Computational Complexity Studies of Synchronous Boolean Finite Dynamical Systems
Abstract

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 \(t\)?
 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.
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.
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.
References
 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)CrossRefMATHMathSciNetGoogle Scholar
 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)CrossRefMATHMathSciNetGoogle Scholar
 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.Fellows, M.R., Koblitz, N.: Selfwitnessing polynomialtime complexity and prime factorization. In: Proceedings of the Seventh Annual Conference on Structure in Complexity Theory, pp.107–110 (1992)Google Scholar
 5.Floréen, P., Orponen, P.: Complexity issues in discrete Hopfield networks. NeuroCOLT Technical report Series, NCTR94009 (1994)Google Scholar
 6.Hemaspaandra, L.A., Ogihara, M.: A Complexity Theory Companion. Springer, Berlin (2001)Google Scholar
 7.Kosub, S.: Dichotomy results for fixedpoint existence problems for boolean dynamical systems. Math. Comput. Sci. 1(3), 487–505 (2008)CrossRefMATHMathSciNetGoogle Scholar
 8.Laubenbacher, R., Pareigis, B.: Equivalence relations on finite dynamical systems. Adv. Appl. Math. 26(3), 237–251 (2001)CrossRefMATHMathSciNetGoogle Scholar
 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.Parberry, I.: Circuit Complexity and Neural Networks. MIT Press, Cambridge (1994)MATHGoogle Scholar
 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