An Application of an Analogue of the Partition Function to the Evolution of Diglossia

John Wyburn (corresponding author)School of Computing and MathematicsUniversity of South WalesPontypridd, CF37 1DLWales, UK.Email: john.wyburn@southwales.ac.uk

John HaywardSchool of Computing and MathematicsUniversity of South Wales

Abstract

Recent trends in the numbers of minority language speakers have given cause for concern as to the effectiveness of traditional language-acquisition and survival strategies. This paper addresses the establishment of a recognized survival scenario, that of diglossia, the allocation of different languages to complementary social domains. The method is the novel application of an analogue of the partition function of statistical mechanics, embodied in a cellular automaton, to a population of subsets of social domains in which either of two languages may be spoken. The case of modern Wales is discussed in detail. Conclusions are drawn regarding the role of diglossia in the preservation of endangered languages.

KeywordsDiglossia; bilingualism; language group dynamics; quantitative linguistics; partition function; cellular automaton

Introduction

Diglossia is the consistent association of a given language or dialect with a particular social situation (see e.g. Fishman 1967). Diglossia is not the case with English speakers in the United Kingdom. A UK English-speaker would not use a different language or dialect in Church to that used on the racecourse (though the vocabulary would differ). However, in (for instance) Switzerland, High German (Swiss) would be used in the first case, Low (Standard) German in the latter (Ferguson, 1959); in Alsace (France), French would be used in the former, and Alsatian in the latter (Fishman, 1967). In linguistic terms, Alsatian is a “low” tongue, and French a “high”. High language is used formally (i.e. in literature, religion, the law) and in areas of high social prestige (the church, institutes of government and higher education). Low language is used informally (i.e. in conversation and popular entertainment) and in areas of low prestige (primary and secondary education, the home, the market). Structural and corpus distinctions between the two are discussed by e.g. Kaye (2001).

Diglossia is, then, the distribution of two means of interaction over a set of interacting units. Any such distribution is analogous to a particular statistical ensemble (corresponding to a particular free energy) over the molecular components of a gas.

Fishman (1967) identifies four situations in which bilingualism and diglossia interact. Here diglossia is taken to be the consistent association of a given language or dialect with a particular societal domain (social situation).

1. Societies that use both diglossia and bilingualism, i.e. different languages perform different functions within society.

2. Societies that use bilingualism without diglossia, i.e. two languages coexist without social differentiation.

3. Diglossia without bilingualism, i.e. different social functions are served by dialects of the same language.

4. Neither diglossia nor bilingualism.

Neither Case 3 nor Case 4 are of interest here. In Case 1 of bilingualism with diglossia, clear societal norms exist that dictate language choice; each language is associated with a particular set of societal domains, such as home, friends, work, place of worship etc., each with its own representative members, roles and topics (Fishman 1972).

This paper is concerned with Case 2, that of bilingualism without diglossia. Here there are no societal norms regarding the choice between two available languages, language A and language B, in a particular situation. Given this case, under which circumstances, or according to which causal mechanism, will a bilingual speaker use language A rather than language B? Assuming that this choice is made consistently, what form of diglossia (i.e. what distribution of languages amongst domains) may come about? This is analogous to the physical scenario in which the statistical ensemble corresponding to scenario 2 evolves into that of scenario 1. Such an evolution may then be modelled by finding those conditions, analogous to macrophysical quantities and microphysical interactions, that promote a change in ensemble.

If those conditions likely to bias new group members towards language A and away from language B are identified on the microsociological level, public venues and media can be organized to encourage or discourage this choice. On the macrosociological level, the topic of language acquisition is more directly relevant to the survival of any language, but this topic may possibly be addressed only where microsociological interactions have conspired to produce stable diglossia. Certainly Fishman (1967) opines that it is under conditions of diglossia that language maintenance is most likely to occur, and this has been given theoretical support by e.g. Kandler (2010). While Kandler does not model the establishment of diglossia as an explicit language-planning strategy, the importance of dedicated domains of use is noted, as is that of language status (Kandler adopts the status model of Minett & Wang, 2008).

An instance of empirical support is given by Mira & Paredes (2005), whose example (Castillian and Galician) is an instance of diglossia (Beswick 2002), although not noted as such. While Hudson (2002) notes that diglossia may aid the resistance of a vernacular to displacement by a rival, rather than affording intrinsic stability to the diglossia per se, the state of diglossia is again advanced as indicative of stability.

The circumstance of bilingualism without diglossia is in this paper considered to apply wherever language revival is evident. That is, in countries that have been

monolingual in language A for some years or generations, and in which there is now interest in reviving a formerly widespread language B. Even if the revival is stimulated by cultural considerations, the population of A-language speakers acquiring language B may not be culturally distinct from those who do not acquire B. The result is a new bilingual population, distinguished by the circumstance of bilingualism without diglossia, since cultural norms regarding circumstances of language use have not yet been established. This is the situation in all Celtic countries where English or French formerly dominated, and where the original Celtic languages (Welsh, Irish and Scottish Gaelic, Breton, Cornish and Manx) are increasingly recognised and promoted.

Related Works

Language death and survival under conditions of bilingualism had been successfully addressed using differential equations by Baggs and Freedman (1990, 1993), before interest was revived by Abrams and Strogatz (2003). Kandler (2009) established the usefulness of reaction-diffusion mathematics, and Kandler et al (2010) extended this approach from temporal evolution to spatial dispersion, and noted the importance of social domains dedicated to minority languages as a survival factor. Alternative methodologies such as Monte-Carlo simulation (Schulze et al, 2008) and system dynamics (Wyburn and Hayward, 2009), came to similar conclusions regarding the viability of bilingual populations. The field has since progressed to more theoretical approaches such as those of Wyburn and Hayward (2010), An et al (2012) and Nie et al (2015), employing the theoretical derivation of stable equilibria in higher-dimensional phase spaces.

All these papers are essentially optimistic as to the potential of conventional language planning measures, and none model the establishment of diglossia as an explicit language-planning strategy. The disappointing performance of present language-preservation strategies in the case of Welsh has been discussed by Wyburn (2018), who attributes this to the increasing prevalence of English-language media. The same paper notes that a Welsh diglossia would be beneficial, but the author was unable to identify likely domains. Prochazka & Vogl (2017) present a cellular automaton over a set of individuals, and demonstrate that local concentrations of speakers of a given language can persist over time. However, this work does not attempt a distribution over explicit social domains, and relates to the microsociology of local language dominance rather than diglossia. Nevertheless, the utility and flexibility of a cellular automaton was strongly established.

It should be noted that the principle concern of studies addressing language death has been that of the nature of language acquisition. This phenomenon has been clearly demonstrated to have its own structural components in e.g. Zhao et al (2018), Lima et al (2018), and Amancio et al (2012). It may therefore be possible to extend the present methodology to an instance in which some proportion of the actors are in the process of acquiring the languages involved.

Considerable work has been done on the broader question of opinion dynamics, and especially that of opinion formation in groups. While most such work assumes rational choice and full information (e.g. Granovetter 1978; Latané, Nowak, and Liu 1994), that of Coleman (1964) examines probabilities of formation and change given

only the preceding distribution of opinion strengths.

It is here assumed that phenomena which temporarily affect the emotional character of whole societies do not impinge on the long-term evolution of diglossia. However, persistent or cyclic phenomena of this type (see e.g. Kristoufek, 2018) may indeed do so, and may be incorporated into a future model as a cyclic fluctuation of e.g. “social temperature”.

Axelrod’s paper (1997, examined in Izquierdo et al, 2009), which models the dissemination of culture, has clear applications in this respect. Perfors (2002) gives a comprehensive review of simulation in the field of language group change. However, it was Beltran et al (2009), in successfully applying a cellular automaton to these questions, which suggested the cellular automaton as a methodology by which the internal structure of the bilingual language group might itself be examined.

Bahr and Passerini (1998a, b) have partially duplicated Coleman, but have introduced quantitative measures of the effects of group volatility, impetus towards different opinions, and noise on the probability of changing opinion. These papers apply the partition function of statistical mechanics to the modelling of the spread and retention of opinions through a population (of individuals). The same papers’ use of a cellular automaton suggested the present application of such a methodology to the problem of the establishment and continuity of diglossia.

The Model

A choice between two languages is likely to be more unconscious, and commitment more context-dependent than the decision to subscribe to, for instance, a political or religious belief, and so the unmodified Bahr and Passerini model would not be appropriate to the present paper. Moreover, the establishment of diglossia requires the application of the methodology to a set of social roles or societal domains, not of individuals. However, a preliminary application of a simplified methodology to the incidental or transitory choice between two languages in a population of individuals will serve to illustrate the basic assumptions and their relation to the formulae used. This will also demonstrate that the Bahr and Passerini model is capable of supporting the results of Prochazka & Vogl (2017) over a comparable set.

Given a context-dependent choice between two languages A and B, the following factors, identified by Bahr and Passerini (1998a, b) are assumed to determine the language that any one individual, modelled as a cell in a two-dimensional array, chooses. The allocation of linguistic factors to these is as follows.

1. The interaction strength of the i,jth individual pij. This is the ability of the i,jth member of an array to persuade a second individual to use the same language. This stands direct comparison with Bahr and Passerini’s “opinion strength”, and is noted to be an element in language choice in Fishman (1972). The assumption is made that pij is initially greater than zero, in that any individual using a language has already exercised the choice to do so.

2. The number of people neighbouring the individual’s cell at the present iteration using either language. For instance, the number of neighbours currently using

language A correlates positively with the probability of switching to language A. Density of neighbours is examined in Fishman (1971a, b).

3. Environmental factors supporting the local prevalence of language A, i.e. tending to influence free choice towards the use of A; Bahr and Passerini’s “social forces”, denoted by hA. For instance, the presence of public information signs in language A, or the intrusion of a public address system employing this language, would constitute social forces in this context. It is here suggested that, while such forces may initially be determined by external factors, it can be expected that they will grow with local dominance of one language; as people speak language A, people will respond in language A. These influences are cited in Cooper (1967, p.157), although the typical motivation for supplying such influences is to encourage both the use and the acquisition of the language of the media.

4. The volatility of the group, i.e. tendency for the average member to change opinion; Bahr and Passerini’s “social temperature” T s. Language switching amongst individuals is more frequent in less formal situations (Fishman 1972), and so temperature amongst such individuals can be broadly taken to be equivalent to informality.

A Simple Illustration

To illustrate the model, consider an assembly of only 25 individuals (Figure 1). All are bilingual in languages A and B, with no preference between the two. Interaction strengths and colour distribution have been generated at random.

Each individual is represented by a cell in an array, either white, representing an individual using language A, or black, for language B. The notation (i, j) is used to identify the cell in the ith row and jth column. The interaction strength of the i,jth individual is indicated by one of the values pij{2,1,+1,+2}, positive values for A-language users and negative for B-language users. Therefore, an individual is either persuasive |1| or very persuasive |2| (i.e. is a “conversation leader”). Individuals interact with those horizontally and vertically connected (the “nearest neighbour” scenario); the outer rows and columns are allowed to interact with their counterparts, forming a closed surface (“periodic boundary conditions”).

Figure 1. An illustrative assembly of 25 bilinguals. White cells (positive pij values) indicate speakers using language A. Black cells (negative pij values) indicate speakers using language B.

Let these 25 persons interact in a waiting room equipped with a television set. The television set can be heard by all twenty-five, and is playing a program in language A. In that the TV furnishes a topic of conversation, it encourages all to use language A, the social force towards A.

As is conventional for CA, time is discrete and changes are synchronized, all cells making their decisions simultaneously at the end of each time step or iteration. Here an iteration is the time taken for all 25 cells to be given the opportunity to change. The interval between iterations is the mean time taken to arrive at a decision, which can here be arbitrarily taken to be of the order of minutes (a lively conversation). Before iteration commences, each of the twenty-five makes the decision to begin speaking in one or the other of their two languages. (It must be assumed that individuals (1,4) and (3,5) are initially speaking to themselves, in that they are isolated.)

On the next iteration, the distribution of choices and interaction strengths influence neighbours to change or retain their initial choice, according to the following interaction equations (1), (2).

Here ∑NA

pij is the sum of interaction strengths summed over all neighbours currently

using language A. The equivalent for language B is ∑NB

−p ij.

The sum ∑N

|p ij| is that of the magnitudes of the interaction strengths over N.

PA to B=

e−h A (n A)

T s (∑N B

−pij

∑N

|pij| )1T s

eh A (n A)

T s ( ∑N A

pij

∑N

|pij|)1T s

+e−hA ( nA )

T s (∑NB

−p ij

∑N

|p ij| )1T s

(1)

PB to A=

ehA ( nA)

T s ( ∑NA

pij

∑N

|pij|)1T s

ehA (n A)

T s ( ∑N A

pij

∑N

|pij|)1T s

+e−h A( nA )

T s (∑N B

−pij

∑N

|p ij| )1T s

(2)

The denominator is the partition function. The exponential form is derived from an analogy with heat diffusion processes in materials, and hence the impetus towards one choice or another is a ratio between factors tending to grow and diminish exponentially. Environmental forces towards the A language hA may be taken to increase with numbers currently speaking that language nA. For illustrative purposes,

this is here identified with the sum of interaction strengths nA=∑N A

pij .

In the following example hA = 0.001nA, hB = 0.001nB, a linear change with increasing numbers, chosen for illustrative purposes so that hA can grow no higher than 0.025. At this point in time nA=nB=16. In the example, as more speak language A, more attention is given to the television; attention may be brought to it by those discussing the programme, or these may turn up the sound volume, so attracting the attention of others.

Social temperature TS is for the sake of illustration taken to be 0.1, found after some experimentation to be low enough to permit long-lasting structures to appear. In context, no-one is inclined to immediately change the language being used.

To illustrate the mechanism, consider cell (4,4). The individual represented here initially chooses to use language B. His interaction strength is of magnitude |p44|= 2, and so he is generally a conversation leader. This interaction strength, and those of the four neighbours, are totalled over the neighbourhood of (4,4) by the element

∑N B

− pij

∑N

|pij|

as (12)/(2+1+1+1) =3/5. Note that p44 is itself excluded from the numerator. This is here referred to as the influence ratio.

Environmental influences are currently at hA = 0.00116 = 0.016, so the individual is encouraged to change to language A by his environment. The environmental effect is locally non-zero but globally zero, a consequence of Bahr and Passerini’s (1998a, b) physical analogy (∑

all ihi=0). Equation (2) now

gives the probability of the individual changing from language B to language A as PBtoA = 0. 9876.

On iteration, a random number 0 r 1 is generated, and if r < 0. 9876, individual (4,4) changes his or her choice of language, and so the cell turns white. Note that the choice does not change the interaction strength of the individual; interaction strength is independent of the choice of language.

Figure 2. A quasi-persistent subset of B-language speakers persists in e.g. 74.3% of cases for over 200 iterations.

Running the model a number of times determines certain regularities in the interactions of the people. Ultimately, the entire 55 block becomes entirely white or black. However, a subset of B-language speakers comprising individuals (3,4), (3,5), the familiar (4,4), and (4,5) persist in speaking language B for many iterations (Figure 2); that is, they demonstrate quasi-persistence. If it so happens that B comes to dominate, this knot is permanent. Over 1000 runs each of 200 iterations, it remains present in 74.3% of cases; over 1000 runs each of 400 iterations, it is present in 59.9%. Over 1000 runs of 2000 iterations each, 19% retain the knot, 8% of these being all black (Figure 3).

Figure 3. The comparative persistence of the subset of Figure 2.

Of the 1000 cases, 780 achieved total cover, with 693 becoming all white, and only 87 all-black. Over the 780 all-white, the average number of iterations to coverage was 402.86. To the all-black, only 94.53 iterations average, and so the all-black case is decided much more rapidly than the all-white. This is apparently due to the number of highly-persuasive individuals in the initial distribution.

In a sense, the distribution of A and B language speakers is a diglossia, in that anyone who becomes interested in a particular conversation will adopt the language being used when they join in: languages A and B now have dedicated uses in this small society.

Pseudocode for the basic automaton is given in Appendix 1.

Macrosociology and Diglossia

The macrosociology is not simply a case of applying the above argument to a larger number of people. Each cell must now represent not a person but a social role. For diglossia to be established, it must be demonstrated that the language choice for a given role does not change over a large number of iterations, i.e. that a quasi-steady state can be established.

Modelling Assumptions

In defining the social roles represented by the cells, the following simple social model of two dimensions is used. The automaton therefore models an abstract space.

The dimension along the i-coordinate of the automaton array indicates increasing topicality or height of topic- the association of a topic, to a greater or lesser degree, with high or low language. Here the distinction between high and low topicality is modelled as a function of frequency of language choice. That is, for the lowest i, the opportunity to change occurs with every iteration; this models conversation, in which languages A and B may be conveniently adopted to follow changes in the subject under discussion. For intermediary values of i, the opportunity to change occurs less frequently, modelling short-term media such as newspapers and periodicals. At the

highest end of the i scale, the language used may not change in the time horizon examined; this models e.g. perennial literature, the “highest” use of language, or language use as stabilized in law or governmental proceeding. This differential iteration is distinguished from temperature, which is a factor that comes into play when an iteration is possible.

Therefore, while an iteration is typically taken to refer to the processing of all cells, it must here be redefined as the time taken to process all j columns once over.

The length of time intended by a single iteration (mean time between language choices) is culturally determined. Intuitively, a role retains use of a language longer than a person might, consistency being one of the things that defines it as a role. This is discussed further in relation to modern Wales below.

The dimension along the j-coordinate indicates “increasing influence”. Cells modelling roles with low j coordinates are connected to fewer cells than those with high j coordinates; that is, they belong to smaller networks. All roles in the automaton are connected to their horizontal and vertical neighbours; neighbouring roles work closely together. As j increases, other links are made, implemented by randomly generating a further x = x(j) links with other cells, modelling random contacts with more distant roles in the course of exercising the present role. The highest value of j might be linked with all other cells, indicating roles of high authority whose views and instructions are widely distributed, and who hear from and must answer to other roles. It must be noted that if such a cell influences more than others, it is likewise influenced more than others, though the reaction to this influence is strongly mitigated by the cell’s slowness to change. The authority in this society is therefore not totalitarian, but willing (in the long term) to adapt to the language preferences of those with which it interacts. An example might be the exchange of vernacular for Latin in the Catholic and High Churches.

The role is therefore defined by varying degrees of height of topic (inversely proportional to iterative frequency) and influence (connectivity), as shown in Figure 4. These parameters of differential iteration and differential connectivity, and how they may be amended for analysis, are discussed further below. Due the imposition of dimensions, the surface is no longer closed.

Figure 4. A simple linguistic model of a society, in which social domains are ordered along two axes.

The notation A=1, B=1 is now adopted to distinguish the macrosociolgical from the microsociological case. The factor h1 is taken to be equivalent to incentive to use language A. Unlike the microsociological case, in which the television was an external stimulus affected by the numbers of A speakers, in the macrosociological case the influence must be generated within the community. Equal numbers of A and B speakers must amount to zero force towards A. A single additional A-language role will have an effect determined partly by its connectivity and iterative frequency, but must also tip the h1 force into the positive. A society in which all roles are fulfilled by language A must make for the highest positive value of h1. Here h1 is taken to be a function of the sum of n1, the sum of the influences of the A-language cells, and n-1, the sum of the influences of the B-language cells, on any iteration, proportional to the total number of cells N,

h1=Cn1−n−1

N.

