Review Article Parallel Dynamical Systems over Graphs and ...downloads.hindawi.com/journals/jam/2015/594294.pdf · in a synchronous manner, the system is called a parallel dynamical

Review ArticleParallel Dynamical Systems over Graphs andRelated Topics A Survey

Juan A Aledo Silvia Martinez and Jose C Valverde

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

Correspondence should be addressed to Jose C Valverde josevalverdeuclmes

Received 10 October 2014 Accepted 10 December 2014

Academic Editor Giuseppe Marino

Copyright copy 2015 Juan A Aledo et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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

1 Introduction

Modeling discrete processes is one of the most importanttasks in modern mathematics In fact several mathematicalconcepts have become fundamental in order to establishdiscretemathematicalmodels for several phenomena comingfrom science and engineering Some of them are Booleanalgebras and functions [1 2] together with other topics asgraphs [1 2] and discrete dynamical systems [3ndash6]

Boolean algebras give symbolic form to Aristotlersquos systemof logic In the second half of the nineteenth century theEnglish mathematician George Boole (1815ndash1864) definedan algebraic structure which encodes several rules of rela-tionship between mathematical quantities limited to twopossible values true or false mathematically modeled by 1or 0 This algebraic structure is currently known as Booleanalgebra The results of the study were published in a survey[7] entitled ldquoAn Investigation of the Laws of Thought onWhich Are Founded the MathematicalTheories of Logic andProbabilitiesrdquo in 1854 Nevertheless after almost a centuryClaude E Shannon was the first one who established howBoolean algebras could be applied to on-off circuits wherethe presence of signal is characterized by 1 and the absence by0 In the forties his masterrsquos thesis [8] entitled ldquoA SymbolicAnalysis of Relay and Switching Circuitsrdquo made Boolersquos

theoretical work become a fundamental mathematical toolfor designing and analyzing digital circuits

Graphs appear in many different fields of science andengineering as a tool to represent the relations among theelements of a system [9ndash15]They constitute a powerful tool toperceive more clearly problems and to face up to them espe-cially when there is a finite number of elements to deal withThis occurs in discrete processes and so they are basic for themathematical formalization of many of these processes

The notion of dynamical system is the mathematical for-malization of the general scientific concept of deterministicprocess [5 6] It involves a set of possible states of the processwhich is named state space or phase space and a law of theevolution or evolution operator in time Thus this notionis fundamental in order to model any phenomenon whosefuture asymptotic state needs to be known

These three concepts are mixed properly to constructa mathematical model named parallel dynamical system(PDS) that allows us to formalize and analyze the dynamicalbehavior of discrete processes

This mathematical model constitutes a generalizationof other relevant ones which appeared previously in theliterature as cellular automata (CA) [16ndash25] or Booleannetworks (BN) [14 26ndash29] (The abbreviations PDS SDS CAand BN will be written for the singular and plural forms of

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2015 Article ID 594294 14 pageshttpdxdoiorg1011552015594294

2 Journal of Applied Mathematics

the corresponding terms since it seems better from an aes-thetic point of view)

A CA has been traditionally conceived as a grid of cellswhere each cell has a state belonging to a finite set (usually0 1) such that all the cells evolve synchronously in discretetime steps The updating of the state of every cell is realizedaccording to a common local function affecting only theneighbors of such a cell Thus if 119909119905

119894is the state value of a cell

119894 at the time 119905 the cell value is updating by applying a localfunction on the cells in the neighborhood of the cell 119894

CA has revealed as a suitable mathematical model to cap-ture the essential features of digital computers synchronicityregular distribution and locality of iterations Although theywere introduced for the first time in the works of Ulam andvon Neumann [30] they became of public interest in 1970when Martin Gardner published [31] with an explanation ofJohnConwayrsquos ldquoGame of Liferdquo in ScientificAmerican (see also[32])

In [21] Wolfram analyzed a set of CA and showed thatdespite their simple construction some of them are capableof a complex behavior Later in [22] based on a deeperresearch he suggested that many one-dimensional cellularautomata fall into four basic behavior classes three of themexhibiting a similar behavior to fixed points periodic orbitsand chaotic attractors and the fourth one such that itsasymptotic properties are undecidable Wolfram publishedother important papers which are compiled in [25]

A BN of size 119899 and connectivity 119896 consists of 119899 inter-connected vertices each one having 119896 inputs The updateof the state of any vertex is determined by the (directed)dependency relations and local rules which are given byBoolean functions Hence BN are a generalization of (finite)Boolean CA

BN appeared almost at the same time in applied modelscreated for the simulation of aspects of the behavior ofbiological systems This occurred in [26] where Kauffmanconstructed molecular automata for modeling a gene as abinary (on-off) device and studied the behavior of largerandomly constructed nets of these binary genes (see also[27]) For this reason BN are also known as Kauffman (net)models The results by Kauffman suggest that if each geneis directly affected by two or three genes then the systembehaves with great order and stability and presents cycles

In the last two decades many works have focused theirattention on mathematical modeling of several computerprocesses The first one of a series of these works was [33]which constituted an important step in the development ofmathematical foundations for the theory of computation Inthis work sequentially updated cellular automata (SCA) overarbitrary graphs are employed as a paradigmatic frameworkThis first work was followed by [34ndash36] where the authorsdeveloped this theory analyzing the asymptotic behavior ofsuch mathematical models The formal definition of PDSappeared for the first time in [9] and it constitutes a funda-mental issue for the posterior development of the results inthis research line

Computer processes involve generation of dynamics byiterating local mappings In fact a computer simulation is amethod for the composition of iterated mappings typically

on local dependency regions [33] It means that themappingshave to be updated in a specific manner that is an updateschedule Update scheduling is also a commonly studiedaspect in discrete event simulations [37ndash39] Mathematicalmodeling of computer processes has resulted very usefulin order to predict what can occur after executing theseprocesses

In the computational context it is common to renamethe local mappings as entities (cells in the language of cellularautomata theory) which are the lowest level of aggregationof the system In computer processes there are many entitiesand each entity has a state at a given time (see [33ndash35]) Theupdate of states of the entities constitutes an evolution intime of the system that is a discrete dynamical system (see[9 40])

In this sense each entity 119894 of the system could be activatedor deactivatedThus it is natural to consider that its state 119909

119894isin

0 1 The evolution or update of the system is implementedby local functions which are restrictions of a global functionor together constitute this global one That is for updatingthe state of any entity the corresponding local function actsonly on the state of that entity itself and the states of theentities related to it The relations among entities are usuallyrepresented by a graph which is called the dependency graphof the system

Thus the update of the states is determined by the depen-dency graph relations of the entities and the local functionswhich together constitute the (global) evolution operator ofthe dynamical system If the states of the entities are updatedin a parallel (or synchronous) manner the system is calleda parallel dynamical system [9 41ndash43] while if they areupdated in a sequential (or asynchronous) order the system isnamed sequential dynamical system (SDS) [9 33ndash36 40 44ndash46]

CA when finite can be considered as a special kind ofPDS by considering cells as entities On the other hand BNare a generalization of (finite) Boolean CA but at the sametime a particular case of PDS by considering nodes as enti-ties One of the main differences with CA is that in BN thestate of each node is not affected necessarily by its neighborsbut potentially by whichever node in the network Thusthe uniform structure of neighborhood in CA disappearsHowever some homogeneity remains since each node isaffected by 119896 connections with other (or the same) entitiesThis homogeneitymakes BN a particular case of PDS since inPDS connections can be totally arbitrary

Since CA and BN are special cases of PDS their applica-tions can be assumed as applications of PDS Some applica-tions of CA and BN that appear in the literature are enu-merated below in order to give an idea of the great interestof this tool for modeling discrete processes from science andengineering

As said before Boolean networks appear for the first timein [26] where Kauffman constructed molecular automatafor modeling a gene as a binary (on-off) device (see also[27]) Up to date these results continue being developedas can be seen in [11] where they consider asynchronousstochastic update [14] where probability is introduced in thedirected dependency graph which they call random Boolean

Journal of Applied Mathematics 3

network or [47] where a Kauffman net model shows thateukaryotic cells are dynamically ordered or critical but notchaotic Since Kauffman nets can be seen as particular casesof directed dependency graphs that are considered herethe results shown in this review article could be useful inthe development of some aspects of biological systems andbiochemical control networks

In [48] the authors show different environments whereCA can be used to model several biological patterns Mostrecently Boolean networks have been applied for modelingepidemics in [15]

In the eighties two important works appeared relatingCA and ecological modeling [18 49] In fact [49] presentsCA as a paradigm for ecological modeling Later in [50] CAwere used to simplify spatial complexity in the geometry ofecological interactions Finally in [51] the authors studied thelinks between CA and population dynamics

More recently two works [10 12] have been dedicated tothe study of the applications of CA in cryptography

On the other hand in [52] a parallel discrete dynamicalmodel with three evolution local functions is used in orderto analyze the structure of a partially ordered set (poset) ofsigned integer partitions whose main properties are actuallynot known The authors affirm that this model is related tothe study of some extremal combinatorial sum problemsBesides in [53] a class of lattices and Boolean functions areemployed to study the truth of a mathematical conjectureOther related interesting works in this research line are [54ndash57]

In the book [58] the use of CA for modeling someproblems of physics is studied in detail while in the book [59]the authors investigate the use of CA in modeling chemicalphenomena

Also in sociology there exist some works where CAare used for modeling the corresponding systems (see forinstance [60 61]) Besides in the literature one can find otherworks where this kind of systems are employed for otherapplications as the simulation of two-lane traffic [62] or thegeneration of rhythmic structure in music [63]

According to [6] one of the main goals in the study ofa dynamical system is to give a complete characterizationof its orbit structure Actually this is the main purposeof the review paper in relation with parallel dynamicalsystems over graphs to show as much information about theorbit structure as possible based on the properties of thedependency graph and the local Boolean functions whichconstitute the evolution law of the system

In this particular case as the state space of the system isfinite every orbit is periodic or eventually periodic Note thatfor a system with 119899 entities the number of possible states isequal to 2

119899 Thus for example if 119899 = 20 then the number ofpossible (initial) states of the system is greater than a millionTherefore computational studies become inefficient when 119899

is big enough and can only analyze small systems or a lownumber of initial states

In view of that the unique way to give general resultsfor these systems is to find out their properties analyticallyas done in [41ndash43 64 65] in order to describe the differentcoexistent periodic orbits of PDS Following Derrida and

Pomeau [66] the most interesting questions about thesesystems concerning their orbit structures are the following

(1) What is the length of the limit cycles(2) What is the number of different limit cycles(3) If one considers two different initial states when do

they arrive in the same limit cycle

Actually these are the main questions reviewed in thispaper for PDS More specifically this paper reviews the fullcharacterization of the orbit structure of PDS over undirectedgraphs covering all the casuistry since it shows the behaviorof any parallel dynamical system regardless the type ofgraph the number of entities and the relationships amongthem Also essential properties of the orbit structure of PDSover directed dependency graphs are revised Besides twoextensions of the concept of PDS are shown

The paper is organized as follows In the next section wepresent theoretical foundations of PDS and compare themwith CA and BN in detail In Section 3 the most importantresults on the dynamics of PDS are reviewed Section 4 isdevoted to reviewing two extensions of this mathematicalmodel A linearization algorithm for the computation oforbits in PDS is addressed in Section 5 The paper finishes bysetting out several interesting future research directions forthe development of these kinds of models

2 Theoretical Foundations of PDS andComparative Study with Related Topics

21 Theoretical Foundations

211 Dependency Graphs In computer processes there aremany entities and each entity has a state at a given time (see[33ndash36]) Entities are related and they get information fromthe entities in their neighborhood Usually in order to geta graphical idea of the situation every entity is representedby a vertex of an undirected graph and two vertices areadjacent if their states influence each other in the updateof the system The undirected graph so built is called the(undirected) dependency graph of the system (see [9])

If we denominate this graph 119866 = (119881 119864) where 119881 =

1 2 119899 is the vertex set and 119864 is the edge set then foreach vertexentity 119894 1 le 119894 le 119899 it is natural to consider thatits state 119909

119894isin 0 1 That is the entity can be activated or

deactivatedOn the other hand for every vertexentity 119894 and every

subset 119882 sub 119881 we will consider all the vertices that interferewith them Thus we denote by

119860119866(119894) = 119895 isin 119881 119895 119894 isin 119864 (1)

the set of vertices that are adjacent to the vertex 119894Nevertheless in many occasions the process of informa-

tion exchange is not bidirectional [67] This situation can berepresented by an arc whose initial vertex is the influencingentity and the final vertex corresponds to the influencedentity obtaining a directed graph or digraph of relationsThedirected graph so built will be called the directed dependency


graph of the system In order to unify the notation it will bealso denoted by 119866 = (119881 119864) although in this case 119864 is a setof arcs instead of edges With the same aim given a directeddependency graph and 119894 isin 119881 119860

119866(119894) will stand for the set of

vertices 119895 isin 119881 such that there exists an arc from 119895 to 119894Recalling that given vertices 119894 119895 isin 119881 the distance 119889(119894 119895)

between them is defined as the length of the shortest pathfrom 119894 to 119895 and the diameter of the digraph as

diam (119866) = max 119889 (119894 119895) 119894 119895 isin 119881 (2)

212 Boolean Algebra and Boolean Functions As the naturalway to consider if the state of any entity is activated or deacti-vated the basic Boolean algebra 0 1will play a fundamentalrole for the theoretical foundations of the model Moreoverthe (general) structure of Boolean algebra is essential tonaturally capture the situation where entities are composedby some subentities such that each of them can be activatedor deactivated

Mathematically aBoolean algebra can be defined [2] as analgebraic structure (119861 and or

1015840 119874 119868) where 119861 is a set and or are

two inner operations defined on 119861 1015840 is an application from119861 to 119861 and 119874 119868 are two distinguished elements of 119861 suchthat they satisfy certain laws (idempotence commutativeassociative absorption distributive existence of neutral anduniversal elements and existence of complement element)

It is common to denote the operation or by + and to callit addition operation of the Boolean algebra Likewise theoperation and is usually denoted by sdot and called multiplicationoperation of the Boolean algebra

It is natural to introduce the algebraic structure of Bool-ean algebra as a Boolean Lattice which is bounded distribu-tive and complemented [1] In fact in [2] a definition whereonly commutative idempotence distributive and comple-ment laws are required is given which fits better the originaldefinition by George Boole

If 119861 is any Boolean algebra then 119861119899

= (1199091 119909

119899)

119909119894isin 119861 forall119894 is also a Boolean algebra where the operations

are performed coordinate by coordinate that is

(i) (1199091 119909

119899) or (119910

1 119910

119899) = (119909

1or 1199101 119909

119899or 119910119899)

(ii) (1199091 119909

119899) and (119910

1 119910

119899) = (119909

1and 1199101 119909

119899and 119910119899)

(iii) (1199091 119909

119899)1015840= (1199091015840

1 119909

1015840

119899)

In this Boolean algebra the neutral and universal elementsare given respectively byO = (119874 119874) and I = (119868 119868)

Two Boolean algebras (1198611 and or1015840 119874 119868) and (119861

2 ⋏ ⋎1015841

119874 119868) are isomorphic (as Boolean algebras) [2] if there is abijection 120593 119861

1rarr 1198612such that for all 119909 119910 isin 119861

1 it satisfies

(i) 120593(119909 or 119910) = 120593(119909) ⋎ 120593(119910)(ii) 120593(119909 and 119910) = 120593(119909) ⋏ 120593(119910)

(iii) 120593(1199091015840) = 120593(119909)1015841

In particular we have that 120593(119874) = 119874 and 120593(119868) = 119868One of themost important facts in the algebraic structure

of Boolean algebra is the existence of some basic elementsnamed atoms of the Boolean algebra which allowus to control

the rest of the elements of the structured and focus on themthe essence of the structure An atom is every element 119886 isin 119861different from the neutral one119874 which verifies that 119909and119886 = 119909

if and only if 119909 = 119886 or 119909 = 119874The atoms allow us to express any element of the Boolean

algebra different from the neutral one 119874 as a disjunction ofthem in a unique way (up to the order) Moreover the well-known Stonersquos theorem (see [2]) asserts that all the Booleanalgebras with the same number of atoms are isomorphic Inparticular every Boolean algebra with 119901 atoms is isomorphicto the Cartesian product of119901 copies of the basic algebra 0 1what is essential for obtaining more general results on thismodel [65] As a corollary if a Boolean algebra119861 has 119901 atomsthen 119861 has 2119901 elements

Another important feature is the duality principle inBoolean algebras which means that if a statement is theconsequence of the definition of Boolean algebra then thedual statement is also true The dual of a statement in aBoolean algebra is the statement obtained by interchangingthe operations or and and and the elements 119874 and 119868 in theoriginal statement [2]

ConcerningBoolean algebras another important conceptthroughout this work is the following

Definition 1 Let 119861 be a Boolean algebra and 119899 isin N A Booleanfunction of 119899 variables is a function of the form

119871 119861119899997888rarr 119861 (3)

where119871(1199091 1199092 119909

119899) isin 119861 is obtained from119909

1 1199092 119909

119899isin 119861

using the inner operations and or defined on 119861 the application 1015840from 119861 to 119861 and the elements 119874 119868 isin 119861

A Boolean function describes how to determine a Bool-ean output from some Boolean inputs Thus such functionsplay a fundamental role in questions as design of circuitsor computer processes [68] In our context they correspondto components of the evolution operator of the dynamicalsystem

An important point is that the set of all the Booleanfunctions of 119899 variables is also a Boolean algebra (see [2])

Maxterms andminterms are both special cases of Booleanfunctions

Definition 2 ABoolean function of 119899 variables1199091 1199092 119909

119899

that only uses the disjunction operator or where each of the 119899variables appears once in either its direct or its complementedform is called amaxterm

In this sense the simplest maxterm corresponds to theone where each of the 119899 variables appears once in its directform that is

OR (1199091 1199092 119909

119899) = 1199091or 1199092or sdot sdot sdot or 119909

119899 (4)

With all the variables in their complemented form wehave the maxterm NAND

NAND (1199091 1199092 119909

119899) = 1199091015840

1or 1199091015840

2or sdot sdot sdot or 119909

1015840

119899 (5)

Minterm is the dual concept of maxterm changing thedisjunction operator for the conjunction one


Definition 3 A Boolean function of 119899 variables that only usesthe conjunction operator and where each of the 119899 variablesappears once in either its direct or its complemented formis called aminterm

The simplest minterm corresponds to the one where eachof the 119899 variables appears once in its direct form that is

AND (1199091 1199092 119909

119899) = 1199091and 1199092and sdot sdot sdot and 119909

119899 (6)

With all the variables in their complemented form wehave the minterm NOR

NOR (1199091 1199092 119909

119899) = 1199091015840

1and 1199091015840

2and sdot sdot sdot and 119909

1015840

119899 (7)

As one can check there exist exactly 2119899 maxterms of 119899

variables since a variable in a maxterm expression can beeither in its direct or in its complemented form and dually 2119899minterms Since the minterms are the atoms of the Booleanalgebra which is constituted by all the Boolean functionsof 119899 variables there are exactly 2

2119899

Boolean functions of 119899variables

In particular see [2 69] any Boolean function except119865 equiv 119868 (resp 119865 equiv 119874) can be expressed in a canonicalform as a conjunction (resp disjunction) of maxterms (respminterms) Therefore it is natural to begin the study of thedynamics with these basic Boolean functions

213 Dynamical Systems Mathematically a dynamical sys-tem is defined as a triple (119883 119879 119865) where 119883 is the state space119879 is the time set and 119865 is the evolution operator verifying thefollowing

(1) 119883 is a set(2) 119879 is a number set(3) 119865 119879 times 119883 rarr 119883 is a map satisfying

119865 (0 119909) = 119909 forall119909 isin 119883

119865 (119905 119865 (119904 119909)) = 119865 (119905 + 119904 119909) forall119905 119904 isin 119879 forall119909 isin 119883(8)

The set 119883 is often a metric space in order to determine thedistances between two different states and it is also calledphase space due to classical mechanics

The most general way in which the evolution operator isgiven is by means of a family of maps depending on 119905

119865119905 119883 997888rarr 119883 (9)

which transforms any initial state1199090 into some state119909119905 at time119905 that is 119865119905(1199090) = 119909

119905Depending on the time set 119879 it is distinguished between

discrete and continuous dynamical systems

119879 = Z or N cup 0 or Zminuscup 0 999492999484 Discrete System

119879 = R or R+cup 0 or

Rminuscup 0 999492999484 Continuous System

(10)

In this review we deal with a special kind of discretedynamical systems A discrete dynamical system is fullyspecified by defining only one map 119865 named the time-onemap of the system since for any other 119905

119865119905= 119865∘

119905)

sdot sdot sdot ∘119865 (11)

In our particular case as the state space is finite the time-one map can be represented by a table which is called lookuptable for the state updating

When the evolution operator is defined for both negativeand positive values of the time the system is called invertibleIn such a system the initial state defines not only the futurestates but its past behavior as well In our context the invert-ibility of the system appears associated to the reversibilityproblem [70 71]

We actually determine a dynamical system over a(directed or undirected) dependency graph 119866 = (119881 119864) byassociating to each vertex 119894 isin 119881 a state 119909

119894isin 0 1 and a local

map 119891119894defined on the states of the vertices in 119860

119866(119894) cup 119894 and

which returns its new state 119910119894isin 0 1

In fact the evolution or update of a system over a(directed or undirected) dependency graph is implementedby local (Boolean) functions These local functions are eitherrestrictions of a global function or jointly constitute the(global) evolution operator of the system

If the states of the entities are updated in a parallelmanner the system is called a parallel dynamical system(PDS) [9 41 42] while if they are updated in a sequentialorder the system is named a sequential dynamical system(SDS) (see [9 40 46]) More precisely

Definition 4 Let 119866 = (119881 119864) be a (directed or undirected)graph with 119881 = 1 2 119899 Then a map

119865 0 1119899997888rarr 0 1

119899

119865 (1199091 1199092 119909

119899) = (119910

1 1199102 119910

119899)

(12)

where 119910119894is the updated state of the entityvertex 119894 by applying

locally the function 119865 over the states of the vertices in 119894 cup

119860119866(119894) constitutes a discrete dynamical system called a par-

allel (discrete) dynamical system (PDS) over 119866 with evolutionoperator 119865 which will be denoted by [119866 119865] or 119865-PDS whenspecifying the dependency graph is not necessary

Definition 5 Let 119866 = (119881 119864) be a (directed or undirected)graph with 119881 = 1 2 119899 and 120587 = 120587

11205872sdot sdot sdot 120587119899a

permutation on 119881 Then a map

[119865 120587] = 119865120587119899

∘ sdot sdot sdot ∘ 1198651205872

∘ 1198651205871

0 1119899997888rarr 0 1

119899 (13)

where 119865120587119894

0 1119899

rarr 0 1119899 is the update function on the

state vector (1199091 1199092 119909

119899) which updates the state of the

vertex 120587119894while keeping the other states unchanged consti-

tutes a discrete dynamical system called a sequential (discrete)dynamical system (SDS) over 119866 with evolution operator[119865 120587] which will be denoted by [119866 119865 120587] or [119865 120587]-SDS whenspecifying the dependency graph is not necessary


Basic objects associated to a dynamical system are itsorbits and consequently the phase portrait that is the parti-tioning of the state space into its orbits

Definition 6 The orbit of a dynamical system (119883 119879 119865)

starting at 1199090 isin 119883 is the ordered subset of the state space 119883given by

Orb (1199090) = 119909 isin 119883 119909 = 119865119905(1199090) forall119905 isin 119879 sub 119883 (14)

Orbits of a discrete dynamical system are orderedsequences of states in the phase space that can be enumeratedby increasing integers

It is worthwhile to recall that given a discrete dynamicalsystem with global evolution function 119865 and an initial state1199090 the following terminology is used

(i) 119865(1199090) is called the successor of the state 1199090 Observe

that since a dynamical system is deterministic thereexists exactly one successor for each initial state butthis initial state could be the successor of more thanone state of the system that are called its predecessorsWhen this occurs the system is often called a dissi-pative system and the in-degree of the initial state 119909

0

is the number of its predecessors (out-degree is 1 forevery initial state)

(ii) If 119865(1199090) = 119909

0 then 1199090 is called a fixed point of

119865 or equilibrium state of the system The notation119865119868119883[119865] represents the set of all the fixed points of thedynamical system with global evolution function 119865

(iii) If there exists an integer 119901 gt 1 such that 119865119901(1199090) = 1199090

and for any integer 0 lt 119897 lt 119901 119865119897(1199090) = 1199090 then 119909

0

is called a periodic point of 119865 or a periodic state of thesystem and 119901 is called the period of the orbit of 1199090The notation 119875119864119877[119865] is adopted to denote the set ofall the periodic points of 119865 Usually a periodic orbitis called a cycle but we do not use this name in orderto avoid confusions with the cycles of the dependencygraph

(iv) When an initial state 1199090 notin 119865119868119883[119865] cup 119875119864119877[119865] then 1199090

is called a transient point of119865 or a transient state of thesystem In the case of a finite space state these pointsare also called eventually fixed points (resp eventuallyperiodic points) if their orbits finally arrive in a fixedpoint (resp periodic orbit)

(v) If there does not exist any state 1199090 such that 119865(1199090) =119910 then 119910 is said to be a Garden-of-Eden (GOE) of 119865That is 119910 is a state without predecessors The set ofGOEs of 119865 is denoted by GOE[119865]

According to [6 72] one of the main goals in the studyof a dynamical system is to give a complete characterizationof its orbit structure In this particular case as the state spaceof the system is finite every orbit is periodic or eventuallyperiodic This means that the evolution from any initial statealways reaches an attractor [73] If the attractor consists of onestate it is called an attractor point whereas if it consists of twoor more states it is said to be an attractor cycle All the states

2 3

1

Figure 1 Graphic representation of the dependency digraph119866withvertex set 119881 = 1 2 3 and arc set 119864 = (1 2) (2 3) (3 1)

100

010

001

111 000

110

101

011

Figure 2 Transition diagram of the PDS [119866 119865]

Table 1 Lookup table for the time-one-map of the PDS [119866 119865]

119909111990921199093

111 110 101 100 011 010 001 000119910111991021199103

000 101 011 111 110 111 111 111

that flow towards an attractor constitute what is called itsbasin of attraction The time that an initial state takes to reachan attractor is named its transient

The representation of the orbit structure of a system iscalled phase portrait phase diagram or transition diagramof the system The phase portrait when possible gives acomplete visual idea of the asymptotic behavior of the systemfrom any initial state

Example 7 In the next simple example the different notionsrelated to PDS that were introduced before will be studied inorder to clarify them Let us consider the PDS [119866 119865] with

119865 0 13997888rarr 0 1

3 119865 (119909

1 1199092 1199093) = (119910

1 1199102 1199103)

(15)

over the directed graph 119866 shown in Figure 1 where the time-one-map of the system is given by the following lookup table(see Table 1)

This lookup table corresponds to the application of themaxterm NAND as evolution operator and the transitiondiagram or phase portrait of the system is the one in Figure 2

Note that the system is not invertible since there are 3initial states that have not any predecessor 001 010 and 100They constitute the GOE of the system and all of them arein the basin of attraction of the attractor cycle 111 000 Infact they are predecessors of the state 111 Thus these threestates are eventually 2-periodic and the transient to arrive inthe attractor cycle is equal to one

On the other hand the system does not present anyfixed point although it has a 3-periodic orbit 110 101 011As expected the Sharkovsky order [4] for the periodic orbit


structure of a discrete dynamical system where the evolutionoperator is a continuous function of the interval does notwork here

The absence of fixed points and the presence of periodicorbits of period greater than two in this kind of PDS overdirected dependency graphs are two important breakpointswith respect to the pattern followed by those defined overundirected graph (see [41]) as explained in [42]

As shown before in spite of their relation with classicaldynamical systems the theory and analysis of PDS (and SDS)are based on techniques from algebra combinatorics anddiscrete mathematics in general In fact due to the especialdiscrete per excellence character of PDS some dynamicaltopics used to measure the intrinsic stability of a dynamicalsystem as sensitivity dependence on initial conditions or chaos[4 72] are difficult to be translated into this context sincethey are established according to the topological propertiesof the corresponding metric state space

22 Comparative Study with Related Topics In the specificliterature several topics which are related to PDS appear Inthis section an overview about them is given The list ofrelated topics presented is reduced to the most similar onesThe purpose is only to provide an introduction in order tocompare them with PDS Specifically the nearest conceptsnamely cellular automata (CA) [21ndash25] and Boolean networks(BN) [14 28] are reviewed Other topics related to PDS arefor instance finite-state machines [74 75] and Petri nets [76ndash78] but they will not be discussed here

221 Cellular Automata Following Wolfram [21] a CAconsists of a regular uniform lattice (or array) usually infinitein extent with a discrete variable at each site called cell Thestate of a CA at a time 119905 is completely determined by thestate value 119909

119894of every cell 119894 at such time 119905 The evolution of

this system consists in the discrete time updating of the statevalue of its cells in a synchronous manner The updating ofthe state value of each cell depends on a local function definedon the state values at the previous time step of other cells inits neighborhood The neighborhood of a site in the uniformlattice is typically constituted by the site itself and all theimmediately adjacent sites These immediately adjacent cellsare determined by the lattice structure and a homogeneousrule for the selection of neighbors

Thus CA can be contemplated as a particular case of(infinite) PDS (the concept of PDS can be easily extended toinvolve an infinite number of entities) if one considers cells asentities Nevertheless CAhave amore restricted and uniformstructure and in contrast to PDS CA are usually consideredover inf of the system are fixed mple Z119889 where 119889 ge 1 is saidto be the dimension of the CA

The more restricted and uniform structure of CA isreflected by both the neighborhood 119860(119894) of any cell 119894 andthe local functions 119891

119894acting on it which are determined

homogeneously Certainly in classical CA every cell 119894 has aneighborhood 119860(119894) which is some sequence of lattice siteswhich displays a homogeneous structure If the CA is definedoverZ119889 this homogeneous structure in the neighborhood of

Figure 3 Three-neighborhood structure for one-dimensional CA

Figure 4 Von Neumann neighborhood structure for two-dimen-sional CA

any cell 119894 is obtained bymeans of a (finite) subset 1198951 119895

119896 sub

Z119889 which provides a rule to have

119860 (119894) = 119894 + 1198951 119894 + 1198952 119894 + 119895

119896 (16)

Observe that in a CA over an infinite structure like Z119889 anystate of the system is given by a sequence (119909

119894)119894isinZ119889 usually

named a configuration At this point one can state thedefinition of a 119889-dimensional cellular automata taking intoaccount the (finite) set 119861 of the state values that any cell canhave and the rule to define the neighborhood 119860(119894) of any cell119894 over an infinite lattice Z119889 where the same local function isapplied When the set 119861 is equal to the basic Boolean algebra119861 = 0 1 the CA is said to be a Boolean cellular automata

One-dimensional (Boolean) CA with two possible valuesfor the state of the cells 0 and 1 in which the neighborhoodof a given site is simply the site itself and the sites immediatelyadjacent to it on its left and right are studied in [21] In sucha case the subset to get a homogeneous neighborhood isminus1 0 1 sub Z and it is called a three-neighborhood structure(see Figure 3)

For two-dimensional CA two of themost used neighbor-hood structures are the von Neumann neighborhood [30] andtheMoore neighborhood [79] (see Figures 4 and 5) The finitesubset to obtain the von Neumann neighborhood is

(0 0) (minus1 0) (0 minus1) (1 0) (0 1) sub Z2 (17)

while the one to get the Moore one is

(0 0) (minus1 0) (0 minus1) (1 0) (0 1) (1 1) (minus1 minus1)

(1 minus1) (minus1 1) sub Z2

(18)

CAover finite lattices with 119899 sites can also be defined Twotraditional ways to do that are as follows

(i) by imposing periodic boundary conditions which con-sists in identifying the cells separated by 119899 positions


Figure 5 Moore neighborhood structure for two-dimensional CA

giving a CA over Z119899Z [80] an idea which can beextended to higher dimensions giving a CA over a 119896-dimensional tori

(ii) by imposing zero boundary conditions which consistsin using a line graph as a lattice and adding twoadditional extreme vertices whose states are alwaysequal to zero

Local rules for the updating of Boolean CA are in factBoolean functions of 119896 variables where 119896 is the number ofneighbors in any neighborhood119860(119894) Most of the research onCA has been concerned with one-dimensional cases whichare commonly known as elementary cellular automata (see[10 12 21ndash23 30 49 71 79 81ndash83])

For a one-dimensional three-neighborhood BooleanCA a local function has to determine the (output) statevalue of the central cell from the (input) state values of thecentral cell and its two immediately adjacent ones As we areconsidering Boolean values there are only 2

3 possible inputstate values for any neighborhood in the lattice namely

111 110 101 100 011 010 001 000 (19)

As from any input state values the output state value canbe 0 or 1 there are 22

3

= 256distinct local rules for elementarythree-neighborhood CA Traditionally each of these localrules has a decimal number belonging to 0 1 2 255

assigned and it is known as the Wolfram enumeration ofelementary CA rules The procedure to enumerate such rulesconsists in considering the (output) value states of the triplesabove in that order and then translating the correspondingbinary number into a decimal one That is if one has theoutput value states of the triples above as in Table 2 then thenumber of the local rule is

7

sum119894=0

120572119894sdot 2119894 (20)

This enumeration procedure can be generalized to otherclasses of neighborhoods in CA

Thus the elementary CA rule 0 erases any initial configu-ration while 204 does not change any initial configurationthat is it corresponds to the identity transformation Oneof the most important elementary three-neighborhood CA

Table 2 Denomination of the updating of the state of any entity 119894 toestablish the procedure to enumerate elementary CA rules

119909119894minus1

119909119894119909119894+1

111 110 101 100 011 010 001 000119910119894

1205727

1205726

1205725

1205724

1205723

1205722

1205721

1205720

Table 3 Updating of the state of any entity 119894 corresponding to theelementary CA rule number 150

119909119894minus1

119909119894119909119894+1

111 110 101 100 011 010 001 000119910119894

1 0 0 1 0 1 1 0

is the one given by the local function 119883119874119877 that is theelementary CA rule number 150 (see Table 3)

Its importance is due to its applications to cryptographicprotocols (see [10] or [12])

CA when finite can be considered as a special kind ofPDS by considering cells as entities It is easy to understandthat the lattice of a CA allows us to infer a PDS dependencygraph by considering the cells as entities being any entityadjacent to those that are its neighbors in the lattice Evenwhen infinite CA can be consideredPDS since the concept ofPDS can be easily extended for an infinite number of entities

Also CA are updated in a parallel or synchronousmannerby applying local functions on a subset that contains the (statevalue of the) cell Nevertheless in the last few years someextensions of the concept of CA considering sequential orasynchronous updating have appeared in the literature (see[84ndash86]) In fact the concept of sequential dynamical system[9 33ndash35 40 44 45] constitutes a generalization of such CAextension

However CA are restricted cases of PDS in several waysFirst of all for a CA seen as a PDS the dependency graphwhich is derived from the lattice and the neighborhoodstructure is regular whereas the graph of a general PDS canbe arbitrary Secondly CA have the same local function orrule associated with every cell while general PDS can havedistinct local functions to update different entities which canbe the restriction of a global one (see [9 41]) or independentlydefined (see [43])Thus general PDS can have more involvedupdate schemes

Although theoretically the set 119861 of state values of any cellcan be whichever finite set most of works in the literaturefocus on Boolean CA [10 12 21ndash23 30 49 71 79 81ndash83]where 119861 = 0 1 In [87] and more recently in [88] a firstapproach in the study of CA where 119861 is any finite set is givenOn the other hand in [65] we study completely the problemwhen the set 119861 is any Boolean algebra in the more generalcontext of PDS

Summing up PDS are a generalization of CA where thestate value of each entity is not affected necessarily by itsneighbors (see Figure 6) but potentially by any entity in thenetwork and the updating function 119891

119894associated with an

entity 119894 can be given by any arbitrary local function

222 Boolean Networks A Boolean network (BN) of size119899 and connectivity 119896 consists of 119899 interconnected verticeseach one having 119896 inputs The update of the state of any


CA BN PDS

Figure 6Graphic representation of comparative dependency graphsfor CA BN and PDS

vertex is determined by the (directed) dependency relationsand local rules which are given by Boolean functions HenceBN are a generalization of (finite) Boolean CA but at thesame time a particular case of PDS by considering nodesas entities One of the main differences with CA is that inBN the state of each node is not affected necessarily by itsneighbors but potentially by any node in the network Thusthe uniform structure of neighborhood in CA disappearsNevertheless some homogeneity remains since each node isaffected by 119896 connections with other (or the same) entitiesThis homogeneity makes BN a particular case of PDS wherethe connections can be totally arbitrary (see Figure 6)

Another important difference with CA is that localBoolean functions of 119896-variables are generated randomlywhich provides different lookup tables for each entity for theupdating processThis idea has been carried out and extendedfor PDS in this work in two directions Firstly as can be seenin [41] local Boolean functions acting on each entity can havedifferent number of variables (which cannot occur for BN)and secondly they can be totally independent for each entity

BN with 119899 nodes and 119896 connectivity are known as 119899-119896models or Kauffman models since they were originally devel-oped by him in [26] when modeling regulatory networks of(on-off) genes For this reason nodes of BN are often calledgenes In particular Kauffman models on complete networksare usually called random maps

As in PDS the state space is finite with 2119899 possible states

Therefore whichever the initial state is eventually a state willbe repeated However the number of 119899-119896models when 119899 and119896 are fixed is lower than the number of possible dependencygraphs that determine a PDS

As for the case of CA originally the updating of BNwas synchronous but also in this case several recent worksprovide some extensions of the concept considering asyn-chronous updating schedule [11 85 86] Also in this casethe concept of sequential dynamical system constitutes ageneralization of such asynchronous Boolean networks

Although first Kauffman models were deterministic inthe last decade some authors have studied an extension ofthe concept considering a probabilistic updating schedule [1466 89] In this context they are named probabilistic Booleannetworks and cannot be described from the perspective ofdynamical systems with Markov chains being the naturalframework to set out their evolution This idea of extensionwas provided for the first time in [66] and it will be explainedin the following paragraphs

Let 0 le 119901 le 1 119894 a gene or node of a BN and denoteby 119891119894a local function to update the state value of 119894 for all

119894 = 1 119899 On the other hand let 1198911015840

119894be another local

function related to the updating of the node 119894 If any node119894 is updating by means of 119891

119894with probability 119901 and by means

of 1198911015840119894with probability (1 minus 119901) then a basic probabilistic BN is

obtained Thus this stochastic construction may be viewedas a weighted superposition of two deterministic BN thatis the one obtained with the global evolution operator 119865 =

(1198911 1198912 119891

119899) and the other one achieved with the time-

one-map 1198651015840= (1198911015840

1 1198911015840

2 119891

1015840

119899) That is by updating the BN

with the evolution operator 119865 a transition diagram couldbe determined namely Γ while updating with 119865

1015840 anothertransition diagram namely Γ1015840 could be also obtained Thenby superposing the transition diagrams (which are in factdirect graphs) one can construct a digraph assigning theweight119901 to the arcs which are only in Γ the weight 1minus119901 to thearcswhich are only in Γ1015840 and theweight 1 to the arcswhich arein both diagramsThis fused weighted digraph is often calledthe probabilistic phase space and its weighted matrix can beconsidered as the corresponding Markov chain matrix The(probabilistic) evolution of the BN from an initial state canbe then identified with a random walk over this probabilisticphase space that starts at this initial state of the BN

The construction explained before is basic in the sensethat only two kinds of functions with two probabilities areconsidered to update the nodes of the BN Obviously morefunctions could be considered and then the probabilisticphase space would be the result of the superposition of allthe transitions diagrams obtained deterministically with anyglobal operator considering for any arc that appears in morethan one of this transitions diagrams the weight resulting ofthe sumof all the probabilities assigned to any of the diagramswhere this arc appears

Finally observe that the PDS notion could be extended inorder to capture probabilistic updating in the same sense thatprobabilistic BN do Nevertheless in this case they wouldlose the structure of (deterministic) dynamical systems

3 Analytical Results on the OrbitalStructure of PDS

31 PDS over Undirected Dependency Graphs This sectionis devoted to review the orbit structure of parallel discretedynamical systems with maxterms and minterms Booleanfunctions as global evolution operators As a result it is shownthat the orbit structure does not remain when the system isperturbed

In [9] it is proved that for a parallel dynamical systemOR-PDS associated with the maxterm OR all the orbits ofthe system are fixed points or eventually fixed points Moreprecisely the system has exactly two fixed points and themaximumnumber of iterations needed by an eventually fixedpoint to reach the corresponding fixed one is at most as largeas the diameter of the dependency graph

Dually for a parallel dynamical system AND-PDS asso-ciated with the minterm AND all the orbits of the systemare fixed points or eventually fixed points and the systemhas exactly two fixed points and the maximum number of


iterations needed by an eventually fixed point to reach thecorresponding fixed one is at most as large as the diameterof the dependency graph

Observe that the orbit structures of OR-PDS andAND-PDS do not depend on the dependency graphs overwhich they are defined covering all the possible casuistry ofthem

In [41] it is proved that for a parallel dynamical systemMAX-PDS associated with an arbitrary maxterm MAX allthe periodic orbits of this system are fixed points or 2-periodic orbits while the rest of the orbits are eventuallyfixed points or eventually 2-periodic orbits In fact from thedemonstration the following can be deduced

(i) If all the variables are in their direct form only fixedpoints or eventually fixed points can appear

(ii) If all the variables are in their complemented formonly 2-periodic orbits or eventually 2-periodic orbitscan appear

(iii) Otherwise both kinds of situations coexist

As before dually for a parallel dynamical systemMIN-PDS associated with an arbitrary mintermMIN all theperiodic orbits of this system are fixed points or 2-periodicorbits while the rest of the orbits are eventually fixed pointsor eventually 2-periodic orbits

These results provide a complete characterization of theorbit structure of parallel discrete dynamical systems withmaxterms and minterms Boolean functions as global evolu-tion operators In particular one can infer that when othermaxterms (resp minterms) different from OR (resp AND)are considered as global functions to define the evolution of asystem the dynamics can change giving as a result a certainabsence of structural stability of the system so perturbed

32 PDS over Directed Dependency Graphs In this sectionthe orbit structure of parallel discrete dynamical systems overdirected dependency graphs (PDDS) (the abbreviation PDDSwill be written for the singular and plural forms of the corre-sponding term since it seems better from an aesthetic pointof view) with Boolean functions as global evolution operatorsis reviewed In this sense for the cases corresponding to thesimplest Boolean functions AND and OR it is explainedthat only fixed or eventually fixed points appear as occursover undirected dependency graphs However for generalBoolean functions it is shown that the pattern found for theundirected case breaks down

In [42] it is proved that for a parallel dynamical systemOR-PDDS over a directed dependency graph associatedwith the maxterm OR all the orbits of this system arefixed points or eventually fixed points Dually for a paralleldynamical system AND-PDDS over a directed dependencygraph associated with the minterm AND we have also thatall the orbits of this system are fixed points or eventually fixedpoints

Nevertheless comparingwith the undirected case studiedin the previous section for the directed one the number offixed points can increase significantly except if the digraphis strongly connected (ie if for all pairs of vertices 119894 119895 isin 119881

there exists a path from 119894 to 119895 and another one from 119895 to 119894)where only two equilibrium states are possible namely thestate with all the entities activated and the one with all theentities deactivated In general this number of possibilitiesdepends on the structure of the directed dependency graphof the system and it is not possible to give a generic result asin the undirected case

Besides also in [42] it is demonstrated that for the gen-eral case of PDDS associated withmaxterms andminterms asglobal evolution operators any period can appear breakingthe pattern found for the undirected case where only fixedpoints or 2-periodic orbits can exist [41] In order to provethat amethod is developed to provide for every given perioda PDDS which presents a periodic orbit of such a period

Subsequent research [64] allows us to check that thisbreakdown is due to the existence of directed cycles in thedependency digraph while for PDDS over acyclic depen-dency digraphs the periodic orbits continue being only fixedpoints or 2-periodic ones Moreover the orbit structure ofspecial digraph classes as line digraphs arborescences and stardigraphs is similar and it is possible to specify the number ofiterations needed by an eventually periodic point to reach thecorresponding periodic orbit

However it is not easy to control the orbit structure whenthe dependency graph has cycles In such a case the orbitalstructure depends on both the global evolution operator andthe structure of the digraph In particular in [64] it is shownthat NAND-PDDS over circle digraphs can present periodicorbits of any period except fixed points and periods 4 and 6and the same occurs for NOR-PDDS

4 Extensions of the Concept of PDS

In this section two extensions on parallel dynamical systemsare reviewedThe first one is related to themanner of definingthe evolution update and the second one to consider that thestates of the entities can take values in an arbitrary Booleanalgebra

41 PDS on Independent Local Functions When Kauffmanintroduced BN for the first time [26] he contemplatedthe possibility of updating any entity using independentlocal functions This idea is extended to PDS in [43] sinceBoolean functions acting on each entity can have differentnumber of variables (which cannot occur for BN) due to therandom (connections) structure of the dependency graphThis extension of the update method widely generalizes thetraditional one where only a global Boolean function isconsidered for establishing the evolution operator of thesystem Besides this analysis allows us to show a richerdynamics in these new kinds of parallel dynamical systems

