Skip to main content

Bounded variability of metric temporal logic

Abstract

Deciding validity of Metric Temporal Logic (MTL) formulas is generally very complex and even undecidable over dense time domains; bounded variability is one of the several restrictions that have been proposed to bring decidability back. A temporal model has bounded variability if no more than v events occur over any time interval of length V, for constant parameters v and V. Previous work has shown that MTL validity over models with bounded variability is less complex—and often decidable—than MTL validity over unconstrained models. This paper studies the related problem of deciding whether an MTL formula has intrinsic bounded variability, that is whether it is satisfied only by models with bounded variability. The results of the paper are mainly negative: over dense time domains, the problem is mostly undecidable (even if with an undecidability degree that is typically lower than deciding validity); over discrete time domains, it is decidable with the same complexity as deciding validity. As a partial complement to these negative results, the paper also identifies MTL fragments where deciding bounded variability is simpler than validity, which may provide for a reduction in complexity in some practical cases.

References

  1. Abadi, M., Lamport, L.: An old-fashioned recipe for real-time. ACM Trans. Program. Lang. Syst. 16(5), 1543–1571 (1994)

    Article  Google Scholar 

  2. Alur, R., Feder, T., Henzinger, T.A.: The benefits of relaxing punctuality. J. ACM 43(1), 116–146 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  3. Alur, R., Henzinger, T.A.: Real-time logics: Complexity and expressiveness. Inf. Comp. 104(1), 35–77 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  4. Alur, R., Henzinger, T.A.: A really temporal logic. J. ACM 41(1), 181–204 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  5. Bauland, M., Schneider, T., Schnoor, H., Schnoor, I., Vollmer, H.: The complexity of generalized satisfiability for linear temporal logic. Logical Methods in Computer Science 5(1) (2009)

  6. Bouyer, P., Markey, N., Ouaknine, J., Worrell, J.: The Cost of Punctuality. In: ACM/IEEE Symposium on Logic in Computer Science, pp 109–120 (2007)

  7. Bresolin, D., Della Monica, D., Goranko, V., Montanari, A., Sciavicco, G.: The dark side of interval temporal logic: Sharpening the undecidability border. In: International Symposium on Temporal Representation and Reasoning, pp 131–138 (2011)

  8. Bresolin, D., Della monica, D., Goranko, V., Montanari, A., Sciavicco, G.: Metric propositional neighborhood logics on natural numbers. Softw. Syst. Model. 12(2), 245–264 (2013)

    Article  MATH  Google Scholar 

  9. Bresolin, D., Monica, D. D., Montanari, A., Sala, P., Sciavicco, G.: Interval temporal logics over strongly discrete linear orders: Expressiveness and complexity. Theor. Comput. Sci. 560, 269–291 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  10. Bresolin, D., Montanari, A., Sala, P., Sciavicco, G.: Optimal decision procedures for MPNL over finite structures, the natural numbers, and the integers. Theor. Comput. Sci. 493, 98–115 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  11. Chaochen, Z., Hansen, M.R., Sestoft, P.: Decidability and Undecidability Results for Duration Calculus. In: Enjalbert, P., Finkel, A., Wagner, K.W. (eds.) STACS 93, 10th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, vol. 665, pp 58–68. Springer (1993)

  12. Demri, S., Schnoebelen, P.: The complexity of propositional linear temporal logics in simple cases. Inf. Comput. 174(1), 84–103 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  13. D’Souza, D., Prabhakar, P.: On the expressiveness of MTL in the pointwise and continuous semantics. STTT 9(1), 1–4 (2007)

    Article  Google Scholar 

  14. D’Souza, D., Prabhakar, P.: On the expressiveness of MTL in the pointwise and continuous semantics. STTT 9(1), 1–4 (2007)

    Article  Google Scholar 

  15. Emerson, E.A.: Temporal and Modal Logic. In: Handbook of Theoretical Computer Science, vol. B, pp 996–1072. Elsevier Science (1990)

  16. Fränzle, M.: Model-checking dense-time duration calculus. Formal Asp. Comput. 16(2), 121–139 (2004)

    Article  MATH  Google Scholar 

  17. Furia, C.A., Mandrioli, D., Morzenti, A., Rossi, M.: Modeling Time in Computing. Monographs in Theoretical Computer Science. An EATCS series Springer (2012)

  18. Furia, C.A., Rossi, M.: MTL with bounded variability: Decidability and complexity. In: FORMATS, LNCS. Extended version in [19], vol. 5215, pp 109–123. Springer (2008)

  19. Furia, C.A., Rossi, M.: MTL with bounded variability: Decidability and complexity. Tech. Rep. 2008.10, Dipartimento di Elettronica e Informazione, Politecnico di Milano. Available at http://bugcounting.net/publications.html#MTLwBoundedVar-TR08 (2008)

  20. Furia, C.A., Rossi, M.: A theory of sampling for continuous-time metric temporal logic. ACM Transactions on Computational Logic 12(1), 1–40 (2010). Article 8

