Skip to main content
SpringerLink
Log in

Synchronous Boolean Finite Dynamical Systems on Directed Graphs over XOR Functions

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

In this paper, we investigate the complexity of a number of computational problems defined on a synchronous boolean finite dynamical system, where update functions are chosen from a template set of exclusive-or and its negation. We first show that the reachability and path-intersection problems are solvable in logarithmic space-uniform AC1 if the objects execute permutations, while the reachability problem is known to be in P and the path-intersection problem to be in UP in general. We also explore the case where the reachability or intersection are tested on a subset of objects, and show that this hardens complexity of the problems: both problems become NP-complete, and even \({\Pi }^{p}_{2}\)-complete if we further require universality of the intersection. We next consider the exact cycle length problem, that is, determining whether there exists an initial configuration that yields a cycle in the configuration space having exactly a given length, and show that this problem is NP-complete. Lastly, we consider the t-predecessor and t-Garden of Eden problem, and prove that these are solvable in polynomial time even if the value of t is also given in binary as part of instance, and the two problems are in logarithmic space-uniform NC2 if the value of t is given in unary as part of instance.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Barrett, C.L., Hunt, IIIH.B., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: Complexity of reachability problems for finite discrete dynamical systems. J. Comput. Syst. Sci. 72(8), 1317–1345 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barrett, C.L., Hunt IIIH, B., Marathe, M.V., Ravi, D. J., Rosenkrantz S.S., Stearns, R.E., Thakur, M.: Predecessor existence problems for finite discrete dynamical systems. Theor. Comput. Sci. 386(1–2), 3–37 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barrett, C.L., 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. In: Proceedings of Discrete Mathematics and Theoretical Computer Science, pp 69–80 (2003)

  5. Barrett, C.L., Hunt III, 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: Proceedings of Discrete Mathematics and Theoretical Computer Science, pp 95–110 (2001)

  6. Barrett, C.L., Mortveit, H.S., Reidys, C.M.: Elements of a theory of simulation II: Sequential dynamical systems. Appl. Math. Comput. 107 (2-3), 121–136 (2000)

    MathSciNet  MATH  Google Scholar 

  7. Hardy, G.H., Wrigth, E.M.: An Introduction to the Theory of Numbers. In: Heath-brown, R, et al. (eds.) . 6th edn. Oxford University Press, Oxford (2008)

  8. Kawachi, A.: Personal communication (2016)

  9. Kawachi, A., Ogihara, M., Uchizawa, K.: Generalized predecessor existence problems for boolean finite dynamical systems on directed graphs. Theor. Comput. Sci. 762, 25–40 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kosub, S.: Dichotomy results for fixed-point existence problems for Boolean dynamical systems. Math. Comput. Sci. 1(3), 487–505 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kosub, S., Homan, C.M.: Dichotomy results for fixed point counting in boolean dynamical systems. In: Proceedings of the 10th Italian Conference on Theoretical Computer Science, pp 163–174 (2007)

  12. Ogihara, M., Uchizawa, K.: Computational complexity studies of synchronous boolean finite dynamical systems on directed graphs. Inf. Comput. 256, 226–236 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rosenkrantz, D.J., Marathe, M.V., Hunt III, H.B., Ravi, S.S., Stearns, R.E.: Analysis problems for graphical dynamical systems A unified approach through graph predicates. In: Proceedings of the International Conference on Autonomous Agents and Multiagent Systems, pp 1501–1509 (2015)

  14. Sipser, M.: Introduction to the Theory of Computation. PWS Publishing Company (1997)

  15. von zur Gathen, J.: Parallel linear algebra. In: Reif, J.H. (ed.) Synthesis of Parallel Algorithms, pp 574–615. Morgan Kaufmann Publishers Inc. (1993)

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Miami, 1365 Memorial Drive, Coral Gables, FL, 33146, USA

    M. Ogihara

  2. Graduate School of Science and Engineering, Yamagata University, Jonan 4-3-16, Yonezawa-shi Yamagata, 992-8510, Japan

    K. Uchizawa

Authors
  1. M. Ogihara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Uchizawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Ogihara.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article belongs to the Topical Collection: Commemorative Issue for Alan L. Selman

Guest Editors: Mitsunori Ogihara, Elvira Mayordomo, Atri Rudra

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ogihara, M., Uchizawa, K. Synchronous Boolean Finite Dynamical Systems on Directed Graphs over XOR Functions. Theory Comput Syst 67, 569–591 (2023). https://doi.org/10.1007/s00224-022-10111-x

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-022-10111-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation