Computer Physics Communications 147 (2002) 724–728

www.elsevier.com/locate/cpc

Mapping applications of cellular automata into applications ofcellular automata networks

C.R. Calidonnaa,∗, S. Di Gregoriob, M. Mango Furnaria

a Istituto di Cibernetica C.N.R “Eduardo Caianiello”, via Campi Flegrei 34, Pozzuoli (NA) I-80078, Italyb Dipartimento di Matematica Università della Calabria, Arcavacata-Rende (CS) I-87036, Italy

Abstract

Cellular automata proved to be a promising model to simulate several complex systems: the requirement is that space andtime, taken into account, have to be discretizable, while the system to be simulated has to satisfy locality and uniformity inthe evolutionary space. Often, in dealing with the simulation of real complex systems some properties of locality are lost andconsequently standard CA model application is very difficult. For this reason it is useful to extend the classical CA modeland introduce feasible mechanisms in order to take advantage of the parallelism source of this computational model. With thisaim the Cellular Automata Network (CAN) model was conceived that includes the advantages of classical CA models andintroduces a new source of parallelism, i.e. the network of cellular automata. In this paper we deal with a sort of heuristics inorder to map CA applications into CANs. This mapping can also be extremely useful as a proposal of a methodology to drivethe modeling and simulation activity of complex phenomena that can be easily fragmented according to local interaction andcomponents. 2002 Elsevier Science B.V. All rights reserved.

Keywords:Cellular automata; Programming model; Complex system simulation; Data and task parallelism

1. Introduction

Cellular Automata(CA) proved to be a suitablemodel to simulate spatio-temporal complex phenom-ena, especially when the composition of processes ofa different nature are involved, e.g., physical, chemi-cal, biological [1]. In fact CA provide a framework fora large class of discrete models with homogeneous in-teractions. The CA approach can be both studied forits theoretical aspects and used as a method for sim-ulation and modeling. The CA programming modelpresents an inner data parallelism and for this reason it

* Corresponding author.E-mail address:c.calidonna@cib.na.cnr.it (C.R. Calidonna).

inspired several parallel computing architectures, pro-gramming environments and CA programming modelextensions [1,2].

Here we deal with an extension of the classical ap-proach called theCellular Automata Network(CAN)model [3]. The CAN approach consists in substitutingthe single automaton into the cell with a network (anacyclically oriented graph) of automata, where a phe-nomenological component is computed by a node ofthe graph.

In particular, in this paper we stress some pointsconcerning a methodological approach to model or totransform CA applications into CAN model compo-nents. These aspects hold especially in dealing withmacroscopic phenomena characterized by transfers of

0010-4655/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0010-4655(02)00385-5

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flows from a cell to its neighboring cells. The paperis organized as follows. Section 2 introduces the cellu-lar automata model. Section 3 is a brief introduction tothe Cellular Automata Network model. Finally, in Sec-tion 4 an example of an application regarding landslidesimulation [4,5] in which heuristics is given in order tomap CA applications into CAN applications.

2. CA computational model

CA are based on a regular spatial grid of cells, eachcell embedding identicalFinite Automata(FA), whoseinput is the states of neighboring cells; FA have anidentical transition function applied simultaneously toeach cell. The neighboring cell is defined in terms ofa spatial pattern, invariant in time and space. At thetime t = 0, FA are in an arbitrary state and the CAevolve changing the state of all the FA simultaneouslyat discrete time steps, according to the FA transitionfunction.

The following informal properties characterize aCA:

• a regular discrete lattice of cells, with a discretevariable at each cell assuming a finite set of states,

• an evolution law, calledtransition function, thattakes place in discrete time steps,

• each cell evolves according to the same ruledepending only on the state of the cell and a finitenumber of neighboring cells,

• the neighbourhood relation is local and uniform.

In implementing CA applications, due to the finitestorage and finite computational power of existingmachines, many choices about the components of aCA have to be made. Here we give a brief commenton possible choices regarding lattice geometry andneighbourhood size:

• lattice geometry, i.e. selection of a specific latticegeometry can be one-, two- or three-dimensional;

• neighbourhood size, i.e. the choice of the neigh-bourhood type, for instance, for a two-dimen-sional lattice, generally includes the Von Neu-mann and the Moore one; for a cell(i, j) they are,respectively, represented as follows:

Nij = {(k, l) ∈ L | (|k − i| + |l − j |) � r

}; (1)

Nij = {(k, l) ∈ L | |k − i| � r and|l − j | � r

},

(2)

wherer = 1 is the mostly used configuration.

3. CAN computational model

The CAN model extends the standard CA model in-troducing the possibility of having anetwork of cellu-lar automata, where each automaton represents a com-ponent of a physical system and connections amongnetwork automata represent a disjoinable evolutivelaw that characterizes the physical system to be sim-ulated.

The CAN model can be applied when the construc-tion of complex physical phenomenon models can beobtained by means of a reduction process in whichthe main model components are identified throughan abstraction mechanism together with the interac-tions among components. So according to the CANmodel it is possible to simulate a two-level evolution-ary process in which the local cellular interaction rulesevolve together with cellular automata connections. Inthe CAN model an automaton is denoted by a name,and its behaviour is described by a set ofproperties,by atransition function, and by a neighbourhood type.A property corresponds either to a physical propertyof the system to be simulated such as temperature, vol-ume and so on, or to some other feature of the systemsuch as the probability of a particle to move and so on.In this schema, a cell of an automaton is considered asa composition of the cells of the automaton properties.A necessary requirement is that the cells of propertiesmust be in correspondence with each other. Accordingto our model, it is not always possible of represent-ing all the system physical components as propertiesof a single automaton. This is why it is necessary tohave the possibility to represent a physical system ascomposed of more than one automaton. In CAN thisis possible through the network of cellular automataabstraction. When a property of an automaton, thatrepresents a component of a physical system, needsto know the values of the cells of another property tobe evolved, this means that these components are par-tially coupled: the case in which it is necessary to usea network of cellular automata. In fact, this is the caseif a property P1 needs to know the cell values of the

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property P2 to evolve at each time step, a network oftwo automata,A andB, has to be defined each havingrespectively the property P1 and P2, whereA is theowner of the property P1, andB is the owner of theproperty P2. In this case we say that the execution ofA must precede the execution ofB. It should be notedthat only the owner of a property canwrite the prop-erty’s values, while these values can bereadby all theautomata in the network. The determination of com-ponents and precedence, between automata, allows anacyclic graph to be derived (at each time step) whosenodes are the automata and whose arcs are the prece-dences between them. The possibility of expressing,in CAN model, the different components of a complexsystem in terms of CA also allows for the exploitationof another source of parallelism, that can improve ap-plication performances; this is calledtask parallelism.Task parallelism comes from the possibility of concur-rently executing network automata (i.e. different sys-tem components) when properties to be updated do notrequire properties of other automata or at least are al-ready updated.

4. An example of mapping: SCIDDICA model

SCIDDICA [4,5] is a cellular automata model forthe simulation of mud/debris flow type landslides,viewed as a dynamic system that is subdivided intoparts whose components evolve exclusively on the ba-sis of local interactions. It was successfully applied tothe Tessina landslide in Italy (1992), Ontake volcanodebris avalanche in Japan (1984), and to the Sarnolandslides in Italy (Campania 1998) [5,6].

Here we propose a sort of heuristic refinementmethod in order to map the CA SCIDDICA applica-tion into CAN.

According to the characteristics of the CA model,SCIDDICA is a quadruple including a lattice, a set ofsubstates (each including the state of embedded FA),a transition function, and a neighbor set.

The first step is to define the lattice, i.e. the regionwhere the landslide evolves, and the same results forCA and CAN: i.e. defined as:

R = {(x, y): x, y ∈ N,0 � x � lx,0� y � ly

}

that represents the bounded set of points with integercoordinates andN is the set of natural numbers.

The definition and mapping of the neighbor set,transition function, substates set with properties setrequire more accurate considerations.

The possibility of fragmenting the complex phe-nomenon, according to local interactions and the in-ternal transformations, can simplify the model, but re-quires different kinds of neighbourhood type.Internaltransformationsare variations of substates inside thecell, where the neighbourhood is the cell itself, whilelocal interactionsare the determination of outflows(e.g., debris/mud outflows) to the neighboring cells,depending on the values of some substate values inits own neighbourhood. In order to determine the to-tal variation of the properties cells, each cell must ap-ply the procedures not only to compute internal trans-formations of substates and their own outflowsof, butalso the neighboring cells’ outflows (corresponding totheir own inflowsif ). So the overall cell neighbour-hood must include not only cells necessary to calcu-late their own outflows but also cells necessary to cal-culate the inflows. This involves, according to the clas-sical CA model, a more extended neighbourhood anda heavy repetition of the same computations.

