Terapaper the Use of Cellular Automata I-5884

8/12/2019 Terapaper the Use of Cellular Automata I-5884

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The use of Cellular Automata in the learning of emergence

G. Faraco a,*, P. Pantano a, R. Servidio b

a Dipartimento di Matematica, Universitadella Calabria, Via P. Bucci, Cubo 30/B, 87036 Rende, Italyb Dipartimento di Linguistica, Universitadella Calabria, Via P. Bucci, Cubo 17/B, 87036 Rende, Italy

Abstract

In recent years, research efforts on complex systems have contributed to improve our ability in investi-

gating, at different levels of complexity, the emergent behaviour showed by system in the course of its evo-

lution. The study of emergence, an intrinsic property of a large number of complex systems, can be tackled

by making use of Cellular Automata (CA): these enable researchers to identify the emergent dynamics of a

complex system, whose behaviour is determined by local rules that define the way in which the elementary

parts interact with each other. This work presents the results of an experimentation aimed to investigate the

efficacy of a methodology which uses the simulation and CA in the learning of emergence. As results, the

93% of the students that which took parts to the experimentation is able to recognize characteristics of com-

plex system.

2004 Elsevier Ltd. All rights reserved.

PACS:87.10. +e

Keywords: Cellular Automata; Emergence; Gliders; Regular dominions; Glider interaction; Learning/teaching models

1. Introduction

It is a common notion that the function of science should not be reduced to a cognitiveactivity. Over the years scientists have realised the need to combine theoretical study with

0360-1315/$ - see front matter 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compedu.2004.10.005

* Corresponding author.

E-mail addresses: [email protected] (G. Faraco),[email protected] (P. Pantano),[email protected] (R. Servidio).

www.elsevier.com/locate/compedu

Computers & Education 47 (2006) 280297
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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the creation of the necessary tools to explore natural phenomena. There are many examples of

this scientific tradition whose contributions have led to the creation of systems and instrumentswhich have deepened our understanding of the physical world (Resnick, Berg, & Eisenberg,

2000).Those engaged in scientific education recognise the importance of getting the students used to

using the instruments needed to study both the simple aspects and the more complex ones of how

our world behaves. Sometimes the right instruments do not exist and, especially when dealing withnew phenomena, have to be designed and constructed for the particular occasion. This practicehas a long history; one only has to think of Galileo, who built his own telescope, or of Boyleand Hooke, who designed the air-pump for experimentation with low pressure. As regards the

study of the phenomenon of emergence, an intrinsic property of many complex systems, the tra-ditional instruments are often found to be insufficient: innovative methods and new instruments

are required.

Yet what is emergence? For example emergence is found when complex behaviours emergespontaneously from out of simple local interactions. Emergence, in fact, appeared with the com-plex systems (systems which cannot be reduced to their elementary components). Interaction be-

tween different components of a system, therefore, operates in such a way that new phenomenaand mechanisms emerge. There are various examples in nature of behaviours that exhibit complex

dynamics. The processes underlying different phenomena, such as the social behaviour of animals,the learning process, biological evolution, the organisation of ant-nests are all examples of behav-iours regulated by complex and adaptive dynamics.

Traditionally, a major part of science teaching has been geared towards the study of arguments,which have already been well structured, utilising well known proofs and experiments. In order toreproduce and investigate systems which manifest emergent dynamics, on the other hand, an inno-

vative methodological approach is required.On the Mathematical Methods for Engineering course at the University of Calabria, a decision

was made to try an innovative approach: the students were encouraged to utilise models and

instruments for the study and a successive analysis of complex phenomenon. Simulation was cho-sen as an investigative tool: through simulation the student acquires the skills to recognise a prob-lem, to identify the relations between the components of a system and is able, at a later stage, tomonitor the system itself. At the same time such a system, however, needs to be accompanied by

traditional tools such as theory and practical exercises/texts. Most important of all, it must serveas an opportunity for a moment of reflection on a given question, stimulating the students abilityto observe closely. The point of this is to overcome the limitations of passive learning, which other

research has underlined (Steinberg, 2000).For the investigation into complexity, Cellular Automata (CA) were chosen as models. In this

model a system can be represented as being composed of many simple parts: the evolution of eachof these parts evolves by interacting with its neighbours. This is why the overall evolution of asystem emerges from the evolutions of all the elementary parts.

