Abstract
Among the approximation methods for the verification of counter systems, one of them consists in model-checking their flat unfoldings. Unfortunately, the complexity characterization of model-checking problems for such operational models is not always well studied except for reachability queries or for Past LTL. In this paper, we characterize the complexity of model-checking problems on flat counter systems for the specification languages including first-order logic, linear mu-calculus, infinite automata, and related formalisms. Our results span different complexity classes (mainly from PTime to PSpace) and they apply to languages in which arithmetical constraints on counter values are systematically allowed. As far as the proof techniques are concerned, we provide a uniform approach that focuses on the main issues.
Work partially supported by the EU Seventh Framework Programme under grant agreement No. PIOF-GA-2011-301166 (DATAVERIF).
A version with proofs is available as [5].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Boigelot, B.: Symbolic methods for exploring infinite state spaces. PhD thesis, Université de Liège (1998)
Bozga, M., Iosif, R., Konečný, F.: Fast acceleration of ultimately periodic relations. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 227–242. Springer, Heidelberg (2010)
Comon, H., Jurski, Y.: Multiple counter automata, safety analysis and PA. In: Vardi, M.Y. (ed.) CAV 1998. LNCS, vol. 1427, pp. 268–279. Springer, Heidelberg (1998)
Demri, S., Dhar, A.K., Sangnier, A.: Taming Past LTL and Flat Counter Systems. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS (LNAI), vol. 7364, pp. 179–193. Springer, Heidelberg (2012)
Demri, S., Dhar, A.K., Sangnier, A.: On the complexity of verifying regular properties on flat counter systems (2013), http://arxiv.org/abs/1304.6301
Demri, S., Finkel, A., Goranko, V., van Drimmelen, G.: Model-checking \(\textsf{CTL}^{*}\) over flat Presburger counter systems. JANCL 20(4), 313–344 (2010)
Finkel, A., Leroux, J.: How to compose presburger-accelerations: Applications to broadcast protocols. In: Agrawal, M., Seth, A.K. (eds.) FSTTCS 2002. LNCS, vol. 2556, pp. 145–156. Springer, Heidelberg (2002)
Jančar, P., Sawa, Z.: A note on emptiness for alternating finite automata with a one-letter alphabet. IPL 104(5), 164–167 (2007)
Jurdziński, M.: Deciding the winner in parity games is in UP ∩ co-UP. IPL 68(3), 119–124 (1998)
Kuhtz, L., Finkbeiner, B.: Weak kripke structures and LTL. In: Katoen, J.-P., König, B. (eds.) CONCUR 2011. LNCS, vol. 6901, pp. 419–433. Springer, Heidelberg (2011)
Kupferman, O., Vardi, M.: Weak alternating automata are not that weak. ACM Transactions on Computational Logic 2(3), 408–429 (2001)
Kučera, A., Strejček, J.: The stuttering principle revisited. Acta Informatica 41(7-8), 415–434 (2005)
Laroussinie, F., Markey, N., Schnoebelen, P.: Temporal logic with forgettable past. In: LICS 2002, pp. 383–392. IEEE (2002)
Leroux, J., Sutre, G.: Flat counter systems are everywhere! In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 489–503. Springer, Heidelberg (2005)
Markey, N., Schnoebelen, P.: Model checking a path. In: Amadio, R.M., Lugiez, D. (eds.) CONCUR 2003. LNCS, vol. 2761, pp. 251–265. Springer, Heidelberg (2003)
Markey, N., Schnoebelen, P.: Mu-calculus path checking. IPL 97(6) (2006)
Minsky, M.: Computation, Finite and Infinite Machines. Prentice Hall (1967)
Miyano, S., Hayashi, T.: Alternating finite automata on ω-words. Theor. Comput. Sci. 32, 321–330 (1984)
Piterman, N.: Extending temporal logic with ω-automata. Master’s thesis, The Weizmann Institute of Science (2000)
Pottier, L.: Minimal Solutions of Linear Diophantine Systems: Bounds and Algorithms. In: Book, R.V. (ed.) RTA 1991. LNCS, vol. 488, pp. 162–173. Springer, Heidelberg (1991)
Sistla, A., Clarke, E.: The complexity of propositional linear temporal logic. JACM 32(3), 733–749 (1985)
Stockmeyer, L.J.: The complexity of decision problems in automata and logic. PhD thesis, MIT (1974)
Vardi, M.: A temporal fixpoint calculus. In: POPL 1988, pp. 250–259. ACM (1988)
Vardi, M., Wolper, P.: Reasoning about infinite computations. I&C 115 (1994)
Wolper, P.: Temporal logic can be more expressive. I&C 56, 72–99 (1983)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Demri, S., Dhar, A.K., Sangnier, A. (2013). On the Complexity of Verifying Regular Properties on Flat Counter Systems,. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds) Automata, Languages, and Programming. ICALP 2013. Lecture Notes in Computer Science, vol 7966. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39212-2_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-39212-2_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39211-5
Online ISBN: 978-3-642-39212-2
eBook Packages: Computer ScienceComputer Science (R0)