Predecessors Existence Problems and Gardens of Eden in Sequential Dynamical Systems

Abstract

In this paper, we deal with one of the main computational questions in network models: the predecessor-existence problems. In particular, we solve algebraically such problems in sequential dynamical systems on maxterm and minterm Boolean functions. We also provide a description of the Garden-of-Eden configurations of any system, giving the best upper bound for the number of Garden-of-Eden points.

Predecessors and Gardens of Eden in sequential dynamical systems over directed graphs

“…In particular, changing slightly the arguments in the demonstrations, we get results, similar as those for SDS, for the predecessor and GOE problems in SDDS. The results here obtained not only suppose an algebraic solution for such problems in SDDS, but they also complete those achieved for PDS in [5], for PDDS in [6], and for SDS in [7]. This paper is organized as follows.…”

“…From the algebraic perspective, we solved the four predecessor problems for PDS whose updating is performed by means of a general maxterm or minterm Boolean function in [5], and for its counterpart over directed graphs PDDS in [6]. Recently, we have also solved these four problems and characterize the GOE states for SDS on these Boolean functions in [7]. In view of these previous works, here we extend our results to the case of SDS over directed graphs.…”

“…Given a generic state (vector) (y 1 , ..., y n ) of a [D, F, π]-SDDS over the directed graph D = (V, A), as in [7], we will consider the following subsets of V , which will be useful throughout this document:…”

“…Observe that, although the notation employed is the same as in [7], the meaning of each of these two subsets is different from the one with the same notation in [7]. Indeed, in contrast with the case of SDS in [7], now, each element i belonging to P 0 (resp. Q 0 ) is influencing a vertex j ∈ V 0 which is updated, according the order expressed in π, before (resp.…”

“…This motivates the study in both contexts separately to extract such differences, as done in [2,5,6,8,9,22,23,26,27]. Thus, in this work, once the situation in the undirected context is known [7], we deal with the predecessors existence problems for SDS over directed graphs (SDDS). a For our purposes, we consider that the digraph D = (V, A) could be arbitrary, although it cannot have more arcs than the corresponding complete digraph.…”