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Automata Terms in a Lazy WSkS Decision Procedure

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Automated Deduction – CADE 27 (CADE 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11716))

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Abstract

We propose a lazy decision procedure for the logic WS\(k\)S. It builds a term-based symbolic representation of the state space of the tree automaton (TA) constructed by the classical WS\(k\)S decision procedure. The classical decision procedure transforms the symbolic representation into a TA via a bottom-up traversal and then tests its language non-emptiness, which corresponds to satisfiability of the formula. On the other hand, we start evaluating the representation from the top, construct the state space on the fly, and utilize opportunities to prune away parts of the state space irrelevant to the language emptiness test. In order to do so, we needed to extend the notion of language terms (denoting language derivatives) used in our previous procedure for the linear fragment of the logic (the so-called WS1S) into automata terms. We implemented our decision procedure and identified classes of formulae on which our prototype implementation is significantly faster than the classical procedure implemented in the Mona tool.

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Notes

  1. 1.

    Intuitively, the operator can be seen as a generalization of the Kleene star to tree languages. The symbol  is the Chinese character for a tree, pronounced mù, as in English moo-n, but shorter and with a falling tone, staccato-like.

  2. 2.

    Note that our definition of projection differs from the usual one, which would in the example produce a single symbol \(\{Y \mapsto 0\}\) over a different alphabet (the alphabet of symbols over \(\{Y\}\)).

  3. 3.

    The implementation is available at https://github.com/vhavlena/lazy-wsks.

  4. 4.

    Building an optimised and overall competitive implementation is a subject of our further work. Our results with an implementation of a lazy decision procedure for WS1S from [29] suggest that this is possible.

References

  1. Møller, A., Schwartzbach, M.: The pointer assertion logic engine. In: PLDI 2001. ACM Press (2001). Also in SIGPLAN Notices 36(5) (2001)

    Google Scholar 

  2. Glenn, J., Gasarch, W.: Implementing WS1S via finite automata. In: Raymond, D., Wood, D., Yu, S. (eds.) WIA 1996. LNCS, vol. 1260, pp. 50–63. Springer, Heidelberg (1997). https://doi.org/10.1007/3-540-63174-7_5

    Chapter  Google Scholar 

  3. Elgaard, J., Klarlund, N., Møller, A.: MONA 1.x: new techniques for WS1S and WS2S. In: Hu, A.J., Vardi, M.Y. (eds.) CAV 1998. LNCS, vol. 1427, pp. 516–520. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0028773

    Chapter  Google Scholar 

  4. Klarlund, N., Møller, A.: MONA Version 1.4 User Manual. BRICS, Department of Computer Science, Aarhus University, January 2001. Notes Series NS-01-1. http://www.brics.dk/mona/. Revision of BRICS NS-98-3

  5. Madhusudan, P., Parlato, G., Qiu, X.: Decidable logics combining heap structures and data. In: POPL 2011, pp. 611–622. ACM (2011)

    Google Scholar 

  6. Madhusudan, P., Qiu, X.: Efficient decision procedures for heaps using STRAND. In: Yahav, E. (ed.) SAS 2011. LNCS, vol. 6887, pp. 43–59. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-23702-7_8

    Chapter  Google Scholar 

  7. Chin, W., David, C., Nguyen, H.H., Qin, S.: Automated verification of shape, size and bag properties via user-defined predicates in separation logic. Sci. Comput. Program. 77(9), 1006–1036 (2012)

    Article  Google Scholar 

  8. Zee, K., Kuncak, V., Rinard, M.C.: Full functional verification of linked data structures. In: POPL 2008, 349–361. ACM (2008)

    Google Scholar 

  9. Tateishi, T., Pistoia, M., Tripp, O.: Path- and index-sensitive string analysis based on monadic second-order logic. ACM Trans. Comput. Log. 22(4), 33 (2013)

    Google Scholar 

  10. Zhou, M., He, F., Wang, B., Gu, M., Sun, J.: Array theory of bounded elements and its applications. J. Autom. Reasoning 52(4), 379–405 (2014)

    Article  MathSciNet  Google Scholar 

  11. Baukus, K., Bensalem, S., Lakhnech, Y., Stahl, K.: Abstracting WS1S systems to verify parameterized networks. In: Graf, S., Schwartzbach, M. (eds.) TACAS 2000. LNCS, vol. 1785, pp. 188–203. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-46419-0_14

    Chapter  MATH  Google Scholar 

  12. Bodeveix, J.-P., Filali, M.: FMona: a tool for expressing validation techniques over infinite state systems. In: Graf, S., Schwartzbach, M. (eds.) TACAS 2000. LNCS, vol. 1785, pp. 204–219. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-46419-0_15

    Chapter  MATH  Google Scholar 

  13. Bozga, M., Iosif, R., Sifakis, J.: Structural invariants for parametric verification of systems with almost linear architectures. Technical report arXiv:1902.02696 (2019)

  14. Klarlund, N., Nielsen, M., Sunesen, K.: A case study in verification based on trace abstractions. In: Broy, M., Merz, S., Spies, K. (eds.) Formal Systems Specification. LNCS, vol. 1169, pp. 341–373. Springer, Heidelberg (1996). https://doi.org/10.1007/BFb0024435

    Chapter  Google Scholar 

  15. Smith, M.A., Klarlund, N.: Verification of a sliding window protocol using IOA and MONA. In: Bolognesi, T., Latella, D. (eds.) Formal Methods for Distributed System Development. ITIFIP, vol. 55, pp. 19–34. Springer, Boston, MA (2000). https://doi.org/10.1007/978-0-387-35533-7_2

    Chapter  Google Scholar 

  16. Basin, D., Klarlund, N.: Automata based symbolic reasoning in hardware verification. In: CAV 1998. LNCS, pp. 349–361. Springer (1998)

    Google Scholar 

  17. Sandholm, A., Schwartzbach, M.I.: Distributed safety controllers for web services. In: Astesiano, E. (ed.) FASE 1998. LNCS, vol. 1382, pp. 270–284. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0053596

    Chapter  Google Scholar 

  18. Hune, T., Sandholm, A.: A case study on using automata in control synthesis. In: Maibaum, T. (ed.) FASE 2000. LNCS, vol. 1783, pp. 349–362. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-46428-X_24

    Chapter  Google Scholar 

  19. Hamza, J., Jobstmann, B., Kuncak, V.: Synthesis for regular specifications over unbounded domains. In: FMCAD 2010, pp. 101–109. IEEE Computer Science (2010)

    Google Scholar 

  20. Morawietz, F., Cornell, T.: The MSO logic-automaton connection in linguistics. In: Lecomte, A., Lamarche, F., Perrier, G. (eds.) LACL 1997. LNCS (LNAI), vol. 1582, pp. 112–131. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48975-4_6

    Chapter  MATH  Google Scholar 

  21. Wies, T., Muñiz, M., Kuncak, V.: An efficient decision procedure for imperative tree data structures. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS (LNAI), vol. 6803, pp. 476–491. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22438-6_36

    Chapter  Google Scholar 

  22. Doyen, L., Raskin, J.-F.: Antichain algorithms for finite automata. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 2–22. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12002-2_2

    Chapter  MATH  Google Scholar 

  23. De Wulf, M., Doyen, L., Henzinger, T.A., Raskin, J.-F.: Antichains: a new algorithm for checking universality of finite automata. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 17–30. Springer, Heidelberg (2006). https://doi.org/10.1007/11817963_5

    Chapter  Google Scholar 

  24. Bouajjani, A., Habermehl, P., Holík, L., Touili, T., Vojnar, T.: Antichain-based universality and inclusion testing over nondeterministic finite tree automata. In: Ibarra, O.H., Ravikumar, B. (eds.) CIAA 2008. LNCS, vol. 5148, pp. 57–67. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70844-5_7

    Chapter  MATH  Google Scholar 

  25. Abdulla, P.A., Chen, Y.-F., Holík, L., Mayr, R., Vojnar, T.: When simulation meets antichains (on checking language inclusion of NFAs). In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 158–174. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12002-2_14

    Chapter  MATH  Google Scholar 

  26. Habermehl, P., Holík, L., Rogalewicz, A., Šimáček, J., Vojnar, T.: Forest automata for verification of heap manipulation. Formal Methods Syst. Des. 41(1), 83–106 (2012)

    Article  Google Scholar 

  27. De Wulf, M., Doyen, L., Maquet, N., Raskin, J.-F.: Antichains: alternative algorithms for LTL satisfiability and model-checking. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 63–77. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78800-3_6

    Chapter  MATH  Google Scholar 

  28. De Wulf, M., Doyen, L., Raskin, J.-F.: A lattice theory for solving games of imperfect information. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 153–168. Springer, Heidelberg (2006). https://doi.org/10.1007/11730637_14

