Advertisement

Abstract

We present a new method for the constraint-based synthesis of termination arguments for linear loop programs based on linear ranking templates. Linear ranking templates are parametrized, well-founded relations such that an assignment to the parameters gives rise to a ranking function. This approach generalizes existing methods and enables us to use templates for many different ranking functions with affine-linear components. We discuss templates for multiphase, piecewise, and lexicographic ranking functions. Because these ranking templates require both strict and non-strict inequalities, we use Motzkin’s Transposition Theorem instead of Farkas Lemma to transform the generated ∃ ∀-constraint into an ∃-constraint.

Keywords

Function Symbol Ranking Function Boolean Combination Linear Arithmetic Linear Loop 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Albert, E., Arenas, P., Genaim, S., Puebla, G.: Closed-form upper bounds in static cost analysis. J. Autom. Reasoning 46(2), 161–203 (2011)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Alias, C., Darte, A., Feautrier, P., Gonnord, L.: Multi-dimensional rankings, program termination, and complexity bounds of flowchart programs. In: Cousot, R., Martel, M. (eds.) SAS 2010. LNCS, vol. 6337, pp. 117–133. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  3. 3.
    Ben-Amram, A.M.: Size-change termination, monotonicity constraints and ranking functions. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 109–123. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Ben-Amram, A.M., Genaim, S.: Ranking functions for linear-constraint loops. In: POPL (2013)Google Scholar
  5. 5.
    Bradley, A.R., Manna, Z., Sipma, H.B.: Linear ranking with reachability. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 491–504. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Bradley, A.R., Manna, Z., Sipma, H.B.: The polyranking principle. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1349–1361. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Colón, M.A., Sankaranarayanan, S., Sipma, H.B.: Linear invariant generation using non-linear constraint solving. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 420–432. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Cook, B., Fisher, J., Krepska, E., Piterman, N.: Proving stabilization of biological systems. In: Jhala, R., Schmidt, D. (eds.) VMCAI 2011. LNCS, vol. 6538, pp. 134–149. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Cook, B., Podelski, A., Rybalchenko, A.: Terminator: Beyond safety. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 415–418. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Grigor’ev, D.Y., Vorobjov Jr., N.N.: Solving systems of polynomial inequalities in subexponential time. Journal of Symbolic Computation 5(1-2), 37–64 (1988)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Gulwani, S., Zuleger, F.: The reachability-bound problem. In: PLDI, pp. 292–304 (2010)Google Scholar
  12. 12.
    Gupta, A., Henzinger, T.A., Majumdar, R., Rybalchenko, A., Xu, R.G.: Proving non-termination. In: POPL, pp. 147–158 (2008)Google Scholar
  13. 13.
    Harris, W.R., Lal, A., Nori, A.V., Rajamani, S.K.: Alternation for termination. In: Cousot, R., Martel, M. (eds.) SAS 2010. LNCS, vol. 6337, pp. 304–319. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Heizmann, M., Hoenicke, J., Leike, J., Podelski, A.: Linear ranking for linear lasso programs. In: Van Hung, D., Ogawa, M. (eds.) ATVA 2013. LNCS, vol. 8172, pp. 365–380. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  15. 15.
    Jech, T.: Set Theory, 3rd edn. Springer (2006)Google Scholar
  16. 16.
    Jovanović, D., de Moura, L.: Solving non-linear arithmetic. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 339–354. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  17. 17.
    Kroening, D., Sharygina, N., Tonetta, S., Tsitovich, A., Wintersteiger, C.M.: Loop summarization using abstract transformers. In: Cha, S(S.), Choi, J.-Y., Kim, M., Lee, I., Viswanathan, M. (eds.) ATVA 2008. LNCS, vol. 5311, pp. 111–125. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  18. 18.
    Kroening, D., Sharygina, N., Tsitovich, A., Wintersteiger, C.M.: Termination analysis with compositional transition invariants. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 89–103. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  19. 19.
    Leike, J.: Ranking function synthesis for linear lasso programs. Master’s thesis, University of Freiburg, Germany (2013)Google Scholar
  20. 20.
    Podelski, A., Rybalchenko, A.: A complete method for the synthesis of linear ranking functions. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 239–251. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  21. 21.
    Podelski, A., Rybalchenko, A.: Transition invariants. In: LICS, pp. 32–41 (2004)Google Scholar
  22. 22.
    Podelski, A., Rybalchenko, A.: Transition predicate abstraction and fair termination. In: POPL, pp. 132–144 (2005)Google Scholar
  23. 23.
    Podelski, A., Wagner, S.: A sound and complete proof rule for region stability of hybrid systems. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 750–753. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  24. 24.
    Rybalchenko, A.: Constraint solving for program verification: Theory and practice by example. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 57–71. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  25. 25.
    Sankaranarayanan, S., Sipma, H.B., Manna, Z.: Constraint-based linear-relations analysis. In: Giacobazzi, R. (ed.) SAS 2004. LNCS, vol. 3148, pp. 53–68. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  26. 26.
    Schrijver, A.: Theory of linear and integer programming. Wiley-Interscience series in discrete mathematics and optimization. Wiley (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Jan Leike
    • 1
    • 2
  • Matthias Heizmann
    • 1
  1. 1.University of FreiburgGermany
  2. 2.Max Planck Institute for Software SystemsGermany

Personalised recommendations