Parallel dynamical systems over special digraph classes

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Parallel dynamical systems over specialdigraph classesJuan A. Aledoa, Silvia Martineza & Jose C. Valverdea

a Department of Mathematics, University of Castilla-La Mancha,Albacete, 02071, SpainAccepted author version posted online: 26 Oct 2012.Publishedonline: 11 Feb 2013.

To cite this article: Juan A. Aledo, Silvia Martinez & Jose C. Valverde (2013) Parallel dynamicalsystems over special digraph classes, International Journal of Computer Mathematics, 90:10,2039-2048, DOI: 10.1080/00207160.2012.742191

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International Journal of Computer Mathematics, 2013Vol. 90, No. 10, 2039–2048, http://dx.doi.org/10.1080/00207160.2012.742191

Parallel dynamical systems over special digraph classes

Juan A. Aledo, Silvia Martinez* and Jose C. Valverde

Department of Mathematics, University of Castilla-La Mancha, Albacete 02071, Spain

(Received 25 August 2012; accepted 15 October 2012)

In a previous work, for parallel dynamical systems over digraphs corresponding to the simplest Booleanfunctions AND and OR, we proved that only fixed or eventually fixed points appear, as it occurs overundirected dependency graphs. Nevertheless, for general Boolean functions, it was shown that any periodcan appear, depending on the Boolean function that infers the global evolution operator of the system andon the structure of the dependency digraph. Motivated by these results, in this work, we analyse the orbitstructure of parallel discrete dynamical systems over some special digraph classes.

Keywords: parallel discrete dynamical systems; periodic orbits; cellular automata; dependency digraphs;Boolean functions

2000 AMS Subject Classifications: 37B99; 37E15; 37N99; 68R10; 94C10

1. Introduction

Mathematical foundations for a theory of computer processes have been developed by severalworks in the last few years. One of a series of these works begins with [3], where sequentiallyupdated cellular automata over arbitrary graphs are employed as a paradigmatic framework.This work was followed by several authors [4–6], where this theory is developed, analysing theasymptotic behaviour.

Computer processes have been usually modelled by cellular automata. In this sense, in [16], aset of cellular automata is analysed and this shows that, despite their simple construction, someof them are capable of complex behaviour. Besides, a deeper research [17] suggests that manyone-dimensional cellular automata fall into four basic behaviour classes: three of them exhibitinga similar behaviour to fixed points, periodic orbits and chaotic attractors, while the asymptoticproperties of the fourth one are undecidable. The concept of parallel dynamical system that wetreat in this paper generalizes the one of cellular automaton.

In every process, there are many entities (cells in the language of cellular automata theory) andeach entity has a state at a given time (see [3–5]). The update of states of all the entities constitutesa discrete dynamical system (see [7,12]).

The update of the states is determined by dependency relations of the entities and local ruleswhich together constitute the (global) evolution operator of the dynamical system (see [15]).If the states of the entities are updated in a synchronous manner, the system is called a parallel

*Corresponding author. Email: [email protected]

© 2013 Taylor & Francis

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dynamical system (PDS), while if they are updated in an asynchronous order, the system is nameda sequential dynamical system (SDS) [7,12,13].

In [7], PDSs and SDSs are studied, considering OR (resp. AND) and NOR (resp. NAND) asglobal evolution operators. Inspired by these ideas, we extended these results for the parallel case,giving a complete characterization of the orbit structure of any PDS with any maxterm (resp.minterm) as a global function in [1]. As a result, for the simplest maxterm (resp. minterm), OR(resp. AND), only fixed or eventually fixed points can appear, while for a general maxterm (resp.minterm), uniquely periodic or eventually periodic orbits of period lower than or equal to two canbe found.

In computer processes, entities are related and they get information from their related ones.Thus, it is very interesting to study non-reciprocal relations between two related entities, becauseit could occur that an entity influences another one, but not vice versa, as it happens in practice [8].This could be modelled by an arc whose initial vertex would be the influencing entity and the finalvertex would correspond to the influenced entity, hence obtaining a directed graph or digraph ofrelations. The directed graph so built will be called the directed dependency graph of the system.

We actually determine a dynamical system over a directed dependency graph (see [14] fora similar approach that considers cellular automata over Cayley graphs) by associating to eachvertex i, a state xi ∈ {0, 1} and a local map fi defined on the states of the influencing vertices and thevertex/entity i, which returns its new state yi ∈ {0, 1}. We shall denote this digraph D = (V , A),where V = {1, 2, . . . , n} is the vertex set and A is the arc set.

For every vertex/entity 1 ≤ i ≤ n, we shall consider all the vertices that influence it in an updateof the system. Thus, we denote ID(i) = {j ∈ V |(j, i) ∈ A}.

These non-reciprocal relations have appeared in other applied models generated for the sim-ulation of aspects of the behaviour of biological systems, as it occurs in [10], where Kauffmanconstructed molecular automata for modelling a gene as a binary (on–off) device and studied thebehaviour of large, randomly constructed nets of these binary genes (see also [11]). A Kauffmannet of size n and connectivity k consists of n interconnected vertices, each one having k inputsand one output. The update of any gene is determined by the (directed) dependency relations andlocal rules which are given by random Boolean functions.

In this context, the evolution or update of the system is implemented by local functions whichare the restrictions of a global one. Then, for updating the state of an entity i, the correspondinglocal function acts only on the state of that entity itself and the states of the entities in ID(i) whichinfluence i.

More precisely, given a digraph D = (V , A) with a vertex set V = {1, 2, . . . , n}, then a map

F : {0, 1}n −→ {0, 1}n, F(x1, x2, . . . , xi, . . . , xn) = (y1, y2, . . . , yi, . . . , yn),

where yi is the updated state of the entity/vertex i by applying a local function fi over the statesof the entities in {i} ∪ ID(i), constitutes a discrete dynamical system called a parallel directeddynamical system (PDDS) over {0, 1}n.

In this work, the global evolution operator F of the system will be induced by Boolean functions,which will describe how to determine a Boolean output from some Boolean inputs. Such functionsplay a fundamental role in questions as design of circuits or computer processes [9]. In our context,they correspond to components of the evolution operator of the dynamical system.

In a previous paper [2], for PDSs over digraphs corresponding to the simplest Boolean functionsAND and OR, we proved that only fixed or eventually fixed points appear, as it occurs overundirected dependency graphs. Nevertheless, for general Boolean functions, it was shown thatany periodic orbit can exist, depending on both the Boolean function that infer the global evolutionoperator of the system and on the structure of the dependency digraph. Motivated by these results,in this work, we analyse the orbit structure of parallel discrete dynamical systems over some

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International Journal of Computer Mathematics 2041

special digraph classes as line digraphs, arborescences, star digraphs, acyclic digraphs and circledigraphs.

To be more precise, the following sections analyse the orbit structure of parallel discrete dynam-ical systems, particularly on the Boolean function NAND, over these special digraph classes.The results for the Boolean function NOR are completely dual. The paper finishes with someconclusions and future research directions.

2. PDSs over line digraphs

A line digraph of order n, Linen, is a combinatorial digraph with vertex set V = {1, 2, . . . , n} andarc set A = {(k, k + 1) : k = 1, . . . , n − 1}. We shall denote by xj

i the state of the entity i after thejth iteration of the system for j ≥ 1, while x0

i stands for the initial state of the entity i. It can berepresented as shown in Figure 1.

First of all, observe these generic situations:

Lemma 1 Given a PDDS on the Boolean function NAND over any dependency digraph:

• If xji = 0 for any i ∈ V , j ≥ 0, then xj+1

i = 1.• If xj

i = 1 for any i ∈ V , j ≥ 0, then xj+1i = 0 if and only if xj

k = 1 for all k ∈ ID(i).• If an entity i is a source of the dependency digraph (i.e. ID(i) = ∅), then its states are 0 and 1

alternatively.

Taking this into account, we have the following result for line digraphs:

Proposition 1 Let NAND-PDDS be a PDS over a line digraph associated with the maxtermNAND. Then all the orbits of this system are 2-periodic orbits or eventually 2-periodic orbits.Moreover, if the dependency line digraph D = (V , A) has n ∈ N vertices, then the number ofiterations needed to arrive to the corresponding 2-periodic orbit is:

• less than or equal to 2n − 3, if the source of the line graph is deactivated.• less than or equal to 2n − 2, if the source of the line graph is activated.

Proof First, let us assume that x01 = 0. Then, by applying NAND to the states of the

entities/vertices we have:

• Iteration 1: x11 = 1, x1

2 = 1.• Iteration 2: x2

1 = 0, x22 = 0.


2 = 1, x33 = 1.


2 = 0, x43 = 0.


2 = 1, x53 = 1, x5

4 = 1, and so on.

Therefore, after 2n − 3 iterations the states of the n entities are equal to 1, and so they are aftertwo more iterations, constituting a period-2 orbit.

On the other hand, if x01 = 1 then x1

1 = 0 after the first iteration, and the situation becomes thesame than before. �

Figure 1. Line digraph Linen.

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Dually, we have:

Proposition 2 Let NOR-PDDS be a PDS over a line digraph associated with the minterm NOR.Then all the orbits of this system are 2-periodic orbits or eventually 2-periodic orbits. Moreover, ifthe dependency line digraph D = (V , A) has n ∈ N vertices, then the number of iterations neededto arrive to the corresponding 2-periodic orbit is:

• less than or equal to 2n − 3, if the source of the line graph is activated and• less than or equal to 2n − 2, if the source of the line graph is deactivated.

3. PDSs over arborescences

An out-arborescence with root r is a digraph where there exists one and only one (directed) pathfrom r to each vertex of the digraph (Figure 2).

Dually, it can be defined an in-arborescence with root r as a digraph where there exists oneand only one (directed) path from each vertex of the digraph to r.

For this kind of digraphs, the situation is very similar to the one regarding line digraphs. Inparticular, we have:

Proposition 3 Let NAND-PDDS be a PDS over an out-arborescence (resp. in-arborescence)associated with the maxterm NAND. Then all the orbits of this system are 2-periodic orbits oreventually 2-periodic orbits. Moreover, if the longest branch of the arborescence has m ∈ N

vertices, then the number of iterations needed to arrive to the corresponding 2-periodic orbit is:

Figure 2. Out-arborescence.

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• less than or equal to 2m − 3, if the root of the arborescence graph is deactivated (resp. the firstvertex of every branch of length m is deactivated) and

• less than or equal to 2m − 2, if the root of the arborescence graph is activated (resp. the firstvertex of a branch of length m is activated).

Proof It is an immediate consequence of Proposition 1. �

As in the section before, we have the dual result for NOR-PDDS:

Proposition 4 Let NOR-PDDS be a PDS over an out-arborescence (resp. in-arborescence)associated with the minterm NOR. Then all the orbits of this system are 2-periodic orbits oreventually 2-periodic orbits. Moreover, if the longest branch of the arborescence has m ∈ N

vertices, then the number of iterations needed to arrive to the corresponding 2-periodic orbit is:

• less than or equal to 2m − 3, if the root of the arborescence graph is activated (resp. the firstvertex of every branch of length m is activated) and

• less than or equal to 2m − 2, if the root of the arborescence graph is deactivated (resp. the firstvertex of a branch of length m is deactivated).

4. PDSs over star digraphs

An out-star with central node c is a digraph consisting of some arcs starting from the vertex c, asdepicted in Figure 3.

Dually, it can be defined an in-star with central node c, where now c is a sink (i.e. ID(c) =V \ {c}).

As one can easily check, these digraphs are particular cases of arborescences, where the numberof vertices of any branch is always 2. Thus, bearing in mind Propositions 3 and 4, one can easilyobtain:

Corollary 1 Let NAND-PDDS be a PDS over an out-star (resp. in-star) associated with themaxterm NAND. Then all the orbits of this system are 2-periodic orbits or eventually 2-periodic

Figure 3. Out-Star graph.

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Figure 4. Generalized-Star digraph.

orbits. Moreover, the number of iterations needed to arrive to the corresponding 2-periodicorbit is:

• less than or equal to 1, if the central node of the star graph is deactivated (resp. all thenon-central vertices are deactivated) and

• less than or equal to 2, if the central node of the star graph is activated (resp. one of thenon-central vertices is activated).

Corollary 2 Let NOR-PDDS be a PDS over an out-star (resp. in-star) associated with theminterm NOR. Then all the orbits of this system are 2-periodic orbits or eventually 2-periodicorbits. Moreover, the number of iterations needed to arrive to the corresponding 2-periodicorbit is:

• less than or equal to 1, if the central node of the star graph is activated (resp. all the non-centralvertices are activated) and

• less than or equal to 2, if the central node of the star graph is deactivated (resp. one of thenon-central vertices is deactivated).

Nevertheless, this star digraph class can be generalized, considering those star digraphs wherethe central node is neither a source nor a sink necessarily, as shown in Figure 4

Proposition 5 Let NAND-PDDS be a PDS over a generalized-star associated with the maxtermNAND. Then all the orbits of this system are 2-periodic orbits or eventually 2-periodic orbits.

Proof If none of the non-central vertices is a source, then the digraph is an out-star digraph andthe result follows from Corollary 1.

If every non-central vertex which is a source has its state equal to 0 (resp. equal to 1), thenthe system behaves in the same way that the out-arborescence which results if we remove all thenon-central vertices which are sources except one, which will be considered the root. In particular,note that the states of the removed vertices alternate 1’s and 0’s.

