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An Automata-Theoretic Approach to Model-Checking Systems and Specifications Over Infinite Data Domains

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Abstract

Data-parameterized systems model systems with finite control over an infinite data domain. VLTL is an extension of LTL that uses variables in order to specify properties of computations over infinite data, and as such, VLTL is suitable for specifying properties of data-parameterized systems. We present alternating variable Büchi word automata (AVBWs), a new model of automata over infinite alphabets, capable of modeling a significant fragment of VLTL. While alternating and non-deterministic Büchi automata over finite alphabets have the same expressive power, we show that this is not the case for infinite data domains, as we prove that AVBWs are strictly stronger than the previously defined non-deterministic variable Büchi word automata (NVBWs). However, while the emptiness problem is easy for NVBWs, it is undecidable for AVBWs. We present an algorithm for translating AVBWs to NVBWs in cases where such a translation is possible. Additionally, we characterize the structure of AVBWs that can be translated to NVBWs with our algorithm. We then rely on the natural iterative behavior of our translation algorithm to describe a bounded model-checking procedure for the logic that we consider. Furthermore, we present several fragments of the logic that can be expressed by NVBWs, as well as a fragment that cannot be expressed by NVBWs, yet whose satisfiability is decidable.

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Notes

  1. In particular, the negation operator is not included.

  2. Comments are given in bold.

  3. Note that AVBWtoNVBW does not halt when given \(\mathcal {A}\) as an input.

  4. Note that absence of cycles means, in particular, that the algorithm AVBWtoNVBW does not halt.

  5. In [29] the authors conjecture without proof that the formula \(\textsf {G}\,\exists x: a.x\) does not have an equivalent in PNF In Lemma 1 we show that \(\textsf {G}\,\exists x(b.x\wedge \textsf {F}\,a.x)\) does not have an equivalent NVBW, and therefore does not have an equivalent \(\exists ^*_{PNF}\)-VLTL formula. This is a different formula from \(\textsf {G}\,\exists x a.x\), but the conclusion remains the same.

  6. As we show in 5.1.3, these latter two formulas are equivalent.

  7. Note that the negation of the formulas \(B_i\) is of the form \(\textsf {F}\,\textsf {G}\,\forall x \lnot a.x\). The semantics of this formula is that from some point of the computation, a does not appear at all, with any value. Although this is a \(\forall \)-VLTL formula, it is easy to construct an NVBW expresses it.

  8. Every \(\exists ^*\)-VLTL has an equivalent in this form.

  9. The set of computations satisfy \(\psi \) is exactly the language of the AVBW \(\mathcal {A}_1\) from Fig. 3.

References

  1. Barringer, H., Falcone, Y., Havelund, K., Reger, G., Rydeheard, D.E.: Quantified event automata: Towards expressive and efficient runtime monitors. In: Giannakopoulou, D., Méry, D. (eds.) FM 2012: Formal Methods—18th International Symposium, Paris, France, August 27–31, 2012. Proceedings, Lecture Notes in Computer Science, vol. 7436, pp. 68–84. Springer, Berlin (2012). https://doi.org/10.1007/978-3-642-32759-9_9

    Chapter  Google Scholar 

  2. Basin, D.A., Klaedtke, F., Müller, S., Zalinescu, E.: Monitoring metric first-order temporal properties. J. ACM 62(2), 15:1–15:45 (2015). https://doi.org/10.1145/2699444

    Article  MathSciNet  MATH  Google Scholar 

  3. Bauer, A., Küster, J., Vegliach, G.: From propositional to first-order monitoring. In: Legay, A., Bensalem, S. (eds.) Runtime Verification—4th International Conference, RV 2013, Rennes, France, September 24–27, (2013). Proceedings, Lecture Notes in Computer Science, vol. 8174, pp. 59–75. Springer, Berlin (2013). https://doi.org/10.1007/978-3-642-40787-1_4

    Chapter  Google Scholar 

  4. Bauer, A., Leucker, M., Schallhart, C.: Runtime verification for LTL and TLTL. ACM Trans. Softw. Eng. Methodol. 20(4), 14:1–14:64 (2011). https://doi.org/10.1145/2000799.2000800

    Article  Google Scholar 

  5. Biere, A., Cimatti, A., Clarke, E.M., Zhu, Y.: Symbolic model checking without BDDs. In: Cleaveland, R. (ed.) Tools and Algorithms for Construction and Analysis of Systems, 5th International Conference, TACAS’99, Held as Part of the European Joint Conferences on the Theory and Practice of Software, ETAPS’99, Amsterdam, The Netherlands, March 22–28, 1999, Proceedings, Lecture Notes in Computer Science, vol. 1579, pp. 193–207. Springer, Berlin (1999). https://doi.org/10.1007/3-540-49059-0_14

    Chapter  Google Scholar 

  6. Bojańczyk, M., Muscholl, A., Schwentick, T., Segoufin, L., David, C.: Two-variable logic on words with data. In: 21th IEEE Symposium on Logic in Computer Science (LICS 2006), 12–15 August 2006, Seattle, WA, USA, Proceedings, pp. 7–16. IEEE Computer Society (2006). https://doi.org/10.1109/LICS.2006.51

  7. Bouajjani, A., Habermehl, P., Jurski, Y., Sighireanu, M.: Rewriting systems with data. In: Csuhaj-Varjú, E., Ésik, Z. (eds.) Fundamentals of Computation Theory, 16th International Symposium, FCT 2007, Budapest, Hungary, August 27–30, 2007, Proceedings, Lecture Notes in Computer Science, vol. 4639, pp. 1–22. Springer, Berlin (2007). https://doi.org/10.1007/978-3-540-74240-1_1

    Chapter  Google Scholar 

  8. Brambilla, M., Ceri, S., Comai, S., Fraternali, P., Manolescu, I.: Specification and design of workflow-driven hypertexts. J. Web Eng. 1(2), 163–182 (2003)

    Google Scholar 

  9. Buechi, 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)

  10. Burch, J.R., Clarke, E.M., McMillan, K.L., Dill, D.L., Hwang, L.J.: Symbolic model checking: 1020 states and beyond. Inf. Comput. 98(2), 142–170 (1992). https://doi.org/10.1016/0890-5401(92)90017-A

    Article  MATH  Google Scholar 

  11. Ceri, S., Matera, M., Rizzo, F., Demaldé, V.: Designing data-intensive web applications for content accessibility using web marts. Commun. ACM 50(4), 55–61 (2007). https://doi.org/10.1145/1232743.1232748

    Article  Google Scholar 

  12. Clarke, E.M., Biere, A., Raimi, R., Zhu, Y.: Bounded model checking using satisfiability solving. Formal Methods Syst. Des. 19(1), 7–34 (2001). https://doi.org/10.1023/A:1011276507260

    Article  MATH  Google Scholar 

  13. Colin, S., Mariani, L.: Run-time verification. In: Broy, M., Jonsson, B., Katoen, J., Leucker, M., Pretschner, A. (eds.) Model-Based Testing of Reactive Systems, Advanced Lectures [The Volume is the Outcome of a Research Seminar That was Held in Schloss Dagstuhl in January 2004], Lecture Notes in Computer Science, vol. 3472, pp. 525–555. Springer, Berlin (2004). https://doi.org/10.1007/11498490_24

    Chapter  Google Scholar 

  14. Decker, N., Leucker, M., Thoma, D.: Monitoring modulo theories. STTT 18(2), 205–225 (2016). https://doi.org/10.1007/s10009-015-0380-3

    Article  Google Scholar 

  15. Emerson, E.A., Halpern, J.Y.: “sometimes” and “not never” revisited: on branching versus linear time temporal logic. J. ACM 33(1), 151–178 (1986). https://doi.org/10.1145/4904.4999