According to the CAN model this difficulty can beovercome specializing the role of each automaton, andby composing the network, according to internal trans-formation and local interaction, where each automa-ton has its own neighbourhood different from otherautomata. The assignment is not so immediate. Eachsubstate is a property in the CAN model, but the roleof each automaton (transition function) and the assign-ment precedences between automata have to be deter-mined.

According to SCIDDICA, the main mechanisms ofthe landslide are described by one internal transforma-tion, and two local interactions, as follows:

mobilisation effect (σT ): it takes into account the ef-fects after the mobilization of the cell region,it influence the thickness of the cell, itsaltitude, the detrital cover, in the cell, and theenergy that can be added to the amount of theconsidered debris/mud.

debris/mud and run-up outflows (σI1): it takes intoaccount the ability of the amount of the con-sidered debris/mud that can overcome an ob-stacle.

C.R. Calidonna et al. / Computer Physics Communications 147 (2002) 724–728 727

mobilisation propagation(σI2): it takes into accountthe possibility of movement if one of theneighbors is already moving.

SCIDDICA substates are the following:

• Qa that is correlated to the altitude of the cell;• Qth that is correlated to the thickness of de-

bris/mud in the cell;• Qr that is correlated to a measure of the energy

of the cell debris/mud given by the product ofdebris/mud thickness withrun-up highness;

• Qdc that is correlated to the type of detrital coverof the cell and it individuates the maximum depthof detrital cover that can be transformed by theerosion in debris/mud;

• Qm that is correlated to themobilization acti-vation of the detrital cover which becomes de-bris/mud; that depends on the altitude differencebetween the central cell and its neighbors.

The passing between CA SCIDDICA and the net-work precedences, according to the CAN model, ismainly driven by the outflows of an automaton that arethe inflows of another automaton. Outflows, in CAN,represent automata properties They are written as out-flows by a cell of an automaton, that owns it, and thencan be read as inflows from the adjacent cells of otherautomata. All the other properties are associated to au-tomata according to theowning rule: this results in anetwork of 5 automata according to the CAN model.The main advantages in using a network of automata isthat: it does not require the repetition of the same com-putation as in the classical model, and this allows out-flows to be specialized according to different neigh-bourhood. In this way all outflows become propertiesof the CAN model and a new property,QE, is intro-duced as follows:

• QE that is correlated to the energy of the celldebris/mud that can overcome an obstacle;

• of th individuates a debris/mud outflow from thecentral cell;

• of r individuates the run-up outflows;• ofm individuates amobilization outflow, that ac-

counts for mobilization propagation from the cen-tral cell.

Network automata are identified as follows:

• A1 is composed by one property, its transitionfunction computes part of theσI1 debris/mudoutflows; it readsQth, Qr and Qa and it writesof4

th andof4th.

• A2 is composed by one property, its transitionfunction computes the remainder part ofσI1; itreadsQa andQE, and the inflowif 4

th, it writesof4r .

• A3 is composed by one property and its transitionfunction computesσI2; it readsQth, Qr, Qa, QrandQm and it writesof8

m.• A4 is composed by five properties, its transition

function readsQa, Qr and Qdc and the inflowsif 4

m and it writes:of5th andof5

r . If the mobilizationoccurs, the detrital cover disappears partially be-cause it is transformed into debris/mud, so that aflow of debris/mud plus run-up is generated insidethe cell and the altitude is reduced.

• A5 is composed by two properties, its transitionfunction computes new values ofQr and Qthadding inflows to and subtracting outflows fromtheir old values; it reads the propertiesQa andQrand the outflowsof5

th andof5r .

Once the new values of all properties are deter-mined, a new evolutionary step of the landslide is sim-ulated.

5. Conclusion

In this paper we deal with a methodological ap-proach to model phenomena, presenting a two-levelevolutionary law, according to the CAN model. Thestarting point is the classical CA approach. In themapping, the crucial point is the determination andsimplification, in terms of interconnection of partialcomponents and effects, of a landslide phenomenon.A refinement methodology in order to individuatethe precedence execution between CAN automata isbriefly described here. We expect that this methodol-ogy can be applied for more general models that in-volve mixed components different by nature.

Acknowledgements

The authors C.R. Calidonna and S. Di Gregorioare partially supported by CNR Project: “Sviluppo di

728 C.R. Calidonna et al. / Computer Physics Communications 147 (2002) 724–728

una Modellistica Sperimentale Spazio-Temporale deiProcessi Evolutivi dell’Ambiente per la Mitigazionedei Rischi”.

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