This work presents the results of the experimental project whose general objective is the use ofCA for teaching purposes and whose specific objectives is the use of CA and simulations in the

learning of the concept of emergence.

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2. Cellular Automata and emergence

A CA is a dynamical deterministic system built up of a finite number of elementary units (cells

or sites) arranged in space of one or more dimensions, which interact together locally and manifestoverall behaviours of notable complexity.

Each cell of the automata can be viewed as a unit of elaboration, which receives as input its own

state and that of its neighbours at time t and emits as output its own state at time t + 1. The stateof every cell varies according to a simple local rule which is common to all the cells: at any givenmoment, this depends on the state of the cell in question and the states of all the other cells in thevicinity in the previous instant (Weimar, 1997).

The number of cells involved varies according to the so-called range of influence; for example,forr= 2 the state of a cell at the successive instant is determined, as well as by the current value of

the cell itself, also by that of the two on its left, and the two on its right. It is necessary, therefore,

to consider a neighbourhood of five cells 1 (see Fig. 1).The evolution of the system is determined by an initial state that assigns an initial value to thecells. The laws regulating this behaviour are called rules of evolution, and indicate how the values

of the central cells vary depending on the values of the cell in question and those of the other cellsin its range of influence. The updating of the states of all the cells is synchronised and, for this

reason, these are considered, on a massive scale, parallel systems. The state of the CA is the setof the states of the cells involved and evolves in temporal steps, which are discrete.

If the space in which the cells are placed is one dimensional, there will be a set of cells indicated

by 1 an, wheren is the number of cells. This space can be self-enclosed in a circular configuration,that is to say the index iand the index i+n refer to the same cell. If the space is two dimensionalthere will be a matrix n m of cells, which are conventionally referred to by a couple of indices

(i,j). Here again, one can consider the space as self-enclosed in a toroidal configuration. This rep-resentation can be extended to an arbitrary number of dimensions.

In general the number of rules of evolution is very high; for automata of a one dimensional

nature the number is given as

kk

2r1

, 1

wherekindicates the number of states of the automata and (2r+ 1) represents the dimension ofthe neighbourhood falling within the range of influence. For example, for a two state one dimen-sional CA with a surrounding area of three cells there are

223

256,

possible rules.There are three characteristic properties of CA: massive parallelism (the cells evolve simulta-

neously and independently); location (the new state of a cell depends only on its current state and

Fig. 1. Neighbood of CA withr = 2.

282 G. Faraco et al. / Computers & Education 47 (2006) 280297


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the state of the cells in its surrounding area); homogeneity (the laws are universal applying to the CA

cells). FollowingWolfram (1984)one dimensional CA can be classified into four main groups:

class I: evolution leads to a stable and homogeneous state (all the cells have the same value); class II: evolution leads to a set of separated simple stable or periodic structures; class III: evolution leads to chaotic patterns; class IV: evolution leads to localised and complex structures, which can, on occasions, live a

long time.

The first two classes are comprised of ordered systems, the third chaotic systems, and the forth

is made up of the so-called complex systems. Through simple evolutionary mechanisms, CA pro-duce diversified patterns with regular nested structures on some occasions, and chaotic ones on

others. The pattern can represent a regular dominion or a quiescent state. A regular dominion

occurs when, given a determined initial state, the CA evolves with the properties of temporalinvariance (or periodicity) and spatial homogeneity 2 (seeFig. 2). A quiescent state occurs when,given a particular initial state, it remains completely unchanged even at successive instances.

Fig. 2. The figure shows some of the regular dominions highlighted by the evolutionary pattern of a CA withk= 4 and

r= 2 (complex rule k4r2 fam 1/2).

Fig. 3. The figure shows the evolution of a three state CA with a range of influence of 2 (complex rulek3r2/b9). The

pattern shows the existence of regular dominions and gliders.



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In complex systems the behaviour of the CA presents certain characteristics which did not ap-

pear in ordered and chaotic regimes. In the particular types of CA belonging to Class IV in Wol-frams categorisation (Wolfram, 1984), emergent structures can be observed; the dynamics of

these structures may evolve in different ways. It is quite easy to observe these structures on ahomogeneous background and they propagate in a periodic way in space and time. These latterperiodic structures are called gliders (Conway, 1982; Wuensche, 1999). Gliders are particles that

become stabilised in the relations between cells, and manage to maintain their own structure for acertain period of time 3 (see Figs. 3 and 4).

Interaction, caused by collision, of these periodic structures can lead to different consequences,ranging from extinction to the generation of new gliders. A glider can also be seen as a defect on a

background (regular dominion), as the zone that reconciles two confused backgrounds out of stepwith each other, it can separate two completely different backgrounds by acting as a kind of frame

(Hordijk, Shalizi, & Crutchfield, 2001; Wuensche, 1999).

The analysis of the interaction of particles and the interaction between the particles and otheremergent particles, provides a great deal of information on the system under investigation. Thepresence of gliders, for example, in systems of identical particles subject to local interaction, al-

lows for the transportation of information from one part of a system to another (Langton,1990; Hordijk et al., 2001; Wolfram, 1984).

Scientific interest in the study of gliders is above all linked to the behavioural rules that thesecarry with them during their evolution; these rules help us to interpret numerous biological phe-nomena. The glider in its evolution transports information and, at the moment of its collision

with another glider, processes the information, and then conserves it (if it continues to exist), orloses it (if it becomes extinct).

Fig. 4. The figure shows gliders on regular dominions (a) and(b) belong to a CApattern with k= 3 and r= 2 (complex rule

k3r2b/9) and quiescent state gliders (c) and (d) belong to a CA pattern with k= 5 andr = 1 (complex rulek5rl fam1/3).



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But how much information is contained within a glider? How can it be measured? The studies

of Crutchfield (Hordijk et al., 2001) have led to a better estimation of this information byattributing diverse factors to it such as: glider and background phases. Another useful factorfor measuring is linked to the quantity of rules which the glider consults in the table of rules in

its evolution (Bilotta, Lafusa, & Pantano, 2003).Through gliders it is possible to analyse emergent behaviours and study the rules that characte-

rise them. The greater distance between the two cells, which give rise to interaction, the higher the

number of phases the glider undergoes and the more complex they become (see Fig. 5).The need arises, therefore, to catalogue these particles and their interactions. This problem is

confronted in the literature by Crutchfield (Hordijk et al., 2001), who while studying Wolframsrule 54, observed and categorised different types of particles identified by the pattern characteris-ing them, from the period of the patterns repetition to the speed of its propagation. The study of a

CA, therefore, also passes through the creation of a catalogue of the automata itself, in otherwords through the study of regular dominions, quiescent states, gliders, other emergent particlesand the interactions between particles. While it is obvious that the construction of a catalogue isnot the whole story, it does, however, provide useful information. For the creation of the CA cat-

alogue, the simulation tool is of great help because it allows researchers to visualise the evolutionof automata by representing the different elements in the process.

3. Experimentation

3.1. Objectives

The experimentation described in this paper is aimed to investigate the efficacy of a methodol-ogy developed by ourselves which uses software tools in the learning of emergence. It has been

carried out within the Mathematical Methods for Engineering course, which was part of the pro-gramme for a degree in Informatic Engineering. The goal of such a course which is worth 12 cred-

its, was to provide mathematical instruments and methods, to the students, useful in applied

Fig. 5. The figure shows, in space-temporal diagrams, two particular gliders of the pattern of the CA inFig. 3(complex

rule k3r2b/9). The glider shown in (a) is big as regards time, whereas the one in (b) is big as regards space.



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contexts. The students were asked to perform a number of computer simulations for carrying out

exercises both of a theoretical and an experimental nature.

3.2. Participants

Data for this study came from the 153 undergraduate students attending Mathematical Methods

for Engineering course. The prerequisites necessary to face our experimentation with CA consistedjust in some basic theoretical notions and ability in using software instruments. The examinationsnecessary for the registration to the course have widely checked all these prerequisites.

3.3. Methodology

The proposed methodology entails that the student acquires a few theoretical notions on thesubject, and that he uses and/or develops by himself software tools for the representation of pat-terns and the production of the catalogue of a CA characterized by a rule assigned by the tea-cher. In the catalogue the student must indicate how many and which (number and code)

quiescent states, regular dominions and gliders are present in the pattern of evolution of the rule.At the end of the course and after the autonomous creation of the catalogue, each student has

answered a questionnaire. This questionnaire, shown inAppendix A, consisted of a series of ques-tions requiring open answers. Its aim is to check the understanding of the subject and of the meth-odologies and tools used for the production of the catalogue.

In order to verify the effectiveness of the proposed methodology the questionnaire and cata-logue were analysed taking into account the following aspects:

Understanding of the subject. An evaluation was made of the student s description of the the-oretical concepts acquired, of his ability in recognizing and classifying the CA by means of therepresentation of its evolution, of his ability in recognizing and classifying the zone of pat-

terns which show emergent characteristics. Use of tools to analyse the CA. An evaluation was made of the students ability in using soft-

wares of simulation for representing and analysing a CA. Innovative results. An evaluation was made of the student s ability in modifying the method-

ology of analysis of the CA, relatively to the instruments used by him.

3.4. Research description

The students were allotted 6 h for the study of CA 3 h of lessons and 3 h of exercises. Theteachers input involved providing:

a historical introduction to the issue (Codd, 1968; von Neumann, 1966); this was important inorder to identify the problem which von Neumann addressed by developing the Theory of CA

at the beginning of the 1950s, in the attempt to formulate models to simulate the complexity ofbiological phenomena and, in particular, to formalize the problem of reproduction;

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an introduction to Wolframs classification of classes of CA (Wolfram, 1984); this was impor-tant for a clear understanding of which CA manifest gliders and/or other emergent particles;

a guided exploration of Conways Life CA (Conway, 1982); this CA presents, with simple

rules, interesting emergent characteristics, such as the examples of gliders with a periodicand progressive structure;

the exploration of the emergent characteristics of CA.

As a back-up to the lessons the students were advised to read made available on a hypertextlink on the course website the relevant scientific literature, in particular the work of Wolframand Crutchfield (Hordijk et al., 2001; Wolfram, 1984) and Emmeches text on the phenomenon

of emergence (Emmeche, 1994). During the hours dedicated to exercises/practice the software pro-grammeGlidAn (Section 3.5) together with its user guide have been presented from a theoretical

point of view. Such a programme was designed and constructed for the specific purpose of sim-

ulating and representing the evolution of a one dimensional CA.The pattern of different rules can represented by means ofGlidAn. In such a way the studentscan visually singled out the existing differences between:

patterns of rules that indefinitely keeps the system in the initial state; pattern of rules that lead the system in regular configurations; pattern of rules that cause emergence of new particles.

In explaining how to use the software in order to represent the evolution of a rule, it has beensuggested the following methodology:

(1) to supply the programme with the data univocally identifying the CA;(2) to generate the pattern of the CA and to follow carefully its evolution;(3) to select and to mark the zone of pattern where the presence of emergent characteristic is

recognized or suspected;(4) to select the code of this zone of pattern;(5) to use this code as evolution rule of the CA;(6) to recognize in the new pattern so generated the emergent characteristics and to catalogue

them.

However, the students have been stimulated to develop both their own methodologies of anal-

ysis and software instruments for pattern generation. Afterward, the students were supplied withtheGlidAn programme which allowed them to perform pattern generation and analysis of emer-

gence of an assigned rule to verify and to consolidate, at the same time, the acquired theoreticalknowledge.

3.5. The software programme suggested

The GlidAn software programme was designed and constructed specifically by the course tea-cher on the basis of a theoretical model which treats CA as dynamical deterministic systems. It



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provides a space-time diagram that represents the evolution of the CA, giving each state of the site

a different colour (Fig. 6).Its main functions are:

pattern generation;

analysis of emergence.

In order to generate the pattern of the CA, it is necessary to provide the programme with dataidentifying the automata itself. From the menu Rete, the user must choose the following param-eters: number of cells, number of states, range of influence, evolutionary time-scale. From the

menu Regole it is possible to insert the rule code in the field Regole a caso multistrato from thewindow shown inFig. 6.

Having done this, you return to the main page from where you can start the simulation, clickingon the switch shown inFig. 7.

According to the used rule, the pattern can reproduce the same state represented by the same

colour, or to evolve towards different configurations represented by different colours. The studentinterprets these configurations as emergent characteristics to be analyzed and classified. The stu-dent should classify the portion of code keeping the system in the same initial state as quiescentstate, the portion of code which shows regular configurations as regular dominions. If the used

rule is a complex rule, then the pattern can show configurations which are repeated in the time:the corresponding codes should be classified as glider.

Once the presence of a glider has been recognised in the pattern, to analyse its characteristics

more closely, it is necessary to utilise the instrument, called Lente, which allows the student to en-large the zone of the selected pattern (Fig. 8).

Fig. 6. The rule menu. A complex rule has been inserted withk= 3 and r= 2.

Fig. 7. The window identifies the pattern resulting from rulek3r2/b30, whose code was inserted from the Regolemenu.



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The next step consists in selecting the code associated with the glider and copying it in the cor-

rect window of the glider menu (Fig. 9).Once this has been done, the pattern of the glider will appear in the main window (Fig. 10).

Moreover the software allows students to check the interactions of the gliders and monitor theappearance of new particles. Other facets of the programme are Filtro, which enables students tovisualise the patterns with different colours, andStatistiche, which enables them to analyse the en-tropy and population values.

Through theComplexmenu, it is possible to simulate the evolution of the complex rules foundin the literature, for exampleWolfram (1984)rules 20, 52, 54, and 110, and the rules discovered byWuensche (1999).

The GlidAn software programme can be found at the website: http://galileo.cincom.unical.it

Fig. 9. Once the glider has been identified go to the menuSearch glider to insert the code.

Fig. 10. The glider, whose code has been inserted in the Search glider menu.

Fig. 8. In the window on the right, the figure shows the enlarged glider, and in the window below its code.

http://galileo.cincom.unical.it/http://galileo.cincom.unical.it/


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4. Production, selection and assignment of the rules

An evolutionary approach was used for exploring the set of the rules in multistate CA in order

to identify the rules to be assigned. By adopting a genetic algorithm (Bilotta et al., 2003) it waspossible to find complex rules which present behaviours belonging to Wolfram s class IV.

The use of the Genetic Algorithms (GA) allowed the teacher to obtain an enormous quantity of

rules manifesting complex behaviour. In particular, selectingkstates andr range of influence (sizeof surrounding areas) 40 rules were generated for every assignment ofkand r.

A CA identified by a rule code and by an evolutionary rule was assigned to each student. Therule consisting of a string of natural numbers (Table 1) identifies the initial state of automata

itself.The rule code provides information about the number kof states (values) that the cell can as-

sume, about the rangerof influence of the CA and about the family the automata is assumed to

belong to.For example, given the rule

k4r2 fam 2=6,

valuek= 4 indicates that every cell can assume, at instant t, value 0, 1, 2, 3. Valuer = 2 indicates

that the state of a cell at the next instant of time is determined by the current value of the cell, bythe value of the two cells to the left and by that of the two cells to the right of the cell in question.

The label fam2/6 indicates the sixth rule of the second evolution.Having considered CA of type k3r1, k3r2, k4r1, k4r2, k5r1 and k8r1, 240 rules were obtained

through the genetic algorithm. Amongst these 137 were chosen of a type shown in the Table 2.

Table 1

Some of the rules assigned

Rule code Rule

k4r1fam1/1 3013131221200231212200112220211113110113032231303012211012330330

k4rfam1/2 3013101221202231212100112220211113110113032231303012111012330330

k4r1fam1/3 3013301201203231222110110220313132110113032231303012211002330330

k4r1fam1/4 3013111221202231212100112220211113110113032231303032211012330330

k4r1fam1/5 3013101221202231212100112220211113110113032231303012211012330330

Table 2

Number and type of rules assigned to the students on the course

Number assigned rules Rules code assigned

5 k3r1

22 k3r2

69 k4r1

1 k4r2

21 k5r1

19 k8r1



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The choice was dictated by criteria of the complexity and presence of gliders. The teachers have

created the catalogue of all the assigned rules and have indicated the number and the code of qui-escent states, of regular dominions and of gliders of every rule. The aim is to verify the feasibility

of the catalogue. They discarded the rules that contained no gliders or too many, and they as-signed only the rules whose the catalogue may be realized in a short time and without difficulty.

The 137 assigned rules demonstrated altogether 190 quiescent states, 970 regular dominions and

1094 gliders.The assigned task involved creating with a software programme the catalogue of CA indicating

quiescent states (total number and code), regular dominions (total number and code) and gliders(total number and code).

5. Data and results

The data for the analysis have been extracted both from the questionnaires and the cata-logues produced by the students. The questionnaire and the catalogue realized by each student

have been analyzed together to check the correspondence between deep knowledge of the topicand completeness at the catalogue, and vice versa. By complete catalogue we mean a catalogue

showing that the student has recognized all the emergent characteristics of the CA, using asoftware programme which treats CA as dynamical deterministic systems and using a correctmethodology.

The data, concerning the emergent characteristics identified by the students with respect to theemergent characteristics actually present, are shown in Table 3.

Although it was not required explicitly, many students also analyzed a great amount of inter-actions (434) between emergent particles, which, in turn, gave birth to new particles (158) and, in

some cases, gliders (24).Not all catalogues turned out to be complete: a small percentage of students did not find any, or

more than one, characteristic pattern in the data. Namely,

1% of the students did not find any glider; 4% of the students did not find any quiescent states; 8% of the students did not find any regular dominions.

From our analysis it follows that the suggested methodological procedure was followed by 52%

of the students: the remaining 48% have modified the part of the code where they recognized

an emergent characteristics, in order to highlight extremely complex gliders. Again, for the

Table 3

Emergent characteristics identified by the students versus those actually present

Emergent characteristics of CA Identified Actual

Quiescent states 182 190

Regular dominions 957 970

Gliders 1087 1094



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identification of particular regular dominions, many students obtained new regular dominions by

adapting the code of the regular dominions already found.GlidAn was used by 69% of the students; as regards the others, 19% used other software pro-

grammes such as MCell General Monte Carlo Simulator of Cellular Microphysiology (http://www.mcell.cnl.salk.edu/) and DDLab Discrete Dynamics Lab (http://www.ddlab.com) and12% planned and implemented their own simulation tools autonomously.

Furthermore, the questionnaire demonstrated the difficulties encountered both in the Mathe-matical Methods for Engineering course and in realizing the catalogue. The 4% of the studentshas declared difficulty in finding material made available by the teacher; the 20% of the studentshas declared difficulty in the utilization of the software. The remaining percentage (76%) of the

students has non-pointed out difficulties.The comparison between data from the questionnaire and from the catalogue demonstrated

that:

93% of the students understood what a discrete dynamical system and what a CA is; they alsounderstood Wolframs classification and, when analysing a pattern, were able to recognise the

emergent characteristics; 7% were lacking sufficient theoretical knowledge.

Nevertheless, the students which have obtained an at least sufficient evaluation (93%), exhibiteda non-uniform level of knowledge:

14% showed a profound knowledge of the topic: their answers were accurate and their cata-logue complete;

52% showed a good knowledge of the theoretical contents and produced a satisfactorycatalogue;

27% showed a reasonable grasp of the theoretical side and produced an incomplete catalogue (anumber of gliders, quiescent states and/or regular dominions had been missed).

6. Discussion and conclusion

As emphasized in the previous sections, the aim of our experimentation is not to improve an

existing didactic praxis, but to propose and verify a suggested methodology that introduce theexploration of emergence in engineering courses. For this reason we cannot directly compare this

experimentation with others about the same subject with a population of students having the sameprerequisites.

In designing the suggested methodology we chose a tool (the simulation) and a model (CellularAutomata).

The experimentation demonstrated that by means of the use of computer simulation and CA

the 93% of the students developed a satisfactory the ability of recognizing the emergent char-acteristics of a complex system. The simulation tool plays a fundamental role, because it makes

possible to represent the evolution of each single cell of the CA and to show the effect of the

http://www.mcell.cnl.salk.edu/http://www.mcell.cnl.salk.edu/http://www.ddlab.com/http://www.ddlab.com/http://www.mcell.cnl.salk.edu/http://www.mcell.cnl.salk.edu/


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local interactions. This confirms the importance of extending the use of computer simulation to

the study of new paradigms of Applied Mathematics such as non-linear dynamical systems,chaos theory and fractals. (The validity of this instrument is already well known for the learning

of classical topics such as Cinematics (Grayson & McDermott, 1996; Hewson, 1985), Optics(Eylon, Ronen, & Ganiel, 1996; Goldberg, 1997) and Modern Physics (Steinberg, Oberem, &Mc Dermott, 1996)).

