Abstract
This paper introduces a class of register machines whose registers can be updated by polynomial functions when a transition is taken, and the domain of the registers can be constrained by linear constraints. This model strictly generalises a variety of known formalisms such as various classes of Vector Addition Systems with States. Our main result is that reachability in our class is PSPACE-complete when restricted to one register. We moreover give a classification of the complexity of reachability according to the type of polynomials allowed and the geometry induced by the range-constraining formula.
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
Babić, D., Cook, B., Hu, A.J., Rakamarić, Z.: Proving termination of nonlinear command sequences. Formal Aspects of Computing, 1–15 (2012)
Bell, P., Potapov, I.: On undecidability bounds for matrix decision problems. Theoretical Computer Science 391(1-2), 3–13 (2008)
Bonnet, R.: The reachability problem for vector addition system with one zero-test. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 145–157. Springer, Heidelberg (2011)
Bradley, A.R., Manna, Z., Sipma, H.B.: Termination of polynomial programs. In: Cousot, R. (ed.) VMCAI 2005. LNCS, vol. 3385, pp. 113–129. Springer, Heidelberg (2005)
Cucker, F., Koiran, P., Smale, S.: A polynomial time algorithm for Diophantine equations in one variable. Journal of Symbolic Computation 27(1), 21–29 (1999)
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)
Fearnley, J., Jurdziński, M.: Reachability in two-clock timed automata is PSPACE-complete. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part II. LNCS, vol. 7966, pp. 212–223. Springer, Heidelberg (2013)
Finkel, A., McKenzie, P., Picaronny, C.: A well-structured framework for analysing Petri net extensions. Information and Computation 195(1-2), 1–29 (2004)
Fremont, D.: The reachability problem for affine functions on the integers (2012), http://web.mit.edu/~dfremont/www/reachability.pdf
Göller, S., Haase, C., Ouaknine, J., Worrell, J.: Branching-time model checking of parametric one-counter automata. In: Birkedal, L. (ed.) FOSSACS 2012. LNCS, vol. 7213, pp. 406–420. Springer, Heidelberg (2012)
Haase, C.: On the Complexity of Model Checking Counter Automata. PhD thesis, University of Oxford, UK (2012)
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)
Haase, C., Ouaknine, J., Worrell, J.: On the relationship between reachability problems in timed and counter automata. In: Finkel, A., Leroux, J., Potapov, I. (eds.) RP 2012. LNCS, vol. 7550, pp. 54–65. Springer, Heidelberg (2012)
Hirst, H.P., Macey, W.T.: Bounding the roots of polynomials. The College Mathematics Journal 28(4), 292–295 (1997)
Lambert, J.-L.: A structure to decide reachability in petri nets. Theoretical Computer Science 99(1), 79–104 (1992)
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)
Minsky, M.L.: Recursive Unsolvability of Post’s Problem of “Tag” and other Topics in Theory of Turing Machines. The Annals of Mathematics 74(3), 437–455 (1961)
Reichert, J.: Personal communication (2013)
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
Finkel, A., Göller, S., Haase, C. (2013). Reachability in Register Machines with Polynomial Updates. In: Chatterjee, K., Sgall, J. (eds) Mathematical Foundations of Computer Science 2013. MFCS 2013. Lecture Notes in Computer Science, vol 8087. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40313-2_37
Download citation
DOI: https://doi.org/10.1007/978-3-642-40313-2_37
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40312-5
Online ISBN: 978-3-642-40313-2
eBook Packages: Computer ScienceComputer Science (R0)