Skip to main content
SpringerLink
Log in

On Computational Complexity of Counting Fixed Points in Symmetric Boolean Graph Automata

  • Conference paper
Unconventional Computation (UC 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3699))

Included in the following conference series:

Abstract

We study computational complexity of counting the fixed point configurations (FPs) in certain classes of graph automata viewed as discrete dynamical systems. We prove that both exact and approximate counting of FPs in Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively) are computationally intractable, even when each node is required to update according to a symmetric Boolean function. We also show that the problems of counting exactly the garden of Eden configurations (GEs), as well as all transient configurations, are in general intractable, as well. Moreover, exactly enumerating FPs or GEs remains hard even in some severely restricted cases, such as when the nodes of an SDS or SyDS use only two different symmetric Boolean update rules, and every node has a neighborhood size bounded by a small constant.

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. Arora, S., Rabani, Y., Vazirani, U.: Simulating quadratic dynamical systems is PSPACE-complete. In: Proc. 26th. Annual ACM Symposium on the Theory of Computing (STOC), Montreal, Canada, pp. 459–467 (May 1994)

    Google Scholar 

  2. Barrett, C., Bush, B., Kopp, S., Mortveit, H., Reidys, C.: Sequential Dynamical Systems and Applications to Simulations, Technical Report. Los Alamos National Laboratory (September 1999)

    Google Scholar 

  3. Barrett, C., Marathe, M., Mortveit, H., Reidys, C., Smith, J., Ravi, S.S.: Adhop- NET: Advanced Simulation-based Analysis of Ad-Hoc Networks. Los Alamos Unclassified Internal Report (2000)

    Google Scholar 

  4. Barrett, C., Hunt III, H.B., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: Dichotomy Results for Sequential Dynamical Systems. Los Alamos National Laboratory Report, LA-UR-00-5984 (2000)

    Google Scholar 

  5. Barrett, C., Hunt III, H.B., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: Predecessor and Permutation Existence Problems for Sequential Dynamical Systems. Los Alamos National Laboratory Report, LA-UR-01-668 (2001)

    Google Scholar 

  6. Barrett, C., Hunt III, H.B., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: Reachability problems for sequential dynamical systems with threshold functions. Theoretical Comp. Sci. 295(1-3), 41–64 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Barrett, C.L., Hunt, H.B., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E., Tosic, P.T.: Gardens of Eden and Fixed Points in Sequential Dynamical Systems. in: Proc. AA (DM-CCG), in Discrete Mathematics Theoretical Computer Science (spec. ed.) (July 2001)

    Google Scholar 

  8. Barrett, C., Mortveit, H., Reidys, C.: Elements of a theory of simulation II: sequential dynamical systems. Applied Math. and Comput. 107/2-3, 121–136 (2000)

    Google Scholar 

  9. Barrett, C., Mortveit, H., Reidys, C.: Elements of a theory of computer simulation III: equivalence of SDS. Applied Math. and Comput. 122, 325–340 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Barrett, C., Reidys, C.: Elements of a theory of computer simulation I: sequential CA over random graphs. Applied Mathematics and Computation 98, 241–259 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Beckman, R.J., et al.: TRANSIMS: Case Study. Dallas Ft-Worth. Los Alamos National Laboratory, LA UR 97-4502 (1999)

    Google Scholar 

  12. Buss, S., Papadimitriou, C., Tsitsiklis, J.: On the predictability of coupled automata: An allegory about Chaos. Complex Systems 1(5), 525–539 (1991)

    MathSciNet  Google Scholar 

  13. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. W. H. Freeman and Co., San Francisco (1979)

    MATH  Google Scholar 

  14. Garzon, M.: Models of Massive Parallelism: Analysis of Cellular Automata and Neural Networks. Springer, Heidelberg (1995)

    MATH  Google Scholar 

  15. Goles, E., Martinez, S.: Neural and Automata Networks: Dynamical behavior and Applications. In: Math. and Its Applications series, vol. 58. Kluwer, Dordrecht (1990)

    Google Scholar 

  16. Goles, E., Martinez, S. (eds.): Cellular Automata and Complex Systems’, Nonlinear Phenomena and Complex Systems series. Kluwer, Dordrecht (1999)

    Google Scholar 

  17. Green, F.: NP-Complete Problems in Cellular Automata. Complex Systems 1(3), 453–474 (1987)

    MATH  MathSciNet  Google Scholar 

  18. Huberman, B., Glance, N.: Evolutionary games and computer simulations. Proc. Nat’l Academy of Sciences (1999)

    Google Scholar 

  19. Hunt III, H.B., Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: Designing Approximation Schemes using L-reductions. In: Proc. 14th Conference on Foundations of Software Technology and Theoretical Computer Science (FST&TCS 1994), Madras, India, pp. 342–353 (December 1994)

