Skip to main content
SpringerLink
Log in

Reachability Problem for Polynomial Iteration Is PSPACE-complete

  • Conference paper
  • First Online:
Reachability Problems (RP 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10506))

Included in the following conference series:

Abstract

In the reachability problem for polynomial iteration, we are given a set of polynomials over integers and we are asked whether a particular integer can be reached by a non-deterministic application of polynomials. This model can be seen as a generalisation of vector addition systems. Our main result is that the problem is PSPACE-complete for single variable polynomials. On the other hand, the problem is undecidable for multidimensional polynomials, already starting with three dimensions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Bell, P., Potapov, I.: On undecidability bounds for matrix decision problems. Theor. Comput. Sci. 391(1–2), 3–13 (2008). http://dx.doi.org/10.1016/j.tcs.2007.10.025

    Article  MathSciNet  MATH  Google Scholar 

  2. Ben-Amram, A.M.: Mortality of iterated piecewise affine functions over the integers: decidability and complexity. Computability 4(1), 19–56 (2015). https://doi.org/10.3233/COM-150032

    MathSciNet  MATH  Google Scholar 

  3. Bournez, O., Kurganskyy, O., Potapov, I.: Reachability problems for one-dimensional piecewise affine maps. Manuscript (2017)

    Google Scholar 

  4. Claus, V.: Some remarks on PCP\((k)\) and related problems. Bull. EATCS 12, 54–61 (1980)

    Google Scholar 

  5. Cucker, F., Koiran, P., Smale, S.: A polynomial time algorithm for diophantine equations in one variable. J. Symb. Comput. 27(1), 21–29 (1999). https://doi.org/10.1006/jsco.1998.0242

    Article  MathSciNet  MATH  Google Scholar 

  6. 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). doi:10.1007/978-3-642-40313-2_37

    Chapter  Google Scholar 

  7. Halava, V., Harju, T., Hirvensalo, M.: Undecidability bounds for integer matrices using Claus instances. Int. J. Found. Comput. Sci. 18(5), 931–948 (2007). http://doi.org/10.1142/s0129054107005066

    Article  MathSciNet  MATH  Google Scholar 

  8. Halava, V., Hirvensalo, M.: Improved matrix pair undecidability results. Acta Inf. 44(3), 191–205 (2007). http://dx.doi.org/10.1007/s00236-007-0047-y

    Article  MathSciNet  MATH  Google Scholar 

  9. Koiran, P., Cosnard, M., Garzon, M.H.: Computability with low-dimensional dynamical systems. Theor. Comput. Sci. 132(2), 113–128 (1994). http://doi.org/10.1016/0304-3975(94)90229-1

    Article  MathSciNet  MATH  Google Scholar 

  10. Kurganskyy, O., Potapov, I.: Reachability problems for PAMs. In: Freivalds, R.M., Engels, G., Catania, B. (eds.) SOFSEM 2016. LNCS, vol. 9587, pp. 356–368. Springer, Heidelberg (2016). doi:10.1007/978-3-662-49192-8_29

    Chapter  Google Scholar 

  11. Kurganskyy, O., Potapov, I., Sancho-Caparrini, F.: Reachability problems in low-dimensional iterative maps. Int. J. Found. Comput. Sci. 19(4), 935–951 (2008). https://doi.org/10.1142/S0129054108006054

    Article  MathSciNet  MATH  Google Scholar 

  12. Kuroda, S.Y.: Classes of languages and linear-bounded automata. Inf. Control 7(2), 207–223 (1964). http://doi.org/10.1016/s0019-9958(64)90120-2

    Article  MathSciNet  MATH  Google Scholar 

  13. Landweber, P.S.: Three theorems on phrase structure grammars of type 1. Inf. Control 6(2), 131–136 (1963). http://doi.org/10.1016/s0019-9958(63)90169-4

    Article  MathSciNet  MATH  Google Scholar 

  14. Neary, T.: Undecidability in binary tag systems and the post correspondence problem for five pairs of words. In: STACS 2015. LIPIcs, pp. 649–661 (2015). http://doi.org/10.4230/LIPIcs.STACS.2015.649

  15. Reichert, J.: Reachability Games with Counters: Decidability and Algorithms. Doctoral thesis, Laboratoire Spécification et Vérification, ENS Cachan, France (2015)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Liverpool, Liverpool, UK

    Reino Niskanen

Authors
  1. Reino Niskanen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Reino Niskanen .

Editor information

Editors and Affiliations

  1. Royal Holloway University of London, London, United Kingdom

    Matthew Hague

  2. University of Liverpool, Liverpool, United Kingdom

    Igor Potapov

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Niskanen, R. (2017). Reachability Problem for Polynomial Iteration Is PSPACE-complete. In: Hague, M., Potapov, I. (eds) Reachability Problems. RP 2017. Lecture Notes in Computer Science(), vol 10506. Springer, Cham. https://doi.org/10.1007/978-3-319-67089-8_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-67089-8_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-67088-1

  • Online ISBN: 978-3-319-67089-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation