The Language of Chaos

Papers

International Journal of Bifurcation and Chaos, Vol. 16, No. 3 (2006) 523–557c© World Scientific Publishing Company

THE LANGUAGE OF CHAOS

ELEONORA BILOTTADepartment of Linguistics, University of Calabria, Italy

[email protected]

PIETRO PANTANODepartment of Mathematics, University of Calabria, Italy

[email protected]

Received July 8, 2005; Revised July 8, 2005

This work presents a linguistic approach to the understanding of chaos. The idea comes fromour work on translating into sounds and music the complexity of chaotic systems, as withChua’s attractors. Therefore, working with sounds, we have used a standardization criterionin order to detect only some of the features of the richness of chaos. This method involvesthe selection of a limited number of sounds, which, in turn, allow for the creation of othercomponents of the system at different levels of organization, as with natural language. Thus,phonetic, morphological, syntactic and semantic levels, linked by a grammar of an “artificialchaotic language” are defined. Each linguistic unit of the artificial language presents networks ofinterrelated phenomena since for each of them, it is possible to detect its path length, tracing agraph of the mutual relationships of the system’s components. We found that there is interactionamong the different levels of the chaotic language. Furthermore, some traits of the dynamics ofthe evolution of language in human infants are found in the main routes to chaos. In this emergentdynamics, mutations, struggle for the fittest, natural selection, and the relative distribution oflinguistic entities in genetic landscapes are observed. On the basis of this, a possible bridge ofconnection between the physical and the mental worlds may be developed.

Keywords : Chua’s oscillator; natural language; artificial languages; evolution.

1. Introduction

It is well known that chaos presents different typesof complex behavior. Many methods of study havebeen developed with the aim of comprehendingits main features. This exploration of chaotic sys-tems, as with Chua’s attractors, through musicand sounds, has permitted us to offer new per-spectives in the analysis of such systems [Bilottaet al., 2005]. The aim was to provide Chua’soscillators with a semantics (sound and music)through which the behavior of such complex sys-tems can be understood from an acoustic perspec-tive. This form of experimentation, which favorsthe auditory sense and therefore sonification ratherthan scientific visualization, combines the search

for significant elements within complex systems, ormore structured categories (their global behavior),with semiotic languages which, by means of a com-putational process, transform numerical meaninginto sound and musical meaning.

The hypothesis developed from these studies isthat it is possible to interpret chaos as a language,which has the same features as all other languages[Hockett, 1963]:

1. Discreteness. The messages in any language arebuilt out of a limited number of units (i.e.phonemes).

2. Semanticity. An element in a language ismeaningful because it stands for somethingelse.

523

524 E. Bilotta & P. Pantano

3. Arbitrariness. There is no physical resemblancebetween the language unit and what it standsfor.

4. Openness. Linguistic messages can be con-structed without limit. There is no limit to thenumber of sentences.

5. Duality of patterning. Language is the rela-tion between sound and meaning, the connectionbetween sequences of physical energy and orga-nized meaning in consciousness.

The features listed above imply that the soundsystem is made up of a small set of meaninglesssounds. These sounds, combined in different ways,form meaningful units and these units can be fur-ther combined to produce an unlimited set of mes-sages. The first part of the duality of patterningfeatures refers to the phonological system, while theother refers to the semantic system, or the rules thatattach meaning to the phonological units. Betweenthe phonological and the semantic systems there isthe syntactic, i.e. the rules that restrict the waysin which the meaning units can be combined intomessages.

In this paper, a linguistic approach is pre-sented, which permits the decomposition of chaoticbehavior and its reduction to a limited inventory ofsounds, which we have called chaosphemes (i.e. anabstract linguistic unit, corresponding to a sound,coming from a chaotic system, which has a uniqueset of physical parameters and has contrasting rela-tionships with other components of the system). Italso produces an incredible variety of combinationsof structures at different levels of organization, asin natural languages.

2. Chua’s Oscillator

For the analysis of chaos according to this approach,Chua’s dynamic systems have been used. Chua’soscillator belongs to the great family of dynamicsystems which manifest chaotic behavior. The cir-cuit (see Fig. 1) devised by Professor Leon O. Chua[Chua, 1993; Chua et al., 1993; Madan, 1993] hasa characteristic nonlinear element, the well-knownChua’s diode. The nonlinearity is expressed by thefollowing piecewise-linear function:

I = f(V )

= Gb · V +12· (Ga − Gb) · (|V + E| − |V − E|)

A diagram of Chua’s oscillator is shown in Fig. 1.

Fig. 1. A scheme of Chua’s oscillator.

This circuit can be described by the followingsystem of differential equations:

dV1

dt=

1C1

[(V2 − V1)G − f(V1)]

dV2

dt=

1C2

[(V1 − V2)G + i3]

diLdt

= − 1L

(V2 + R0i3)

where G = 1/R.

3. Method

In the following section, we explain the method forthe extraction of linguistic features, organized atdifferent levels of a functional hierarchy, in order toanalyze the chaotic behavior of Chua’s attractors.

Starting a simulation for 32,000 steps of aChua’s attractor, we have defined the time of originof the chaotic signal (t = 1000), when the system,after a transient phase, settles to its characteristicbehavior, setting windows of 500 steps. The choiceof this window size involves two considerations:

(i) the window must be short enough to avoid sig-nificant change of the time waveform proper-ties of interest within it;

(ii) the window must be long enough to pro-duce sufficient samples for the calculation ofthe desired parameters. The number of stepsfor the simulation can be chosen withoutrestriction.

For the translation into sounds of the sys-tem’s time series, a standardization criterion wasadopted. Each window was analyzed using thetypical method for the estimation of formant centerfrequencies and their bandwidths [O’Shaughnessy,

The Language of Chaos 525

Fig. 2. On the left, a sample of the waveform of Chua’s attractor is shown. On the right, the formants for the same sampleare visible, organized in F1 near 336 Hz, F2 near 528 Hz, F3 about 1691 Hz.

1987] and the detection of peaks in spectral repre-sentations due to Fast Fourier Transforms (Fig. 2).We have assumed that the distinctive features ofchaos sound are three formants for each window.

We considered only specific rearranged sets offormants. For example, the set of natural numbers(9, 10, 11) which identifies a sound extracted byone of the system’s time-series, was considered onlyonce, despite the fact that permutations of thesevalues could be present.

The value in Hz of these sets of formants isobtained multiplying each figure by 88.2 Hz. Sixtytwo intervals for each time waveform of the sys-tem were obtained. Each piece of sound is sam-pled as mono at a 44,100 frequency rate, 16 bit,lasting 0.0113605 seconds, but conventionally, dura-tions are normalized to unity. From a formal pointof view, we have generated the sets of XF , YF andZF , composed respectively of three formants foreach sound, where:

XF = Oi|i = 1, . . . , 62 ,

YF = Qi|i = 1, . . . , 62 ,

ZF = Ri|i = 1, . . . , 62 .

(1)

From these sets it is possible to extract theshared formants, which form a novel combinationof sounds. We have called this set P , which standsfor phonemes: P = XF ∪ YF ∪ ZF . This is thephonological inventory of this chaotic system (seeFig. 3).

3.1. Level 1: Phonemes

While humans can produce an infinite number ofsounds, each language has a small set of abstractlinguistic units, called phonemes, to describe itssounds. A phoneme is the smallest meaningful

contrastive unit in the phonology of a language.The sounds associated with each phoneme usuallyhave some articulatory configuration in common.Each language typically has 30–60 phonemes, whichprovide an alphabet of sounds uniquely to describewords in the language. The alphabet of phonemesis just large enough to allow such differentiation.For each phoneme, articulatory phonetics relates itsfeatures to the position and configurations of vocaltract articulators that produce them. The physicalsound produced when a phoneme is articulated iscalled a phone. Since the vocal tract is not a dis-crete system, and can vary in infinite ways, a count-less number of phones can correspond to a singlephoneme. The most common scheme for describ-ing speech sound is to characterize each sound asa unique bundle of articulatory features. Consider-ing the great diversity of languages, it is surprisingthat the following articulatory differences have beenfound universally:

(a) the manner of articulation, which distinguishesconsonants;

(b) the manner of voicing, voiced or unvoiced;(c) the place of articulation, which distinguishes

between vowels and among consonants thathave the same manner of articulation and thathave voicing.

Each sound is therefore described in terms of itsvalue with regard to the three articulatory features.The inventory of sounds of Chua’s attractor, theparameters of which are given in the caption forFig. 3, is reported in Table 1. In the phonolog-ical inventory of the artificial language of chaos,the unit of analysis is the phoneme, or as we havecalled it, the chaospheme, an ideal representationof a real occurrence of sound, while physical sounds


Fig. 3. Arbitrary method for creating a set of shared sounds among the curves x, y and z of Chua’s chaotic system. Heredata refer to a circuit with the following parameters: C1 = −13, 33 nF, C2 = 1 µ F, G = 1kΩ−1, L = 31mH, R0 = −100 Ω,Ga = −0.98 mS, Gb = −2.4mS.

give rise to the time waveforms of the system weare investigating (phonetic level of the artificial lan-guage of chaos). Differences in the physical variablesof phonemes bring differences in meaning. After fix-ing the list of classes of sounds for the artificiallanguage, it is necessary to establish their mutualdifferences or resemblances on a functional basis,describing the sounds and the order in which theyoccur in higher units, as in morphemes and words.

3.2. Level 2: Morphemes

Combinations of phonemes give rise to morphemes.Morphemes are the basic meaningful elements thatmake up words.

The set of phonemes P = Pi|i = 1, . . . , npermits us to sequence the curves x, y and z ofChua’s system, by substituting Oi, Qj and Rk withthe points (P ∗

i , P ∗j and P ∗

k) of the P set:

XP = P ∗i |i = 1, . . . , 62

Yp = P ∗j |j = 1, . . . , 62

Zp = P ∗k|k = 1, . . . , 62 .

(2)

These sequences can be expressed as:

S = Si|i = 1, . . . , 62, (3)

where

Si =

P ∗xi

P ∗yi

P ∗zi

(4)

with P ∗xi ∈ XP , P ∗

yi ∈ YP , P ∗zi ∈ ZP . The element

Si can be thus represented as a triplet of phonemesor a matrix of 3 ∗ 3 formants:

Si =

f i11 f i

12 f i13

f i21 f i

22 f i23

f i31 f i

32 f i33

.

As seen in Fig. 4, a color code has been used for thesequencing of Chua’s attractor.

Differences in the physical composition of mor-phemes also entail differences in meaning. Forexample, the first interval of the structure inFig. 4 (going from left to right) produced by P1,

P6, P6, with S1 =

(P1

P6

P6

)and the ninth interval, pro-

duced by P6, P6, P6, with S9 =

(P6

P6

P6

), are very sim-

ilar from the acoustic perspective, yet there is no


Table 1. Inventory of chaosphemes for Chua’s attractor.

functional distinction since, as in natural language,this difference does not distinguish two separatephonemes in the morpheme (they are variations ofthe same sounds with very little differentiation).On the contrary, we maintain that morphemes havediverse meaning when the phonemes that makethem up have very different sets of formants orwhen they distinguish minimum couples in thislanguage. For example, the tenth interval of thesound structure produced by the phonemes P10,

P1, P1 with S10 =

(P10

P1

P1

)and the twenty-sixth

interval, produced by the phonemes P12, P1, P1,

with S26 =

(P12

P1

P1

), though they may be very simi-

lar from the acoustic point of view, they differ inthat they give rise to morphemes with differentmeanings.

Morphemes can be made of both consonantsand vowels. We consider the growth of a chaotic sys-tem as a discourse made up of words and sentences:chaos communicates. We can extract from thesequence, S, Mi elements which are not repeated.This procedure gives us another set, M , where Mstands for morphemes.

At this level, it is possible to look at the possi-ble organization in the structure we have obtained.Thus we have found that morphemes can be joinedtogether to obtain words, made up of two or moremorphemes:

W 1 = M

W 2 = M1Mj |Mi ∈ M,Mj ∈ MW 3 = MiMjMk|Mi ∈ M,Mj ∈ M,Mk ∈ M. . .

W N = MiMj , . . . ,Mk|Mi ∈ M,

Mj ∈ M, . . . ,Mk ∈ M.(5)

The set of words will emerge from the infinite com-bination of all the elements present in (5):

W = W 1 ∪ W 2 ∪ W 3 ∪ · · · ∪ W N · · · .

An example of the composition of words is visual-ized in Fig. 5.

3.3. Level 3: Syntax

The possible combinations of morphemes can beanalyzed in a chaotic system, and it is possibleto establish the rules of a new kind of grammar,

Fig. 4. In this image, the graphical representation of the time-series sequencing method, with the color code we have used,is displayed. The structure has been achieved by sequencing the curves of Chua’s attractor.


Fig. 5. In this image, the sequence of the same attractor, with a different color code, is represented at the top. At the bottom,blocks of two morphemes are present in the sequence. In the center, blocks of two morphemes, occurring at different points inthe structure, are visible.

according to the following criteria:

• explaining the emergence of linguistic structures;• defining the functional roles of each linguistic

unit;• measuring their relative complexity;• quantifying the laws of order and organization

present in these systems;• defining the syntagmatic and paradigmatic rela-

tionships of each linguistic structure.

During the evolution of natural languages, theemergence of grammar allows for the realization ofa set of conventional rules which in turn make itpossible to manage the complex amount of linguis-tic structures. In this context, the grammar definesthe networks of links that the system can activate,as possible routes in the general structure of com-ponents of the system’s occurrences.

For example, the random occurrence (M3M39)might never appear, or occur once, or N times, with(N > 1). We can extract the combinatorial rules,according to the place each occurrence occupies inthe general sequence, according to the functionaland hierarchical roles of each single unit, according

to the grade of order — complexity — chaos of thesystem. So we might obtain small-world networksphenomena [Watts & Strogatz, 1998] for each lin-guistic unit (Fig. 6).

Following the links of these networks meansdetecting their path lengths, tracing the graph ofthe system components’ mutual relationship: thegrammar. By means of this grammar it is possi-ble to use this artificial language as a communica-tion system. In Fig. 7, an artificial chaos sentence isshown, realized by using some of the linguistic con-figurations which are present in the chaotic systemunder analysis. As can be seen, we have used per-mitted words for writing a sentence, adopting thearrangement of natural language syntax [Chomsky,1957, 1965].

Thus, the Noun Phrase (NP) M11M12M61M62 ∈W 4 is made of the words M11M12 ∈ W 2 andM61M62 ∈ W 2, in which the first part is com-posed of two functional units, the Article or Deter-miner and the Noun, while the second representsan Adjective. The Verb Phrase is composed of twomorphemes M38M39, while an Article and a NounM17M18 make the final NP.


Fig. 6. In this image, “small worlds networks phenomena” visualized for morphemes are illustrated. As can be seen, eachmorpheme can be linked with other elements of the network and it is possible to trace the relative length of its path and itsconnectivity level.

Fig. 7. Here, a simple phrase-structure grammar Sentence (S) is represented according to Chomsky, where the Noun Phrase(NP) is composed of the structure (M11M12 & M61M62) in which the first part is made up of two functional units, the Articleor Determiner and the Noun, while the second represents an Adjective. The verb is composed of two morphemes (M38M39)while the final NP is composed again of an Article and a Noun (M17M18).


It will be necessary to investigate whether ornot the rules of transformational grammar fit wellwith this approach. For further details on thistopic, see Appendix A reports on Formal Languagestheory.

3.4. Level 4: Semantics

The meaning of this structure has been constructedusing the structural semantic approach, conceivedby De Sausurre [1916], with the aim of arrivingat an exclusively linguistic definition of meaning.The starting point is the idea of natural languageas a system made up of signs, which is an orga-nized structure, to be studied autonomously, remov-ing any relationships with the real world and withthe conceptual organization of signs in the mindof human subjects. How can we arrive at a purelylinguistic meaning? De Sausurre introduces the the-ory of meaning as a value, that is, the possibility foreach word to be confronted with and put in oppo-sition to any other word in the same language. Sothe meaning of dog is not given in a positive senseby the real or conceptual identity of dog (what itis), but is defined in a negative manner, thanks tomatching with all the other opposable words of agiven linguistic system. The word dog extracts itsmeaning from the position the term occupies in aspecific language, where no other word can exist.So the meaning of a word initiates from the intra-linguistic relationship with the other words and itconsists of a system of differences that exist betweenthat word (for example dog) and all the other words.De Sausurre says that “the more exact character-istic (of the values of a word) is to be what theother words are not”. From this approach, emergesa differential and positional conception of meaning.More precisely, the meaning is given by syntagmaticand paradigmatic relationships (see Fig. 8). On thesyntagmatic axis, the various linguistic elements ofa sentence are linked among them by a relationshipof contiguity. For example, the sentence, Robert eatsan apple has a phonetic, morphological and syntac-tic concatenation: the subject is expressed in a sin-gular form, and so is the verb. It is a relationshipdue to the fact that the terms are all present inthe sentence. On the other hand, in the paradig-matic relationship, the various linguistic elementsof a sentence are linked by an association of equiva-lence (or resemblance). Using the above mentionedsentence, we can have William, or Ellen instead ofRobert. William and Ellen are equivalent terms of

Fig. 8. Networks of links that can be activated in the arti-ficial language on the syntagmatic and paradigmatic axes,according to the structural semantics approach.

Robert since they are appropriate to take its placethanks to a process of commutation. The sameobservations are valid for the verbs eats, devours,consumes, while the word apple can be replaced bypear, apricot, mango, etc. The paradigmatic rela-tionship links terms that are not present since oneterm can be replaced with another on the basis ofequivalence. These horizontal and vertical dimen-sions of a language explain in a relevant manner theconnections that link the linguistic elements amongthem, describing the semantic networks of theconsidered language as the set of words connectedon the syntagmatic and paradigmatic axes of thelinguistic system.

4. Evolution of the Artificial Language:Inside the Routes to Chaos

Language evolution is one of the most inspiringtopics in science. The framework of contempo-rary research on language acquisition is very dif-ferent from the positions held by Skinner [1957]and Chomsky [1959]. Skinner argued that language,like animal behavior, was due to an “operant”learning process, developed in children as a func-tion of external reinforcement and shaping, whileChomsky proposed the existence of a “language fac-ulty” [Chomsky, 1968], with innately determinedconstraints, including a specification of a Univer-sal Grammar and a Universal Phonetics. Recently,research has shown that infants have specific per-ceptual biases that segment phonetic units, usingunexpected learning strategies, which endow themwith a powerful discovery procedure for language.The process they produce is made up of six main


tenets [Kuhl, 2000]:

(a) Infants parse the basic units of speech,a process that allows them to acquirehigher order units created by their combina-tion;

(b) the developmental process does not selectinnately specified options. Instead, theseoptions are selected on the basis of experience;

(c) with exposure to language, infants activatea perceptual learning process in which theydetect patterns, exploit statistical properties,and they are themselves in turn modified bythis perceptual experience;

(d) in vocal imitation, infants link speech percep-tion and production, together with visual andmotor information;

(e) adults communicating with infants modify theirspeech to support infants’ learning strategies,and this is very helpful in allowing infants’ ini-tial mapping of speech;

(f) the critical period for language acquisition isinfluenced not only by time, but by the neuralgrowth that results from experience.

As we have seen, the artificial language of Chaoshas the same structures as natural languages. Yetwhat happens if we study the evolution of languageinside Chaos? Is it possible to extend this methodof detecting linguistic structures inside the routeto chaos? Let us consider the values of the controlparameters of attractor number 1, which determinesthe period doubling route to chaos (Fig. 9):

C1 = 10.443 nF, C2 = 100nF, G = 1kΩ−1,

L = 6.25mH, R0 = 0Ω, Ga = −1.143mS,

Gb = −0.714mS.

The behavior of this Chua attractor varies, chang-ing the control parameter G from 0.901 kΩ−1 to1.023 kΩ−1and maintaining unchanged the initialdata. Figure 10 represents this path, where thescheme of the main changes of the system’s behavioris the following:

Fixed point → Limit cycle → Chua’s spiral → Double scroll.

In the top left-hand side of the map in Fig. 10 thesystem settles on a limit cycle. Afterwards, a seriesof bifurcations take place which, repeating them-selves over time, go through Chua’s spiral to thedouble scroll and lead to the generation of unstablecycles, arriving at chaos.

By analogy, starting from the evolution ofChua’s attractor number 1, we at first fixed the

interval to be analyzed in the space of this dynam-ical system, between G = 0.901 kΩ−1 to G =1.023 kΩ−1. Each structure was then analyzed sin-gularly, and the inventory of each system consid-ered. Common phonemes were then extracted fromdifferent inventory systems, considering the sharedsounds (Fig. 11).

Fig. 9. Chua’s attractor number 1.


Fig. 10. In this image, the Bifurcation Map of Chua’s attractor is displayed. On axis y we have represented the values of V1

comprised between 3000 and 4000 steps. Axis x reports the values of the control parameter between the minimum and themaximum values of G.

4.1. Quantitative results

The experiment was run 123 times, from G =0.901 kΩ−1 to G = 1.023 kΩ−1 in the space ofparameters. The chosen interval between param-eters was 0.001. Results show that there are 129phonemes and 1829 morphemes shared among 123systems (Figs. 12–15).

In the following section, some of the most fre-quent phonemes and their relative distribution inthe evolution process are presented (see Tables 2and 3).

As can be seen, the number of phonemes seemsrelatively small with respect to the number of mor-phemes in the process of evolution. This can beexplained by the fact that phonemes are stablestructures, which change over long periods of time,while morphemes are changing structures that aremodified each time language is used. It is possi-ble to follow the evolution of phonemes as if theywere a genetic population, in which genetic forces,such as mutation and natural selection, act to drive

the system towards evolution in specific directionsrather than in others, indicating a specific strat-egy for survival. Contrary to evolutionary dynamics[Crutchfield & Schuster, 2003], which uses simula-tion and many genetic techniques, with populationschosen at random and many candidate solutions,evaluated according to fitness function, the dynam-ics of evolution is emergent, since nothing is ran-dom and there are no candidate solutions or fitnessfunctions at work. We merely detect the process asit appears. In particular, it is possible to observehow a group of phonemes (the first to appear andto be selected), continues to be present as theparameters of the system are changed (Fig. 16).While the process continues, new phonemes appear,but they are not so persistent. When they do notappear, it means that particular sounds have beendiscarded, to privilege the strongest group of ele-ments. So, in the genetic landscape we have created,sounds that emerge on few occasions create islandswhile sounds that are stable create mountain-likechains.


Fig. 11. In this image, the method we have adopted for analyzing the evolution of language is visualized. First of all, eachsystem is analyzed singularly, and the inventory is considered. Common phonemes are then extracted from different inventorysystems, considering the shared sounds.

Fig. 12. Number of phonemes distributed in the evolution realized inside the period-doubling route to chaos.


Fig. 13. Number of morphemes distributed in the evolution realized inside the period-doubling route to chaos.

Fig. 14. This graph shows the related distribution of phonemes and morphemes. What happens to morphemes in relation tothe phonological evolution? First of all, it is important to note that there is a jump from one level to another, ranging fromelements that can be represented as a 1 ∗ 3 matrix (phonemes) to morphemes (combination of more than 1 phoneme), whichcan be represented as a matrix 3 ∗ 3.

We have created a computer programme thatis capable of repeating the process of phonemeidentification, described above, in the space of theparameters of Chua’s dynamical systems, and of

collecting an inventory of phonemes for each pointin all systems. The elements of each inventory havebeen marked with 1 when they are present in theprocess of evolution and with 0 when they are not


Fig. 15. Number of occurrences of phonemes after 123 runs. As can be seen, the group of sounds that emerges early in theevolution has the highest number of occurrences. On the other hand, the sounds that emerge late in the evolution have a lowernumber of occurrences.

Table 2. This table reports the distribution of a single phoneme in the evolution process.

present. So it has been possible to create a “geneticlandscape” of this evolution and to visualize thebifurcation route to chaos of this artificial language,as shown in Fig. 17.

The map should be read from left to right.It is an ideal temporal line in Chua’s param-eters’ space, where each blue dot in the maprepresents a phoneme. Thus, it is possible to


Table 3. This table reports the distribution of a single morpheme in the evolution process.

read the number of phonemes for each pointof the evolution in the space of the parame-ters. This map has been visualized using MatLab(Fig. 18).

The same method has been used for visualizingthe evolution of morphemes (Fig. 19). From theseresults, it is possible to draw some preliminary con-clusions. At each change in the space of the param-eters, it seems that:

(a) The mutation and the emergence of newphonemes occur with a slow tendency;

(b) The mutation and the emergence of new mor-phemes happen abruptly, where clusters ofmorphemes are discarded while new clusters arecreated;

(c) Phoneme dynamics creates mountain-likechains;

(d) Morpheme dynamics creates islands;(e) The structures that phonemes and morphemes

produce in the genetic landscape are concen-trated near the start, becoming sparse towardsthe end of the process.

4.2. Qualitative analysis

The following points illustrate schematically theevolution of the artificial language of chaos:

→ Identification of the attractor number 1 in thespace of parameters;

→ Start

→ 1 phoneme, which lasts for eighteen pointsin the space of the parameters (from G =0.901 kΩ−1 to G = 0.917 kΩ−1);

→ 2 phonemes, which last for fifteen points in thespace of the parameters (from G = 0.918 kΩ−1

to G = 0.931 kΩ−1);

→ 3 phonemes, which last for ten points in thespace of the parameters (from G = 0.932 kΩ−1

to G = 0.941 kΩ−1);

→ 5 phonemes (from G = 0.942 kΩ−1 to G =0.972 kΩ−1), which during the evolution con-tinues oscillating among a number of phonemesequal to: (5), (4), (5), (6), (6), (6), (6), (7), (8),(6), (8), (8), (8), (9), (9), (9), (10), (9), (9), (10),


Fig. 16. Schematic representation of the evolution of phonemes. As is evident from the figure, phonemes that compete withothers that emerge and then disappear are the strongest group. On the horizontal line, we report the variation of the param-eters of Chua’s attractor number 1. On the vertical line, the phonemes with their names, as they emerge for each point in thespace of the parameters.

(9), (9), (9), (11), (10), (11), (10), (12), (12),(12), (14);

→ 11 phonemes (from G = 0.973 kΩ−1 to G =0.984 kΩ−1), the evolution continues to increasethe number of phonemes in an unpredictableway, jumping from 11 to 24 different sounds,with the following occurrences: (11), (13), (16),(16), (16), (20), (14), (21), (22), (25), (25), (24);

→ 33 phonemes (from G = 0.985 kΩ−1 to G =0.998 kΩ−1), the process of development ofphonemes continues with the following numberof sounds: (33), (30), (29), (34), (25), (28), (29),(34), (32), (30), (37), (44), (39), (38);

→ more than 50 phonemes (from G =0.999 kΩ−1 to G = 1.023 kΩ−1) (50), (48), (49),(43), (42), (48), (48), (46), (51), (47), (51), (52),

(45), (55), (49), (53), (50), (39), (50), (51), (48),(53), (43), (49), (53),

→ If we run the experiment for more thanG = 1.023kΩ−1 (from G = 1.024 kΩ−1 to G =1.079 kΩ−1) (46), (48), (45), (44), (55), (59),(50), (52), (44), (51), (50), (57), (55), (49), (43),(44), (50), (43), (52), (46), (48), (43), (49), (46),(51), (42), (54), (44), (32), (31), (35), (10), (6),(38), (51), (44), (28), (45), (37), (34), (8), (35),(37), (37), (36), (34), (28), (37), (30), (30), (29),(15), (30), (30), (29), (25) the number ofphonemes moves towards stability.

The evolution of chaosphemes starts with one soundwhich is repeated and it is always the same foreighteen steps of space of parameters (from G =0.9010 kΩ−1 to G = 0.9017 kΩ−1), where the pro-cess presents two phonemes (Fig. 20).


Fig. 17. This image represents the evolution of phonemes in the period-doubling bifurcation route to chaos. On the horizontalaxis, we report the number of runs. On the vertical axis, the emergence of the phonemes.

Chua’s attractors behave like a limit cycle.These two sounds allow the generation of differentpatterns in sequencing chaos structures. In Fig. 21a collection of four sequences from G = 0.918 kΩ−1

to G = 0.922 kΩ−1 can be seen.At G = 0.932 kΩ−1, the sounds become three:

P1 (9, 10, 11), P2 (5, 9, 10), and P3 (8, 9, 10).These three phonemes last until G = 0.941 kΩ−1.The sequences they produce are quite different, asthree sounds allow the generation of very artic-ulated patterns. At G = 0.942 kΩ−1, the soundsbecome five: P1 (9, 10, 11), P2 (5, 9, 10), P3

(8, 9, 10), P4 (6, 9, 10) and P5 (7, 9, 10) (seeFig. 22).

From this point, evolution increases by num-ber of phonemes. In fact, from G = 0.942 kΩ−1 to

G = 0.972 kΩ−1, the system presents successionsof chaotic sequences, whose number of phonemesranges from 5 to 14. Figure 23 illustrates the chaossequence for G = 0.972 kΩ−1.

From G = 0.973 kΩ−1 to G = 0.984 kΩ−1,the evolution continues to increase the number ofphonemes in an unpredictable way, jumping from11 to 24 different sounds. The product of this evo-lution is the first scroll of Chua’s dynamical sys-tems. Figure 24 shows two chaos sequences, one forG = 0.978 kΩ−1, with 20 phonemes, and the otherfor G = 0.984 kΩ−1, with 24 phonemes.

From G = 0.985 kΩ−1 to G = 0.998 kΩ−1,the process of development of phonemes continuesas Chua’s attractor finishes completion of the firstscroll, with sequences which range from 30 to 44


Fig. 18. 3D representation of the evolution of phonemes in Chua’s circuit number 1 parameter space. On the vertical axis,the number of occurrences of phonemes.

phonemes. In Fig. 25 two chaos sequences are pre-sented. The first refers to G = 0.988 kΩ−1, with 34phonemes, the second is about G = 0.996 kΩ−1 with44 sounds.

From G = 0.999 kΩ−1 to G = 1.023 kΩ−1, thesystem goes on the double scroll. The number ofphonemes ranges from 50 to 59. In this region ofthe space of parameters, the system reaches its max-imum in complexity, as can be seen from Fig. 26,where the first sequence presents 55 phonemes andthe second 59.

If we run the experiment for more thanG= 1.023 kΩ−1 (from G = 1.034 kΩ−1 to G =1.079 kΩ−1), the number of phonemes continuesto be stable ranging from 40 to 50 sounds. Theattractor behavior continues to present two scrolls,until the system reaches the parameter value G =1.052 kΩ−1, where it is possible to observe a firstdiminution of the number of phonemes, whicharrives at 32 (Fig. 27).

This trend continues at G = 1.055 kΩ−1 andG = 1.056 kΩ−1 where the number of phonemesabruptly falls to 10 and 6, respectively. Figure 28shows the sequence for G = 1.055 kΩ−1.

There is a strict correlation between the struc-ture of the attractor and the number of phonemes.The more complex the attractor configuration,the higher the number of phonemes. Figure 29,shows attractor 1 for G = 1.052 kΩ−1 and G =1.055 kΩ−1. The configurations are dissimilar, so thenumber of phonemes for each varies from 32 to 10,respectively. Then the process of evolution producesa higher number of phonemes, ranging from 38 to 51(Fig. 30).

At G = 1.1060 kΩ−1, the process of evolutionreaches 28 (number of phonemes), while for thisvalue of the G parameter, the system maintains thedouble-scroll configuration. At G = 1.1061 kΩ−1,Chua’s attractor goes from double to single scrollconfiguration and the number of phonemes falls


Fig. 19. This image represents the evolution of morphemes in the period-doubling bifurcation route to chaos. On thehorizontal axis, we report the number of runs. On the vertical axis, the emergence of the morphemes.

Fig. 20. Chaos sequence obtained for G = 0.918. As can be seen, P1 (9, 10, 11) is relevant in percentage, while P2 (5, 9, 10)appears only five times.


Fig. 21. This image shows sequences of chaos from G = 0.919 to G = 0.922 where one can see how with two phonemes it ispossible to create different patterns, in the same way as a human infant can create different sound groupings.

Fig. 22. In this chaos sequence, obtained at G = 0.942, the presence of five phonemes allows for the articulation of verycomplex configurations. The relatively simple model from the beginning of this evolution has been lost.


Fig. 23. Chaos sequence for G = 0.972. This organization presents 14 phonemes.

Fig. 24. This image shows two chaos sequences, one for G = 0.978, with 20 phonemes, and the other for G = 0.984, with24 phonemes.

Fig. 25. This image shows two chaos sequences, one for G = 0.988, with 34 phonemes, and the other for G = 0.996, with44 phonemes.


Fig. 26. This image shows two chaos sequences, belonging to the period of evolution that ranges from G = 0.999 to G = 1.033.The first sequence, with G = 1.012 has 55 phonemes, while the second, with G = 1.029 has 59 phonemes.

Fig. 27. This image shows the chaos sequence for G = 1.052 which has 32 phonemes. The number of phonemes diminishesin a range that goes from 40 to 50 to 32.

Fig. 28. The number of phonemes rapidly falls to 10 sounds at G = 1.055. In this image, the sequence for this value of theparameter G is shown.

again to 15. From G = 1.1076 kΩ−1 to G =1.1079 kΩ−1, the process goes through a series ofsequences which have about 30–25 phonemes, andthe system appears to become stable.

Table 4 shows a collection of 123 sequen-ces of this evolution (from G = 0.0901 kΩ−1 toG = 1.023 kΩ−1). We have collected in a sin-gle sequence all the preceding ones, and again we


Fig. 29. This image shows attractor 1 for G = 1.052 (on the left) and G = 1.055 (on the right). As can be seen, theconfigurations are very different.

Fig. 30. This image shows the chaos sequence for G = 1.057, with 38 phonemes and the sequence for G = 1.058, with51 phonemes.

have obtained the same structure made up of manypieces.

In the last table (Table 5), three images arepresented. The first describes the evolution of thetorus breakdown route to chaos. The second image

illustrates the collection of sequences obtained byevolving Chua’s attractor, that leads to the inter-mittency route to chaos. The third image displaysthe evolution of a Chua’s 4-scrolls system. EachChua system can be analyzed with this approach.


Table 4. Collection of all the sequences obtained by using the linguistic approach for interpreting chaos evolution. The firstimage is the sequence that contains all the sequences we have considered (123 evolutive runs). The other images representmagnifications of part of the entire sequence.

123 runs

Magnification of the first 43 runs


Table 4. (Continued )

Magnification of the second block of 43 runs

Magnification of 37 runs

Table

5.

Inth

efirs

tim

age,

the

collec

tion

ofth

eto

rus

bre

akdow

nro

ute

toch

aos

sequen

ces

(a).

Inth

ese

cond

image,

the

collec

tion

ofth

ein

term

itte

ncy

route

toch

aos

sequen

ces

(b).

As

itis

poss

ible

tose

e,in

both

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tions,

the

num

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ofphonem

esis

not

sore

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the

configura

tions

that

thes

ero

ute

sre

alize

are

org

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aco

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xw

ayas

wel

l.T

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third

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displa

ys

the

evolu

tion

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hua’s

four-

scro

lls

syst

em(c

).

(a)C

ollecti

on

ofse

quences

for

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sbre

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ute

toch

aos

547

Table

5.

(Continued

)

(b)C

ollecti

on

ofse

quences

for

inte

rmit

tency

route

toch

aos

548

Table

5.

(Continued

)

(c)C

ollecti

on

ofse

quences

obta

ined

by

Chua’s

4-s

cro

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syst

em

549


5. Conclusions

From empirical observations made on translatingchaotic systems into sound and music [Bilotta et al.,2005], we have explored the analogy between humanlanguage and chaos. We hypothesize that in chaoticsystems there are different levels of sound organiza-tion, which can be considered analogous to the wayhumans both interpret sound and produce it (seeTable 6).

The first level of analysis is chaos. We thinkthat it is possible to understand chaos. We conceivechaos not as a single unitary process but as a prod-uct of many types of different processes, yet madeup of the same simple units put together. We do nothave the opportunity to look at the process in itsdynamical development, but only to see the result-ing chaotic behavior. We do not understand how itis possible to arrive at such an organization, becausewe have not been able to observe the structure for-mation process. The main features of chaos are nov-elty, ambiguity, implicit meaningfulness, emergence,incongruity, divergence, and so on. Not all chaos ischaracterized by all these properties; rather, theseproperties form a kind of family resemblance struc-ture [Wittgenstein, 1958].

Most chaotic behavior will possess most of theproperties, but other, different chaotic behavior willcontain different subsets of these properties andno single property is uniquely necessary. Whenviewed in this way, chaos can be conceived interms of a graded structure. The more of theseproperties chaos contains, the more likely it isthat chaotic behavior will result. Due to thisconception of family resemblance and because weview chaotic and nonchaotic behavior as resultingfrom the same types of underlying process, or the

Table 6. Possible scheme of sound organizationin Chua’s oscillator.

First level: Chaos Indistinguishable sound

Second level:Inside chaos

Identification of regularregions from which it

is possible to detect

timbres or pitches.

Third level:Organized sound

Natural sounds:Animal Languages such

as Bird SongsPhonemes of natural

languagesPhonemes organization

Music

same components, organized in different ways, weconsider them as essentially lying along the samecontinuum.

In fact, the second level of analysis permitsthe exploration and identification of regular regionsfrom which it is possible to detect some of themain structures of sound organization. This levelof analysis considers both the generative and theexploratory chaotic processes, with two distinct pro-cessing components: a generative phase, followedby an exploratory phase. In the initial generativephase, one constructs a chaotic structure by usingsimulation and translating it into sound. In thisphase, the creation of tools for “reading” chaos and“translating” it into sound is very important anddemanding. We have realized some applications forthe pursuit of this aim.

These results manifest the various propertiesof chaotic behavior. These properties are thenexploited during the exploratory phase in which oneseeks to interpret these structures in meaningfulways. Then, by means of FFT analysis it will bepossible to extract all the harmonics of the chaoticwaveforms. Vice versa, by means of sound synthe-sis, it is possible to recreate predefined waveforms,starting from chosen frequencies.

In this way, we have disentangled chaos fromits complexity and now it is possible to look at it asif it were organized sound (third level of analysis).We believe that inside the constructive and creativepotential of chaos there are sounds that we can findin natural environments, like onomatopoeia, animallanguages such as bird songs, phonemes of natu-ral languages and organized pitches, as in musicalmelodies. It is as if there were many different scalelevels in sound, with the same components orga-nized with different patterns or methods of orga-nization but we do not yet understand the sharedfeatures when we make the jump from one level tothe other.

We have also taken the following points intoconsideration:

(a) Overcoming the mental/physical dichotomy. Ifchaos is organized according to the same cri-teria as natural languages, we can make thehypothesis that the essential physical natureof things possesses the same organization asbiological systems. We can consider the men-tal/physical binomial as a gradual continuumin which expressive languages differ only super-ficially, and remain substantially analogous in


their organizational structures and essentialcomponents, which comprise living matter.

(b) Creation of computational systems based onanalogical rather than digital modalities as aprelude to the application of cryptographic sys-tems for secure communication protocols.

(c) Creation of different evolution models boundto chaos. Inside chaos, it is possible to observeDarwinian models of evolution, struggle for sur-vival phenomena, selection or death of struc-tures, and these give origin to the components,which, on occasion, are the most fit for thesystem.

The three routes to chaos exemplify the differ-ent types of evolution of language. They are notin contradiction, but complementary and, in time,they manifest different growth phenomena, follow-ing single modalities or changed compositions ofeach modality.

Acknowledgments

The authors would like to thank Dr. Michel Cronin,Department of Linguistics, University of Calabria,Italy, for his valuable help in translating the paperfrom Italian.

References

Bilotta, E., Gervasi, S. & Pantano, P. [2005] “Readingcomplexity in Chua’s oscillator through music. Part I:A new way of understanding chaos,” Int. J. Bifurca-tion and Chaos 15, 253–382.

Chomsky, N. [1957] Syntactic Structures (Mouton, TheHague).

Chomsky, N. [1959] “A review of B. F. Skinner’s verbalbehaviour,” Language 35, 26–58.

Chomsky, N. [1965] Aspects of the Theory of the Syntax(MIT Press, Cambridge, MA).

Chomsky, N. [1968] Language and Mind (BraceJovanovich, Inc., Hartcourt, NY).

Chua, L. O. [1993] “Global unfolding of Chua circuits,”IEICE Trans. Fundam. Electron. Commun. Comput.Sci. E76-A, 704–734.

Chua, L. O., Wu, C. W., Huang, A. & Zhong, G. Q.[1993] “A universal circuit for studying and generatingchaos. II. Strange attractors,” IEEE Trans. CircuitsSyst. I: Fundam. Th. Appl. 40, 745–761.

Crutchfield, J. P. & Schuster, P. (eds.) [2003] Evolution-ary Dynamics: Exploring the Interplay of Selection,Accident, Neutrality, and Function (Oxford Univer-sity Press, NY).

De Sausurre, F. [1916] Course de Linguistique General,eds. Bally, C. & Sechehaye, A. (Paris).

Hockett, C. F. [1963] “The problem of universals inlanguage,” in Universals of Language, ed. Greenberg,J. H. (MIT Press, Cambridge, MA), pp. 1–22.

Kuhl, P. K. [2000] “Language, mind and brain: Experi-ence alters perception,” in The New Cognitive Neuro-sciences, ed. Gazzaniga, M. (MIT Press, Cambridge,MA).

Madan, R. N. [1993] Chua’s Circuit: A Paradigm forChaos (World Scientific, Singapore).

O’Shaughnessy, D. [1987] Speech Communication,Human and Machine (Addison-Wesley, NY).

Skinner, B. F. [1957] Verbal Behaviour (Prentice-Hall,Englewood Cliffs, NJ).

Watts, D. J. & Strogatz, S. H. [1998] “Collective dynam-ics of small-world networks,” Nature 393, 440–442.

Wittgenstein, L. [1958] Philosophical Investigations, 3rdedition, Trans. Anscombe, G.E.M. (Oxford Univ.Press).

Appendix AFormal Languages Primer

A.1. Language

In contemporary research, a formal language is a setof finite-length words (i.e. a string of characters),obtained from a finite alphabet, while the scientifictheory which deals with these entities is known asformal language theory.

A typical alphabet would be a, b, and a typ-ical string over that alphabet would be ababba.

A language over that alphabet, containing thatstring, would be the set of all strings which containthe same number of symbols a and b.

For example, the space of the phonemes P =Pi|i = 1, . . . , n, visualized as a sequence of thecolors could be considered as an alphabet,and the three sequences (2) of phonemes

XP = P ∗

i |i = 1, . . . , 62YP = P ∗

i |i = 1, . . . , 62ZP = P ∗

i |i = 1, . . . , 62can be considered as strings of the same alphabet.

At the same time, the set of morphemes M ,represented by a sequence of colored symbols, asin Fig. 31 could be considered as an alphabet,while the sequence (3), composed of 62 symbols:S = Si|i = 1, . . . , 62 and visualized in Fig. 32could be considered as a string over the alphabet ofmorphemes.

The empty string is allowed and often denotedby λ. While the alphabet is a finite set and everystring has finite length, a language may have aninfinite number of strings.


Fig. 31. Graphical representation of a set of six morphemes.

From a mathematical point of view, startingfrom the alphabet V , we can define the set V ∗, com-posed of all the strings of symbols of V . We denotewith λ an empty string and we indicate with thesymbol V + all the finite strings contained in V ∗,except that V + does not contain the empty stringV + = V ∗ − λ. A language L over an alphabetV is a subset of V ∗. A language L ⊆ V ∗ that doesnot contain λ is called λ−free. The length of a stringl ∈ V ∗ is represented by |l|. The number of occur-rences of a given symbol a ∈ V in a string l ∈ V ∗ isindicated by |l|a.

In the case of strings of phonemes or mor-phemes, emerging from the attractors and obtained

following the approach described in this paper, |l| =62. For a language L ⊆ V ∗, it is possible to definethe Length(L) = |l|/l ∈ L of the set. The set ofsymbols occurring in a string l ∈ V ∗ is indicated byalph(l).

For example, let us consider the toroidal attrac-tor illustrated in Fig. 33, that emerges for the fol-lowing values of the control parameters:

C1 = 0.099 nF, C2 = 10nF, G = −1 kΩ−1,

L = 0.00667mH, R0 = 0.000651Ω,

Ga = 0.856mS, Gb = 1.1mS.

Following the same reasoning introduced above, wenotice that the attractor realizes five phonemes.

P1 = (5 6 7), P2 = (1 6 7), P3 = (6 7 8),P4 = (1 2 6), P5 = (1 2 6).

If we consider the three strings (2), we canobtain alph(XP YP ZP ) = alph(XP ) ∪ alph(YP ) ∪alph(ZP ) = P1, P2, P3, P4, P5 = P that could berepresented by the string of colors .

The sequencing, represented by the string l andoutlined in Fig. 32, identifies the following six mor-phemes M1 = (1 1 4), M2 = (1 1 2), M3 = (1 1 5),M4 = (2 1 5), M5 = (1 1 1), M6 = (2 3 5). Thus

alph(S) = M1,M2,M3,M4,M5,M6 = M

and is represented in Fig. 31.Let us now introduce a series of abstract sym-

bols M = a, b, c, d, e, f, establishing a biunivocalcorrespondence between each morpheme and eachelement of M . If we think of M as an alphabet, thenthe following string will correspond to the sequenc-ing of Fig. 32:

l = abbcabadeaadaaafaaefacedabacbaccaacaaaceaacaabdabadbcadaaecaaa

and obviously |l| = 62, |l|a = 31, |l|b = 8, |l|c = 10,|l|d = 6, |l|e = 5, |l|f = 2,

while alph(l) = M .

Fig. 32. Graphical representation of a string of 62 morphemes.


Fig. 33. Toroidal attractor.

To summarize, we can illustrate a formal lan-guage with Fig. 34, using Chomsky’s approach.

In fact, a formal language can be specified in agreat variety of ways, such as:

• strings produced by formal grammars (such asChomsky’s hierarchy);

• strings produced by a regular expression;• strings accepted by some automaton such as a

Turing machine or a finite state automaton.

Many operations (concatenation, intersection,union, complementation, the right quotients,Kleene∗, reverse) can be used to produce new lan-guages from given ones.

Let us suppose that L1 and L2 are languagesover a common alphabet. The concatenation L1L2

is made of all strings of the form νw where ν ∈ L1

and w ∈ L2.The intersection of L1 and L2 consists of all

strings which are contained in L1 and also in L2.The union of L1 and L2 is made of all strings whichare contained in L1 or in L2. The complement ofthe language L1 involves all strings over the alpha-bet which are not contained in L1.

The Kleene* L∗1 consists of all strings which

can be written in the form w1w2 · · ·wn with stringswi in L1 and n ≥ 0. This operation includes theempty string λ because n = 0 is allowed; namely

L0 = λLi+1 = LiL1, i ≥ 1

. . .

L∗ =∞⋃i=0

Li.

Taking the previous example as a reference, we canset

L0 = λL1 = M

Li+1 = LiL1, i ≥ 1. . .

L∗ =∞⋃i=0

Li

thus obtaining a regular language. We shouldremember that a language that can be obtainedfrom the letters of an alphabet V and λ, using a


Fig. 34. In this diagram, the basic objects of formal lan-guage theory (alphabet, sentence, language and grammar)are represented. Grammars consist of rewriting rules: a par-ticular string can be rewritten as another string. Such rulescontain symbols of the alphabet (here a and b), and so-called“nonterminals” (here S, A, B and F ), and a null-element,ε, according to Chomsky’s approach. The grammar in thisfigure works as follows: each sentence begins with the symbolS. S is rewritten as aA. Then there are two choices: A canbe rewritten as bA or aB. B can be rewritten as bB or aF .F always goes to ε.

very large but finite number of operations of union,concatenations and Kleene*, is called regular.

The reverse LR1 contains the reversed versions

of all the strings in L1. A typical question askedabout a formal language is how difficult it is todecide whether a given word or sentence belongsto it.

A.2. Grammar

A formal grammar is an abstract structure that gen-erates a formal language. More precisely a gram-mar is a set of rules that mathematically defines a(usually infinite) set of finite-length strings over a(usually finite) alphabet, analogously to the processof producing sentences according to a grammar inhuman languages. Formal grammars can be dividedinto two main categories: generative and analytic.A generative grammar is a set of rules by which allpossible strings in the language can be generated byusing sequentially rewriting strings, starting froma designated start symbol. A generative grammaris described by an algorithm that generates stringsin the language. On the contrary, an analytical

grammar is a set of rules that assumes an arbitrarystring to be given as input, and which successivelyexamines that input string yielding a final Boolean“yes/no” result, indicating whether or not theinput string is a member of the language describedby the grammar. An analytic grammar formallydescribes a parser for a language. Basically, ananalytic grammar explains how to interpret a lan-guage, a generative grammar gives details on how towrite it.

Generative grammars

A generative grammar consists of a set of rulesfor transforming strings. To generate a string ina language, one begins with a string consist-ing of a single “start” symbol, and then succes-sively applies the rules (any number of times,in any order) to rewrite this string. The lan-guage consists of all the strings that can be gen-erated in this manner. Any particular sequence oflegal choices taken during this rewriting processyields one particular string in the language, andif there are multiple different ways of generatinga single string, then the grammar is said to beambiguous.

Formal definition

Generally speaking, a grammar is a (finite) devicethat generates, in a very specific sense, the stringsof a language, therefore defining a set of syntacti-cally correct strings. Many kinds of grammars arerewriting systems. In order to formally introducethe concept of grammar, we should recall the notionof relation.

