., Sturm, T.: Redlog user manual. Technical Report MIP-9905, FMI, Universität Passau, D-94030 Passau, Germany (1999); Edition 2.0 for Version 2.0Google Scholar
  3. 3.
    Hearn, A.C.: Reduce User’s Manual for Version 3.8, Santa Monica, CA (2004), http://reduce-algebra.com/
  4. 4.
    Hearn, A.C.: Reduce: The first forty years. In: Dolzmann, A., Seidl, A., Sturm, T. (eds.) Algorithmic Algebra and Logic, pp. 19–24. BoD, Norderstedt (2005)Google Scholar
  5. 5.
    Sturm, T.: Linear problems in valued fields. Journal of Symbolic Computation 30(2), 207–219 (2000)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dolzmann, A., Sturm, T.: P-adic constraint solving. In: Dooley, S. (ed.) Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation (ISSAC 1999), Vancouver, BC, pp. 151–158. ACM Press, New York (1999)CrossRefGoogle Scholar
  7. 7.
    Seidl, A.M., Sturm, T.: Boolean quantification in a first-order context. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing. Proceedings of the CASC 2003, Institut für Informatik, Technische Universität München, Passau, pp. 329–345 (2003)Google Scholar
  8. 8.
    Sturm, T., Weispfenning, V.: Quantifier elimination in term algebras. The case of finite languages. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing. Proceedings of the CASC 2002. TUM München, pp. 285–300 (2002)Google Scholar
  9. 9.
    Weispfenning, V.: Simulation and optimization by quantifier elimination. Journal of Symbolic Computation 24(2), 189–208 (1997); Special issue on applications of quantifier eliminationGoogle Scholar
  10. 10.
    Dolzmann, A., Sturm, T., Weispfenning, V.: A new approach for automatic theorem proving in real geometry. Journal of Automated Reasoning 21(3), 357–380 (1998)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Seidl, A., Sturm, T.: A generic projection operator for partial cylindrical algebraic decomposition. In: Sendra, R. (ed.) Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation (ISSAC 2003), Philadelphia, Pennsylvania, pp. 240–247. ACM Press, New York (2003)CrossRefGoogle Scholar
  12. 12.
    Dolzmann, A., Gilch, L.A.: Generic Hermitian quantifier elimination. In: Buchberger, B., Campbell, J.A. (eds.) AISC 2004. LNCS (LNAI), vol. 3249, pp. 80–93. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  13. 13.
    Dolzmann, A., Weispfenning, V.: Local quantifier elimination. In: Traverso, C. (ed.) Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation (ISSAC 2000), St. Andrews, Scotland, pp. 86–94. ACM Press, New York (2000)CrossRefGoogle Scholar
  14. 14.
    Lasaruk, A., Sturm, T.: Weak quantifier elimination for the full linear theory of the integers. A uniform generalization of presburger arithmetic. Technical Report MIP-0604, FMI, Universität Passau, D-94030 Passau, Germany (2006)Google Scholar
  15. 15.
    Dolzmann, A., Sturm, T., Weispfenning, V.: Real quantifier elimination in practice. In: Matzat, B.H., Greuel, G.M., Hiss, G. (eds.) Algorithmic Algebra and Number Theory, pp. 221–247. Springer, Berlin (1998)Google Scholar
  16. 16.
    Loos, R., Weispfenning, V.: Applying linear quantifier elimination. THE Computer Journal 36(5), 450–462 (1993); Special issue on computational quantifier eliminationGoogle Scholar
  17. 17.
    Dolzmann, A.: Algorithmic Strategies for Applicable Real Quantifier Elimination. Doctoral dissertation, Department of Mathematics and Computer Science. University of Passau, Germany, D-94030 Passau, Germany (2000)Google Scholar
  18. 18.
    Sturm, T.: Reasoning over networks by symbolic methods. Applicable Algebra in Engineering, Communication and Computing 10(1), 79–96 (1999)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Dolzmann, A., Sturm, T., Weispfenning, V.: Automatic theorem proving in geometry. In: Grabmeier, J., Kaltofen, E., Weispfenning, V. (eds.) Computer Algebra Handbook, pp. 201–207. Springer, Berlin (2003)Google Scholar
  20. 20.
    Sturm, T., Weispfenning, V.: Computational geometry problems in Redlog. In: Wang, D. (ed.) ADG 1996. LNAI (Subseries), vol. 1360, pp. 58–86. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  21. 21.
    Sturm, T., Weispfenning, V.: Rounding and blending of solids by a real elimination method. In: Sydow, A. (ed.) Proceedings of the 15th IMACS World Congress on Scientific Computation, Modelling, and Applied Mathematics (IMACS 1997), vol. 2, pp. 727–732. Wissenschaft & Technik Verlag, Berlin (1997)Google Scholar
  22. 22.
