Quantitative Monitoring of STL with Edit Distance

  • Stefan JakšićEmail author
  • Ezio Bartocci
  • Radu Grosu
  • Dejan Ničković
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10012)


In cyber-physical systems (CPS), physical behaviors are typically controlled by digital hardware. As a consequence, continuous behaviors are discretized by sampling and quantization prior to their processing. Quantifying the similarity between CPS behaviors and their specification is an important ingredient in evaluating correctness and quality of such systems. We propose a novel procedure for measuring robustness between digitized CPS signals and Signal Temporal Logic (STL) specifications. We first equip STL with quantitative semantics based on the weighted edit distance (WED), a metric that quantifies both space and time mismatches between digitized CPS behaviors. We then develop a dynamic programming algorithm for computing the robustness degree between digitized signals and STL specifications. We implemented our approach and evaluated it on an automotive case study.


Field Programmable Gate Array Edit Distance Quantization Step Discrete Signal Continuous Behavior 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We would like to thank Oded Maler, Mario Klima and the anonymous reviewers for their comments on the earlier drafts of the paper.

We acknowledge the support of the IKT der Zukunft of Austrian FFG project HARMONIA (nr. 845631), the ICT COST Action IC1402 Runtime Verification beyond Monitoring (ARVI), the Austrian National Research Network S 11405-N23 and S 11412-N23 (RiSE/SHiNE) of the Austrian Science Fund (FWF) and the Doctoral Program Logical Methods in Computer Science of the Austrian Science Fund (FWF).


