Advertisement

A Linear-Time Algorithm for the Orbit Problem over Cyclic Groups

  • Anthony Widjaja Lin
  • Sanming Zhou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8704)

Abstract

The orbit problem is at the heart of symmetry reduction methods for model checking concurrent systems. It asks whether two given configurations in a concurrent system (represented as finite sequences over some finite alphabet) are in the same orbit with respect to a given finite permutation group (represented by their generators) acting on this set of configurations. It is known that the problem is in general as hard as the graph isomorphism problem, which is widely believed to be not solvable in polynomial time. In this paper, we consider the restriction of the orbit problem when the permutation group is cyclic (i.e. generated by a single permutation), an important restriction of the orbit problem. Our main result is a linear-time algorithm for this subproblem.

Keywords

Model Check Cyclic Group Permutation Group Arithmetic Progression Concurrent System 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Primorial Numbers (The On-Line Encyclopedia of Integer Sequences), http://oeis.org/A002110
  2. 2.
    Babai, L., Beals, R., Cai, J.-Y., Ivanyos, G., Luks, E.M.: Multiplicative equations over commuting matrices. In: SODA, pp. 498–507 (1996)Google Scholar
  3. 3.
    Babai, L., Luks, E.M.: Canonical labeling of graphs. In: STOC, pp. 171–183 (1983)Google Scholar
  4. 4.
    Bach, E., Shallit, J.: Algorithmic Number Theory. Foundations of Computing, vol. 1. MIT Press (1996)Google Scholar
  5. 5.
    Bostan, A., Gaudry, P., Schost, É.: Linear Recurrences with Polynomial Coefficients and Application to Integer Factorization and Cartier-Manin Operator. SIAM J. Comput. 36(6), 1777–1806 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Brualdi, R.A.: Combinatorial matrix classes. Encyclopedia of Mathematics and Its Applications, vol. 108. Cambridge University Press (2006)Google Scholar
  7. 7.
    Cameron, P.J.: Permutation Groups. London Mathematical Society Student Texts. Cambridge University Press (1999)Google Scholar
  8. 8.
    Chrobak, M.: Finite automata and unary languages. Theor. Comput. Sci. 47(3), 149–158 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Clarke, E.M., Emerson, E.A., Jha, S., Sistla, A.P.: Symmetry reductions in model checking. In: Hu, A.J., Vardi, M.Y. (eds.) CAV 1998. LNCS, vol. 1427, pp. 147–158. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  10. 10.
    Clarke, E.M., Jha, S., Enders, R., Filkorn, T.: Exploiting symmetry in temporal logic model checking. Formal Methods in System Design 9(1/2), 77–104 (1996)CrossRefGoogle Scholar
  11. 11.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press (2009)Google Scholar
  12. 12.
    Costa, E., Harvey, D.: Faster deterministic integer factorization. CoRR, abs/1201.2116 (2012)Google Scholar
  13. 13.
    Donaldson, A.F., Miller, A.: On the constructive orbit problem. Ann. Math. Artif. Intell. 57(1), 1–35 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Emerson, E.A., Sistla, A.P.: Symmetry and model checking. Formal Methods in System Design 9(1/2), 105–131 (1996)CrossRefGoogle Scholar
  15. 15.
    Erdös, P., Graham, R.L.: On a linear diophantine problem of Frobenius. Acta Arithm. 21, 399–408 (1972)zbMATHGoogle Scholar
  16. 16.
    Göller, S., Mayr, R., To, A.W.: On the computational complexity of verifying one-counter processes. In: LICS, pp. 235–244 (2009)Google Scholar
  17. 17.
    Hardy, G.H., Wright, E.M.: An Introduction to The Theory of Numbers, 6th edn. OUP Oxford (2008)Google Scholar
  18. 18.
    Ip, C.N., Dill, D.L.: Better verification through symmetry. Formal Methods in System Design 9(1/2), 41–75 (1996)Google Scholar
  19. 19.
    Kannan, R., Lipton, R.J.: Polynomial-time algorithm for the orbit problem. J. ACM 33(4), 808–821 (1986)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Sedgewick, R., Flajolet, P.: An Introduction to the Analysis of Algorithms, 2nd edn. Addison-Wesley Professional (2013)Google Scholar
  21. 21.
    Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time: Preliminary report. In: STOC, pp. 1–9 (1973)Google Scholar
  22. 22.
    To, A.W.: Unary finite automata vs. arithmetic progressions. Inf. Process. Lett. 109(17), 1010–1014 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Wahl, T., Donaldson, A.F.: Replication and abstraction: Symmetry in automated formal verification. Symmetry 2, 799–847 (2010)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Anthony Widjaja Lin
    • 1
  • Sanming Zhou
    • 2
  1. 1.Yale-NUS CollegeSingapore
  2. 2.Department of Mathematics and StatisticsThe University of MelbourneAustralia

Personalised recommendations