Given a set X = X1 ∗ X2 ∗ · · · ∗ Xn, a relationR is a subset of X. The notation R(x1, x2, . . . , xn)is often used instead of (x1, x2, . . . , xn) ∈ R. Obvi-ously if X = X1 ∗ X1 the relation is called binaryrelation and will be written as R(x, y). Let us givesome definitions which apply to binary relations:

• R is reflective if xRx• R is symmetrical if xRy implies yRx• R is transitive if xRy and yRz imply xRz

The transitive and reflective closure R∗ of R isdefined as xR ∗ y if ∃ x0, x1, x2, . . . , xn with n ≥ 0so that x0 = x, xn = y and ∀ i = 1, . . . , n resultsin xi−1Rxi. Many kinds of grammars are rewritingsystems. A rewriting system is a couple γ = (V, P )where V is an alphabet and P is a finite subset ofV ∗ ∗ V ∗. The elements (u, v) ∈ P are written inthe form u → v and are called rewriting rules or


production rules. For x, y ∈ V ∗ we write x⇒γ y ifx = x1ux2, y = x1vx2 for some u → v ∈ Pe x1, x2 ∈V ∗. If the rewriting system γ is understood, we willsimply write ⇒ instead of x⇒γ y. The reflectiveand transitive closure is indicated by ⇒ ∗. If anaxiom is added to a rewriting system and all therules u → v have u = λ, then we obtain the notionof pure (formal) grammar.

For a pure grammar G = V, s, P, wheres ∈ V ∗ is the axiom, we define the language gen-erated by G as

L(G) = l ∈ V ∗|s ⇒ ∗l.In the classic formalization of generative grammarsfirst proposed by Noam Chomsky in the 1950s, agrammar G is a quadruple G = (N,T, P, s) where:

• N is a finite set of nonterminal symbols.• T is a finite set of terminal symbols that is dis-

jointed from N :N ∩ T = φ,• P is a finite set of production rules, thus P is a

finite subset of (N ∪ T )∗N(N ∪ T ) ∗ (N ∪ T )∗,with the restriction that the left-hand side of arule (i.e. the part on the left of the →) must con-tain at least one nonterminal symbol.

• s ∈ N is a particular element of N and is calledaxiom and is indicated as the start symbol.

The language of a formal grammar G (N,T, P, s)denoted as L(G), is defined as the set of all stringsover T that can be generated from the starting sym-bol s and by applying recursively the productionrules in P , until no more nonterminal symbols arepresent.

The production rules (u, v) ∈ P of G are writ-ten in the form u → v.

It should be noted that |u|N ≥ 1, which is thegenerating string contains at least one nonterminalsymbol.

Consider, for example, the grammar G withN = s,B, T = a, b, c, P consisting of the fol-lowing production rules:1. s → aBsc2. s → abc3. Ba → aB4. Bb → bb

and the nonterminal symbol s as the start symbol.Some examples of the derivation of strings in L(G)are:S → (2) abcS → (1) aBsc → (2) aBabcc → (3) aaBbcc →

(4) aabbcc

S → (1) aBsc → (1) aBaBscc → (2) aBaBabccc →(3) aaBBabccc → (3) aaBaBbccc →(3) aaaBBbccc → (4) aaaBbbccc → (4)aaabbbccc

(where the used production rules are indicated inbrackets and the replaced part is each time indi-cated in bold).

Generative formal grammars are identical toLindenmayer systems (L-systems), except that L-systems are not affected by a distinction betweenterminals and nonterminals, L-systems have restric-tions on the order in which the rules are applied,and L-systems can run forever, generating an infi-nite sequence of strings. Typically, each string isassociated with a set of points in space, and the“output” of the L-system is defined as being thelimit of those sets.

The Chomsky hierarchy

There is an infinite number of languages over a par-ticular alphabet. So, any finite method of describinglanguages cannot include all of them. Formal lan-guage theory gives us techniques for defining somelanguages over an alphabet.

In the 1950s, Noam Chomsky classified gram-mars into four types:

Type 0, Recursively Enumerable Language.

Type 1,Monotonous(i) if ∀u → v ∈ P , |u| ≤ |v|Context-sensitive(ii) if each u → v ∈ P has u = u1Au2,

v = u1xu2 for u1, u2 ∈ (N ∪ T )∗, A ∈ N andx ∈ (N ∪ T )+.

Type 2,Context-free(iii) A → β where A is in N , β is in (N ∪ T )∗Linear(iv) A → x or A → xBy, where A and B are in

N and x and y are in T ∗.

Type 3,Left-linear(v) A → Ba or A → a, where A and B are in

N and a is in T ∗.Right-linear(vi) A → aB or A → a, where A and B are in

N and a is in T ∗Regular


Fig. 35. This figure shows a hierarchy of related classes of machines (automata) and formal languages which are equivalentat each level across the whole hierarchy. The grammars generate a continuum that ranges from simple to complex. This meansthat they have increasingly rigorous production rules.

(vii) A → B, where A is in N and B is inT ∪ TN ∪ λ.This taxonomy is now known as the Chom-sky hierarchy. The difference between these kindsof grammars is that they have increasingly rigor-ous production rules and can express fewer formallanguages. Particularly important are context-freegrammars and regular grammars, and the languagesthat can be described by them, respectively, calledcontext-free and regular languages. The importanceof these kinds of grammars relies on the fact thatthey can express any language that can be acceptedby a Turing machine and parsers for them can becompletely implemented. We can identify a hier-archy of related classes of machines (automata, amathematical model of a computer which can deter-mine whether a particular string is in the language)and formal languages which are equivalent at eachlevel across the whole hierarchy (Fig. 35).

In the following, we present some examples ofhow it is possible to use the language of chaos forspecifying a formal language, in the framework ofChomsky’s hierarchy.

A.3. Example 1

Let us consider the toroidal attractor illustrated inFig. 33. We define:

N = sT = M

Fig. 36. Graphical representation of two strings producedby the grammar described in Example 1.

where M = a, b, c, d, e, f is compound byabstract symbols corresponding to six mor-phemes of Fig. 31, with the following productionrules:

s → a s → as s → b s → bs s → c s → cs

s → d s → ds s → e s → es s → f s → fs.

This grammar results as context-free and here fol-low some productions:

befdfcacfd to which the string of morphemescorrespond [see Fig. 36(a)].

def to which the following string of morphemescorrespond [see Fig. 36(b)].


Fig. 37. Network of morphemes. The nodes represent the morphemes that in Fig. 32 appear in counter-clockwise order,starting from the top. The arcs link repeated morphemes.

Fig. 38. Graphical representation of a string produced by the grammar described in Example 2.

A.4. Example 2

Let us consider again the toroidal attractor illus-trated in Fig. 33. We define:

N = s,A,B,C,D,E, FT = M

Production rules:

s → A s → B s → C s → D s → E s → F.

The other rules may be generated studying thesequence of morphemes in the network they

produce (Figs. 32–37):

A → a A → Ab A → aC A → aD

A → aE A → aF B → bA B → b

B → bC B → bD C → cA C → cB

C → c C → cE D → dA D → dB

D → d D → dE E → eA E → eC

E → eD E → e E → eF F → fA F → f

In Fig. 38, an example of production is visualized.

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The Language of Chaos