  1. 1.
    Akazaki, T., Hasuo, I.: Time robustness in MTL and expressivity in hybrid system falsification. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9207, pp. 356–374. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-21668-3_21 CrossRefGoogle Scholar
  2. 2.
    Allauzen, C., Mohri, M.: Linear-space computation of the edit-distance between a string and a finite automaton. CoRR abs/0904.4686 (2009)Google Scholar
  3. 3.
    Annpureddy, Y., Liu, C., Fainekos, G., Sankaranarayanan, S.: S-TaLiRo: a tool for temporal logic falsification for hybrid systems. In: Abdulla, P.A., Leino, K.R.M. (eds.) TACAS 2011. LNCS, vol. 6605, pp. 254–257. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-19835-9_21 CrossRefGoogle Scholar
  4. 4.
    Abbas, H., Hoxha, B., Fainekos, G.: Benchmarks for temporal logic requirements for automotive systems. In: Proceedings of Applied Verification for Continuous and Hybrid Systems (2014)Google Scholar
  5. 5.
    Bartocci, E., Bortolussi, L., Sanguinetti, G.: Data-driven statistical learning of temporal logic properties. In: Legay, A., Bozga, M. (eds.) FORMATS 2014. LNCS, vol. 8711, pp. 23–37. Springer, Heidelberg (2014). doi: 10.1007/978-3-319-10512-3_3 Google Scholar
  6. 6.
    Brim, L., Dluhos, P., Safránek, D., Vejpustek, T.: STL*: extending signal temporal logic with signal-value freezing operator. Inf. Comput. 236, 52–67 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Deshmukh, J.V., Majumdar, R., Prabhu, V.S.: Quantifying conformance using the Skorokhod metric. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9207, pp. 234–250. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-21668-3_14 CrossRefGoogle Scholar
  8. 8.
    Deshmukh, J.V., Majumdar, R., Prabhu, V.S.: Quantifying conformance using the Skorokhod metric (full version). CoRR abs/1505.05832 (2015)Google Scholar
  9. 9.
    Donzé, A.: Breach, a toolbox for verification and parameter synthesis of hybrid systems. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 167–170. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-14295-6_17 CrossRefGoogle Scholar
  10. 10.
    Donzé, A., Ferrère, T., Maler, O.: Efficient robust monitoring for STL. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 264–279. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-39799-8_19 CrossRefGoogle Scholar
  11. 11.
    Donzé, A., Maler, O.: Robust satisfaction of temporal logic over real-valued signals. In: Chatterjee, K., Henzinger, T.A. (eds.) FORMATS 2010. LNCS, vol. 6246, pp. 92–106. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-15297-9_9 CrossRefGoogle Scholar
  12. 12.
    Fainekos, G.E., Pappas, G.J.: Robustness of temporal logic specifications for continuous-time signals. Theor. Comput. Sci. 410(42), 4262–4291 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fainekos, G.E., Sankaranarayanan, S., Ivancic, F., Gupta, A.: Robustness of model-based simulations. In: Proceedings of the 30th IEEE Real-Time Systems Symposium, RTSS 2009, Washington, DC, USA, 1–4 December 2009, pp. 345–354 (2009)Google Scholar
  14. 14.
    Gerth, R., Peled, D., Vardi, M.Y., Wolper, P.: Simple on-the-fly automatic verification of linear temporal logic. In: Protocol Specification, Testing and Verification XV, Proceedings of the Fifteenth IFIP WG6.1 International Symposium on Protocol Specification, Testing and Verification, Warsaw, Poland, pp. 3–18 (1995)Google Scholar
  15. 15.
    Konstantinidis, S.: Computing the edit distance of a regular language. Inf. Comput. 205(9), 1307–1316 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Krause, E.F.: Taxicab Geometry: An Adventure in Non-Euclidean Geometry. Courier Corporation, North Chelmsford (2012)Google Scholar
  17. 17.
    Levenshtein, V.I.: Binary codes capable of correcting deletions, insertions, reversals. Sov. Phys. Dokl. 10, 707 (1966)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Maler, O., Nickovic, D.: Monitoring properties of analog and mixed-signal circuits. STTT 15(3), 247–268 (2013)CrossRefGoogle Scholar
  19. 19.
    Mohri, M.: Edit-distance of weighted automata: general definitions and algorithms. Int. J. Found. Comput. Sci. 14(6), 957–982 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nguyen, T., Ničković, D.: Assertion-based monitoring in practice–checking correctness of an automotive sensor interface. In: Lang, F., Flammini, F. (eds.) FMICS 2014. LNCS, vol. 8718, pp. 16–32. Springer, Heidelberg (2014). doi: 10.1007/978-3-319-10702-8_2 Google Scholar
  21. 21.
    Pnueli, A., Zaks, A.: On the merits of temporal testers. In: Grumberg, O., Veith, H. (eds.) 25 Years of Model Checking. LNCS, vol. 5000, pp. 172–195. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-69850-0_11 CrossRefGoogle Scholar
  22. 22.
    Rizk, A., Batt, G., Fages, F., Soliman, S.: On a continuous degree of satisfaction of temporal logic formulae with applications to systems biology. In: Heiner, M., Uhrmacher, A.M. (eds.) CMSB 2008. LNCS (LNBI), vol. 5307, pp. 251–268. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-88562-7_19 CrossRefGoogle Scholar
  23. 23.
    Samanta, R., Deshmukh, J.V., Chaudhuri, S.: Robustness analysis of string transducers. In: Van Hung, D., Ogawa, M. (eds.) ATVA 2013. LNCS, vol. 8172, pp. 427–441. Springer, Heidelberg (2013). doi: 10.1007/978-3-319-02444-8_30 CrossRefGoogle Scholar
  24. 24.
    Schulz, K.U., Mihov, S.: Fast string correction with Levenshtein automata. Int. J. Doc. Anal. Recogn. 5(1), 67–85 (2002)CrossRefzbMATHGoogle Scholar
  25. 25.
    Veanes, M., Bjørner, N., de Moura, L.: Symbolic automata constraint solving. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR-17. LNCS, vol. 6397, pp. 640–654. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-16242-8_45 CrossRefGoogle Scholar
  26. 26.
    Wagner, R.A.: Order-n correction for regular languages. Commun. ACM 17(5), 265–268 (1974)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Stefan Jakšić
    • 1
    • 2
    Email author
  • Ezio Bartocci
    • 2
  • Radu Grosu
    • 2
  • Dejan Ničković
    • 1
  1. 1.AIT Austrian Institute of TechnologySeibersdorfAustria
  2. 2.Faculty of InformaticsVienna University of TechnologyViennaAustria

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