Reachability Problem for Polynomial Iteration Is PSPACE-complete

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10506)

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.

Keywords

Reachability problem Polynomial iteration Decidability 

References

  1. 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 MathSciNetCrossRefMATHGoogle Scholar
  2. 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 MathSciNetMATHGoogle Scholar
  3. 3.
    Bournez, O., Kurganskyy, O., Potapov, I.: Reachability problems for one-dimensional piecewise affine maps. Manuscript (2017)Google Scholar
  4. 4.
    Claus, V.: Some remarks on PCP\((k)\) and related problems. Bull. EATCS 12, 54–61 (1980)Google Scholar
  5. 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 MathSciNetCrossRefMATHGoogle Scholar
  6. 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 CrossRefGoogle Scholar
  7. 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 MathSciNetCrossRefMATHGoogle Scholar
  8. 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 MathSciNetCrossRefMATHGoogle Scholar
  9. 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 MathSciNetCrossRefMATHGoogle Scholar
  10. 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 CrossRefGoogle Scholar
  11. 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 MathSciNetCrossRefMATHGoogle Scholar
  12. 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 MathSciNetCrossRefMATHGoogle Scholar
  13. 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 MathSciNetCrossRefMATHGoogle Scholar
  14. 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. 15.
    Reichert, J.: Reachability Games with Counters: Decidability and Algorithms. Doctoral thesis, Laboratoire Spécification et Vérification, ENS Cachan, France (2015)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK

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