Computational Complexity Studies of Synchronous Boolean Finite Dynamical Systems

  • Mitsunori Ogihara
  • Kei Uchizawa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9076)


The finite dynamical system is a system consisting of some finite number of objects that take upon a value from some domain as a state, in which after initialization the states of the objects are updated based upon the states of the other objects and themselves according to a certain update schedule. This paper studies the subclass of finite dynamical systems the synchronous boolean finite dynamical system (synchronous BFDS, for short), where the states are boolean and the state update takes place in discrete time and at the same on all objects. The present paper is concerned with some problems regarding the behavior of synchronous BFDS in which the state update functions (or the local state transition functions) are chosen from a predetermined finite basis of boolean functions \({\mathcal {B}}\). Specifically the following three behaviors are studied:
  • Convergence.   Does a system at hand converge on a given initial state configuration?

  • Path Intersection.   Will a system starting in given two state configurations produce a common configuration?

  • Cycle Length.   Since the state space is finite, every BFDS on a given initial state configuration either converges or enters a cycle having length greater than 1. If the latter is the case, what is the length of the loop? Or put more simply, for an integer \(t\), is the length of loop greater than \(t\)?

The paper studies these questions in terms of computational complexity (in the case of Cycle Length using the decision version of the problem) and shows the following:
  1. 1.

    The three problems are each \(\mathrm {PSPACE}\)-complete if the boolean function basis contains \(\mathrm {NAND}\), \(\mathrm {NOR}\) or both \(\mathrm {AND}\) and \(\mathrm {OR}\).

  2. 2.

    The Convergence Problem is solvable in polynomial time if the set \(B\) is one of \(\{\mathrm {AND}\}\), \(\{\mathrm {OR}\}\) and \(\{\mathrm {XOR}, \mathrm {NXOR}\}\).

  3. 3.

    If the set \(B\) is chosen from the three sets as in the case of the Convergence Problem, the Path Intersection Problem is in UP, and the Cycle Length Problem is in \(\mathrm {UP}\cap \mathrm {coUP}\); thus, these are unlikely to be \(\mathrm {NP}\)-hard.



  1. 1.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Barrett, C.L., Hunt III, H.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)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Barrett, C.L., Hunt III, H.B., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E., Tošić, P.T.: Gardens of eden and fixed points in sequential dynamical systems. In: Proceedings of Discrete Models: Combinatorics, Computation, and Geometry, pp. 95–110 (2001)Google Scholar
  4. 4.
    Fellows, M.R., Koblitz, N.: Self-witnessing polynomial-time complexity and prime factorization. In: Proceedings of the Seventh Annual Conference on Structure in Complexity Theory, pp.107–110 (1992)Google Scholar
  5. 5.
    Floréen, P., Orponen, P.: Complexity issues in discrete Hopfield networks. Neuro-COLT Technical report Series, NC-TR-94-009 (1994)Google Scholar
  6. 6.
    Hemaspaandra, L.A., Ogihara, M.: A Complexity Theory Companion. Springer, Berlin (2001)Google Scholar
  7. 7.
    Kosub, S.: Dichotomy results for fixed-point existence problems for boolean dynamical systems. Math. Comput. Sci. 1(3), 487–505 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Laubenbacher, R., Pareigis, B.: Equivalence relations on finite dynamical systems. Adv. Appl. Math. 26(3), 237–251 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Kosub, S., Homan, C.M.: Dichotomy results for fixed point counting in boolean dynamical systems. In: Proceedings of the Tenth Italian Conference on Theoretical Computer Science (ICTCS 2007), pp. 163–174 (2007)Google Scholar
  10. 10.
    Parberry, I.: Circuit Complexity and Neural Networks. MIT Press, Cambridge (1994)zbMATHGoogle Scholar
  11. 11.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the Tenth ACM Symposium on Theory of Computing, pp. 216–226 (1978)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of MiamiCoral GablesUSA
  2. 2.Faculty of EngineeringYamagata UniversityYonezawaJapan

Personalised recommendations