Abstract
Variable automata with arithmetic enable the specification of reactive systems with variables over an infinite domain of numeric values and whose operation involves arithmetic manipulation of these values [9]. We study the synthesis problem for such specifications. While the problem is in general undecidable, we define a fragment, namely semantically deterministic variable automata with arithmetic, for which the problem is decidable. Essentially, an automaton is semantically deterministic if the restrictions on the possible assignments to the variables that are accumulated along its runs resolve its nondeterministic choices. We show that semantically deterministic automata can specify many interesting behaviors – many more than deterministic ones, and that the synthesis problem for them can be reduced to a solution of a two-player game. For automata with simple guards, the game has a finite state space, and the synthesis problem can be solved in time polynomial in the automaton and exponential in the number of its variables.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bloem, R., Chatterjee, K., Jobstmann, B.: Graph games and reactive synthesis. In: Clarke, E., Henzinger, T., Veith, H., Bloem, R. (eds.) Handbook of Model Checking, pp. 921–962. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-10575-8_27
Bojańczyk, M., Muscholl, A., Schwentick, T., Segoufin, L.: Two-variable logic on data trees and XML reasoning. J. ACM 56(3), 1–48 (2009)
Bouajjani, A., Habermehl, P., Mayr, R.R.: Automatic verification of recursive procedures with one integer parameter. TCS 295, 85–106 (2003)
Ceri, S., Fraternali, P., Bongio, A., Brambilla, M., Comai, S., Matera, M.: Designing Data-Intensive Web Applications. Morgan Kaufmann Publishers Inc., San Francisco (2002)
Church, A.: Logic, arithmetics, and automata. In: Proceedings of the International Congress of Mathematicians, 1962, pp. 23–35. Institut Mittag-Leffler (1963)
Delzanno, G., Sangnier, A., Traverso, R.: Parameterized verification of broadcast networks of register automata. In: Abdulla, P.A., Potapov, I. (eds.) RP 2013. LNCS, vol. 8169, pp. 109–121. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-41036-9_11
Ehlers, R., Seshia, S.A., Kress-Gazit, H.: Synthesis with identifiers. In: McMillan, K.L., Rival, X. (eds.) VMCAI 2014. LNCS, vol. 8318, pp. 415–433. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54013-4_23
Exibard, L., Filiot, E., Reynier, P.-A.: Synthesis of data word transducers. In: Proceedings of the 30th CONCUR (2019)
Faran, R., Kupferman, O.: LTL with arithmetic and its applications in reasoning about hierarchical systems. In: Proceedings of the 22nd LPAR. EPiC, vol. 57, pp. 343–362 (2018)
Grumberg, O., Kupferman, O., Sheinvald, S.: An automata-theoretic approach to reasoning about parameterized systems and specifications. In: Van Hung, D., Ogawa, M. (eds.) ATVA 2013. LNCS, vol. 8172, pp. 397–411. Springer, Cham (2013). https://doi.org/10.1007/978-3-319-02444-8_28
Henzinger, T.A., Piterman, N.: Solving games without determinization. In: Ésik, Z. (ed.) CSL 2006. LNCS, vol. 4207, pp. 395–410. Springer, Heidelberg (2006). https://doi.org/10.1007/11874683_26
Khalimov, A., Kupferman, O.: Register bounded synthesis. In: Proceedings of the 30th CONCUR (2019)
Khalimov, A., Maderbacher, B., Bloem, R.: Bounded synthesis of register transducers. In: Lahiri, S.K., Wang, C. (eds.) ATVA 2018. LNCS, vol. 11138, pp. 494–510. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-01090-4_29
Kupferman, O., Safra, S., Vardi, M.Y.: Relating word and tree automata. Ann. Pure Appl. Logic 138(1–3), 126–146 (2006)
Kupferman, O., Vardi, M.Y.: Safraless decision procedures. In: Proceedings of the 46th FoCS, pp. 531–540 (2005)
Neven, F., Schwentick, T., Vianu, V.: Towards regular languages over infinite alphabets. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 560–572. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44683-4_49
Niwiński, D., Walukiewicz, I.: Relating hierarchies of word and tree automata. In: Morvan, M., Meinel, C., Krob, D. (eds.) STACS 1998. LNCS, vol. 1373, pp. 320–331. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0028571
Pnueli, A., Rosner, R.: On the synthesis of a reactive module. In: Proceedings of the 16th POPL, pp. 179–190 (1989)
Safra, S.: On the complexity of \(\omega \)-automata. In: Proceedings of the 29th FoCS, pp. 319–327 (1988)
Schrijver, A.: Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, Hoboken (1999)
Shemesh, Y., Francez, N.: Finite-state unification automata and relational languages. Inf. Comput. 114, 192–213 (1994)
Vianu, V.: Automatic verification of database-driven systems: a new frontier. In: ICDT 2009, pp. 1–13 (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Faran, R., Kupferman, O. (2020). On Synthesis of Specifications with Arithmetic. In: Chatzigeorgiou, A., et al. SOFSEM 2020: Theory and Practice of Computer Science. SOFSEM 2020. Lecture Notes in Computer Science(), vol 12011. Springer, Cham. https://doi.org/10.1007/978-3-030-38919-2_14
Download citation
DOI: https://doi.org/10.1007/978-3-030-38919-2_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-38918-5
Online ISBN: 978-3-030-38919-2
eBook Packages: Computer ScienceComputer Science (R0)