    MathSciNet  Article  MATH  Google Scholar 

  21. Furia, C.A., Spoletini, P.: On Relaxing Metric Information in Linear Temporal Logic. In: International Symposium on Temporal Representation and Reasoning, pp 72–79. IEEE (2011)

  22. Furia, C.A., Spoletini, P.: Automata-Based Verification of Linear Temporal Logic Models with Bounded Variability. In: International Symposium on Temporal Representation and Reasoning, pp 89–96. IEEE (2012)

  23. Furia, C.A., Spoletini, P.: Bounded Variability of Metric Temporal Logic. In: Cesta, A., Combi, C., Laroussinie, F. (eds.) Proceedings of the 21st International Symposium on Temporal Representation and Reasoning (TIME’14), pp 155–163. IEEE Computer Society (2014)

  24. Gabbay, D.M., Hodkinson, I., Reynolds, M.: Temporal Logic (vol. 1): mathematical foundations and computational aspects, Oxford Logic Guides, vol. 28. Oxford University Press (1994)

  25. Gabbay, D.M., Pnueli, A., Shelah, S., Stavi, J.: On the Temporal Basis of Fairness. In: Conference Record of the 7Th Annual ACM Symposium on Principles of Programming Languages (POPL’80), pp 163–173 (1980)

  26. Hirshfeld, Y., Rabinovich, A.: Logics for real time: Decidability and complexity. Fundam. Inf. 62(1), 1–28 (2004)

    MathSciNet  MATH  Google Scholar 

  27. Hirshfeld, Y., Rabinovich, A.: Continuous time temporal logic with counting. Inf. Comput. 214, 1–9 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  28. Hirshfeld, Y., Rabinovich, A.M.: Logics for real time: Decidability and complexity. Fundam. Inform. 62(1), 1–28 (2004)

    MathSciNet  MATH  Google Scholar 

  29. Hunter, P., Ouaknine, J., Worrell, J.: Expressive Completeness for Metric Temporal Logic. In: LICS, pp 349–357. IEEE (2013)

  30. Kamp, J.A.W.: Tense Logic and the Theory of Linear Order. Ph.D. Thesis. University of California, Los Angeles (1968)

    Google Scholar 

  31. Koymans, R.: Specifying real-time properties with metric temporal logic. Real-Time Systems 2(4), 255–299 (1990)

    Article  Google Scholar 

  32. Lamport, L.: Proving the correctness of multiprocess programs. IEEE Trans. Softw. Eng. SE-3(2), 125–143 (1977)

    MathSciNet  Article  MATH  Google Scholar 

  33. Lutz, C., Walther, D., Wolter, F.: Quantitative temporal logics over the reals: PSPACE and below. Inf. Comput. 205(1), 99–123 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  34. Maler, O., Nickovic, D., Pnueli, A.: Real Time Temporal Logic: Past, present, future. In: Petterson, P., Yi, W. (eds.) Proceedings of the 3rd International Conference on Formal Modeling and Analysis of Timed Systems (FORMATS’05), Lecture Notes in Computer Science, vol. 3829, pp 2–16. Springer-Verlag (2005)

  35. Maler, O., Nickovic, D., Pnueli, A.: From MITL to Timed Automata. In: Asarin, E., Bouyer, P. (eds.) Proceedings of the 4th International Conference on Formal Modeling and Analysis of Timed Systems (FORMATS’06), Lecture Notes in Computer Science, vol. 4202, pp 274–289. Springer-Verlag (2006)

  36. Maler, O., Nickovic, D., Pnueli, A.: Checking Temporal Properties of Discrete, Timed and Continuous Behaviors. In: Pillars of Computer Science, Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85Th Birthday, Lecture Notes in Computer Science, vol. 4800, pp 475–505. Springer (2008)

  37. Manna, Z., Pnueli, A.: A Hierarchy of Temporal Properties. In: Proceedings of the 9Th Annual ACM Symposium on Principles of Distributed Computing, pp 377–410. ACM (1990)

  38. Minsky, M.L.: Computation: Finite and infinite machines prentice hall (1967)

  39. Montanari, A., Pazzaglia, M., Sala, P.: Metric Propositional Neighborhood Logic with an Equivalence Relation. In: 21St International Symposium on Temporal Representation and Reasoning, (TIME), pp 49–58. IEEE Computer Society (2014)

  40. Montanari, A., Puppis, G., Sala, P.: Decidability of the Interval Temporal Logic \(\mathsf {A}\bar {\mathsf {A}}\mathsf {B}\bar {\mathsf {B}}\) over the Rationals. In: Mathematical Foundations of Computer Science 2014 - 39Th International Symposium, MFCS 2014, Budapest, Hungary, August 25-29, 2014. Proceedings, Part I, pp 451–463 (2014)