    Google Scholar 

  20. Laubenbacher, R., Pareigis, B.: Finite Dynamical Systems. Technical report, Department of Mathematical Sciences, New Mexico State University, Las Cruces (2000)

    Google Scholar 

  21. Martin, B.: A Geometrical Hierarchy of Graphs via Cellular Automata. In: Proc. MFCS 1998 Satellite Workshop on Cellular Automata, Brno, Czech Republic (August 1998)

    Google Scholar 

  22. Mortveit, H., Reidys, C.: Discrete, sequential dynamical systems. Discrete Mathematics 226, 281–295 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. Nichitiu, C., Remila, E.: Simulations of Graph Automata. In: Proc. MFCS 1998 Satellite Workshop on Cellular Automata, Brno, Czech Republic (August 1998)

    Google Scholar 

  24. Papadimitriou, C.: Computational Complexity. Addison-Wesley, Reading (1994)

    MATH  Google Scholar 

  25. Roka, Z.: One-way cellular automata on Cayley graphs. Theoretical Computer Science 132(1-2), 259–290 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  26. Roth, D.: On the Hardness of Approximate Reasoning. Artificial Intelligence 82, 273–302 (1996)

    Article  MathSciNet  Google Scholar 

  27. Sutner, K.: On the computational complexity of finite cellular automata. Journal of Computer and System Sciences 50(1), 87–97 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  28. Sutner, K.: Computation theory of cellular automata. In: MFCS 1998 Satellite Workshop on Cellular Automata Bruno, Czech Republic (August. 1998)

    Google Scholar 

  29. Schittenkopf, C., Deco, G., Brauer, W.: Finite automata-models for the investigation of dynamical systems. Information Processing Letters 63(3), 137–141 (1997)

    Article  MathSciNet  Google Scholar 

  30. Tosic, P.: On Complexity of Counting Fixed Point Configurations in Certain Classes of Graph Automata, Electronic Colloquium on Computational Complexity, ECCC- TR05-051 (revision 1) (April 2005)

    Google Scholar 

  31. Tosic, P., Agha, G.: Concurrency vs. Sequential Interleavings in 1-D Threshold Cellular Automata. In: APDCM Workshop within IPDPS 2004, Santa Fe, New Mexico, USA (April 2004) (in Proc. IEEE-IPDPS 2004)

    Google Scholar 

  32. Tosic, P., Agha, G.: Characterizing configuration spaces of simple threshold cellular automata. In: Sloot, P.M.A., Chopard, B., Hoekstra, A.G. (eds.) ACRI 2004. LNCS, vol. 3305, pp. 861–870. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  33. Vadhan, S.: The Complexity of Counting in Sparse, Regular and Planar Graphs. SIAM J. Computing 31(2), 398–427 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  34. Valiant, L.: The Complexity of Computing the Permanent. Theoretical Comp. Sci. 8, 189–201 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  35. Valiant, L.: The Complexity of Enumeration and Reliability Problems. SIAM J. Computing 8(3), 410–421 (1979)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Open Systems Laboratory, Department of Computer Science, University of Illinois at Urbana-Champaign, Thomas Siebel Center for Computer Science, 201 N. Goodwin Avenue, Urbana, IL, 61801, USA

    Predrag T. Tošić & Gul A. Agha

Authors
  1. Predrag T. Tošić
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gul A. Agha
    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. Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand

    Michael J. Dinneen

  3. Institute of Mathematics of the Romanian Academy, PO Box 1-764, 014700, Bucureşti, Romania

    Gheorghe Păun

  4. Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Seville, Avda. Reina Mercedes, 41012, Sevilla, Spain

    Mario J. Pérez-Jímenez

  5. Leiden Center of Advanced Computer Science (LIACS), Leiden University, Niels Bohrweg 1, 2333, Leiden, CA, The Netherlands

    Grzegorz Rozenberg

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Tošić, P.T., Agha, G.A. (2005). On Computational Complexity of Counting Fixed Points in Symmetric Boolean Graph Automata. In: Calude, C.S., Dinneen, M.J., Păun, G., Pérez-Jímenez, M.J., Rozenberg, G. (eds) Unconventional Computation. UC 2005. Lecture Notes in Computer Science, vol 3699. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11560319_18

Download citation

  • DOI: https://doi.org/10.1007/11560319_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-29100-8

  • Online ISBN: 978-3-540-32022-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation