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.
Chapter PDF
Similar content being viewed by others
Keywords
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
Albert, E., Arenas, P., Genaim, S., Puebla, G.: Closed-form upper bounds in static cost analysis. J. Autom. Reasoning 46(2), 161–203 (2011)
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)
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)
Ben-Amram, A.M., Genaim, S.: Ranking functions for linear-constraint loops. In: POPL (2013)
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)
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)
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)
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)
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)
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)
Gulwani, S., Zuleger, F.: The reachability-bound problem. In: PLDI, pp. 292–304 (2010)
Gupta, A., Henzinger, T.A., Majumdar, R., Rybalchenko, A., Xu, R.G.: Proving non-termination. In: POPL, pp. 147–158 (2008)
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)
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)
Jech, T.: Set Theory, 3rd edn. Springer (2006)
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)
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)
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)
Leike, J.: Ranking function synthesis for linear lasso programs. Master’s thesis, University of Freiburg, Germany (2013)
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)
Podelski, A., Rybalchenko, A.: Transition invariants. In: LICS, pp. 32–41 (2004)
Podelski, A., Rybalchenko, A.: Transition predicate abstraction and fair termination. In: POPL, pp. 132–144 (2005)
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)
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)
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)
Schrijver, A.: Theory of linear and integer programming. Wiley-Interscience series in discrete mathematics and optimization. Wiley (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Leike, J., Heizmann, M. (2014). Ranking Templates for Linear Loops. In: Ábrahám, E., Havelund, K. (eds) Tools and Algorithms for the Construction and Analysis of Systems. TACAS 2014. Lecture Notes in Computer Science, vol 8413. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54862-8_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-54862-8_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-54861-1
Online ISBN: 978-3-642-54862-8
eBook Packages: Computer ScienceComputer Science (R0)