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A practical approach to model checking Duration Calculus using Presburger Arithmetic

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Abstract

This paper investigates the feasibility of reducing a model-checking problem K ⊧ ϕ for discrete time Duration Calculus to the decision problem for Presburger Arithmetic. Theoretical results point at severe limitations of this approach: (1) the reduction in Fränzle and Hansen (Int J Softw Inform 3(2–3):171–196, 2009) produces Presburger formulas whose sizes grow exponentially in the chop-depth of ϕ, where chop is an interval modality originating from Moszkowski (IEEE Comput 18(2):10–19, 1985), and (2) the decision problem for Presburger Arithmetic has a double exponential lower bound and a triple exponential upper bound. The generated Presburger formulas have a rich Boolean structure, many quantifiers and quantifier alternations. Such formulas are simplified using so-called guarded formulas, where a guard provides a context used to simplify the rest of the formula. A normal form for guarded formulas supports global effects of local simplifications. Combined with quantifier-elimination techniques, this normalization gives significant reductions in formula sizes and in the number of quantifiers. As an example, we solve a configuration problem using the SMT-solver Z3 as backend. Benefits and the current limits of the approach are illustrated by a family of examples.

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References

  1. Bjørner, N.: Linear quantifier elimination as an abstract decision procedure. In: The 6th International Joint Conference on Automated Reasoning, IJCAR. Lecture Notes in Computer Science, vol. 6173, pp. 316–330. Springer, Heidelberg (2010)

    Google Scholar 

  2. Bolander, T., Hansen, J., Hansen, M.R.: Decidability of a hybrid Duration Calculus. Electron. Notes Theor. Comput. Sci. 174, 113–133 (2007)

    Article  Google Scholar 

  3. Bouajjani, A., Lakhnech, Y., Robbana, R.: From Duration Calculus to linear hybrid automata. In: 7th International Conference on Computer Aided Verification, CAV. Lecture Notes in Computer Science, vol. 939, pp. 196–210. Springer, Heidelberg (1995)

    Google Scholar 

  4. Bresolin, D., Goranko, V., Montanari, A., Sciavicco, G.: Right propositional neighborhood logic over natural numbers with integer constraints for interval lengths. In: 7th IEEE International Conference on Software Engineering and Formal Methods, SEFM, pp. 240–249. IEEE Computer Society Press, Los Alamitos (2009)

    Google Scholar 

  5. Bresolin, D., Della Monica, D., Goranko, V., Montanari, A., Sciavicco, G.: The dark side of interval temporal logic: sharpening the undecidability border. In: 18th International Symposium on Temporal Representation and Reasoning, TIME, pp. 131–138. IEEE Press, Piscataway (2011)

    Google Scholar 

  6. Bresolin, D., Monica, D., Montanari, A., Sciavicco, G.: The light side of interval temporal logic: the Bernays–Schönfinkel’s fragment of CDT. In: 18th International Symposium on Temporal Representation and Reasoning, TIME, pp. 123–130. IEEE Press, Piscataway (2011)

    Google Scholar 

  7. Cooper, D.: Theorem proving in arithmetic without multiplication. In: Machine Intelligence, pp. 91–100. Edinburgh University Press (1972)

  8. de Moura, L., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) 14th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS. Lecture Notes in Computer Science, vol. 4963, pp. 337–340. Springer, Heidelberg (2008)

    Google Scholar 

  9. Esparza, J.: Petri nets, commutative context-free grammars, and basic parallel processes. Fundam. Inf. 31(1), 13–25 (1997)

    MATH  MathSciNet  Google Scholar 

  10. Esparza, J., Melzer, S.: Verification of safety properties using integer programming: beyond the state equation. Form. Method. Syst. Des. 16(2), 159–189 (2000)

    Article  Google Scholar 

  11. Fischer, M.J., Rabin, M.O.: Super-exponential complexity of Presburger Arithmetic. In: Karp, R. (ed.) Complexity of Computation, pp. 27–41. American Mathematical Society (1974)

  12. Fränzle, M.: Model-checking dense-time Duration Calculus. Form. Asp. Comput. 16(2), 121–139 (2004)

    Article  MATH  Google Scholar 

  13. Fränzle, M., Hansen, M.R.: Deciding an interval logic with accumulated durations. In: 13th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS. Lecture Notes in Computer Science, vol. 4424, pp. 201–215. Springer, Heidelberg (2007)

    Google Scholar 

  14. Fränzle, M., Hansen, M.R.: Efficient model checking for Duration Calculus based on branching-time approximations. In: 6th IEEE International Conference on Software Engineering and Formal Methods, SEFM, pp. 63–72. IEEE Computer Society Press, Los Alamitos (2008)

    Google Scholar 

  15. Fränzle, M., Hansen, M.R.: Efficient model checking for Duration Calculus. Int. J. Softw. Inform. 3(2–3), 171–196 (2009)

    Google Scholar 

  16. Goranko, V., Montanari, A., Sciavicco, G.: Propositional interval neighborhood temporal logics. J. Univ. Comput. Sci. 9(9), 1137–1167 (2003)

    MathSciNet  Google Scholar 

  17. Halpern, J.Y., Shoham, Y.: A propositional modal logic of time intervals. J. ACM 38(4), 935–962 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hansen, M.R.: Model-checking discrete Duration Calculus. Form. Asp. Comput. 6(6A), 826–845 (1994)

    Article  MATH  Google Scholar 

  19. Hansen, M.R., Brekling, A.W.: On tool support for Duration Calculus on the basis of Presburger arithmetic. In: 18th International Symposium on Temporal Representation and Reasoning, TIME, pp. 115–122. IEEE Press, Piscataway (2011)