The criticisms ofSteinberg et al. (1996)on the risk that simulation can induce passivity in thelearning does not apply to our experimented methodology. Indeed, in our work the simulation isused not only for visualizing the pattern of CA, its main role to is confirm the actual existence ofthe emergent characteristics already single out in the pattern of CA.

The experimented methodology showed the importance of developing tools to acquire scientificconcepts tied to complex behaviours that generally are difficult to grasp without adequate learning

tools.

Our results say that 12% of the students have constructed by themselves software instrumentsdemonstrating the acquisition of the mastery of the subjects. Nineteen percent of the studentswhich have used software different from GlidAn demonstrated the ability of constructing by

themselves plan of study in relation to the topics of the experimentation. Moreover, also thestudents (48%) which have changed some parameters of the CA to obtain both particular glid-

ers and/or regular dominions in the pattern, have shown the ability of using the software instru-ments in order to confirm or to reject their predictions about existence of emergentcharacteristics.

To sum up, can we state that the students have understood what emergence really is? Clearly wecan say that they are able to recognize it, formulate hypotheses and analyse the behaviour of asystem presenting emergent characteristics. They are also able to discover the relationships which

are at the root of complex behaviours, and they can develop methodological strategies and candevise tools for studying the phenomenon.

7. The catalogues

The catalogues (Art Galleries Section) of the rules assigned, the list of rule and the complex

objects can be found on the website: http://galileo.cincom.unical.it.

Appendix A. Questionnaire

Part A: Knowledge of the topic

(1) What is a dynamical system?..............................................................................................................................................................................................................................................................................

.......................................................................................................................................

.......................................................................................................................................

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(2) When is a dynamical system said discrete?..............................................................................................................................................................................................................................................................................

.......................................................................................................................................

.......................................................................................................................................(3) What is a Cellular Automata (CA)?

.......................................................................................................................................

.......................................................................................................................................

.......................................................................................................................................

.......................................................................................................................................

.......................................................................................................................................(4) The Cellular Automata are autonomous systems?

.......................................................................................................................................

.......................................................................................................................................

.......................................................................................................................................

.......................................................................................................................................(5) What is the rule of evolution of a Cellular Automata (CA)?

.......................................................................................................................................

.......................................................................................................................................

.......................................................................................................................................

.......................................................................................................................................(6) What is the classification of the Cellular Automata (CA) by Wolfram?

.......................................................................................................................................

.......................................................................................................................................

.......................................................................................................................................

.......................................................................................................................................(7) In what class of Wolfram would you put the following Cellular Automata (CA)? Class I Class II Class III Class IV

(8) Whats the name conventionally assigned by Wolfram to the Cellular Automata (CA) shown

in the below figure? Figure a Figure b Figure c Figure d Figure e Figure f Figure g Figure h Figure i Figure l Figure m Figure n



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(9) What are the characteristics of the gliders?..............................................................................................................................................................................................................................................................................

.......................................................................................................................................

.......................................................................................................................................

(10) If you analyze a pattern, how do you determine regular dominions and gliders?

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.......................................................................................................................................(11) Single out gliders and regular dominions present in the following patterns:



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Part B: Methodologies and tools used for the production of the catalogue

(1) Internet resources..............................................................................................................................................................................................................................................................................

.......................................................................................................................................(2) Books or papers

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.......................................................................................................................................(3) Software

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..............................................................................................................................................................................................................................................................................(4) Methodological procedure

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(5) Suggestion and comments.......................................................................................................................................

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Part C: General information

(1) Sources of the bibliographical material used

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.......................................................................................................................................(2) Problem in using simulator

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(3) Problem in analyzing of the Cellular Automata (CA)..............................................................................................................................................................................................................................................................................

.......................................................................................................................................(4) How the problems have been overcome?

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References

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Documents

Terapaper the Use of Cellular Automata I-5884