# An Over-Approximation Forward Analysis for Nested Timed Automata

## Abstract

*Nested timed automata (NeTAs)*, proposed by Li et al. are a pushdown system whose stack symbols are *timed automata (TAs)*. With this formal models, we can model and analyze complex real-time frameworks with recursive context switches. The reachability problem of NeTAs is proved to be decidable, via encoding NeTAs to *dense timed pushdown automata (DTPDAs)*. This paper gives a forward algorithm for reachability problem of NeTAs, by dividing the problem into two phases and integrating these two corresponding results. One phase is the reachability checking for the stack contents (i.e. TAs) and another is the state reachability problem for the TAs nested in an NeTA. The algorithm neglects time accumulation during context switches and thus an over-approximation of the original problem. As the result, the algorithm gains soundness in the sense that there exists one corresponding timed trace in the NeTA when the approximation has a timed trace to the state in less time-complexity.

## Notes

### Acknowledgment

The authors thank Prof. Mizuhito Ogawa for fruitful discussions on the research. This research is supported by the National Nature Science Foundation of China (61100052, 61472240, 91318301, 61261130589), JSPS Kakenhi Grant-in-Aid for Scientific Research(B) (25280023) and Challenging Exploratory Research(26540026).

## References

- 1.Li, G., Cai, X., Ogawa, M., Yuen, S.: Nested timed automata. In: Braberman, V., Fribourg, L. (eds.) FORMATS 2013. LNCS, vol. 8053, pp. 168–182. Springer, Heidelberg (2013) CrossRefGoogle Scholar
- 2.Abdulla, P.A., Atig, M.F., Stenman, J.: Dense-Timed pushdown automata. In: Proceedings of the 27th Annual IEEE Symposium on Logic in Computer Science (LICS’12). IEEE Computer Society (2012), pp. 35–44 (2012)Google Scholar
- 3.Alur, R., Dill, D.L.: A theory of timed automata. Theor. Comput.Sci.
**126**, 183–235 (1994)CrossRefMATHMathSciNetGoogle Scholar - 4.Henzinger, T.A., Nicollin, X., Sifakis, J., Yovine, S.: Symbolic model checking for real-time systems. Inf. Comput.
**111**, 193–244 (1994)CrossRefMATHMathSciNetGoogle Scholar - 5.Larsen, K.G., Pettersson, P., Yi, W.: Uppaal in a nutshell. Int. J. Softw. Tools Technol. Transfer
**1**, 134–152 (1997)CrossRefMATHGoogle Scholar - 6.Bengtsson, J.E., Yi, W.: Timed automata: semantics, algorithms and tools. In: Desel, J., Reisig, W., Rozenberg, G. (eds.) Lectures on Concurrency and Petri Nets. LNCS, vol. 3098, pp. 87–124. Springer, Heidelberg (2004) CrossRefGoogle Scholar
- 7.Schwoon, S.: Model-checking pushdown system. Ph.D. thesis, Technical University of Munich (2000)Google Scholar
- 8.Bouyer, P., Dufourd, C., Fleury, E., Petit, A.: Updatable timed automata. Theor. Comput. Sci.
**321**, 291–345 (2004)CrossRefMATHMathSciNetGoogle Scholar - 9.Bouyer, P.: Forward analysis of updatable timed automata. Formal Methods Syst. Design
**24**, 281–320 (2004)CrossRefMATHGoogle Scholar - 10.Ausmees, K.: Zone-based reachability analysis of dense-timed pushdown automata. Student thesis, IT 12 034, Department of Information Technology, Uppsala University (2012)Google Scholar
- 11.Trivedi, A., Wojtczak, D.: Recursive timed automata. In: Bouajjani, A., Chin, W.-N. (eds.) ATVA 2010. LNCS, vol. 6252, pp. 306–324. Springer, Heidelberg (2010) CrossRefGoogle Scholar