Abstract
This paper studies reachability, coverability and inclusion problems for Integer Vector Addition Systems with States (ℤ-VASS) and extensions and restrictions thereof. A ℤ-VASS comprises a finite-state controller with a finite number of counters ranging over the integers. Although it is folklore that reachability in ℤ-VASS is NP-complete, it turns out that despite their naturalness, from a complexity point of view this class has received little attention in the literature. We fill this gap by providing an in-depth analysis of the computational complexity of the aforementioned decision problems. Most interestingly, it turns out that while the addition of reset operations to ordinary VASS leads to undecidability and Ackermann-hardness of reachability and coverability, respectively, they can be added to ℤ-VASS while retaining NP-completeness of both coverability and reachability.
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.
References
Bell, P., Potapov, I.: On undecidability bounds for matrix decision problems. Theor. Comput. Sci. 391(1-2), 3–13 (2008)
Borosh, I., Treybing, L.B.: Bounds on positive integral solutions of linear Diophantine equations. Proc. AMS 55, 299–304 (1976)
Buckheister, P., Zetzsche, G.: Semilinearity and context-freeness of languages accepted by valence automata. In: Chatterjee, K., Sgall, J. (eds.) MFCS 2013. LNCS, vol. 8087, pp. 231–242. Springer, Heidelberg (2013)
Dufourd, C., Finkel, A., Schnoebelen, P.: Reset nets between decidability and undecidability. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 103–115. Springer, Heidelberg (1998)
Esparza, J.: Petri nets, commutative context-free grammars, and basic parallel processes. Fundam. Inform. 31(1), 13–25 (1997)
Finkel, A., Göller, S., Haase, C.: Reachability in register machines with polynomial updates. In: Chatterjee, K., Sgall, J. (eds.) MFCS 2013. LNCS, vol. 8087, pp. 409–420. Springer, Heidelberg (2013)
Fraca, E., Haddad, S.: Complexity analysis of continuous petri nets. In: Colom, J.-M., Desel, J. (eds.) PETRI NETS 2013. LNCS, vol. 7927, pp. 170–189. Springer, Heidelberg (2013)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)
Ginsburg, S., Spanier, E.H.: Semigroups, Presburger formulas and languages. Pac. J. Math. 16(2), 285–296 (1966)
Grädel, E.: Dominoes and the complexity of subclasses of logical theories. Ann. Pure Appl. Logic 43(1), 1–30 (1989)
Greibach, S.A.: Remarks on blind and partially blind one-way multicounter machines. Theor. Comput. Sci. 7(3), 311–324 (1978)
Haase, C.: Subclasses of Presburger arithmetic and the weak EXP hierarchy. In: Proc. CSL-LICS (to appear, 2014)
Haase, C., Kreutzer, S., Ouaknine, J., Worrell, J.: Reachability in succinct and parametric one-counter automata. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 369–383. Springer, Heidelberg (2009)
Hack, M.: The equality problem for vector addition systems is undecidable. Theor. Comput. Sci. 2(1), 77–95 (1976)
Hague, M., Lin, A.W.: Model checking recursive programs with numeric data types. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 743–759. Springer, Heidelberg (2011)
Huynh, D.T.: The complexity of equivalence problems for commutative grammars. Inform. Control 66(1-2), 103–121 (1985)
Huynh, D.T.: A simple proof for the \({\Sigma}^p_2\) upper bound of the inequivalence problem for semilinear sets. Elektron. Inform. Kybernet. 22(4), 147–156 (1986)
Jančar, P.: Nonprimitive recursive complexity and undecidability for Petri net equivalences. Theor. Comput. Sci. 256(1-2), 23–30 (2001)
Kopczyński, E., To, A.W.: Parikh images of grammars: Complexity and applications. In: Proc. LICS, pp. 80–89 (2010)
Lipton, R.: The reachability problem is exponential-space-hard. Technical report, Yale University, New Haven, CT (1976)
Mayr, E.W.: An algorithm for the general Petri net reachability problem. In: Proc. STOC, pp. 238–246. ACM, New York (1981)
Plandowski, W., Rytter, W.: Complexity of language recognition problems for compressed words. In: Karhumäki, J., Maurer, H.A., Păun, G., Rozenberg, G. (eds.) Jewels are Forever, pp. 262–272 (1999)
Rackoff, C.: The covering and boundedness problems for vector addition systems. Theor. Comput. Sci. 6(2), 223–231 (1978)
Reichert, J.: On the complexity of counter reachability games. In: Abdulla, P.A., Potapov, I. (eds.) RP 2013. LNCS, vol. 8169, pp. 196–208. Springer, Heidelberg (2013)
Schnoebelen, P.: Revisiting ackermann-hardness for lossy counter machines and reset petri nets. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 616–628. Springer, Heidelberg (2010)
Seidl, H., Schwentick, T., Muscholl, A., Habermehl, P.: Counting in trees for free. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1136–1149. Springer, Heidelberg (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Haase, C., Halfon, S. (2014). Integer Vector Addition Systems with States. In: Ouaknine, J., Potapov, I., Worrell, J. (eds) Reachability Problems. RP 2014. Lecture Notes in Computer Science, vol 8762. Springer, Cham. https://doi.org/10.1007/978-3-319-11439-2_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-11439-2_9
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11438-5
Online ISBN: 978-3-319-11439-2
eBook Packages: Computer ScienceComputer Science (R0)