    Chapter  Google Scholar 

  29. Fiedor, T., Holík, L., Janků, P., Lengál, O., Vojnar, T.: Lazy automata techniques for WS1S. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10205, pp. 407–425. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-54577-5_24

    Chapter  MATH  Google Scholar 

  30. Havlena, V., Holík, L., Lengál, O., Vojnar, T.: Automata terms in a lazy WS\(k\)S decision procedure (technical report). Technical report arXiv:1905.08697 (2019)

  31. Comon, H., et al.: Tree automata techniques and applications (2008)

    Google Scholar 

  32. Büchi, J.R.: On a decision method in restricted second-order arithmetic. In: International Congress on Logic, Methodology, and Philosophy of Science, pp. 1–11. Stanford University Press (1962)

    Google Scholar 

  33. Rabin, M.O.: Decidability of second order theories and automata on infinite trees. Trans. Am. Math. Soc. 141, 1–35 (1969)

    MathSciNet  MATH  Google Scholar 

  34. Thatcher, J.W., Wright, J.B.: Generalized finite automata theory with an application to a decision problem of second-order logic. Math. Syst. Theory 2(1), 57–81 (1968)

    Article  MathSciNet  Google Scholar 

  35. Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time (preliminary report). In: Fifth Annual ACM Symposium on Theory of Computing, STOC 1973, pp. 1–9. ACM, New York (1973)

    Google Scholar 

  36. Klarlund, N., Møller, A., Schwartzbach, M.I.: MONA implementation secrets. Int. J. Found. Comput. Sci. 13(4), 571–586 (2002)

    Article  MathSciNet  Google Scholar 

  37. Klarlund, N.: A theory of restrictions for logics and automata. In: Halbwachs, N., Peled, D. (eds.) CAV 1999. LNCS, vol. 1633, pp. 406–417. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48683-6_35

    Chapter  Google Scholar 

  38. Topnik, C., Wilhelm, E., Margaria, T., Steffen, B.: jMosel: a stand-alone tool and jABC plugin for M2L(Str). In: Valmari, A. (ed.) SPIN 2006. LNCS, vol. 3925, pp. 293–298. Springer, Heidelberg (2006). https://doi.org/10.1007/11691617_18

    Chapter  Google Scholar 

  39. D’Antoni, L., Veanes, M.: Minimization of symbolic automata. In: POPL 2014, pp. 541–554 (2014)

    Google Scholar 

  40. Margaria, T., Steffen, B., Topnik, C.: Second-order value numbering. In: GraMoT 2010. Volume 30 of ECEASST, pp. 1–15. EASST (2010)

    Google Scholar 

  41. Fiedor, T., Holík, L., Lengál, O., Vojnar, T.: Nested antichains for WS1S. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 658–674. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46681-0_59

    Chapter  MATH  Google Scholar 

  42. Traytel, D.: A coalgebraic decision procedure for WS1S. In: 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Volume 41 of Leibniz International Proceedings in Informatics (LIPIcs), pp. 487–503. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2015)

    Google Scholar 

  43. Ganzow, T., Kaiser, Ł.: New algorithm for weak monadic second-order logic on inductive structures. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 366–380. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15205-4_29

    Chapter  MATH  Google Scholar 

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Acknowledgement

We thank the anonymous reviewers for their helpful comments on how to improve the exposition in this paper. This work was supported by the Czech Science Foundation project 17-12465S, the FIT BUT internal project FIT-S-17-4014, and The Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPU II) project IT4Innovations excellence in science—LQ1602.

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Authors and Affiliations

  1. FIT, IT4I Centre of Excellence, Brno University of Technology, Brno, Czech Republic

    Vojtěch Havlena, Lukáš Holík, Ondřej Lengál & Tomáš Vojnar

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  1. Vojtěch Havlena
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  2. Lukáš Holík
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  3. Ondřej Lengál
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  4. Tomáš Vojnar
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Correspondence to Ondřej Lengál .

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  1. University of Lorraine, Villers-lès-Nancy, France

    Pascal Fontaine

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Havlena, V., Holík, L., Lengál, O., Vojnar, T. (2019). Automata Terms in a Lazy WSkS Decision Procedure. In: Fontaine, P. (eds) Automated Deduction – CADE 27. CADE 2019. Lecture Notes in Computer Science(), vol 11716. Springer, Cham. https://doi.org/10.1007/978-3-030-29436-6_18

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