If there is a non-central vertex which is a source whose state is 0 and a another one whosestate is 1, then after the first iteration the central vertex becomes activated and remains so forever,while the rest of the entities change their state alternatively.

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Figure 5. Circle graph.

In any case, a period-2 orbit is reached. �

Dually, we have:

Proposition 6 Let NOR-PDDS be a PDS over a generalized-star associated with the mintermNOR. Then all the orbits of this system are 2-periodic orbits or eventually 2-periodic orbits.

5. PDSs over acyclic digraphs

An acyclic digraph is a digraph without (directed) cycles. Bearing in mind, the results providedin the sections before, we obtain the following results:

Proposition 7 Let NAND-PDDS be a PDS over an acyclic digraph associated with the maxtermNAND. Then all the orbits of this system are 2-periodic orbits or eventually 2-periodic orbits.

Proof Let us take an arbitrary entity i. Thus, if i is a source, then it alternates 0’s and 1’s. Noticethat, as the digraph is acyclic, there is at least one source. On the contrary, if i is not a source, wecan consider the in-arborescence subgraph with root i. Then, from Proposition 3, it follows that,from a certain iteration, the state of i alternates 1’s and 0‘s or attains a fix value 1. With all ofthese, the result follows immediately. �

Dually, one can easily obtain the corresponding result for NOR-PDDS.

Proposition 8 Let NOR-PDDS be a PDS over an acyclic digraph associated with the mintermNOR. Then all the orbits of this system are 2-periodic orbits or eventually 2-periodic orbits.

6. PDSs over circle digraphs

A circle digraph of n nodes, Circn, is a digraph consisting of a directed cycle, as shown in Figure 5.We will assume that the vertices of a circle digraph Circn are ordered according the direction ofthe circle, i.e. A = {(k, k + 1) : k = 1, . . . , n − 1} ∪ {(n, 1)}. In particular, we shall denote byxj

1xj2 · · · xj

i · · · xjn the vector state of the n entities after the jth iteration of the system.

In contrast with the digraph classes analysed so far, in this case, there is no unique pattern ofthe orbital structure. In this sense, we have achieved some partial results regarding them.

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Lemma 2 Let NAND-PDDS be a PDS over a circle digraph of n nodes, n ≥ 4. Then xj+2i+1 = xj

i ,i ∈ V , j ≥ 1. In other words, the n-tuple of states moves one position to the right (in a cyclicsense) each two iterations.

Proof First of all, observe that if the vector state is a n-tuple of 1’s (resp. 0’s), the result followsimmediately, since in the next iteration all the states will be 0 (resp. 1).

Now, let us see that after the first iteration, there does not exist any fragment of the n-tuplewith structure xj

i−1xjix

ji+1 = 010. Certainly, suppose, by reduction to the absurd, that there exists

such a fragment after the jth iteration. Let us consider the preceding and subsequent states in then-tuple xj

i−2010xji+2. Note that if n = 4 then xj

i−2 and xji+2 stand for the state of the same vertex,

which is not a problem because of the cyclic structure of the n-tuple. Now, observe that in orderto obtain the fragment xj

i−2010xji+2, the previous states should be 110xj−1

i+1xji+2. But none of the

states xj−1i+1 = 0 or xj−1

i+1 = 1 leads to the states xji−2010xj

i+2, as we wanted to prove.Taking this into account and assuming that the vector state contains both 1’s and 0’s, after the

first iteration, we can decompose the n-tuple of states in pieces (not necessarily disjoint) of oneof these two types:

• Type 1: 100 · · · 01, where the central part has α 0’s, α ≥ 2.• Type 2: 011 · · · 10, where the central part contains β 1’s, β ≥ 2.

Now, it is a straightforward computation to check that, in both cases, the α 0’s or the β 1’s moveone position to the right each two iterations. �

As a consequence, we have:

Corollary 3 Let NAND-PDDS be a PDS over a circle digraph of n nodes, n ≥ 4. Then thelongest periodic orbit has period 2n. Actually, the state of the system given by the n-tuple 110 · · · 0originates a 2n-periodic orbit in a circle digraph Circn.

Proposition 9 Let NAND-PDDS be a PDS over a circle digraph of n nodes, n ≥ 5 odd. Thenthere exists a periodic orbit of period n.

Proof Considering the initial state of the system given by x01 = x0

2 = · · · = x0(n+1)/2 = 1 and

x0(n+1)/2+1 = · · · = x0

n = 0, one can easily check that this initial state generates a periodic orbit ofperiod n. �

Finally, using the previous results we obtain:

Theorem 1 NAND-PDDS over circle digraphs can present periodic orbits of any period exceptfixed points and periods 4 and 6.

