Abstract
This paper proves the NP-completeness of the reachability problem for the class of flat counter machines with difference bounds and, more generally, octagonal relations, labeling the transitions on the loops. The proof is based on the fact that the sequence of powers \(\{R^i\}_{i=1}^\infty\) of such relations can be encoded as a periodic sequence of matrices, and that both the prefix and the period of this sequence are \(2^{{O}(||R||_2)}\) in the size of the binary encoding ||R||2 of a relation R. This result allows to characterize the complexity of the reachability problem for one of the most studied class of counter machines [6,10], and has a potential impact on other problems in program verification.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
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.
References
Bansal, K., Koskinen, E., Wies, T., Zufferey, D.: Structural counter abstraction. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 62–77. Springer, Heidelberg (2013)
Boigelot, B.: Symbolic Methods for Exploring Infinite State Spaces. PhD, Univ. de Liège (1999)
Bouajjani, A., Bozga, M., Habermehl, P., Iosif, R., Moro, P., Vojnar, T.: Programs with lists are counter automata. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 517–531. Springer, Heidelberg (2006)
Bozga, M., Gîrlea, C., Iosif, R.: Iterating octagons. In: Kowalewski, S., Philippou, A. (eds.) TACAS 2009. LNCS, vol. 5505, pp. 337–351. Springer, Heidelberg (2009)
Bozga, M., Habermehl, P., Iosif, R., Konečný, F., Vojnar, T.: Automatic verification of integer array programs. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 157–172. Springer, Heidelberg (2009)
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)
Bozga, M., Iosif, R., Konečný, F.: Safety problems are NP-complete for flat integer programs with octagonal loops. Tech. Rep. arXiv 1307.5321 (2013), http://arxiv.org/abs/1307.5321
Bozga, M., Iosif, R., Lakhnech, Y.: Flat parametric counter automata. Fundamenta Informaticae 91(2), 275–303 (2009)
Bozzelli, L., Pinchinat, S.: Verification of gap-order constraint abstractions of counter systems. In: Kuncak, V., Rybalchenko, A. (eds.) VMCAI 2012. LNCS, vol. 7148, pp. 88–103. Springer, Heidelberg (2012)
Comon, H., Jurski, Y.: Multiple counters automata, safety analysis and presburger arithmetic. In: Hu, A.J., Vardi, M.Y. (eds.) CAV 1998. LNCS, vol. 1427, pp. 268–279. Springer, Heidelberg (1998)
Demri, S., Dhar, A.K., Sangnier, A.: On the complexity of verifying regular properties on flat counter systems, In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part II. LNCS, vol. 7966, pp. 162–173. Springer, Heidelberg (2013)
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, vol. 7364, pp. 179–193. Springer, Heidelberg (2012)
Demri, S., Jurdzinski, M., Lachish, O., Lazic, R.: The covering and boundedness problems for branching vector addition systems. J. Comput. Syst. Sci. 79(1), 23–38 (2013)
Gawlitza, T.M., Monniaux, D.: Invariant generation through strategy iteration in succinctly represented control flow graphs. Logical Methods in Computer Science 8(3) (2012)
Gurari, E.M., Ibarra, O.H.: The complexity of the equivalence problem for simple programs. J. ACM 28(3), 535–560 (1981)
Ibarra, O.H.: Reversal-bounded multicounter machines and their decision problems. J. ACM 25(1), 116–133 (1978)
Leroux, J.: Vector addition system reachability problem: a short self-contained proof. In: POPL, pp. 307–316 (2011)
Minsky, M.: Computation: Finite and Infinite Machines. Prentice-Hall (1967)
Rackoff, C.: The covering and boundedness problems for vector addition systems. Theor. Comput. Sci. 6, 223–231 (1978)
Revesz, P.Z.: A closed-form evaluation for Datalog queries with integer (gap)-order constraints. Theor. Comput. Sci. 116(1&2), 117–149 (1993)
Schutter, B.D.: On the ultimate behavior of the sequence of consecutive powers of a matrix in the max-plus algebra. Linear Algebra and its Applications 307, 103–117 (2000)
Verma, K.N., Seidl, H., Schwentick, T.: On the complexity of equational Horn clauses. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 337–352. Springer, Heidelberg (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bozga, M., Iosif, R., Konečný, F. (2014). Safety Problems Are NP-complete for Flat Integer Programs with Octagonal Loops. In: McMillan, K.L., Rival, X. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2014. Lecture Notes in Computer Science, vol 8318. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54013-4_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-54013-4_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-54012-7
Online ISBN: 978-3-642-54013-4
eBook Packages: Computer ScienceComputer Science (R0)