  41. Montanari, A., Sala, P.: An Optimal Tableau System for the Logic of Temporal Neighborhood over the Reals. In: 19Th International Symposium on Temporal Representation and Reasoning, TIME 2012, Leicester, United Kingdom, September 12-14, 2012, pp 39–46 (2012)

  42. Nickovic, D., Piterman, N.: From MTL to Deterministic Timed Automata. In: Chatterjee, K., Henzinger, T.A. (eds.) Formal Modeling and Analysis of Timed Systems – 8th International Conference, FORMATS 2010, Lecture Notes in Computer Science, vol. 6246, pp 152–167. Springer (2010)

  43. Ouaknine, J., Rabinovich, A., Worrell, J.: Time-Bounded Verification. In: Bravetti, M. , Zavattaro, G. (eds.) CONCUR 2009 – Concurrency Theory, 20th International Conference, Lecture Notes in Computer Science, vol. 5710, pp 496–510. Springer (2009)

  44. Ouaknine, J., Worrell, J.: On Metric Temporal Logic and Faulty Turing Machines. In: FoSSaCS, LNCS, vol. 3921, pp 217–230. Springer (2006)

  45. Ouaknine, J., Worrell, J.: On the decidability and complexity of metric temporal logic over finite words. Logical Methods in Computer Science 3(1) (2007)

  46. Ouaknine, J., Worrell, J.: Some Recent Results in Metric Temporal Logic. In: FORMATS, LNCS, vol. 5215, pp 1–13. Springer (2008)

  47. Ouaknine, J., Worrell, J.: Towards a Theory of Time-Bounded Verification. In: Abramsky, S., Gavoille, C., Kirchner, C., auf der Heide, F.M., Spirakis, P.G. (eds.) Automata, Languages and Programming, 37th International Colloquium, ICALP 2010, Lecture Notes in Computer Science, vol. 6199, pp 22–37. Springer (2010)

  48. Papadimitriou, C.: Computational complexity Addison-Wesley (1994)

  49. Perrin, D., Pin, J.E.́: Infinite Words, Pure and Applied Mathematics, vol. 141. Elsevier (2004)

  50. Pnueli, A.: The Temporal Logic of Programs. In: Proceedings of the 18Th Annual Symposium on Foundations of Computer Science, SFCS ’77, pp 46–57. IEEE Computer Society (1977)

  51. Rabinovich, A.: Complexity of Metric Temporal Logics with Counting and the Pnueli Modalities. In: FORMATS, Lecture Notes in Computer Science, vol. 5215, pp 93–108. Springer (2008)

  52. Rabinovich, A.: Complexity of metric temporal logics with counting and the Pnueli modalities. Theor. Comput. Sci. 411(22-24), 2331–2342 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  53. Rabinovich, A.M.: Expressive completeness of Duration Calculus. Inf. Comput. 156(1-2), 320–344 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  54. Reynolds, M.: The complexity of temporal logic over the reals. Ann. Pure Appl. Logic 161(8), 1063–1096 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  55. Reynolds, M.: Metric temporal reasoning with less than two clocks. Journal of Applied Non-Classical Logics 20(4), 437–455 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  56. Reynolds, M.: A New Metric Temporal Logic for Hybrid Systems. In: 20Th International Symposium on Temporal Representation and Reasoning (TIME), pp 73–80. IEEE Computer Society (2013)

  57. Rogers, Jr., H.: Theory of recursive functions and effective computability MIT press (1987)

  58. Shepherdson, J.C., Sturgis, H.E.: Computability of recursive functions. J. ACM 10(2) (1963)

  59. Sistla, A.P., Clarke, E.M.: The complexity of propositional linear temporal logics. J. ACM 32(3), 733–749 (1985)

    MathSciNet  Article  MATH  Google Scholar 

  60. Vardi, M.Y.: An Automata-Theoretic Approach to Linear Temporal Logic. In: Logics for Concurrency – Structure versus Automata (8Th Banff Higher Order Workshop), Lecture Notes in Computer Science, vol. 1043, pp 238–266. Springer (1995)

  61. Vardi, M.Y., Wolper, P.: An Automata-Theoretic Approach to Automatic Program Verification. In: LICS, pp 332–344. IEEE (1986)

  62. Wilke, T.: Specifying Timed State Sequences in Powerful Decidable Logics and Timed Automata. In: FTRTFT, LNCS, vol. 863, pp 694–715. Springer (1994)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Carlo A. Furia.

Additional information

The first author’s work was partially done at the Chair of Software Engineering of ETH Zurich, Switzerland. A preliminary version of this work appeared in the 21st International Symposium on Temporal Representation and Reasoning (TIME) in 2014 [23].

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Furia, C.A., Spoletini, P. Bounded variability of metric temporal logic. Ann Math Artif Intell 80, 283–316 (2017). https://doi.org/10.1007/s10472-016-9532-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10472-016-9532-8

Keywords

  • Metric temporal logic
  • Bounded variability
  • Decidability and complexity

Mathematics Subject Classification (2010)

  • 03B70
  • 03B44