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International Workshop on Verification, Model Checking, and Abstract Interpretation

VMCAI 2011: Verification, Model Checking, and Abstract Interpretation pp 371–386Cite as

Decision Procedures for Automating Termination Proofs

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 6538)

Abstract

Automated termination provers often use the following schema to prove that a program terminates: construct a relational abstraction of the program’s transition relation and then show that the relational abstraction is well-founded. The focus of current tools has been on developing sophisticated techniques for constructing the abstractions while relying on known decidable logics (such as linear arithmetic) to express them. We believe we can significantly increase the class of programs that are amenable to automated termination proofs by identifying more expressive decidable logics for reasoning about well-founded relations. We therefore present a new decision procedure for reasoning about multiset orderings, which are among the most powerful orderings used to prove termination. We show that, using our decision procedure, one can automatically prove termination of natural abstractions of programs.

Keywords

  • Decision Procedure
  • Function Symbol
  • Ranking Function
  • Cardinality Constraint
  • Ground Term

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.

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References

  1. Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)

    CrossRef  MATH  Google Scholar 

  2. Bachmair, L., Ganzinger, H.: Resolution theorem proving. In: Handbook of Automated Reasoning, pp. 19–99. MIT Press, Cambridge (2001)

    CrossRef  Google Scholar 

  3. Berdine, J., Cook, B., Distefano, D., O’Hearn, P.W.: Automatic termination proofs for programs with shape-shifting heaps. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 386–400. Springer, Heidelberg (2006)

    CrossRef  Google Scholar 

  4. Colón, M.A., Sipma, H.B.: Synthesis of linear ranking functions. In: Margaria, T., Yi, W. (eds.) TACAS 2001. LNCS, vol. 2031, pp. 67–81. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  5. Cook, B., Podelski, A., Rybalchenko, A.: Abstraction refinement for termination. In: Hankin, C., Siveroni, I. (eds.) SAS 2005. LNCS, vol. 3672, pp. 87–101. Springer, Heidelberg (2005)

    CrossRef  Google Scholar 

  6. 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)

    CrossRef  Google Scholar 

  7. Deng, Y., Sangiorgi, D.: Ensuring termination by typability. Inf. Comput. 204(7), 1045–1082 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Dershowitz, N.: Orderings for term-rewriting systems. In: Symposium on Foundations of Computer Science (SFCS), pp. 123–131 (1979)

    Google Scholar 

  9. Dershowitz, N., Manna, Z.: Proving termination with multiset orderings. Commun. ACM 22(8), 465–476 (1979)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Dick, J., Kalmus, J., Martin, U.: Automating the Knuth Bendix Ordering. Acta Inf. 28(2), 95–119 (1990)

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. Floyd, R.W.: Assigning meanings to programs. In: Proc. Amer. Math. Soc. Symposia in Applied Mathematics, vol. 19, pp. 19–31 (1967)

    Google Scholar 

  12. Ihlemann, C., Jacobs, S., Sofronie-Stokkermans, V.: On local reasoning in verification. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 265–281. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  13. Jacobs, S.: Incremental instance generation in local reasoning. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 368–382. Springer, Heidelberg (2009)

    CrossRef  Google Scholar 

  14. Kuncak, V., Piskac, R., Suter, P.: Ordered sets in the calculus of data structures. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 34–48. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  15. Lee, C.S., Jones, N.D., Ben-Amram, A.M.: The size-change principle for program termination. In: POPL, pp. 81–92 (2001)

    Google Scholar 

  16. Martín-Mateos, F.-J., Ruiz-Reina, J.-L., Alonso, J.-A., Hidalgo, M.J.: Proof pearl: A formal proof of higman’s lemma in ACL2. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 358–372. Springer, Heidelberg (2005)

    CrossRef  Google Scholar 

  17. Narendran, P., Rusinowitch, M., Verma, R.M.: RPO Constraint Solving Is in NP. In: Gottlob, G., Grandjean, E., Seyr, K. (eds.) CSL 1998. LNCS, vol. 1584, pp. 385–398. Springer, Heidelberg (1999)

    CrossRef  Google Scholar 

  18. Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM TOPLAS 1(2), 245–257 (1979)

    CrossRef  MATH  Google Scholar 

  19. Nieuwenhuis, R.: Simple LPO constraint solving methods. Inf. Process. Lett. 47(2), 65–69 (1993)

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. Piskac, R., Kuncak, V.: Decision procedures for multisets with cardinality constraints. In: Logozzo, F., Peled, D.A., Zuck, L.D. (eds.) VMCAI 2008. LNCS, vol. 4905, pp. 218–232. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  21. Piskac, R., Kuncak, V.: Linear arithmetic with stars. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 268–280. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  22. 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)

    CrossRef  Google Scholar 

  23. Podelski, A., Rybalchenko, A.: Transition invariants. In: LICS 2004 (2004)

    Google Scholar 

  24. Podelski, A., Rybalchenko, A.: Transition predicate abstraction and fair termination. ACM TOPLAS 29(3), 15 (2007)

    CrossRef  MATH  Google Scholar 

  25. Sofronie-Stokkermans, V.: Hierarchic reasoning in local theory extensions. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 219–234. Springer, Heidelberg (2005)

    CrossRef  Google Scholar 

  26. Sofronie-Stokkermans, V., Ihlemann, C.: Automated reasoning in some local extensions of ordered structures. In: ISMVL (2007)

    Google Scholar 

  27. Suter, P., Dotta, M., Kuncak, V.: Decision procedures for algebraic data types with abstractions. In: 37th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages, POPL (2010)

    Google Scholar 

  28. Venkataraman, K.N.: Decidability of the purely existential fragment of the theory of term algebras. Journal of the ACM (JACM) 34(2), 492–510 (1987)

    CrossRef  MathSciNet  Google Scholar 

  29. Zarba, C.G.: Combining multisets with integers. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, p. 363. Springer, Heidelberg (2002)

    CrossRef  Google Scholar 

  30. Zhang, T., Sipma, H.B., Manna, Z.: The Decidability of the First-Order Theory of Knuth-Bendix Order. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 131–148. Springer, Heidelberg (2005)

    CrossRef  Google Scholar 

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Author information

Authors and Affiliations

  1. École Polytechnique Fédérale de Lausanne (EPFL), Switzerland

    Ruzica Piskac

  2. Institute of Science and Technology (IST), Austria

    Thomas Wies

Authors
  1. Ruzica Piskac
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  2. Thomas Wies
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Editor information

Editors and Affiliations

  1. UC San Diego, USA

    Ranjit Jhala

  2. Department of Computing and Information Sciences, Kansas State University, 234 Nichols Hall, 66506, Manhattan, KS, USA

    David Schmidt

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Piskac, R., Wies, T. (2011). Decision Procedures for Automating Termination Proofs. In: Jhala, R., Schmidt, D. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2011. Lecture Notes in Computer Science, vol 6538. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18275-4_26

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  • DOI: https://doi.org/10.1007/978-3-642-18275-4_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-18274-7

  • Online ISBN: 978-3-642-18275-4

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