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Degree and Dimension Estimates for Invariant Ideals of \(P\)-Solvable Recurrences

  • Marc Moreno Maza
  • Rong Xiao
Conference paper

Abstract

Motivated by the generation of polynomial loop invariants of computer programs, we study \(P\)-solvable recurrences. While these recurrences may contain non-linear terms, we show that the solutions of any such relation can be obtained by solving a system of linear recurrences. We also study invariant ideals of \(P\)-solvable recurrences (or equivalently of while loops with no branches). We establish sharp degree and dimension estimates of those invariant ideals.

References

  1. 1.
    Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Graduate Text in Mathematics. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ge, G.: Algorithms related to multiplicative representations of algebraic numbers. Ph.D. thesis, U.C. Berkeley (1993).Google Scholar
  3. 3.
    Heintz, Joos: Definability and fast quantifier elimination in algebraically closed fields. Theor. Comput. Sci. 24(3), 239–277 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kapur, D., Rodriguez-Carbonell, E.: Automatic generation of polynomial invariants of bounded degree using abstract interpretation. Sci. Comput. Program. 64(1), 54–75 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kauers, M., Zimmermann, B.: Computing the algebraic relations of c-finite sequences and multisequences. J. Symb. Comput. 43, 787–803 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kovács, L.: Invariant generation for p-solvable loops with assignments. In: Proceedings of the 3rd international conference on computer science: theory and applications. CSR’08, pp. 349–359. Springer, Berlin (2008)Google Scholar
  7. 7.
    Moreno Maza, M., Xiao, R.: Generating program invariants via interpolation. CoRR, abs/1201.5086, 2012.Google Scholar
  8. 8.
    Müller-Olm, M., Seidl, H.: A Note on Karr’s Algorithm. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) Automata, Languages and Programming, vol. 3142 of Lecture Notes in Computer Science, pp. 1016–1028. Springer, Turku (2004)Google Scholar
  9. 9.
    Müller-Olm, M., Seidl, H.: Computing polynomial program invariants. Inf. Process. Lett. 91(5), 233–244 (2004)CrossRefzbMATHGoogle Scholar
  10. 10.
    Osborne, M.J.: Math tutorial: first-order difference equations (2000).Google Scholar
  11. 11.
    Rodríguez-Carbonell, E., Kapur, D.: An Abstract Interpretation Approach for Automatic Generation of Polynomial Invariants. In International Symposium on Static Analysis (SAS 2004), vol. 3148 of Lecture Notes in Computer Science, pp. 280–295. Springer, Heidelberg (2004).Google Scholar
  12. 12.
    Rodríguez-Carbonell, E., Kapur, D.: Automatic generation of polynomial loop invariants: algebraic foundations. ISSAC ’04, pp. 266–273. ACM (2004).Google Scholar
  13. 13.
    von zur Gathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, Cambridge (1999).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.University of Western OntarioLondonCanada

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