    Sturm, T.: An algebraic approach to offsetting and blending of solids. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing. Proceedings of the CASC 2000, pp. 367–382. Springer, Berlin (2000)Google Scholar
  23. 23.
    Weispfenning, V.: Semilinear motion planning among moving objects in REDLOG. In: Ganzha, V.G., Mayr, E.W. (eds.) Computer Algebra in Scientific Computing. Proceedings of the CASC 2001, pp. 541–553. Springer, Berlin (2001)Google Scholar
  24. 24.
    Weispfenning, V.: Semilinear motion planning in REDLOG. Applicable Algebra in Engineering, Communication and Computing 12, 455–475 (2001)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Hong, H., Liska, R., Steinberg, S.: Testing stability by quantifier elimination. Journal of Symbolic Computation 24(2), 161–187 (1997); Special issue on applications of quantifier eliminationGoogle Scholar
  26. 26.
    Kahoui, M., Weber, A.: Deciding hopf bifurcations by quantifier elimination in a software component architecture. Journal of Symbolic Computation 30(2), 161–179 (2000)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Kahoui, M., Weber, A.: Symbolic equilibrium point analysis in parameterized polynomial vector fields. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing. Proceedings of the CASC 2002. TUM München, pp. 71–83 (2002)Google Scholar
  28. 28.
    Seiler, W.M., Weber, A.: Deciding ellipticity by quantifier elimination. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing. Proceedings of the CASC 2003, Institut für Informatik, Technische Universität München, Passau, pp. 347–355 (2003)Google Scholar
  29. 29.
    Brown, C.W., Kahoui, M., Novotni, D., Weber, A.: Algorithmic methods for investigating equilibria in epidemic modeling. Journal of Symbolic Computation (to appear)Google Scholar
  30. 30.
    Ioakimidis, N.I.: Automatic derivation of positivity conditions inside boundary elements with the help of the REDLOG computer logic package. Engineering Analysis with Boundary Elements 23(10), 847–856 (1999)MATHCrossRefGoogle Scholar
  31. 31.
    Ioakimidis, N.I.: REDLOG-aided derivation of feasibility conditions in applied mechanics and engineering problems under simple inequality constraints. Journal of Mechanical Engineering (Strojnícky Časopis) 50(1), 58–69 (1999)MathSciNetGoogle Scholar
  32. 32.
    Jirstrand, M.: Nonlinear control system design by quantifier elimination. Journal of Symbolic Computation 24(2), 137–152 (1997); Special issue on applications of quantifier eliminationGoogle Scholar
  33. 33.
    Lafferriere, G., Pappas, G.J., Yovine, S.: A new class of decidable hybrid systems. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 137–151. Springer, Heidelberg (1999)Google Scholar
  34. 34.
    Größlinger, A., Griebl, M., Lengauer, C.: Quantifier elimination in automatic loop parallelization. In: Dolzmann, A., Seidl, A., Sturm, T. (eds.) Algorithmic Algebra and Logic. Proceedings of the A3L 2005, BoD, Germany, Norderstedt, pp. 123–128 (2005)Google Scholar
  35. 35.
    Snelting, G.: Quantifier elimination and information flow control for software security. In: Dolzmann, A., Seidl, A., Sturm, T. (eds.) Algorithmic Algebra and Logic. Proceedings of the A3L 2005, BoD, Germany, Norderstedt, pp. 237–242 (2005)Google Scholar
  36. 36.
    Snelting, G., Robschink, T., Krinke, J.: Efficient path conditions in dependence graphs for software safety analysis. ACM Transactions on Software Engineering and Methodolody (to appear, 2006)Google Scholar
  37. 37.
    Dolzmann, A., Sturm, T.: Parametric systems of linear congruences. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing. Proceedings of the CASC 2001, pp. 149–166. Springer, Berlin (2001)Google Scholar
  38. 38.
    Weispfenning, V.: The complexity of linear problems in fields. Journal of Symbolic Computation 5(1&2), 3–27 (1988)MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Weispfenning, V.: Complexity and uniformity of elimination in Presburger Arithmetic. In: Küchlin, W.W. (ed.) Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, Maui, HI (ISSAC 1997), pp. 48–53. ACM Press, New York (1997)CrossRefGoogle Scholar
  40. 40.
    Lasaruk, A.: Univariate weak quantifier elimination for the integers. In: Gerdt, V., Spiridonova, M., Nisheva-Pavlova, M. (eds.) ACA 2006. 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June 26–29. Abstracts of Presentations. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria, p. 8 (2006)Google Scholar
  41. 41.
    Straßer, C.: Quantifier elimination for queues. In: Rhine Workshop on Computer Algebra. In: Proceedings of the RWCA 2006. Universität Basel, Basel (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Thomas Sturm
    • 1
  1. 1.Fakultät für Mathematik und InformatikUniversität PassauGermany

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