    Google Scholar 

  20. Heise, W.: An efficient model checker for Duration Calculus. Master’s Thesis, DTU Informatics, Technical University of Denmark. Available at http://etd.dtu.dk/thesis/266706/ (2010). Accessed 10 Sept 2010

  21. Heise, W.P., Fränzle, M., Hansen, M.R.: A prototype model checker for Duration Calculus. In: 21st Nordic Workshop on Programming Theory, NWPT, pp. 26–29 (2009)

  22. Meyer, R., Faber, J., Hoenicke, J., Rybalchenko, A.: Model checking Duration Calculus: a practical approach. Form. Asp. Comput. 20(4–5), 481–505 (2008)

    Article  MATH  Google Scholar 

  23. Moszkowski, B.: A temporal logic for multilevel reasoning about hardware. IEEE Comput. 18(2), 10–19 (1985)

    Article  Google Scholar 

  24. Müller-Olm, M.: A modal fixpoint logic with chop. In: Proc. 16th. Symposium on Theoretical Aspects in Computer Science, STACS. Lecture Notes in Computer Science, vol. 1563, pp. 510–520. Springer, Heidelberg (1999)

    Google Scholar 

  25. Oppen, D.C.: A \(2^{2^{2^{p n}}}\) upper bound on the complexity of Presburger Arithmetic. J. Comput. Syst. Sci. 16(3), 323–332 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  26. Pandya, P.: Specifying and deciding quantified discrete-time Duration Calculus formulae using DCVALID. Technical Report TCS-00-PKP-1, Tata Institute of Fundamental Research. Mumbai, India (2000)

  27. Phan A.-D.: Presburger arithmetic and its use in verification. Master’s Thesis, DTU Informatics, Technical University of Denmark. Available at http://www2.imm.dtu.dk/pubdb/views/publication_details.php?id=6058 (2011). Accessed 1 Aug 2011

  28. Phan A.-D., Hansen, M.R.: From functional programming to multicore parallelism: a case study based on Presburger arithmetic. In: 23rd Nordic Workshop on Programming Theory, NWPT, pp. 5–7 (2011)

  29. Pugh, W.: The omega test: a fast and practical integer programming algorithm for dependence analysis. In: Proceedings of the 1991 ACM/IEEE Conference on Supercomputing, pp. 4–13. ACM (1991)

  30. Reddy, C.R., Loveland, D.W.: Presburger arithmetic with bounded quantifier alternation. In: 10th Annual ACM Symposium on Theory of Computing, STOC, pp. 320–325. ACM (1978)

  31. Seidl, H., Schwentick, T., Muscholl, A., Habermehl, P.: Counting in trees for free. In: 31st International Colloquium on Automata, Languages and Programming, ICALP. Lecture Notes in Computer Science, vol. 3142, pp. 1136–1149. Springer, Heidelberg (2004)

    Google Scholar 

  32. Sharma, B., Pandya, P.K., Chakraborty, S.: Bounded validity checking of interval duration logic. In: Halbwachs, N., Zuck, L.D. (eds.) 11th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS. Lecture Notes in Computer Science, vol. 3440, pp. 301–316. Springer, Heidelberg (2005)

    Google Scholar 

  33. Thai, P.H., Van Hung, D.: Verifying linear duration constraints of timed automata. In: Liu, Z., Araki, K. (eds.) 1st International Colloquium on Theoretical Aspects of Computing, ICTAC. Lecture Notes in Computer Science, vol. 3407, pp. 295–309. Springer, Heidelberg (2004)

    Google Scholar 

  34. Zhang, M., Van Hung, D., Liu, Z.: Verification of linear duration invariants by model checking CTL properties. In: 5th International Colloquium on Theoretical Aspects of Computing, ICTAC. Lecture Notes in Computer Science, vol. 5160, pp. 395–409. Springer, Heidelberg (2008)

    Google Scholar 

  35. Zhou, C., Hansen, M.R.: An adequate first order interval logic. In: Compositionality: the Significant Difference, COMPOS. Lecture Notes in Computer Science, vol. 1536, pp. 584–608. Springer, Heidelberg (1998)

    Google Scholar 

  36. Zhou, C., Hansen, M.R.: Duration Calculus: A Formal Approach to Real-Time Systems. Springer, Heidelberg (2004)

    Google Scholar 

  37. Zhou, C., Hansen, M.R., Sestoft, P.: Decidability and undecidability results for Duration Calculus. In: 10th Annual Symposium on Theoretical Aspects of Computer Science, STACS. Lecture Notes in Computer Science, vol. 665. Springer, Heidelberg (1993)

    Google Scholar 

  38. Zhou, C., Hoare, C.A.R., Ravn, A.P.: A calculus of durations. Inform. Process. Lett. 40(5), 269–276 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  39. Zhou, C., Zang, J., Yang, L., Li, X.: Linear duration invariants. In: Third International Symposium on Formal Techniques in Real-Time and Fault-Tolerant Systems, FTRTFT. Lecture Notes in Computer Science, vol. 863, pp. 86–109. Springer, Heidelberg (1994)

    Google Scholar 

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Authors and Affiliations

  1. DTU Compute, Technical University of Denmark, 2800, Lyngby, Denmark

    Michael R. Hansen & Anh-Dung Phan

  2. cBrain A/S, Dampfærgevej 30, 2100, Copenhagen Ø, Denmark

    Aske W. Brekling

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  1. Michael R. Hansen
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  2. Anh-Dung Phan
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  3. Aske W. Brekling
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Correspondence to Michael R. Hansen.

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Hansen, M.R., Phan, AD. & Brekling, A.W. A practical approach to model checking Duration Calculus using Presburger Arithmetic. Ann Math Artif Intell 71, 251–278 (2014). https://doi.org/10.1007/s10472-013-9373-7

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