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Simulated Evolution in a Linguistic ModelCARSTEN KNUDSENa’* and LYNNE CAMERONb

aDepartment of Physics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark,"bSchool of Education, University of Leeds, Leeds LS2 9JT, England

(Received 15 March 2000)

In this paper we present a simple evolutionary model of childrens’ languagedevelopment, whose central nonlinearity is represented by noninvertible discretedynamical systems. The underlying assumption of the model is that children learnfrom other children through their interactions. The concrete learning mechanism used isbased on imitation, where childrens’ languages evolve through attempting to imitateother childrens’ utterances. The use of imitation in evolutionary models has been used,for instance, in evolution of bird song by Kaneko and Suzuki. The model to bepresented here is similar to Kaneko and Suzuki’s model, the primary difference beingthe continuous nature of bird song, in contrast to the discrete nature of childrens’utterances.

Keywords: Learning; Speech generation; Imitation; Evolution; Noninvertible mappings; Chaos

1. INTRODUCTION

Recent work in a complex systems approach ishighlighting the co-evolution of vocabulary andgrammar in child language [1], and the interde-pendence of vocabulary and syntax in languagelearning modelled by simple recurrent networks ina connectionist approach [5].

Before proceeding we shall introduce somenotation, by the following definitions: A vocabu-lary is a fixed collection of words; an utterance isa sequence of words generated by a child; thelanguage of a child is the set of utterances that canbe generated by the child; and finally, an interac-

*Corresponding author.

189

tion is when two children talk together. It shouldbe noted that in our simple model neither wordsnor utterances have any particular meaningassociated with them (apart from an inherentordering; see later). Improvements on the modelwould obviously have to address the issue ofincluding some meaning, for example, through theinclusion of a grammar.

In order to keep the model as simple as possiblewe will make the following two basic assumptionsabout childrens’ development of simple language.We stress that the assumptions are meant as

suggestive and pedagogical rather than as a fullscale model of the complex phenomena involved in

190 C. KNUDSEN AND L. CAMERON

real life language development. Firstly, it isassumed that children learn through interactionwith other children, that is, the learning is aninstance of unsupervised learning, contrasted byneural networks which normally learn undersupervision [17]. Secondly, the primary learningis through imitation of other children. Our modelis similar to the bird song model by Kaneko andSuzuki [10].The model is reduced from the complications

of real life language use and learning to a fewimportant qualitative features: basically, speechgeneration and learning through reproducing thespeech of others heard in interaction. Currentviews of both first and second language learningemphasise the importance of social interaction inthe development [13, 14]. The model incorporateskey aspects of second language learning in whichboth input to the learner and output produced bythe learner are necesssary [15]. The input to thechild in the model arises through interaction, albeitof a very simplified form that downplays theimportance of comprehensibility or understandingto learning [12, 13].The co-adaptive language development being

modelled might be seen as either (a) first languagedevelopment after about three years of age, such as

might occur in nursery or pre-school classes, or (b)classroom foreign language learning through pairor group work. The major difference between thesetwo contexts of language development, as far asthe constituents of the model are concerned, lies inthe size of the vocabularies brought to the learningtask. In the first language context, the vocabulariesof pre-school children are estimated at 3000 wordsor beyond [14]; in a typical foreign languagecontext, this level of vocabulary would take atleast four or five years of tuition to reach. Thevocabulary used in the present model is very smallrelative to the first language figures, but might beconsidered to represent a subset of vocabulary fora particular group of children, perhaps referring toa relatively unfamiliar semantic field that is metonly though formal schooling, such as weight andmass.

The modelling process will involve defining thedevelopment of the following three elements: amodel of a child; a model of interaction betweentwo children; and a model of the learning process.Having defined these elements we can continuewith exposing the model to simulated evolution.This will consist of letting a group of childreninteract, and letting (some of) the children learnfrom these interactions. After this we can thenevaluate the results qualitatively as well as quan-titatively. First, we shall simply observe char-acteristic features of the evolutionary processes,such as determining what groups of childrenperform well, and describing what stable (andunstable) evolutionary structures are present in thesystem; then we will evaluate the complexity of thelanguages involved using finite state machinesas models of computation.

In the model, a child will be capable of listeningto other children (though only one at a time), tothink about what it has heard, or rather iscurrently hearing, as well as generate speech.Thus, the child is merely a specific instance of a

much more general situation: An entity capableof receiving information, processing information,and communicating information, i.e., the compo-nents characterising a computer.The organisation of this paper is as follows. In

Section 2 the mathematical model is derived andformulated. Section 3 introduces the fitness land-scape and examples are given. Then in Section 4we present the results of simulated evolution in themodel by descriping the evolved structures. InSection 5 we evaluate the complexity of the lan-guages developed and interpret the results in thiscontext. Finally, Section 6 contains a discussionof the presented results.

2. THE MODEL

Let us briefly go through the main model com-

ponents, listening and speaking. The first com-

ponent is the listening device. This determines howmuch we listen to ourselves and others. In one

EVOLUTION IN A LINGUISTIC MODEL 191

end we mainly ignore others and essentially wehave autonomous speach. This is referred to as lowcoupling. At the other end we mainly listen toothers and ignore our own internal state; this iscalled high coupling. For the speech generation weneed to know what we say in the absence of input,i.e., autonomous generation of speech. We shallpick the speech generation process as the model’sprimary nonlinearity, and most interpretations willbe in terms of this parameter. The model of asingle child will be characterised by three mainparameters, one determining the speech process:a parameter p E [0, 1] (which will be referred to asthe parameter); one determining how much thechild listens to others during imitation: a

coupling c E [0, 1]. Finally, it is also necessary tospecify how the child starts to speak autonomouslythrough a state x0 [0, 1]. In the present model thisis always chosen randomly (i.e., Xo in Eq. (2) belowis always chosen by a pseudo random numbergenerator).To simplify matters we will let each child have

access to the exact same vocabulary consisting of,say, 20 words. As an example the vocabulary couldconsist of the following words:

and, dog, not, cat, why, pat, sit, bat, let, say, has,him, her, dot, wet, hot, hat, run, all, win.

The only significance of the words is that theyare ordered as we have just listed them. Thatmeans that we consider the word pairs (and, dog),(sit, bat), and (hot, hat) as being close, andsimilarly the word pair (and, win) is considereddistant. The number of words in the vocabularyis a parameter in the model and the effect ofchanging it will be considered. Although allchildren have the same basic vocabulary theymay not all be able to use all of the words in thesentence, especially the words at the end of the listare not used as much as those in the middle (theexplanation for this lies in the used speech gen-eration process outlined below).

Let us now explain the autonomous discretespeech generation (as in the case of a child beingasked to say "something"). The process must

generate a sequence of discrete elements, i.e.,words taken from the vocabulary. Discrete statesystems are then a natural choice, e.g., finite statemachines, Turing machines and cellular automata.However, we have chosen to use a continuousvariable, namely by the use of the logistic map,and then apply coarse graining (or symbolicdynamics) to generate utterances, as we describein the following. We start by generating a sequenceof fixed length, say, 20, {x0, xl,...,x19} by thelogistic equation

where a is determined by the parameter p throughthe relation a=ao+p/2 where a0= 3.5 and theinitial condition x0. This sequence is converted toan integer sequence {I0, I,... ,I19) by transform-ing each x,. to an integer by I,.--integer part of(20x.), i.e., by a coarse graining of the orbit.Finally, the integer Ii is transformed into a wordby the vocabulary, i.e., Ii-0 gives and, and I; 6gives sit. As an example of the creation of anutterance consider the orbit {0.01, 0.16, 0.49, 0.71,0.82}. This is turned into the integer sequence{0, 3, 9, 14, 16}, which in turn yields the utterance

{and cat say wet hat}. Let us in passing note thatthe choice of the logistic map as the underlyingdynamical system is rather arbitrary; any systemcapable of generating complex dynamics couldpresumably be used.Having defined the autonomous speech genera-

tion we can proceed to define the interactionbetween two children. The interaction between isan imitation process. A child evaluates its lan-guage by attempting to imitate an utterancegenerated by another child and vice versa. This iscarried out as follows. The imitated child talksautonomously (using Eq. (1)). The imitating childlistens to the other child while simultaneously"talking" but not out loud; this is a transientphase. After a while (an utterance of fixed length)the imitating child stops listening and attemptsto imitate the other child by autonomous speechand initial condition as generated by the transient


phase. The roles are reversed and the imitation

process repeated. We then assign a score depend-ing on their ability to imitate each other.The mathematical details of the interaction are

as follows. The child that is being imitated

generates a sequence autonomously by the recur-

rence relation

yn+l ay,(1 -y), n 0, 1,..., 18 (2)

whereas the child attempting to imitate is beingcoupled to the child being imitated. The non-

autonomous recurrence relation used here is

x,+l a((1 e)x, + ey,,)(1 ((1 e)x, + ey,,)),

n=0,1,...,8(3)

For small values of the coupling parameter e

the result very similar to that generated by theautonomous process described in Eq. (2). Afterthe non-autonomous speech the imitating childUSeS

x,+=ax,,(1-x,), n 9,10,11,...,18 (4)

where a ao+p/2 and e is the coupling paramterwith different parameter and coupling values forthe two children.For future reference, Figure shows the

bifurcation diagram for the nonlinear map in Eq.(2). Note that this is a truncated version of theusual logistic map for the chosen parameter range[0, 1]. Also note the periodic windows, e.g., the

period three window near p 0.68. More informa-tion on the behaviour of the logistic map can befound in, e.g., Guckenheimer and Holmes [9] andCollet and Eckmann [2].The learning process is clearly an important part

of the model. If a child performs poorly relativeto other children it will learn; if it performs wellit doesn’t learn from the interactions. We havechosen to let the speech generation process bethe only dynamic quantity during evolution,meaning that the coupling remains fixed over

time. This is done in order to make it easier to

interpretate the results. To make sure that learningis gradual, the learning involves adding or

subtracting a random number between 0 and0.01 to the current value of the nonlinearityparameter. The only exception is when p becomes

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

,,0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

p (nonlinearity)

FIGURE Bifurcation diagram for the reparameterized logistic map.


larger than one we subtract one, and similarlywhen the parameter becomes negative we add one.This effectively places the nonlinearity parameteron the circle, i.e., p E S1.As mentioned above we assign a score to the

performance of the children in the imitation game.This is done by measuring the quality of theimitation. To do this we define the differencebetween two utterances by the formula:

Z (distance between words)2

words in utterance

or more precisely

19

(integer part of(201xk ykl)) (6)k--10

The better speaking child receives 10 points, thepoorer point; in case of a tie both receive 5.5points. As described in Section 4, in the roundrobin tournament, dealing with simulated evolu-tion, the scores are normalized such that the bestoverall gets a unit score.

3. THE FITNESS LANDSCAPE

To represent the performance of the children (orthe individual speech generation processes) we can

produce a so-called fitness landscape. The fitnesslandscape represents the performance or fitness ofall individuals in a population relative to all otherindividuals. To determine the landscape we take a

population of children and let all children interactwith all other children in a round robin tourna-ment. The total score of each child is thennormalized with respect to the best score. Thefitness landscape is plotted by showing the fitnessversus the nonlinearity parameter. A generalfeature of the fitness landscape is that it is ingeneral rugged, see, e.g., Kauffman [11]. We notethat the fitness landscape depends on the popula-tion, and hence when evolution takes place (as wewill include later on), the fitness landscape changesover time. In other words it may be considered thegoal of each individual to move around in thechanging fitness landscape attempting to optimizeits performance.To test the importance of the size of the

vocabulary we have computed the fitness

period-3jl!|1[, perid’5perlod-4 |

0.95

0.9

0.85

0.8

0.750 0.2 0.4 0.6 0.8

p (nonlinearity)

FIGURE 2 Fitness landscape for a vocabulary with two words.


0.95

0.9

0.85

0.8

0.750 0.2 0.4 0.6 0.8

p (nonlinearity)

FIGURE 3 Fitness landscape for a vocabulary with twenty words.

0.95

0.9

0.85

0.8

0.750 0.2 0 4 0.6 0.8

p (nonlinearity)

FIGURE 4 Fitness landscape for a vocabulary with one thousand words.

landscape for different vocabulary sizes, anda population containing 1000 individuals (smallvariations occur depending on the number of

individuals, but the qualitative appearance is thesame). Each child is assigned a randomizedcoupling between 0 and 0.1 which remains fixed.


In Figure 2 we have shown a fitness landscape for 4. SIMULATED EVOLUTIONa vocabulary with only two words. Note theruggedness of the landscape. Figure 3 shows a In this section we will present the results offitness landscape when the vocabulary contains carrying out simulated evolution in the model.twenty words. There are obviously many differ- The recipe is very straightforward. We take a

ences between the fitness landscapes shown in uniformly randomized population of children,Figures 2 and 3, but also similarities are present, i.e., they have random nonlinearity and couplingsuch as the ruggedness. More important simila- parameter. The children then interact with allrities are the peaks in the landscape, for example, other children in a round robin-like tournament.the largest peak is close to 0.7, located near the Through each interaction the children accumulateperiod-3 window (as is also indicated in the a score determining their fitness in the current

figures). Also note the smaller peaks near some population. A ranking is performed and the 10%of the period-4 and period-5 windows. These poorest performers then learn, according to thepeaks are all clearly present in both fitness simple ruled described above. These steps are thenlandscapes. Dramatically increasing the size of repeated over and over. We shall now show some

the vocabulary to one thousand words yields a examples of simulated evolution in the model withfitness landscape as shown in Figure 4. The a population of 100 children and discuss thequalitative similarity with the previous landscapes results.is clear. Hence we conclude that the system As we are only including evolution on thedynamics does not critically depend on the size nonlinearity parameter, we can plot this parameterof the vocabulary. For simplicity we shall there- against time, i.e., for each, say ten time units (orfore fix the size of the vocabulary at twenty words longer depending on the time scale) we show theuntil Section 5 where it will be reduced to two parameter for each individual against time. Inwords. Figure 5 we have shown the evolution taking place

0.9

++0.8

0.7

0.6

+++:. 0.5 _H_+

0.4

0.2 -’0.1

00 O0 200 300 400 500 600 700 800 900 1000

time

FIGURE 5 Simulated evolution in a population of 100 children.


on a time scale from 0 to 1000. The initialpopulation is uniformly distributed as can be seenon the ordinate axis. Around time 50 we note thatthere is a depletion of individuals around theparameter 0.6 which lasts until time 200. Theexplanation for this can be seen in the fitnesslandscape shown in Figure 3. The average fitnessof individuals near this parameter value is ratherpoor, and hence the individuals located hereinitially have learned and moved away from thisregion. Similar depletions are seen many places inthe figure, but is most notable in this region whereit occurs repeatedly. Another feature we can noteis the clustering of individuals, particularly near0.7. Figure 3 reveals that this is where the mainpeak is located. Another concentration is foundaround 0.8 at another peak. This structure alsoseems to persist on the time scale shown. A smallerclustering can be found near 0.95, but this is lessstable, i.e., it can be seen how the individuals heredisappear around time 600 only to return later.The reason for the latter being unstable is foundin Figure 3 where we can see that there areindividuals with poor performance in the region

around 0.95, and hence the noise level (in theevolution) makes this evolutionary structure un-stable. Increasing the time horizon to 10000 weobtain the result shown in Figure 6. Here thestructure near 0.7 becomes even more pronouncedand its stability emphasized. The two otherpreviously mentioned regions are seen to reappearoccasionally but are never stable for long. Aninteresting feature present in this figure but not inthe previous one, is the clear trend of individualsbelow 0.7 moving towards lower values of thenonlinearity parameter. Again Figure 3 providesa clue. It can be seen that for decreasing valuesof the nonlinearity the average fitness increasesslightly, but of course with many fluctuations.Increasing the time horizon once more to 100000yields Figure 7. Here the structure around 0.7is still stable, and we conclude that it is stable forall times. The two other regions of clustering ofchildren can be seen to exist once in a while but notfor long judged on this time scale. The downwardsmoving trend is now barely visible which is due tothe fact that it now moves downwards with a slopeten times larger, i.e., almost vertically.

0.9

0.8

0.7

0.5

0.4-

0.3

0.2

0.1

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

time



0.9

0.8

0.7

0.6

0.5

0.4

0.2

0.1

0

**+

20000 a0000 40000

++

/

+, .,_; -+.++ +’+ +

5oooo oooo 7ooootime


$. COMPLEXITY OF LANGUAGES

In the previous section we saw that simulatedevolution leads to clustering of children, with someclusterings beings stable, some unstable. Theclusterings were located around specific parameterranges, which corresponded to the peaks in theuniformly randomized populations fitness land-scape (for example, as shown in Fig. 3). These highperformance peaks were located near large peri-odic windows for the underlying logistic map (seeFig. 1). Thus, simulated evolution does tend toproduce a clustering around parameter values forthe speech generation process that performs well inthe imitation process. However, it does not answerany questions about what kind of languages are

doing well. To investigate this we shall attempt toevaluate the languages used by the children in themodel.To evaluate the language of a child we need to

decide what is meant by the complexity of a

language. This is obviously not an easy question toanswer satisfactorily, so we shall settle for anoperational definition. We will define the languagein terms of how difficult it is to recognize the set of

utterances generated by a particular child. In otherwords we wish to construct a machine that, givenan utterance, is capable of determining whetherit could have been uttered by that child.The language of a child can be difficult to

describe, and we will approach the problem with a

straightforward operational definition. Many chil-dren in the model are capable of chaotic wordcomposition and hence can generate infinitelymany utterances. Thus to simplify we only considerutterances of fixed length, more specifically we willonly consider utterances of length ten. To deter-mine the language we first have the child generateten utterances, each of length one thousand. Theseutterances are then searched with a template oflength ten for all possible occurrences of strings oflength ten (out of the total possible number whichis 21= 1024). The set of different utterances foundis then defined as the language.As we recall that the fitness landscape does not

depend sensitively on the number of words in thevocabulary, we shall for simplicity only considerthe case of a binary language.A language recognizer is a special case of a finite

state machine, see, e.g., Grimaldi [8] (more


advanced machine models of the computation ofthe logistic map are described in Crutchfield andYoung [4]). It can formally be described as a three-tuple IS, I, f) where S is the set of states of thefinite state machine, I is the input alphabet, in thiscase is the binary alphabet I= {0, 1}, and f: S xI S is the function determining the next stateas a function of the present state and the input.

Figure 8 shows an example of a simple finite statemachine that recognizes the language defined bythe binary strings {000, 001,100, 110}. In the figureit is clear that the two states 4 and 5 are identicalsince they give rise to the same next state on thesame input, i.e., f(4,0) =f(5,0) and f(4,1) =f(5,1).Collapsing these two states to one single state leadsto a smaller finite state machine equivalent to theoriginal. Continuing to collapse identical states isa minimization process that leads to the smallestfinite state machine equivalent to the original one,which is easily constructed. We then define thecomplexity of a language as the number of states inthis minimal finite state machine.To illustrate the minimization procedure we

shall now minimize the finite state machine shownin Figure 8. As mentioned above states 4 and 5 are

equivalent and hence can be collapsed. The same istrue for state 3 and the acceptance state (note thatwe haven’t included the arrows on the error and

acceptance states: the arrows should point to thestates themselves). Reducing the recognizer weobtain the finite state machine shown in Figure 9.Considering states and 4 we can see that theymap to the same states under the same input, andhence they can be collapsed into one single state.This yields the finite state machine depicted inFigure 10. We observe that this machine cannot bereduced any further, and hence it represents theminimal finite state machine capable of recogniz-ing the language {000, 001, 100, 110}.

0

FIGURE 9 Partially reduced finite state machine recognizingthe language {000, 001, 100, 110}.

0

FIGURE 8 Sample finite state machine recognizing thelanguage {000, 001, 100, 110}.

Accc

FIGURE 10 Minimal finite state machine recognizing thelanguage {000, 001, 100, 110}.


180

160

140

120

100

80

60

4O

20

period-5

period-6

period-3

period-,

period-5

0 0.2 0.4 06. 0.8

p (nonlinearity)

FIGURE 11 Complexity of the language of children against the nonlinearity parameter.

In Figure we have shown the computedcomplexity of the languages against the nonlinear-ity parameter for a total of 2049 children,uniformly distributed along the nonlinearity para-meter axis. The most notable feature of the figureis the peaks, which are much more pronouncedthan the performance peaks in the fitness land-scape. We recognize the largest peak as the period-3 window in the logistic map where the mostdominant clustering took place during simulatedevolution. Also we note the period-4 and period-5windows that had visible clustering during evolu-tion. In addition there are a few peaks to be seen inthe left hand side of the figure, most prominently a

significant peak associated with another period-5window. The figure shows that the most complexlanguages are not created in the period-3 windowbut rather in the period-4 and period-5 windows.However, as we saw these are not as importantduring simulated evolution. This is due to thenarrow structure of the peaks. Remember thatthere are a number of inherent noise factors. First,initial conditions before imitation are alwayschosen at random, and secondly the learning

process adds a random number to the individualsnonlinearity parameter when they learn. Especiallythe latter has the effect of destabilizing verynarrow peaks. The most interesting informationwe gain from Figure 11 is that the systemdynamics clearly selects individuals with languagesthat are complex in terms of recognition, althoughthere is nothing in the system dynamics that refersto the complexity of the language (only the abilityto imitate matters).

6. DISCUSSION

We have described a model of childrens’ languagedevelopment whose central element was the speechgeneration process. This process was based upondiscretization of the orbits of noninvertible maps,here logistic maps. The model of language devel-opment was based on imitation, i.e., the childrens’skills were measured by their ability to imitateutterances of other children. The relative abilitieswere presented in terms of the speech generationparameter by computing the fitness landscape. The


peaks in this landscape, which correspond to goodlanguage skills with respect to imitation, were seento be located mainly in periodic windows for thelogistic map, with only the largest ones beingvisible, the largest by far being the period-3window.The model was extended with a learning

mechanism by which the poor performing childrencould learn from the better. This was used insimulating evolution in a model with many indi-viduals interacting through imitation. The dy-namic interactions and hence learning lead to

clustering of children near parameter values whereperiodic windows are located.We introduced the complexity of the language

of a child through the number of states of aminimal finite state machine capable of recogniz-ing the child’s language. It was demonstrated thatthe imitation mechanism during evolution selectedexactly those children with complex languages,in the above sense. This was despite the fact thatthe imitation process contains no informationabout the complexity of the language. We sawthat even more complex languages existed but thatthey were destabilized these in evolutionary termsby noise.

Evolutionary models such as the present repre-sent a qualitative approach to analysing complexdynamical systems, in that they are constructed as

analogous to the real life system, not as a directrepresentation of it [7]. The model of languagelearning that we have explored contains only a fewfeatures characteristic of real life language useand development. In evaluating the usefulness ofcomputational modelling in the investigation ofextremely complicated real life systems, such aslanguage development, the extent to which themodel mimics, or not, the behaviour of real lifesystems will give some indication of usefulness. Ifa simple model performs appropriately, it mightbe gradually developed by introducing furtherfeatures.The features that were selected to be included

in the model-speech generation and learningthrough reproducing the speech of others-are

central to language development in both first andforeign language contexts. Evolving the model ledto the clustering around certain values, and toa preference for the children who succeed in learn-ing to be also those who develop more complexlanguages, but not the most complex languageswhich was due to the noise level as discussedabove. The model produces this as an emergentoutcome; it could not have been predicted fromthe components of the model.How does the emergent behaviour of the model

compare with real life language development? Insome key ways, the model reflects what appliedlinguists and teachers would recognise as intui-tively likely: reproducing the speech of others canlead to the development of a more complexlanguage, as it has been defined here, whichcontains some sense of being original or unex-

pected. The finding that this process is moresuccessful for the good speaker (those around theperiod three window) than for really excellentspeakers, or those who are not so good, is initiallysurprising, but might reflect the need for alter-native ways of learning for children of differentproficiency levels. On the other hand, the lesspositive learning experience of those children withjust slightly less complex languages than the goodlearners is a counter-intuitive outcome that needsfurther study.The model we have presented is clearly extre-

mely simplified and, of course, many aspects havebeen left out. There are obviously many ways inwhich one could develop the model. If we startwith the speech generation mechanism it is impor-tant that the underlying model is capable ofgenerating complex utterances, which was themain reason for our choice of the logistic map. It ispossible to use other maps, for instance, higherdimensional maps that include more parameters,all or some of which could then be included in thelearning process. Since the utterances we requireare composed from a discrete set one could alsoconsider the use of finite state machines or Turingmachines, both of which could immediately deliverutterances without coarse graining. The richness of


the dynamics of the logistic map would requirelarge finite state machines and Turing machinesif we were to mimic these.

Other models of interaction could be tried.For example, it would be better to include a

dynamic coupling rather than the static beingused, in order to reflect how speakers accommo-date to each other in the process of word com-

position [3].It is possible to maintain reproduction as the

main mechanism of learning but rather than thecurrent use of unsupervised learning one couldintroduce supervised learning with a teacher or

other helpful adult who could present morelearnable utterances for reproduction. Likewisethe model of learning, which is extremely simple inthe present model could be replace by somethingmore sophisticated, for instance, a learning me-chanism that takes some account of the child’szone of proximal development [16], i.e., that at anygiven time, certain things are easier to learn thanothers [6]. It would also be interesting to includeadditional effects into the model, such as a

grammar which would constrain the order ofwords in an utterance.

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[15] Swain, M., Three functions of output in second languagelearning, Principles and practices in applied linguistics(Oxford) (Cook, G. and Seidlhofer, B. Eds.), OxfordUniversity Press, 1996, pp. 245-256.

[16] Vygotsky, L., Thought and language, Wiley, New York,1962.

[17] Wasserman, P. D., Neural computing: theory and practice,Van Nostrand Reinhold, New York, 1989.

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