In particular for a PDS [119866 119891119894] over an undirected or

directed graph 119866 where each local function 119891119894is either AND

or OR it is demonstrated that all the orbits of this systemare fixed points or eventually fixed points Although theorbit structure coincides with the corresponding one for aPDS where the updating global Boolean function is eitherAND or OR for a PDS [119866 119891

119894] as the one described above

the number of fixed points of the system (and therefore


the complexity of the system) may increase depending onthe election of the local functions 119891

119894and the structure of the

dependency graph In order to illustrate this we show thefollowing example

Example 8 Consider the undirected dependency graphgiven by 119881 = 1 2 3 and 119864 = 1 2 2 3 If we considerthe PDS defined by the global Boolean function AND (respOR) then the fixed points are (1 1 1) and (0 0 0) On theother hand if we consider the PDS defined by the localfunctions

1198911= AND 119891

2= OR 119891

3= AND (21)

then the fixed points are (1 1 1) (0 0 0) (0 1 0) (1 1 0)and (0 1 1)

Moreover for a PDS [119866 119891119894] over an undirected graph

where each local function 119891119894is either NOR or NAND the

periodic orbits are fixed points or 2-periodic orbits and theonly possible fixed point is the one where 119909

119894= 0 if 119891

119894= NOR

and 119909119894= 1 if119891

119894= NANDNevertheless a PDS over a directed

graph where each local function 119891119894is either NOR or NAND

can present periodic orbits of any period 119899 isin NFinally also in [43] it is demonstrated that for a PDS

[119866 119891119894] over an undirected graph where each local function

119891119894isin ANDORNORNAND the periodic orbits of this

system are fixed points or 2-periodic orbits while over adirected graph periodic orbits of any period 119899 isin N can appear

42 PDS with General Boolean State Values In [65] anotherextension of PDS is provided by considering that the states ofthe vertices can take values in an arbitrary Boolean algebra 119861of 2119901 elements 119901 isin N 119901 ge 1This situation naturally appearsfor instance when each entity is composed of 119901 subentitieswhich can be activated or deactivated and consequently thestates of the entities belong to the Boolean algebra 0 1

119901This definition widely extends the traditional one where itis assumed that every entity can take values in the simplestBoolean algebra 0 1 To study the orbit structure of thesemore general PDS in view of the Stonersquos Theorem (see[2]) an isomorphism as Boolean algebras of 119861 and 0 1

119901

is considered Then since the operations and complementelements are obtained coordinate by coordinate in 0 1

119901it is shown how one can translate all the results previouslyachieved to this new context

5 Algorithms for the Computation of Orbits

As said before in [42] it is demonstrated that for the generalcase of PDDS associated with maxterms and minterms asglobal evolution operators any period can appear breakingthe pattern found for the undirected case Since we havea finite state space it is obvious that every orbit is eitherperiodic or eventually periodic However it is not so easyto determine a priori the different coexistent periods of itsorbits because it depends fundamentally on the structureof the directed dependency graph In this sense in [90]algorithms for computing the (eventually) period of any orbitare given

Since the coexistence of periodic orbits depends on thenumber of entities their connections and the Boolean oper-ator several matrix algorithms for the computation of orbitsin PDDS over directed dependency graph are developed in[90] These algorithms constitute a new tool for the study oforbits of these dynamical systems The algorithms are alsovalid for PDS over undirected dependency graphs Likewisethe methods are extended to the case of SDS even when thelocal functions that are considered for the updating of anyentity are independent

These matrix methods of computation of orbits providean especial mathematical linearization of the system bymeans of particular products of the adapted adjacencymatrixof the dependency graph and the state vectors

We think the algorithms established can help to infersome analytical results on the orbit structure of PDS and SDSover special classes of digraphs since they can be used toreveal some orbit patterns

6 Conclusions and Future Research Directions

The results on PDS reviewed provide a full characterizationof the orbit structure of PDS over undirected graphs coveringall the casuistry since it studies the behavior of any paralleldynamical system regardless the type of graph the number ofentities and the relationships among them Specifically whenthe evolution functions are OR or AND or a combinationof both the orbits are fixed points But when consideringother maxterm or minterms dynamics can change due to theappearance of 2-periodic orbits resulting in a certain absenceof structural stability when systems are slightly perturbed

On the other hand essential properties of the orbit struc-ture of PDS over directed dependency graphs are achieved Infact evolutions with OR and AND continue giving only fixedor eventually fixed points But on the contrary for generalmaxterm or minterm functions it is shown that any periodcan appear so breaking the pattern found for the undirectedcase Actually several PDS over especial digraph class arecharacterized Besides it is shown that this disappearanceof the general pattern is due to presence of cycles in thedependency digraph

Motivated by these findings computation methods forcalculating orbits in PDS are also developed Besides twoextensions of the PDS concept are shown

The study developed so far opens several future researchdirections Most of the problems that can arise after thisstudy of PDS correspond to extensions of the model someof which have already begun to be studied in some works inthe narrower context of CA or BN It would be interesting toextend this study to themore general context of PDS (or SDS)

One of the initial questions posed by this work is to checkif the results for PDS are transferable to the case of SDS Paper[9] shows a first approach to the problem since the authorscarried out a parallel study of the orbital structure of PDS andSDSwith evolutionORandNORoperators obtaining similarresults in both cases that is fixed points when the evolutionoperator is OR and periodic points of period 2 when theevolution operator is NOR By duality these results are alsovalid for the case of AND and NAND operators


Results in [41] solve the problemof the orbital structure ofPDS over undirected graphs for which the evolution operatoris any maxterm or minterm including the former OR ANDNOR and NAND Therefore a possible new line of researchin view of this work is to study if the results obtained for PDScan be translated to the case of SDS fundamentally in the caseinwhich the systems are defined over undirected dependencygraphs

In the case of directed dependency graphs the extensionof the results to SDS seems much more complicated as onecan check with simple examples

When modeling discrete processes it can occur that theentities are updated in a mixed manner between parallel andsequential one This happens when there exists a partition ofthe set of entities 119881 of the system such that every set of thispartition consists of entities that are updated synchronouslybut the updating of these sets is asynchronous That is oncethe partition of the vertex set is known the subsets can beordered by a permutation that indicates how these subsetsare sequentially updated This means that given an orderedpartition 119881

1 1198812 119881

119896sub 119881 firstly the entities of the subset

1198811are updating in a parallel manner then the entities of 119881

2

are also updated in a parallel manner but taking into accountthe new states values of the entities in 119881

1 and so on These

systems can be called mixed dynamical systems (MDS) Thisnew updating scheme could be used as an intermediate stepbetween PDS and SDS In fact SDS can be considered asMDSwhere the partition of the set119881 is given by all the subsets withonly one elementThis notion should be investigated not onlyfor the natural interest to model mixed computer processesbut also for the possible repercussions in the translation ofresults from PDS to SDS

The original definition of CA [21] contemplates the pos-sibility that the cells take state values in a finite set althoughsubsequently the majority of studies have been made in thecase of Boolean CA The studies that have appeared so far inthis direction ([87 88]) consider someuniformity since everyentity can have all the state values in the finite set This canhave the physical sense of assuming that the entities can havedifferent levels of intensity In this sense considering that theentities of a systemmaynot have necessarily the samenumberof states it could be encouraging to establish a consistentupdate schedule when the states of the different entities of thesystem do not belong to the same finite set

In [65] we study completely the problemwhen the set119861 isany Boolean algebra in the more general context of PDSThisconstitutes an important issue since it can model the casewhere entities are composed of a finite number of parts thatcan be activated or deactivated In this context the problemabove concerning entities with different set of state values ismaybe more suitable since any set with structure of Booleanalgebra is isomorphic to a subalgebra of 0 1119899 and then onecould establish a surjection from this algebra to one smallerin order tomake a correspondence among states of this biggerBoolean algebra and the smaller one

On the other hand to give a fair explanation about howa system evolves when the entities can have state values in afinite set would open the door to the definition and posteriorstudy of a fuzzy version of PDS

Of course one can also think in the possibility of consid-ering an infinite set of state values for the entities But againthis is not a natural situation in the processes modeled byPDS

Also in the original definition of CA in [21] it is allowedto have an infinite number of cells in the system Althoughcomputer processes depend on a finite number of entities theconcept of PDS and the results obtained in this work can beextended to the case of infinite number of entities

In the same sense as before one can also think in thepossibility of considering an infinite number of entities anda finite or infinite set of state values for the entities

The notion of dynamically equivalent systems allows us todescribe theoretically the class of all dynamically equivalentsystems Therefore it should be desirable to find out someproperties in the dependency graph and the evolution oper-ator which provide or characterize PDS equivalent to a givenone A first approach in this sense appears in [91] where finitesequential dynamical systems on binary strings are studiedSeveral equivalence relations notions as isomorphic and stablyisomorphic are introduced and the resulting equivalenceclasses are studied In this same sense researching in theconditions that provoke bifurcations is a compiling need

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work has been partially supported by Grants MTM2011-23221

References

[1] C L Liu Elements of Discrete Mathematics McGraw-Hill NewYork NY USA 1985

[2] K Rosen Discrete Mathematics and Its Applications McGraw-Hill Boca Raton Fla USA 2011

[3] D K Arrowsmith and C M Place An Introduction to Dynami-cal Systems Cambridge University Press Cambridge UK 1990

[4] R L Devaney An Introduction to Chaotic Dynamical SystemsAddison-Wesley Ann Arbor Mich USA 1989

[5] M W Hirsch and S Smale Differential Equations DynamicalSystems and Linear Algebra Academic Press New York NYUSA 1974

[6] Y A Kuznetsov Elements of Applied Bifurcation TheorySpringer New York NY USA 2004

[7] G Boole An Investigation of the Laws of Thought onWhich AreFounded the Mathematical Theories of Logic and ProbabilitiesCambridge University Press Cambridge Mass USA 1854

[8] C E ShannonA symbolic analysis of relay and switching circuits[MS thesis] MIT Press Cambridge Mass USA 1940

[9] C L Barrett W Y Chen and M J Zheng ldquoDiscrete dynamicalsystems on graphs and Boolean functionsrdquo Mathematics andComputers in Simulation vol 66 no 6 pp 487ndash497 2004


[10] A Fuster-Sabater and P Caballero-Gil ldquoOn the use of cellularautomata in symmetric cryptographyrdquo Acta Applicandae Math-ematicae vol 93 no 1ndash3 pp 215ndash236 2006

[11] F Greil and B Drossel ldquoDynamics of critical Kauffman net-works under asynchronous stochastic updaterdquo Physical ReviewLetters vol 95 no 4 Article ID 048701 2005

[12] S Nandi B K Kar and P P Chaudhuri ldquoTheory and applica-tions of cellular automata in cryptographyrdquo IEEE Transactionson Computers vol 43 no 12 pp 1346ndash1357 1994

[13] Z RokaCellular automata on cayley graphs [PhD thesis] EcoleNormale Superieure de Lyon Lyon France 1994

[14] I Shmulevich E R Dougherty and W Zhang ldquoFrom booleanto probabilistic boolean networks as models of genetic regula-tory networksrdquo Proceedings of the IEEE vol 90 no 11 pp 1778ndash1792 2002

[15] Z Toroczkai and H Guclu ldquoProximity networks and epi-demicsrdquo Physica A vol 378 no 1 pp 68ndash75 2007

[16] A Ilachinski Cellular Automata A Discrete Universe WorldScientic Singapore 2001

[17] J Kari ldquoTheory of cellular automata a surveyrdquo TheoreticalComputer Science vol 334 no 1ndash3 pp 3ndash33 2005

[18] J Maynard-Smith Evolution and the Theory of Games Cam-bridge University Press Cambridge Mass USA 1982

[19] P Sarkar ldquoA brief history of cellular automatardquoACMComputingSurveys vol 32 no 1 pp 80ndash107 2000

[20] J L Schiff Cellular Automata John Wiley amp Sons New YorkNY USA 2008

[21] S Wolfram ldquoStatistical mechanics of cellular automatardquoReviews of Modern Physics vol 55 no 3 pp 601ndash644 1983

[22] S Wolfram ldquoUniversality and complexity in cellular automatardquoPhysica D Nonlinear Phenomena vol 10 no 1-2 pp 1ndash35 1984

[23] S Wolfram ldquoAlgebraic properties of cellular automatardquo PhysicaD vol 10 pp 1ndash25 1984

[24] S Wolfram Theory and Applications of Cellular Automata vol1 of Advanced Series on Complex Systems World ScientificPublishing Singapore 1986

[25] S Wolfram Cellular Automata and Complexity Addison-Wesley New York NY USA 1994

[26] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22no 3 pp 437ndash467 1969

[27] S A Kauffman Origins of Order Self-Organization and Selec-tion in Evolution Oxford University Press Oxford UK 1993

[28] R Serra M Villani and L Agostini ldquoOn the dynamics of ran-dom Boolean networks with scale-free outgoing connectionsrdquoPhysica A StatisticalMechanics and its Applications vol 339 no3-4 pp 665ndash673 2004

[29] I Shmulevich and S A Kauffman ldquoActivities and sensitivitiesin Boolean networkmodelsrdquo Physical Review Letters vol 93 no4 2004

[30] J von NeumannTheory of Self-Reproducing Automata Univer-sity of Illinois Press Chicago Ill USA 1966

[31] M Gardner ldquoThe fantastic combination of John Conwayrsquos newsolitaire game lsquolifersquordquo Scientific American vol 223 pp 120ndash1231970

[32] J H Conway ldquoWhat is liferdquo in Winning Ways for YourMathematical Plays Academic Press New York NY USA 1982

[33] C L Barrett and C M Reidys ldquoElements of a theory ofcomputer simulation I sequential CA over random graphsrdquoAppliedMathematics and Computation vol 98 no 2-3 pp 241ndash259 1999

[34] C L Barret H S Mortveit and C M Reidys ldquoElements ofa theory of computer simulation IIrdquo Applied Mathematics andComputation vol 107 pp 121ndash136 2002

[35] C L Barrett H S Mortveit and C M Reidys ldquoElementsof a theory of simulation III equivalence of SDSrdquo AppliedMathematics and Computation vol 122 no 3 pp 325ndash3402001

[36] C L Barrett H SMortveit and CM Reidys ldquoETS IV sequen-tial dynamical systems fixed points invertibility and equiva-lencerdquo Applied Mathematics and Computation vol 134 no 1pp 153ndash171 2003

[37] A Bagrodia K M Chandy and W T Liao ldquoA unifyingframework for distributed simulationrdquo ACM Transactions onModeling and Computer Simulation vol 1 no 4 pp 348ndash3851991

[38] D Jefferson ldquoVirtual timerdquoACMTransactions on ProgrammingLanguages and Systems vol 7 no 3 pp 404ndash425 1985

[39] R Kumar and V GargModeling and Control of Logical DiscreteEvent Systems Kluwer Academic Dordrecht The Netherlands1995

[40] H S Mortveit and C M Reidys ldquoDiscrete sequential dynami-cal systemsrdquoDiscreteMathematics vol 226 no 1ndash3 pp 281ndash2952001

[41] J A Aledo S Martınez F L Pelayo and J C Valverde ldquoParalleldiscrete dynamical systems on maxterm and minterm Booleanfunctionsrdquo Mathematical and Computer Modelling vol 55 no3-4 pp 666ndash671 2012

[42] J A Aledo S Martınez and J C Valverde ldquoParallel dynamicalsystems over directed dependency graphsrdquo Applied Mathemat-ics and Computation vol 219 no 3 pp 1114ndash1119 2012

[43] J A Aledo S Martınez and J C Valverde ldquoParallel discretedynamical systems on independent local functionsrdquo Journal ofComputational andAppliedMathematics vol 237 no 1 pp 335ndash339 2013

[44] R Laubenbacher and B Pareigis ldquoDecomposition and simu-lation of sequential dynamical systemsrdquo Advances in AppliedMathematics vol 30 no 4 pp 655ndash678 2003

[45] R Laubenbacher and B Pareigis ldquoUpdate schedules of sequen-tial dynamical systemsrdquo Discrete Applied Mathematics vol 154no 6 pp 980ndash994 2006

[46] H S Mortveit and C M Reidys An Introduction to SequentialDynamical Systems Springer New York NY USA 2007

[47] I Shmulevich S A Kauffman andMAldana ldquoEukaryotic cellsare dynamically ordered or critical but not chaoticrdquo Proceedingsof the National Academy of Sciences of the United States ofAmerica vol 102 no 38 pp 13439ndash13444 2005

[48] A Deutsch and S Dormann Cellular Automaton Modelling ofBiological Pattern Formation Birkhauser Boston Mass USA2004

[49] P Hogeweg ldquoCellular automata as a paradigm for ecologicalmodelingrdquo Applied Mathematics and Computation vol 27 no1 pp 81ndash100 1988

[50] U Dieckman R Law and J A J Metz The Geometry of Eco-logical Interactions Simplifying Spatial Complexity CambridgeUniversity Press Cambridge UK 2000

[51] J Hofbauer and K Sigmund Evolutionary Games and Popula-tion Dynamics Cambridge University Press Cambridge UK2003

[52] G Chiaselotti G Marino P A Oliverio and D PetrassildquoA discrete dynamical model of signed partitionsrdquo Journal ofApplied Mathematics vol 2013 Article ID 973501 10 pages2013


[53] C Bisi and G Chiaselotti ldquoA class of lattices and booleanfunctions related to the Manickam-Miklos-Singhi conjecturerdquoAdvances in Geometry vol 13 no 1 pp 1ndash27 2013

[54] C Bisi G Chiaselotti and P A Oliverio ldquoSand piles models ofsigned partitions with d pilesrdquo ISRN Combinatorics vol 2013Article ID 615703 7 pages 2013

[55] GChiaselotti TGentile GMarino andPAOliverio ldquoParallelrank of two sandpile models of signed integer partitionsrdquoJournal of Applied Mathematics vol 2013 Article ID 292143 12pages 2013

[56] G Chiaselotti T Gentile and P A Oliverio ldquoParallel andsequential dynamics of two discrete models of signed integerpartitionsrdquoApplied Mathematics and Computation vol 232 pp1249ndash1261 2014

[57] G Chiaselotti W Keith and P A Oliverio ldquoTwo self-duallattices of signed integer partitionsrdquo Applied Mathematics ampInformation Sciences vol 8 no 6 pp 3191ndash3199 2014

[58] B Chopard and M Droz Cellular Automata for ModelingPhysics Cambridge University Press Cambridge UK 1998

[59] L B Kier P G Seybold and C-K Cheng Modeling ChemicalSystemsUsing Cellular Automata Springer NewYork NYUSA2005

[60] M Schnegg and D Stauffer ldquoDynamics of networks andopinionsrdquo International Journal of Bifurcation and Chaos vol17 no 7 pp 2399ndash2409 2007

[61] T C Shelling ldquoDynamic models of segregationrdquo Journal ofMathematical Sociology vol 1 pp 143ndash186 1971

[62] M Rickert K Nagel M Schreckenberg and A Latour ldquoTwolane traffic simulations using cellular automatardquo Physica A vol231 no 4 pp 534ndash550 1996

[63] A Dorin ldquoBoolean networks for the generation of rhythmicstructurerdquo in Proceedings of the Australian Computer MusicConference pp 38ndash45 2000

[64] J A Aledo S Martınez and J C Valverde ldquoParallel dynamicalsystems over special digraph classesrdquo International Journal ofComputer Mathematics vol 90 no 10 pp 2039ndash2048 2013

[65] J A Aledo S Martınez and J C Valverde ldquoGraph dynamicalsystems with general boolean statesrdquo Applied Mathematics ampInformation Sciences vol 9 no 4 pp 1ndash6 2015

[66] B Derrida and Y Pomeau ldquoRandomnetworks of automata asimple annealed approximationrdquo Europhysics Letters vol 1 no2 pp 45ndash49 1986

[67] W Y C Chen X Li and J Zheng ldquoMatrix method for linearsequential dynamical systems on digraphsrdquo Applied Mathemat-ics and Computation vol 160 no 1 pp 197ndash212 2005

[68] O Colon-Reyes R Laubenbacher and B Pareigis ldquoBooleanmonomial dynamical systemsrdquo Annals of Combinatorics vol 8no 4 pp 425ndash439 2004

[69] E A Bender and S G Williamson A Short Course in DiscreteMathematics Dover Publications New York NY USA 2005

[70] KMorita ldquoReversible cellular automatardquo Journal of InformationProcessing vol 35 pp 315ndash321 1994

[71] T Toffoli and N H Margolus ldquoInvertible cellular automata areviewrdquo Physica D Nonlinear Phenomena vol 45 no 1ndash3 pp229ndash253 1990

[72] S Wiggins Introduction to Applied Nonlinear Systems andChaos vol 2 of Texts in Applied Mathematics Springer NewYork NY USA 1990

[73] R A Hernandez Toledo ldquoLinear finite dynamical systemsrdquoCommunications in Algebra vol 33 no 9 pp 2977ndash2989 2005

[74] J E Hopcroft and J D Ullman Introduction to AutomataTheory Languages and Computation Addison-Wesley ReadingMass USA 1979

[75] M Sipser Introduction to the Theory of Computation PWSPublishing Company Boston Mass USA 1997

[76] J L Guirao F L Pelayo and J C Valverde ldquoModeling thedynamics of concurrent computing systemsrdquo Computers ampMathematics with Applications vol 61 no 5 pp 1402ndash14062011

[77] F L Pelayo and J CValverde ldquoNotes onmodeling the dynamicsof concurrent computing systemsrdquo Computers amp Mathematicswith Applications vol 64 no 4 pp 661ndash663 2012

[78] W Reisig and G Rozenberg Lectures on Petri Nets I BasicModels Advances in Petri Nets vol 1491 of Lecture Notes inComputer Science Springer New York NY USA 1998

[79] E F Moore ldquoMachine models of self-reproductionrdquo inMathe-matical Problems in the Biological Sciences vol 14 of Proceedingsof Symposia in Applied Mathematics pp 17ndash33 1962

[80] T W Hungerford Algebra vol 73 of Graduate Texts in Mathe-matics Springer New York NY USA 1974

[81] M Gardner ldquoOn cellular automat self-reproduction the Gar-den of Eden and the game liferdquo Scientic American vol 224 pp112ndash117 1971

[82] T Toffoli ldquoCellular automata as an alternative to (rather than anapproximation of) differential equations in modeling physicsrdquoPhysica D vol 10 no 1-2 pp 117ndash127 1984

[83] T Toffoli and N Margolus Cellular Automata Machines MITPress Cambridge Mass USA 1987

[84] H J Blok and B Bergersen ldquoSynchronous versus asynchronousupdating in the lsquogame of Lifersquordquo Physical Review E StatisticalPhysics Plasmas Fluids and Related Interdisciplinary Topicsvol 59 no 4 pp 3876ndash3879 1999

[85] T Mihaela T Matache and J Heidel ldquoAsynchronous randomBoolean network model based on elementary cellular automatarule 126rdquo Physical Review E vol 71 no 2 Article ID 026232 pp1ndash13 2005

[86] B Schonfisch and A de Roos ldquoSynchronous and asynchronousupdating in cellular automatardquo BioSystems vol 51 no 3 pp123ndash143 1999

[87] R V Sole B Luque and S A Kauffman ldquoPhase transitions inrandom networks with multiple statesrdquo Tech Rep 00-02-011Santa Fe Institute 2000

[88] B Luque and F J Ballesteros ldquoRandomwalk networksrdquo PhysicaA vol 342 no 1-2 pp 207ndash213 2004

[89] I Shmulevich and E R Dougherty Probabilistic Boolean Net-works The Modeling and Control of Gene Regulatory NetworksSIAM Philadelphia Pa USA 2010

[90] J A Aledo S Martınez and J C Valverde ldquoUpdating methodfor the computation of orbits in parallel and sequential dynam-ical systemsrdquo International Journal of Computer Mathematicsvol 90 no 9 pp 1796ndash1808 2013

[91] R Laubenbacher and B Pareigis ldquoEquivalence relations onfinite dynamical systemsrdquo Advances in Applied Mathematicsvol 26 no 3 pp 237ndash251 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the corresponding terms since it seems better from an aes-thetic point of view)

A CA has been traditionally conceived as a grid of cellswhere each cell has a state belonging to a finite set (usually0 1) such that all the cells evolve synchronously in discretetime steps The updating of the state of every cell is realizedaccording to a common local function affecting only theneighbors of such a cell Thus if 119909119905

119894is the state value of a cell

119894 at the time 119905 the cell value is updating by applying a localfunction on the cells in the neighborhood of the cell 119894

CA has revealed as a suitable mathematical model to cap-ture the essential features of digital computers synchronicityregular distribution and locality of iterations Although theywere introduced for the first time in the works of Ulam andvon Neumann [30] they became of public interest in 1970when Martin Gardner published [31] with an explanation ofJohnConwayrsquos ldquoGame of Liferdquo in ScientificAmerican (see also[32])

In [21] Wolfram analyzed a set of CA and showed thatdespite their simple construction some of them are capableof a complex behavior Later in [22] based on a deeperresearch he suggested that many one-dimensional cellularautomata fall into four basic behavior classes three of themexhibiting a similar behavior to fixed points periodic orbitsand chaotic attractors and the fourth one such that itsasymptotic properties are undecidable Wolfram publishedother important papers which are compiled in [25]

A BN of size 119899 and connectivity 119896 consists of 119899 inter-connected vertices each one having 119896 inputs The updateof the state of any vertex is determined by the (directed)dependency relations and local rules which are given byBoolean functions Hence BN are a generalization of (finite)Boolean CA

BN appeared almost at the same time in applied modelscreated for the simulation of aspects of the behavior ofbiological systems This occurred in [26] where Kauffmanconstructed molecular automata for modeling a gene as abinary (on-off) device and studied the behavior of largerandomly constructed nets of these binary genes (see also[27]) For this reason BN are also known as Kauffman (net)models The results by Kauffman suggest that if each geneis directly affected by two or three genes then the systembehaves with great order and stability and presents cycles

In the last two decades many works have focused theirattention on mathematical modeling of several computerprocesses The first one of a series of these works was [33]which constituted an important step in the development ofmathematical foundations for the theory of computation Inthis work sequentially updated cellular automata (SCA) overarbitrary graphs are employed as a paradigmatic frameworkThis first work was followed by [34ndash36] where the authorsdeveloped this theory analyzing the asymptotic behavior ofsuch mathematical models The formal definition of PDSappeared for the first time in [9] and it constitutes a funda-mental issue for the posterior development of the results inthis research line

Computer processes involve generation of dynamics byiterating local mappings In fact a computer simulation is amethod for the composition of iterated mappings typically

on local dependency regions [33] It means that themappingshave to be updated in a specific manner that is an updateschedule Update scheduling is also a commonly studiedaspect in discrete event simulations [37ndash39] Mathematicalmodeling of computer processes has resulted very usefulin order to predict what can occur after executing theseprocesses

In the computational context it is common to renamethe local mappings as entities (cells in the language of cellularautomata theory) which are the lowest level of aggregationof the system In computer processes there are many entitiesand each entity has a state at a given time (see [33ndash35]) Theupdate of states of the entities constitutes an evolution intime of the system that is a discrete dynamical system (see[9 40])

In this sense each entity 119894 of the system could be activatedor deactivatedThus it is natural to consider that its state 119909

119894isin

0 1 The evolution or update of the system is implementedby local functions which are restrictions of a global functionor together constitute this global one That is for updatingthe state of any entity the corresponding local function actsonly on the state of that entity itself and the states of theentities related to it The relations among entities are usuallyrepresented by a graph which is called the dependency graphof the system

Thus the update of the states is determined by the depen-dency graph relations of the entities and the local functionswhich together constitute the (global) evolution operator ofthe dynamical system If the states of the entities are updatedin a parallel (or synchronous) manner the system is calleda parallel dynamical system [9 41ndash43] while if they areupdated in a sequential (or asynchronous) order the system isnamed sequential dynamical system (SDS) [9 33ndash36 40 44ndash46]

CA when finite can be considered as a special kind ofPDS by considering cells as entities On the other hand BNare a generalization of (finite) Boolean CA but at the sametime a particular case of PDS by considering nodes as enti-ties One of the main differences with CA is that in BN thestate of each node is not affected necessarily by its neighborsbut potentially by whichever node in the network Thusthe uniform structure of neighborhood in CA disappearsHowever some homogeneity remains since each node isaffected by 119896 connections with other (or the same) entitiesThis homogeneitymakes BN a particular case of PDS since inPDS connections can be totally arbitrary

Since CA and BN are special cases of PDS their applica-tions can be assumed as applications of PDS Some applica-tions of CA and BN that appear in the literature are enu-merated below in order to give an idea of the great interestof this tool for modeling discrete processes from science andengineering

As said before Boolean networks appear for the first timein [26] where Kauffman constructed molecular automatafor modeling a gene as a binary (on-off) device (see also[27]) Up to date these results continue being developedas can be seen in [11] where they consider asynchronousstochastic update [14] where probability is introduced in thedirected dependency graph which they call random Boolean



























119860119866(119894) = 119895 isin 119881 119895 119894 isin 119864 (1)








diam (119866) = max 119889 (119894 119895) 119894 119895 isin 119881 (2)








= (1199091 119909

119899)



(i) (1199091 119909

119899) or (119910

1 119910

119899) = (119909

1or 1199101 119909

119899or 119910119899)

(ii) (1199091 119909

119899) and (119910

1 119910

119899) = (119909

1and 1199101 119909

119899and 119910119899)

(iii) (1199091 119909

119899)1015840= (1199091015840

1 119909

1015840

119899)



2 ⋏ ⋎1015841



1 it satisfies

(i) 120593(119909 or 119910) = 120593(119909) ⋎ 120593(119910)(ii) 120593(119909 and 119910) = 120593(119909) ⋏ 120593(119910)

(iii) 120593(1199091015840) = 120593(119909)1015841









119871 119861119899997888rarr 119861 (3)

where119871(1199091 1199092 119909


1 1199092 119909

119899isin 119861






119899



OR (1199091 1199092 119909


119899 (4)


NAND (1199091 1199092 119909

119899) = 1199091015840

1or 1199091015840


1015840

119899 (5)





AND (1199091 1199092 119909


119899 (6)


NOR (1199091 1199092 119909

119899) = 1199091015840

1and 1199091015840


1015840

119899 (7)



2119899





119865 (0 119909) = 119909 forall119909 isin 119883




119865119905 119883 997888rarr 119883 (9)





119879 = R or R+cup 0 or


(10)


119865119905= 119865∘

119905)







119866(119894) cup 119894 and





119865 0 1119899997888rarr 0 1

119899

119865 (1199091 1199092 119909

119899) = (119910

1 1199102 119910

119899)

(12)






11205872sdot sdot sdot 120587119899a


[119865 120587] = 119865120587119899

∘ sdot sdot sdot ∘ 1198651205872

∘ 1198651205871

0 1119899997888rarr 0 1

119899 (13)

where 119865120587119894

0 1119899


state vector (1199091 1199092 119909













0


(ii) If 119865(1199090) = 119909





0






2 3

1


100

010

001

111 000

110

101

011



119909111990921199093

111 110 101 100 011 010 001 000119910111991021199103

000 101 011 111 110 111 111 111




119865 0 13997888rarr 0 1

3 119865 (119909

1 1199092 1199093) = (119910

1 1199102 1199103)

(15)




















119896 sub


119860 (119894) = 119894 + 1198951 119894 + 1198952 119894 + 119895

119896 (16)


119894)119894isinZ119889 usually




(0 0) (minus1 0) (0 minus1) (1 0) (0 1) sub Z2 (17)




(18)










111 110 101 100 011 010 001 000 (19)


3



7

sum119894=0

120572119894sdot 2119894 (20)




119909119894minus1

119909119894119909119894+1

111 110 101 100 011 010 001 000119910119894

1205727

1205726

1205725

1205724

1205723

1205722

1205721

1205720


119909119894minus1

119909119894119909119894+1

111 110 101 100 011 010 001 000119910119894

1 0 0 1 0 1 1 0












CA BN PDS
















(1198911 1198912 119891


one-map 1198651015840= (1198911015840

1 1198911015840

2 119891

1015840







































1198911= AND 119891

2= OR 119891

3= AND (21)





119894= 0 if 119891

119894= NOR

and 119909119894= 1 if119891









119901


















1 1198812 119881



2


1 and so on These











Acknowledgment


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



































119860119866(119894) = 119895 isin 119881 119895 119894 isin 119864 (1)








diam (119866) = max 119889 (119894 119895) 119894 119895 isin 119881 (2)








= (1199091 119909

119899)



(i) (1199091 119909

119899) or (119910

1 119910

119899) = (119909

1or 1199101 119909

119899or 119910119899)

(ii) (1199091 119909

119899) and (119910

1 119910

119899) = (119909

1and 1199101 119909

119899and 119910119899)

(iii) (1199091 119909

119899)1015840= (1199091015840

1 119909

1015840

119899)



2 ⋏ ⋎1015841



1 it satisfies

(i) 120593(119909 or 119910) = 120593(119909) ⋎ 120593(119910)(ii) 120593(119909 and 119910) = 120593(119909) ⋏ 120593(119910)

(iii) 120593(1199091015840) = 120593(119909)1015841









119871 119861119899997888rarr 119861 (3)

where119871(1199091 1199092 119909


1 1199092 119909

119899isin 119861






119899



OR (1199091 1199092 119909


119899 (4)


NAND (1199091 1199092 119909

119899) = 1199091015840

1or 1199091015840


1015840

119899 (5)





AND (1199091 1199092 119909


119899 (6)


NOR (1199091 1199092 119909

119899) = 1199091015840

1and 1199091015840


1015840

119899 (7)



2119899





119865 (0 119909) = 119909 forall119909 isin 119883




119865119905 119883 997888rarr 119883 (9)





119879 = R or R+cup 0 or


(10)


119865119905= 119865∘

119905)







119866(119894) cup 119894 and





119865 0 1119899997888rarr 0 1

119899

119865 (1199091 1199092 119909

119899) = (119910

1 1199102 119910

119899)

(12)






11205872sdot sdot sdot 120587119899a


[119865 120587] = 119865120587119899

∘ sdot sdot sdot ∘ 1198651205872

∘ 1198651205871

0 1119899997888rarr 0 1

119899 (13)

where 119865120587119894

0 1119899


state vector (1199091 1199092 119909













0


(ii) If 119865(1199090) = 119909





0






2 3

1


100

010

001

111 000

110

101

011



119909111990921199093

111 110 101 100 011 010 001 000119910111991021199103

000 101 011 111 110 111 111 111




119865 0 13997888rarr 0 1

3 119865 (119909

1 1199092 1199093) = (119910

1 1199102 1199103)

(15)




















119896 sub


119860 (119894) = 119894 + 1198951 119894 + 1198952 119894 + 119895

119896 (16)


119894)119894isinZ119889 usually




(0 0) (minus1 0) (0 minus1) (1 0) (0 1) sub Z2 (17)




(18)










111 110 101 100 011 010 001 000 (19)


3



7

sum119894=0

120572119894sdot 2119894 (20)




119909119894minus1

119909119894119909119894+1

111 110 101 100 011 010 001 000119910119894

1205727

1205726

1205725

1205724

1205723

1205722

1205721

1205720


119909119894minus1

119909119894119909119894+1

111 110 101 100 011 010 001 000119910119894

1 0 0 1 0 1 1 0












CA BN PDS
















(1198911 1198912 119891


one-map 1198651015840= (1198911015840

1 1198911015840

2 119891

1015840







































1198911= AND 119891

2= OR 119891

3= AND (21)





119894= 0 if 119891

119894= NOR

and 119909119894= 1 if119891









119901


















1 1198812 119881



2


1 and so on These











Acknowledgment


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














diam (119866) = max 119889 (119894 119895) 119894 119895 isin 119881 (2)








= (1199091 119909

119899)



(i) (1199091 119909

119899) or (119910

1 119910

119899) = (119909

1or 1199101 119909

119899or 119910119899)

(ii) (1199091 119909

119899) and (119910

1 119910

119899) = (119909

1and 1199101 119909

119899and 119910119899)

(iii) (1199091 119909

119899)1015840= (1199091015840

1 119909

1015840

119899)



2 ⋏ ⋎1015841



1 it satisfies

(i) 120593(119909 or 119910) = 120593(119909) ⋎ 120593(119910)(ii) 120593(119909 and 119910) = 120593(119909) ⋏ 120593(119910)

(iii) 120593(1199091015840) = 120593(119909)1015841









119871 119861119899997888rarr 119861 (3)

where119871(1199091 1199092 119909


1 1199092 119909

119899isin 119861






119899



OR (1199091 1199092 119909


119899 (4)


NAND (1199091 1199092 119909

119899) = 1199091015840

1or 1199091015840


1015840

119899 (5)





AND (1199091 1199092 119909


119899 (6)


NOR (1199091 1199092 119909

119899) = 1199091015840

1and 1199091015840


1015840

119899 (7)



2119899





119865 (0 119909) = 119909 forall119909 isin 119883




119865119905 119883 997888rarr 119883 (9)





119879 = R or R+cup 0 or


(10)


119865119905= 119865∘

119905)







119866(119894) cup 119894 and





119865 0 1119899997888rarr 0 1

119899

119865 (1199091 1199092 119909

119899) = (119910

1 1199102 119910

119899)

(12)






11205872sdot sdot sdot 120587119899a


[119865 120587] = 119865120587119899

∘ sdot sdot sdot ∘ 1198651205872

∘ 1198651205871

0 1119899997888rarr 0 1

119899 (13)

where 119865120587119894

0 1119899


state vector (1199091 1199092 119909













0


(ii) If 119865(1199090) = 119909





0






2 3

1


100

010

001

111 000

110

101

011



119909111990921199093

111 110 101 100 011 010 001 000119910111991021199103

000 101 011 111 110 111 111 111




119865 0 13997888rarr 0 1

3 119865 (119909

1 1199092 1199093) = (119910

1 1199102 1199103)

(15)




















119896 sub


119860 (119894) = 119894 + 1198951 119894 + 1198952 119894 + 119895

119896 (16)


119894)119894isinZ119889 usually




(0 0) (minus1 0) (0 minus1) (1 0) (0 1) sub Z2 (17)




(18)










111 110 101 100 011 010 001 000 (19)


3



7

sum119894=0

120572119894sdot 2119894 (20)




119909119894minus1

119909119894119909119894+1

111 110 101 100 011 010 001 000119910119894

1205727

1205726

1205725

1205724

1205723

1205722

1205721

1205720


119909119894minus1

119909119894119909119894+1

111 110 101 100 011 010 001 000119910119894

1 0 0 1 0 1 1 0












CA BN PDS
















(1198911 1198912 119891


one-map 1198651015840= (1198911015840

1 1198911015840

2 119891

1015840







































1198911= AND 119891

2= OR 119891

3= AND (21)





119894= 0 if 119891

119894= NOR

and 119909119894= 1 if119891









119901


















1 1198812 119881



2


1 and so on These











Acknowledgment


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












AND (1199091 1199092 119909


119899 (6)


NOR (1199091 1199092 119909

119899) = 1199091015840

1and 1199091015840


1015840

119899 (7)



2119899





119865 (0 119909) = 119909 forall119909 isin 119883




119865119905 119883 997888rarr 119883 (9)





119879 = R or R+cup 0 or


(10)


119865119905= 119865∘

119905)







119866(119894) cup 119894 and





119865 0 1119899997888rarr 0 1

119899

119865 (1199091 1199092 119909

119899) = (119910

1 1199102 119910

119899)

(12)






11205872sdot sdot sdot 120587119899a


[119865 120587] = 119865120587119899

∘ sdot sdot sdot ∘ 1198651205872

∘ 1198651205871

0 1119899997888rarr 0 1

119899 (13)

where 119865120587119894

0 1119899


state vector (1199091 1199092 119909













0


(ii) If 119865(1199090) = 119909





0






2 3

1


100

010

001

111 000

110

101

011



119909111990921199093

111 110 101 100 011 010 001 000119910111991021199103

000 101 011 111 110 111 111 111




119865 0 13997888rarr 0 1

3 119865 (119909

1 1199092 1199093) = (119910

1 1199102 1199103)

(15)




















119896 sub


119860 (119894) = 119894 + 1198951 119894 + 1198952 119894 + 119895

119896 (16)


119894)119894isinZ119889 usually




(0 0) (minus1 0) (0 minus1) (1 0) (0 1) sub Z2 (17)




(18)










111 110 101 100 011 010 001 000 (19)


3



7

sum119894=0

120572119894sdot 2119894 (20)




119909119894minus1

119909119894119909119894+1

111 110 101 100 011 010 001 000119910119894

1205727

1205726

1205725

1205724

1205723

1205722

1205721

1205720


119909119894minus1

119909119894119909119894+1

111 110 101 100 011 010 001 000119910119894

1 0 0 1 0 1 1 0












CA BN PDS
















(1198911 1198912 119891


one-map 1198651015840= (1198911015840

1 1198911015840

2 119891

1015840







































1198911= AND 119891

2= OR 119891

3= AND (21)





119894= 0 if 119891

119894= NOR

and 119909119894= 1 if119891









119901


















1 1198812 119881



2


1 and so on These











Acknowledgment


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















0


(ii) If 119865(1199090) = 119909





0






2 3

1


100

010

001

111 000

110

101

011



119909111990921199093

111 110 101 100 011 010 001 000119910111991021199103

000 101 011 111 110 111 111 111




119865 0 13997888rarr 0 1

3 119865 (119909

1 1199092 1199093) = (119910

1 1199102 1199103)

(15)




















119896 sub


119860 (119894) = 119894 + 1198951 119894 + 1198952 119894 + 119895

119896 (16)


119894)119894isinZ119889 usually




(0 0) (minus1 0) (0 minus1) (1 0) (0 1) sub Z2 (17)




(18)










111 110 101 100 011 010 001 000 (19)


3



7

sum119894=0

120572119894sdot 2119894 (20)




119909119894minus1

119909119894119909119894+1

111 110 101 100 011 010 001 000119910119894

1205727

1205726

1205725

1205724

1205723

1205722

1205721

1205720


119909119894minus1

119909119894119909119894+1

111 110 101 100 011 010 001 000119910119894

1 0 0 1 0 1 1 0












CA BN PDS
















(1198911 1198912 119891


one-map 1198651015840= (1198911015840

1 1198911015840

2 119891

1015840







































1198911= AND 119891

2= OR 119891

3= AND (21)





119894= 0 if 119891

119894= NOR

and 119909119894= 1 if119891









119901


















1 1198812 119881



2


1 and so on These











Acknowledgment


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























119896 sub


119860 (119894) = 119894 + 1198951 119894 + 1198952 119894 + 119895

119896 (16)


119894)119894isinZ119889 usually




(0 0) (minus1 0) (0 minus1) (1 0) (0 1) sub Z2 (17)




(18)










111 110 101 100 011 010 001 000 (19)


3



7

sum119894=0

120572119894sdot 2119894 (20)




119909119894minus1

119909119894119909119894+1

111 110 101 100 011 010 001 000119910119894

1205727

1205726

1205725

1205724

1205723

1205722

1205721

1205720


119909119894minus1

119909119894119909119894+1

111 110 101 100 011 010 001 000119910119894

1 0 0 1 0 1 1 0












CA BN PDS
















(1198911 1198912 119891


one-map 1198651015840= (1198911015840

1 1198911015840

2 119891

1015840







































1198911= AND 119891

2= OR 119891

3= AND (21)





119894= 0 if 119891

119894= NOR

and 119909119894= 1 if119891









119901


















1 1198812 119881



2


1 and so on These











Acknowledgment


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















111 110 101 100 011 010 001 000 (19)


3



7

sum119894=0

120572119894sdot 2119894 (20)




119909119894minus1

119909119894119909119894+1

111 110 101 100 011 010 001 000119910119894

1205727

1205726

1205725

1205724

1205723

1205722

1205721

1205720


119909119894minus1

119909119894119909119894+1

111 110 101 100 011 010 001 000119910119894

1 0 0 1 0 1 1 0












CA BN PDS
















(1198911 1198912 119891


one-map 1198651015840= (1198911015840

1 1198911015840

2 119891

1015840







































1198911= AND 119891

2= OR 119891

3= AND (21)





119894= 0 if 119891

119894= NOR

and 119909119894= 1 if119891









119901


















1 1198812 119881



2


1 and so on These











Acknowledgment


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










CA BN PDS
















(1198911 1198912 119891


one-map 1198651015840= (1198911015840

1 1198911015840

2 119891

1015840







































1198911= AND 119891

2= OR 119891

3= AND (21)





119894= 0 if 119891

119894= NOR

and 119909119894= 1 if119891









119901


















1 1198812 119881



2


1 and so on These











Acknowledgment


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































1198911= AND 119891

2= OR 119891

3= AND (21)





119894= 0 if 119891

119894= NOR

and 119909119894= 1 if119891









119901


















1 1198812 119881



2


1 and so on These











Acknowledgment


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1198911= AND 119891

2= OR 119891

3= AND (21)





119894= 0 if 119891

119894= NOR

and 119909119894= 1 if119891









119901


















1 1198812 119881



2


1 and so on These











Acknowledgment


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1 1198812 119881



2


1 and so on These











Acknowledgment


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Review Article Parallel Dynamical Systems over Graphs and ...downloads.hindawi.com/journals/jam/2015/594294.pdf · in a synchronous manner, the system is called a parallel dynamical