On periods and equilibria of computational sequential systems

Predecessors Existence Problems and Gardens of Eden in Sequential Dynamical Systems

“…The interactions among elements of a phenomenon do typically not occur simultaneously. When it occurs, it is said that the model updates synchronously or parallelly (see [25,26,[28][29][30][31][32][33][34][35][36]); otherwise it is said that the model updates asynchronously or sequentially (see [20,27,[37][38][39][40][41]). In this last case, an update order is needed to specify the sequence in which the states of the elements evolve.…”

“…The periodic structure of SDS on maxterm and minterm Boolean functions is studied in [37], where it is proved that any period can appear in their phase portrait, although fixed points cannot coexist with other periods. This issue is someway a generalization of the results in PDS on the same Boolean functions [29,33], where only fixed points and 2-periodic orbits can appear but not coexist.…”

Coexistence of Periods in Parallel and Sequential Boolean Graph Dynamical Systems over Directed Graphs

“…In previous works [31][32][33][34][35][36], a complete study of the periodic structure of homogeneous parallel and sequential systems induced by maxterm and minterm Boolean functions over undirected dependency graphs were performed. More specifically, in [36], the periodic structure of the simplest parallel systems, i.e., those induced by the maxterm OR (resp.…”

“…minterm) can only present fixed points or two-periodic orbits, while fixed points and two-periodic orbits cannot coexist. Concerning the sequential case, in [32], it was proved that, in contrast with the parallel case, sequential (homogeneous) systems induced by any maxterm (resp. minterm) can present periodic points of whichever period.…”

An Algorithm for Counting the Fixed Point Orbits of an AND-OR Dynamical System with Symmetric Positive Dependency Graph

“…Some works put an additional restriction which requires the graph of the DS to be symmetric and/or either all the edges to be positive or all of them negative. Among others, the FPOE problem of an {AND}-DS, {OR}-DS, {NAND}-DS and {NOR}-DS is studied in [2][3][4][8][9][10]. In addition, the FPOE problem in an {AND, OR, NAND, NOR}-DS is polynomially solvable as shown in [4,11].…”