Proof One example of period 3 can be obtained in Circ3 taking the initial state of the system110. The case of n ≥ 8, n even, follows from Corollary 3. Analogously, the case n ≥ 5, n odd,follows from Proposition 9.

On the other hand, let us see that the system cannot have fixed points. This is a direct consequenceof Lemma 2, taking also into account that if all the states are 0’s (resp. 1’s), in the followingiterations, all the states become 1’s (resp. 0’s). Moreover, this last case is an example of period 2.

In order to show that there does not exist periodic orbits of period 4, first note that Circ2 andCirc3 do not contain such orbits. Suppose, by reduction to the absurd, that Circn, n ≥ 4, containsa periodic orbit of period 4, and let us consider an arbitrary fragment of length 4 of its n-tuple

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of states xji+1xj

i+2xji+3xj

i+4 once the system has attained the orbit of period 4. From Lemma 2 and

using that the system has an orbit of period 4, it must be xji+1 = xj

i+3 and xji+2 = xj

i+4. Observe that

it cannot be xji+1 �= xj

i+2 and xji+3 �= xj

i+4, since under this assumption, a fragment of the n-tuple ofstates with structure 010, which is not possible, would exist (see the proof of Lemma 2). Hence,xj

i+1 = xji+2 = xj

i+3 = xji+4. Since this reasoning is valid for whichever fragment of length 4 of the

n-tuple of states, then all the states are either 1’s or all of them are 0’s. But in such a case, thesystem presents a periodic orbit of period 2. Therefore, no orbits of period 4 can exist.

Now, let us see that periodic orbits of period 6 are not possible. It is easy to check that Circ2,Circ3, Circ4 and Circ5 do not contain such orbits. Suppose that Circn, n ≥ 6, contains a peri-odic orbit of period 6, and let us consider an arbitrary fragment of length 6 of its vector statexj

i+1xji+2xj

i+3xji+4xj

i+5xji+6 once the system has attained the orbit of period 6. From Lemma 2 and

using that the system has an orbit of period 6, it must be xji+1 = xj

i+4, xji+2 = xj

i+5 and xji+3 = xj

i+6.We distinguish the following possibilities:

• If xji+1xj

i+2xji+3 = 100, xj

i+1xji+2xj

i+3 = 010 or xji+1xj

i+2xji+3 = 001, a fragment of the n-tuple of

states with structure 010, would exist, which is not possible.• If xj

i+1xji+2xj

i+3 = 000 or xji+1xj

i+2xji+3 = 111, reasoning as above all the states are either 1’s or

all of them are 0’s, and in this case the system presents a period-2 orbit.• If xj

i+1xji+2xj

i+3 = 110, xji+1xj

i+2xji+3 = 011 or xj

i+1xji+2xj

i+3 = 101, it can be checked that n is amultiple of 3 and the system present a periodic orbit of period 3.

Therefore, no orbit of period 6 can appear. �

Remark 1 As in the previous sections, all the results regarding circle digraphs can be stated ina dual way for NOR-PDDS

7. Conclusions and future research directions

This work provides some results on the orbital structure of PDSs over special digraph classes.Except for the case of circle digraphs, only period-2 orbits of fixed points appear. Thus, it is thepresence of cycles in the structure of the dependency digraph what provokes the appearance ofperiodic orbits with period greater than two.

As this orbital structure depends on both the global evolution operator and the structure of thedigraph, it would be impossible to cover the whole casuistic. However, as shown along the paper,the study of some special cases allows us to obtain important consequences for more generaldigraphs. Therefore, this work could help to study in the future particular cases, specially relevantfor applied sciences or engineering, where these models appear.

Acknowledgements

This work has been partially supported by the Grants MTM2011-23221, PEII11-0132-7661 and PEII09-0184-7802.

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[12] H.S. Mortveit and C.M. Reidys, Discrete, sequential dynamical systems, Discrete Math. 226 (2002), pp. 281–295.[13] H.S. Mortveit and C.M. Reidys, An Introduction to Sequential Dynamical Systems, Springer, New York, 2007.[14] Z. Róka, Cellular automata on Cayley graphs, Ph.D. thesis, Ecole Normale Superieure de Lyon, France, 1994.[15] S. Wiggins, Introduction to Applied Nonlinear Systems and Chaos, Springer, New York, 1990.[16] S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys. 55(3) (1983), pp. 601–644.[17] S. Wolfram, Universality and complexity in cellular automata, Physica D 10 (1984), pp. 1–35.

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Parallel dynamical systems over special digraph classes