Parallel dynamical systems over special digraph classes

Aledo, Juan A.; Martinez, Silvia; Valverde, Jose C.

doi:10.1080/00207160.2012.742191

Cited by 12 publications

(11 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In such a case, the orbital structure depends on both the global evolution operator and the structure of the digraph. In particular, in [64], it is shown that NAND-PDDS over circle digraphs can present periodic orbits of any period except fixed points and periods 4 and 6, and the same occurs for NOR-PDDS.…”

Section: Pds Over Directed Dependency Graphsmentioning

confidence: 87%

“…Subsequent research [64] allows us to check that this breakdown is due to the existence of directed cycles in the dependency digraph, while for PDDS over acyclic dependency digraphs the periodic orbits continue being only fixed points or 2-periodic ones. Moreover, the orbit structure of special digraph classes as line digraphs, arborescences, and star digraphs is similar and it is possible to specify the number of iterations needed by an eventually periodic point to reach the corresponding periodic orbit.…”

Section: Pds Over Directed Dependency Graphsmentioning

confidence: 99%

See 1 more Smart Citation

Parallel Dynamical Systems over Graphs and Related Topics: A Survey

Aledo¹,

Martinez²,

Valverde

2015

Journal of Applied Mathematics

Self Cite

View full text Add to dashboard Cite

In discrete processes, as computational or genetic ones, there are many entities and each entity has a state at a given time. The update of states of the entities constitutes an evolution in time of the system, that is, a discrete dynamical system. The relations among entities are usually represented by a graph. The update of the states is determined by the relations of the entities and some local functions which together constitute (global) evolution operator of the dynamical system. If the states of the entities are updated in a synchronous manner, the system is called aparallel dynamical system. This paper is devoted to review the main results on the dynamical behavior of parallel dynamical systems over graphs which constitute a generic tool for modeling discrete processes.

show abstract

Section: Pds Over Directed Dependency Graphsmentioning

confidence: 87%

Section: Pds Over Directed Dependency Graphsmentioning

confidence: 99%

Parallel Dynamical Systems over Graphs and Related Topics: A Survey

Aledo¹,

Martinez²,

Valverde

2015

Journal of Applied Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…In particular, in [30], it is proved that any OR-PDS and AND-PDS are fixed point systems, while NAND-PDS and NOR-PDS are 2-periodic point systems, independently of the associated (simple) graph. Furthermore, these results are generalized in [25][26][27][28][29]33,34] where the authors study the periodic structure of MAX-PDS, MAX-PDDS, MIN-PDS, and MIN-PDDS. That is, parallel dynamical systems where the future state of each node is computed using the Boolean maxterm MAX or the Boolean minterm MIN.…”

Section: Introductionmentioning

confidence: 96%

“…Usually, the acronym PDS is used for parallel dynamical systems over undirected graphs, whereas if the associated graph is properly a digraph, F is said to be a parallel directed dynamical system (PDDS). As can be seen in the recent literature [25][26][27][28][29][30][31][32], the dynamics of PDDS are, in general, much more involved than the dynamics of PDS.…”

Section: Introductionmentioning

confidence: 99%

On the Periodic Structure of Parallel Dynamical Systems on Generalized Independent Boolean Functions

et al. 2020

Self Cite

View full text Add to dashboard Cite

In this paper, based on previous results on AND-OR parallel dynamical systems over directed graphs, we give a more general pattern of local functions that also provides fixed point systems. Moreover, by considering independent sets, this pattern is also generalized to get systems in which periodic orbits are only fixed points or 2-periodic orbits. The results obtained are also applicable to homogeneous systems. On the other hand, we study the periodic structure of parallel dynamical systems given by the composition of two parallel systems, which are conjugate under an invertible map in which the inverse is equal to the original map. This allows us to prove that the composition of any parallel system on a maxterm (or minterm) Boolean function and its conjugate one by means of the complement map is a fixed point system, when the associated graph is undirected. However, when the associated graph is directed, we demonstrate that the corresponding composition may have points of any period, even if we restrict ourselves to the simplest maxterm OR and the simplest minterm AND. In spite of this general situation, we prove that, when the associated digraph is acyclic, the composition of OR and AND is a fixed point system.

show abstract

“…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].…”

Section: Introductionmentioning

confidence: 99%