Skip to main content
SpringerLink
Log in

From Post Systems to the Reachability Problems for Matrix Semigroups and Multicounter Automata

  • Conference paper
Developments in Language Theory (DLT 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3340))

Included in the following conference series:

Abstract

The main result of this paper is the reduction of PCP(n) to the vector reachability problem for a matrix semigroup generated by n 4 × 4 integral matrices. It follows that the vector reachability problem is undecidable for a semigroup generated by 7 integral matrices of dimension 4. The question whether the vector reachability problem is decidable for n = 2 and n = 3 remains open. Also we show that proposed technique can be applied to Post’s tag-systems. As a result we define new classes of counter automata that lie on the border between decidability and undecidability.

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bestel, J., Karhumaki, J.: Combinatorics on Words - A Tutorial, Bulletin of the EATCS, February 2003, pp. 178–228 (2003)

    Google Scholar 

  2. Blondel, V., Tsitsiklis, J.: A survey of computational complexity results in systems and control. Automatica 36, 1249–1274 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Comon, H., Jurski, Y.: Multiple counters automata, safety analysis and Presburger arithmetic, Research Report LSV-98-1 (March 1998)

    Google Scholar 

  4. Dufourd, C., Finkel, A., Schnoebelen, Ph.: 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)

    Chapter  Google Scholar 

  5. Finkel, A., Sutre, G.: Decidability of reachability problems for classes of two counters automata. In: Reichel, H., Tison, S. (eds.) STACS 2000. LNCS, vol. 1770, pp. 346–357. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  6. Finkel, A., Purushothaman Iyer, S., Sutre, G.: Well-abstracted transition systems: application to FIFO automata. Inf. Comput. 181(1), 1–31 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gaubert, S., Katz, R.: Reachability problems for products of matrices in semirings. arXiv:math.OC/0310028, v1, 1-21 (2003)

    Google Scholar 

  8. Graham, R., Knuth, D., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison-Wesley, Reading (1994)

    MATH  Google Scholar 

  9. Gouda, M., Manning, E., Yu, Y.: On the progress of communication between two finite state machines. Information and control 63, 200–216 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hayes, B.: On the Teeth of Wheels. American Scientist 88(4), 296–300 (2000)

    Google Scholar 

  11. Ibarra, O.H., Su, J., Bultan, T., Dang, Z., Kemmerer, R.A.: Counter Machines: Decidable Properties and Applications to Verification Problems. In: Nielsen, M., Rovan, B. (eds.) MFCS 2000. LNCS, vol. 1893, pp. 426–435. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  12. Kannan, R., Lipton, R.J.: Polynomial-time algorithm for the orbit problem. Journal of the ACM (JACM) 33(4), 808–821 (1986)

    Article  MathSciNet  Google Scholar 

  13. Kurganskyy, O., Potapov, I.: On the bound of algorithmic resolvability of correctness problems of automaton interaction through communication channels. Cybernetics and System Analysis 3, 49–57 (1999)

    Google Scholar 

  14. Lisitsa, A., Potapov, I.: Membership and reachability problems for row-monomial transformations. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 623–634. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  15. Mergenstern, M.: Frontier between decidability and undecidability: a survey. Theoretical Computer Science 231, 217–251 (2000)

    Article  MathSciNet  Google Scholar 

  16. Matiyasevich, Y.V., Senizergues, G.: Decision problems for semi-Thue systems with a few rules. In: Proc. LICS 1996, pp. 523–531 (1996)

    Google Scholar 

  17. Minsky, M.: Computation: Finite and Infinite Machines. Prentice-Hall, Englewood Cliffs (1967)

    MATH  Google Scholar 

  18. Moore, C.: Unpredictability and Undecidability in Dynamical Systems. Physical Review Letters 64, 2354–2357 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  19. Wang, H.: Tag Systems and Lag Systems. Math. Ann. 152, 65–74 (1963)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Liverpool, Liverpool, U.K.

    Igor Potapov

Authors
  1. Igor Potapov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Auckland, New Zealand

    Cristian S. Calude

  2. Computer Science, Institute of Information and Mathematical Sciences, Massey University Albany, Auckland, New Zealand

    Elena Calude

  3. Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand

    Michael J. Dinneen

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Potapov, I. (2004). From Post Systems to the Reachability Problems for Matrix Semigroups and Multicounter Automata. In: Calude, C.S., Calude, E., Dinneen, M.J. (eds) Developments in Language Theory. DLT 2004. Lecture Notes in Computer Science, vol 3340. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30550-7_29

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-30550-7_29

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-24014-3

  • Online ISBN: 978-3-540-30550-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation