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Abstract Interpretation of CTL Properties

  • Caterina UrbanEmail author
  • Samuel Ueltschi
  • Peter Müller
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11002)

Abstract

CTL is a temporal logic commonly used to express program properties. Most of the existing approaches for proving CTL properties only support certain classes of programs, limit their scope to a subset of CTL, or do not directly support certain existential CTL formulas. This paper presents an abstract interpretation framework for proving CTL properties that does not suffer from these limitations. Our approach automatically infers sufficient preconditions, and thus provides useful information even when a program satisfies a property only for some inputs. We systematically derive a program semantics that precisely captures CTL properties by abstraction of the operational trace semantics of a program. We then leverage existing abstract domains based on piecewise-defined functions to derive decidable abstractions that are suitable for static program analysis. To handle existential CTL properties, we augment these abstract domains with under-approximating operators. We implemented our approach in a prototype static analyzer. Our experimental evaluation demonstrates that the analysis is effective, even for CTL formulas with non-trivial nesting of universal and existential path quantifiers, and performs well on a wide variety of benchmarks.

References

  1. 1.
    Baier, C., Katoen, J.P.: Principles of Model Checking. MIT Press, Cambridge (2008)zbMATHGoogle Scholar
  2. 2.
    Bakhirkin, A., Piterman, N.: Finding recurrent sets with backward analysis and trace partitioning. In: Chechik, M., Raskin, J.-F. (eds.) TACAS 2016. LNCS, vol. 9636, pp. 17–35. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49674-9_2CrossRefGoogle Scholar
  3. 3.
    Bertrane, J., Cousot, P., Cousot, R., Feret, J., Mauborgne, L., Miné, A., Rival, X.: Static analysis and verification of aerospace software by abstract interpretation. In: AIAA, pp. 1–38 (2010)Google Scholar
  4. 4.
    Beyene, T.A., Popeea, C., Rybalchenko, A.: Solving existentially quantified horn clauses. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 869–882. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-39799-8_61CrossRefGoogle Scholar
  5. 5.
    Chen, H.-Y., Cook, B., Fuhs, C., Nimkar, K., O’Hearn, P.: Proving nontermination via safety. In: Ábrahám, E., Havelund, K. (eds.) TACAS 2014. LNCS, vol. 8413, pp. 156–171. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-642-54862-8_11CrossRefGoogle Scholar
  6. 6.
    Clarke, E.M., Emerson, E.A.: Design and synthesis of synchronization skeletons using branching time temporal logic. In: Kozen, D. (ed.) Logic of Programs 1981. LNCS, vol. 131, pp. 52–71. Springer, Heidelberg (1982).  https://doi.org/10.1007/BFb0025774CrossRefGoogle Scholar
  7. 7.
    Clarke, E.M., Emerson, E.A., Sistla, A.P.: Automatic verification of finite-state concurrent systems using temporal logic specifications. ACM Trans. Program. Lang. Syst. 8(2), 244–263 (1986)CrossRefGoogle Scholar
  8. 8.
    Cook, B., Khlaaf, H., Piterman, N.: Faster temporal reasoning for infinite-state programs. In: FMCAD, pp. 75–82 (2014)Google Scholar
  9. 9.
    Cook, B., Khlaaf, H., Piterman, N.: On automation of CTL* verification for infinite-state systems. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9206, pp. 13–29. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-21690-4_2CrossRefzbMATHGoogle Scholar
  10. 10.
    Cook, B., Koskinen, E.: Reasoning about nondeterminism in programs. In: PLDI, pp. 219–230 (2013)Google Scholar
  11. 11.
    Cook, B., Koskinen, E., Vardi, M.: Temporal property verification as a program analysis task. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 333–348. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22110-1_26CrossRefGoogle Scholar
  12. 12.
    Cook, B., Koskinen, E., Vardi, M.Y.: Temporal property verification as a program analysis task - extended version. Formal Methods Syst. Des. 41(1), 66–82 (2012)CrossRefGoogle Scholar
  13. 13.
    Courant, N., Urban, C.: Precise widening operators for proving termination by abstract interpretation. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10205, pp. 136–152. Springer, Heidelberg (2017).  https://doi.org/10.1007/978-3-662-54577-5_8CrossRefGoogle Scholar
  14. 14.
    Cousot, P.: Constructive design of a hierarchy of semantics of a transition system by abstract interpretation. Theoret. Comput. Sci. 277(1–2), 47–103 (2002)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Cousot, P., Cousot, R.: Static determination of dynamic properties of programs. In: Symposium on Programming, pp. 106–130 (1976)Google Scholar
  16. 16.
    Cousot, P., Cousot, R.: Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: POPL, pp. 238–252 (1977)Google Scholar
  17. 17.
    Cousot, P., Cousot, R.: Temporal abstract interpretation. In: POPL, pp. 12–25 (2000)Google Scholar
  18. 18.
    Cousot, P., Cousot, R.: An abstract interpretation framework for termination. In: POPL, pp. 245–258(2012)Google Scholar
  19. 19.
    Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among variables of a program. In: POPL, pp. 84–96 (1978)Google Scholar
  20. 20.
    Dietsch, D., Heizmann, M., Langenfeld, V., Podelski, A.: Fairness modulo theory: a new approach to LTL software model checking. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9206, pp. 49–66. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-21690-4_4CrossRefGoogle Scholar
  21. 21.
    Giacobazzi, R., Ranzato, F.: Incompleteness of states w.r.t. traces in model checking. Inf. Comput. 204(3), 376–407 (2006)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gupta, A., Henzinger, T.A., Majumdar, R., Rybalchenko, A., Xu, R.: Proving non-termination. In: POPL, pp. 147–158 (2008)Google Scholar
  23. 23.
    Gurfinkel, A., Wei, O., Chechik, M.: Yasm: a software model-checker for verification and refutation. In: CAV, pp. 170–174 (2006)Google Scholar
  24. 24.
    Jeannet, B., Miné, A.: Apron: a library of numerical abstract domains for static analysis. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 661–667. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-02658-4_52CrossRefGoogle Scholar
  25. 25.
    Koskinen, E.: Temporal verification of programs. Ph.D. thesis, University of Cambridge, November 2012Google Scholar
  26. 26.
    Kupferman, O., Vardi, M.Y., Wolper, P.: An automata-theoretic approach to branching-time model checking. J. ACM 47(2), 312–360 (2000)MathSciNetCrossRefGoogle Scholar
  27. 27.
    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
  28. 28.
    Manna, Z., Pnueli, A.: A hierarchy of temporal properties. In: PODC, pp. 377–410 (1990)Google Scholar
  29. 29.
    Manna, Z., Pnueli, A.: The Temporal Verification of Reactive Systems: Progress (1996)Google Scholar
  30. 30.
    Miné, A.: The octagon abstract domain. High. Order Symbolic Comput. 19(1), 31–100 (2006)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Miné, A.: Inferring sufficient conditions with backward polyhedral under-approximations. Electron. Notes Theor. Comput. Sci. 287, 89–100 (2012)CrossRefGoogle Scholar
  32. 32.
    Nielson, F., Nielson, H.R., Hankin, C.: Principles of Program Analysis. Springer, (1999)CrossRefGoogle Scholar
  33. 33.
    Pnueli, A.: The temporal logic of programs. In: FOCS, pp. 46–57 (1977)Google Scholar
  34. 34.
    Podelski, A., Rybalchenko, A.: Transition invariants. In: LICS, pp. 32–41 (2004)Google Scholar
  35. 35.
    Rival, X., Mauborgne, L.: The trace partitioning abstract domain. ACM TOPLAS 29(5), 26 (2007)CrossRefGoogle Scholar
  36. 36.
    Song, F., Touili, T.: Efficient CTL model-checking for pushdown systems. Theoret. Comput. Sci. 549, 127–145 (2014)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Ueltschi, S.: Proving temporal properties by abstract interpretation. Master’s thesis, ETH Zurich, Zurich, Switzerland (2017)Google Scholar
  38. 38.
    Urban, C.: Static Analysis by abstract interpretation of functional temporal properties of programs. Ph.D. thesis, École Normale Supérieure, Paris, France, July 2015Google Scholar
  39. 39.
    Urban, C., Miné, A.: A decision tree abstract domain for proving conditional termination. In: SAS, pp. 302–318 (2014)Google Scholar
  40. 40.
    Urban, C., Miné, A.: An abstract domain to infer ordinal-valued ranking functions. In: ESOP, pp. 412–431 (2014)Google Scholar
  41. 41.
    Urban, C., Miné, A.: Inference of ranking functions for proving temporal properties by abstract interpretation. Comput. Lang. Syst. Struct. 47, 77–103 (2017)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Caterina Urban
    • 1
    Email author
  • Samuel Ueltschi
    • 1
  • Peter Müller
    • 1
  1. 1.Department of Computer ScienceETH ZurichZurichSwitzerland

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