    Article  MathSciNet  MATH  Google Scholar 

  16. Frenkel, H., Grumberg, O., Sheinvald, S.: An automata-theoretic approach to modeling systems and specifications over infinite data. In: C. Barrett, M. Davies, T. Kahsai (eds.) NASA Formal Methods—9th International Symposium, NFM 2017, Moffett Field, CA, USA, May 16–18, 2017, Proceedings, Lecture Notes in Computer Science, vol. 10227, pp. 1–18 (2017). https://doi.org/10.1007/978-3-319-57288-8_1

    Google Scholar 

  17. Grumberg, O., Kupferman, O., Sheinvald, S.: Variable automata over infinite alphabets. In: Dediu, A., Fernau, H., Martín-Vide, C. (eds.) Language and Automata Theory and Applications, 4th International Conference, LATA 2010, Trier, Germany, May 24–28, 2010. Proceedings, Lecture Notes in Computer Science, vol. 6031, pp. 561–572. Springer, Berlin (2010). https://doi.org/10.1007/978-3-642-13089-2_47

    Chapter  Google Scholar 

  18. Grumberg, O., Kupferman, O., Sheinvald, S.: Model checking systems and specifications with parameterized atomic propositions. In: Chakraborty, S., Mukund, M. (eds.) Automated Technology for Verification and Analysis—10th International Symposium, ATVA 2012, Thiruvananthapuram, India, October 3–6, 2012. Proceedings, Lecture Notes in Computer Science, vol. 7561, pp. 122–136. Springer, Berlin (2012). https://doi.org/10.1007/978-3-642-33386-6_11

    Chapter  Google Scholar 

  19. Grumberg, O., Kupferman, O., Sheinvald, S.: A game-theoretic approach to simulation of data-parameterized systems. In: Cassez, F., Raskin, J. (eds.) Automated Technology for Verification and Analysis—12th International Symposium, ATVA 2014, Sydney, NSW, Australia, November 3–7, 2014, Proceedings, Lecture Notes in Computer Science, vol. 8837, pp. 348–363. Springer, Berlin (2014). https://doi.org/10.1007/978-3-319-11936-6_25

    Chapter  Google Scholar 

  20. Havelund, K., Peled, D., Ulus, D.: First order temporal logic monitoring with bdds. In: D. Stewart, G. Weissenbacher (eds.) 2017 Formal Methods in Computer Aided Design, FMCAD 2017, Vienna, Austria, October 2–6, 2017, pp. 116–123. IEEE (2017). https://doi.org/10.23919/FMCAD.2017.8102249

  21. Kaminski, M., Francez, N.: Finite-memory automata. Theor. Comput. Sci. 134(2), 329–363 (1994). https://doi.org/10.1016/0304-3975(94)90242-9

    Article  MathSciNet  MATH  Google Scholar 

  22. Meredith, P.O., Jin, D., Griffith, D., Chen, F., Rosu, G.: An overview of the MOP runtime verification framework. STTT 14(3), 249–289 (2012). https://doi.org/10.1007/s10009-011-0198-6

    Article  Google Scholar 

  23. Miyano, S., Hayashi, T.: Alternating finite automata on omega-words. Theor. Comput. Sci. 32, 321–330 (1984). https://doi.org/10.1016/0304-3975(84)90049-5

    Article  MATH  Google Scholar 

  24. Muller, D.E., Schupp, P.E.: Alternating automata on infinite objects, determinacy and rabin’s theorem. In: Nivat, M., Perrin, D. (eds.) Automata on Infinite Words, Ecole de Printemps d’Informatique Théorique, Le Mont Dore, May 14–18, 1984, Lecture Notes in Computer Science, vol. 192, pp. 100–107. Springer, Berlin (1984). https://doi.org/10.1007/3-540-15641-0_27

    Chapter  Google Scholar 

  25. Neven, F., Schwentick, T., Vianu, V.: Towards regular languages over infinite alphabets. In: Sgall, J., Pultr, A., Kolman, P. (eds.) Mathematical Foundations of Computer Science 2001, 26th International Symposium, MFCS 2001 Marianske Lazne, Czech Republic, August 27–31, 2001, Proceedings, Lecture Notes in Computer Science, vol. 2136, pp. 560–572. Springer, Berlin (2001). https://doi.org/10.1007/3-540-44683-4_49

    Chapter  Google Scholar 

  26. Piterman, N., Pnueli, A., Sa’ar, Y.: Synthesis of reactive(1) designs. In: Emerson, E.A., Namjoshi, K.S. (eds.) Verification, Model Checking, and Abstract Interpretation, 7th International Conference, VMCAI 2006, Charleston, SC, USA, January 8–10, 2006, Proceedings, Lecture Notes in Computer Science, vol. 3855, pp. 364–380. Springer, Berlin (2006). https://doi.org/10.1007/11609773_24

    Chapter  Google Scholar 

  27. Rozier, K.Y., Vardi, M.Y.: A multi-encoding approach for LTL symbolic satisfiability checking. In: FM 2011: Formal Methods—17th International Symposium on Formal Methods, Limerick, Ireland, June 20–24, 2011. Proceedings, pp. 417–431 (2011). https://doi.org/10.1007/978-3-642-21437-0_31

    Google Scholar 

  28. Safra, S.: On the complexity of omega-automata. In: 29th Annual Symposium on Foundations of Computer Science, White Plains, New York, USA, 24–26 October 1988, pp. 319–327. IEEE Computer Society (1988). https://doi.org/10.1109/SFCS.1988.21948

  29. Song, F., Wu, Z.: Extending temporal logics with data variable quantifications. In: V. Raman, S.P. Suresh (eds.) 34th International Conference on Foundation of Software Technology and Theoretical Computer Science, FSTTCS 2014, December 15–17, 2014, New Delhi, India, LIPIcs, vol. 29, pp. 253–265. Schloss Dagstuhl–Leibniz–Zentrum fuer Informatik (2014). https://doi.org/10.4230/LIPIcs.FSTTCS.2014.253

  30. Vardi, M.Y.: An automata-theoretic approach to linear temporal logic. In: Moller, F., Birtwistle, G.M. (eds.) Logics for Concurrency - Structure versus Automata (8th Banff Higher Order Workshop, August 27–September 3, 1995, Proceedings), Lecture Notes in Computer Science, vol. 1043, pp. 238–266. Springer, Berlin (1995). https://doi.org/10.1007/3-540-60915-6_6

    Chapter  Google Scholar 

  31. Vardi, M.Y., Wolper, P.: An automata-theoretic approach to automatic program verification (preliminary report). In: Proceedings of the Symposium on Logic in Computer Science (LICS ’86), Cambridge, Massachusetts, USA, June 16–18, 1986, pp. 332–344. IEEE Computer Society (1986)

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Authors and Affiliations

  1. Department of Computer Science, The Technion, Haifa, Israel

    Hadar Frenkel & Orna Grumberg

  2. Department of Software Engineering, ORT Braude Academic College, Karmiel, Israel

    Sarai Sheinvald

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  1. Hadar Frenkel
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  2. Orna Grumberg
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Correspondence to Hadar Frenkel.

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This work was funded in part by the Binational Science Foundation (BSF Grant No. 2012259) and in part by the Israel Science Foundation (ISF Grant No. 979/11).

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Frenkel, H., Grumberg, O. & Sheinvald, S. An Automata-Theoretic Approach to Model-Checking Systems and Specifications Over Infinite Data Domains. J Autom Reasoning 63, 1077–1101 (2019). https://doi.org/10.1007/s10817-018-9494-0

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