Abstract
In this work, we study properties of deterministic finite-state automata with timers, a subclass of timed automata proposed by Vaandrager et al. as a candidate for an efficiently learnable timed model. We first study the complexity of the configuration reachability problem for such automata and establish that it is \(\textsf{PSPACE}\)-complete. Then, as simultaneous timeouts (we call these, races) can occur in timed runs of such automata, we study the problem of determining whether it is possible to modify the delays between the actions in a run, in a way to avoid such races. The absence of races is important for modelling purposes and to streamline learning of automata with timers. We provide an effective characterization of when an automaton is race-avoiding and establish that the related decision problem is in \(\textsf{3EXP}\) and \(\textsf{PSPACE}\)-hard.
This work was supported by the Belgian FWO “SAILor” project (G030020N). Gaëtan Staquet is a research fellow (Aspirant) of the Belgian F.R.S.-FNRS. The research of Frits Vaandrager was supported by NWO TOP project 612.001.852 “Grey-box learning of Interfaces for Refactoring Legacy Software (GIRLS)”.
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Notes
- 1.
Notation \({\textsf {dom}}(f)\) means the domain of the partial function f.
- 2.
The reason for this choice will be clarified at the end of this section.
- 3.
When using the action indices in the blocks, we have \(B_1 = (1 ~3, \bot )\) and \(B_2 = (2 ~ 4, \bot )\).
- 4.
Recall that the sequence of a block can be composed of a single action.
References
Aceto, L., Laroussinie, F.: Is your model checker on time? On the complexity of model checking for timed modal logics. J. Log. Algebraic Meth. Program. 52–53, 7–51 (2002). https://doi.org/10.1016/S1567-8326(02)00022-X
Aichernig, B.K., Pferscher, A., Tappler, M.: From passive to active: learning timed automata efficiently. In: Lee, R., Jha, S., Mavridou, A., Giannakopoulou, D. (eds.) NFM 2020. LNCS, vol. 12229, pp. 1–19. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-55754-6_1
Alur, R.: Timed automata. In: Halbwachs, N., Peled, D. (eds.) CAV 1999. LNCS, vol. 1633, pp. 8–22. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48683-6_3
Alur, R., Dill, D.L.: A theory of timed automata. Theor. Comput. Sci. 126(2), 183–235 (1994). https://doi.org/10.1016/0304-3975(94)90010-8
An, J., Chen, M., Zhan, B., Zhan, N., Zhang, M.: Learning One-Clock Timed Automata. In: TACAS 2020, Part I. LNCS, vol. 12078, pp. 444–462. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45190-5_25
Angluin, D.: Learning regular sets from queries and counterexamples. Inf. Comput. 75(2), 87–106 (1987). https://doi.org/10.1016/0890-5401(87)90052-6
Baier, C., Katoen, J.: Principles of Model Checking. MIT Press (2008)
Bouyer, P., Fahrenberg, U., Larsen, K.G., Markey, N., Ouaknine, J., Worrell, J.: Model checking real-time systems. In: Handbook of Model Checking, pp. 1001–1046. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-10575-8_29
Bruyère, V., Pérez, G.A., Staquet, G., Vaandrager, F.W.: Automata with timers. CoRR abs/2305.07451 (2023). https://doi.org/10.48550/arXiv.2305.07451
Clarke, E.M., Henzinger, T.A., Veith, H., Bloem, R. (eds.): Handbook of Model Checking. Springer, Heidelberg (2018). https://doi.org/10.1007/978-3-319-10575-8
Dill, D.L.: Timing assumptions and verification of finite-state concurrent systems. In: Sifakis, J. (ed.) CAV 1989. LNCS, vol. 407, pp. 197–212. Springer, Heidelberg (1990). https://doi.org/10.1007/3-540-52148-8_17
Grädel, E., Thomas, W., Wilke, T.: Automata, Logics, and Infinite Games: A Guide to Current Research, vol. 2500. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-36387-4
Grinchtein, O., Jonsson, B., Leucker, M.: Learning of event-recording automata. Theor. Comput. Sci. 411(47), 4029–4054 (2010). https://doi.org/10.1016/j.tcs.2010.07.008
Grinchtein, O., Jonsson, B., Pettersson, P.: Inference of event-recording automata using timed decision trees. In: Baier, C., Hermanns, H. (eds.) CONCUR 2006. LNCS, vol. 4137, pp. 435–449. Springer, Heidelberg (2006). https://doi.org/10.1007/11817949_29
Howar, F., Steffen, B.: Active automata learning in practice. In: Bennaceur, A., Hähnle, R., Meinke, K. (eds.) Machine Learning for Dynamic Software Analysis: Potentials and Limits. LNCS, vol. 11026, pp. 123–148. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96562-8_5
Thomas, W.: Languages, automata, and logic. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, pp. 389–455. Springer, Heidelberg (1997). https://doi.org/10.1007/978-3-642-59126-6_7
Tripakis, S., Yovine, S.: Analysis of timed systems using time-abstracting bisimulations. Formal Meth. Syst. Des. 18(1), 25–68 (2001). https://doi.org/10.1023/A:1008734703554
Vaandrager, F., Ebrahimi, M., Bloem, R.: Learning Mealy machines with one timer. Inf. Comput., 105013 (2023). https://doi.org/10.1016/j.ic.2023.105013
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Bruyère, V., Pérez, G.A., Staquet, G., Vaandrager, F.W. (2023). Automata with Timers. In: Petrucci, L., Sproston, J. (eds) Formal Modeling and Analysis of Timed Systems. FORMATS 2023. Lecture Notes in Computer Science, vol 14138. Springer, Cham. https://doi.org/10.1007/978-3-031-42626-1_3
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