ETS IV: sequential dynamical systems: fixed points, invertibility and equivalence

  • Authors:

  • C. L. Barrett;H. S. Mortveit;C. M. Reidys

  • Affiliations:

  • Los Alamos National Laboratory, TSA-2, MS M997, Mailstop TA-0, SM-1237, MS M997, Los Alamos, NM;Los Alamos National Laboratory, TSA-2, MS M997, Mailstop TA-0, SM-1237, MS M997, Los Alamos, NM;Los Alamos National Laboratory, TSA-2, MS M997, Mailstop TA-0, SM-1237, MS M997, Los Alamos, NM

  • Venue:
  • Applied Mathematics and Computation
  • Year:
  • 2003

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Abstract

Sequential dynamical systems (SDSs) are discrete dynamical systems that are obtained from the following data: (a) a finite (labeled) graph Y with vertex set {1,..., n} where each vertex has a binary state, (b) a vertex labeled sequence of functions (Fi,Y : F2n → F2n)i and (c) a permutation π∈ Sn. The function Fi,Y updates the binary state of vertex i as a function of the states of vertex i and its Y-neighbors and leaves the states of all other vertices fixed. The permutation π represents a Y-vertex ordering according to which the functions Fi,Y are applied. By composing the functions Fi,Y in the order given by π we obtain the SDS [FY, π] = Πi=1n Fπ(i),Y : F2n → F2n. In this paper we will generalize a class of results on SDS that have been proven for symmetric Boolean (local) functions to quasi-symmetric local functions. Further, we completely classify invertible SDS and investigate fixed points of sequential and parallel cellular automata (CA). Finally, we show sharpness of a combinatorial upper bound for the number of non-equivalent SDS that can be obtained through rescheduling for a certain class of graphs.