Summary: Sequential dynamical systems (SDSs) are discrete dynamical systems that are obtained from the following data: (a) a finite (labeled) graph \(Y\) with vertex set \(\{1,\dots,n\}\) where each vertex has a binary state, (b) a vertex labeled sequence of functions \((F_{i,Y}: \mathbb{F}^n_2\to \mathbb{F}^n_2)_i\) and (c) a permutation \(\pi\in S_n\). The function \(F_{i,Y}\) 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 \(\pi\) represents a \(Y\)-vertex ordering according to which the functions \(F_{i,Y}\) are applied. By composing the functions \(F_{i,Y}\) in the order given by \(\pi\) we obtain the SDS \([F_Y,\pi]= \prod^n_{i=1}F_{\pi(i),Y}: \mathbb{F}^n_2\to \mathbb{F}^n_2\).
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.
For Part III, see ibid. 122, 325-340 (2001; Zbl 1050.68161).