Here h1=h1, where n1<n1 and the numerator is negative. It should be noted that these will be different for different cells at different iterations. The constant of proportionality C may be considered an “influence factor”. Experimentation showed that increasing C (and therefore the strength of forces towards language A or B, depending on current predominance) decreased the time to single-language domination, but left the end results qualitatively unchanged. Since the relation between number of iterations and time horizon is conjectural, it has been retained.

Temperature Ts must be quite low, else the roles would not be stable. A value of 0.1 was chosen as allowing for evolution to total colour in reasonable real time.

The assumption is made that no profound change occurs in the structure of the society. In particular, no novel social roles are introduced in the time horizon chosen, and therefore the number of cells remains constant. Therefore, the time scale involved cannot be more than a couple of generations, say 50 years maximum. For the sake of the general illustration below, if a work of literature is taken (arbitrarily) to be influential over a generation, then the highest value of i in a 5050 array may be taken to indicate 50 iterations = 25 years. Then the lowest i value iterates once every 6 months. This has a certain intuitive appeal, but in such an early stage of investigation, it would be best to keep an open mind as to what is intended by an iteration.

In that we are now concerned with roles, not people, opinion strength is discounted. All cells have the value 1 (using language A, coloured white) or 1 (using language B, coloured black).

The appropriate number of cells deserves some attention. Social roles are associated with social class, and with the occupations associated with class. For instance, writing of Luxembourg, Davis (1994) notes the tendency of the social elite, identified with “lawyers, doctors, engineers and ‘professors’... upper management, and company or shop owners” to interact and entertain in French. The upper middle class, characterized by “middle management… technicians, and primary school teachers”

use, and have the opportunity to use French much less. The lower middle class, whose members “perform clerical work... or other forms of skilled labour” interact in German and Lëtzebuergesch, whereas “lower class individuals... [who] are semi- or unskilled labourers tend to speak only Lëtzebuergesch” (it should be noted that Davis’ concern in this work is language planning, not an exposition of diglossia). In this instance there are clearly more occupations than social groups and height of topic, but the numbers in any two occupations might suggest proportional numbers of cells. Therefore occupations as social institutions are at least indicative of social roles.

Also, a group of connected cells might be considered to constitute as many roles, or subdivisions of the same role. Finally, as a practical consideration, a small array is more likely to be more subject to random variations than a large. The strategy adopted here is to use a variety of reasonably large arrays, and make connectivity a function of groups of j values (see below). When data is gathered over a number of runs, an attempt at definition will be made as a necessary precursor to application.

It must be acknowledged that this is a very incomplete and simplistic society. Nevertheless, this grass-roots approach was felt to be necessary before more sophisticated attempts be made, perhaps using dedicated software, and taking the above assumptions as read.

Details of the pseudo-code used to implement the automaton are given in Appendix 2.

Running and Testing the Model

A preliminary investigation of the model was made by generating 100 random matrices. Each such matrix was iterated until total coverage was achieved. Ts was set at 0.1.

Each model demonstrated the same tendency for areas of solid colour to begin consolidation in the corner of low i and high j coordinates, where connectivity is high and iteration is rapid. The colour which gains early dominance in this sector pushes the alternative to the high i edge. Ultimately this colour takes over the entire automaton, as expected (Figure 5).

Figure 5. The evolution of a random initial distribution (0 iterations) through 20000 such.

After 5000 of the fastest iterations, the diglossia is firmly established, persisting beyond 20,000 such iterations.

In any such society, therefore, the dedication of some language to specific roles seems to diverge between the low-topic and high-influential, and move towards the high-topic and least-influential. Historically, it is typical for a High language to be used less as a Low is employed for more roles.

The quick changeover of the high-influence squares suggests that these might be termed “high susceptibility”. However, those broadcasting to the majority, and those courting majority support, would certainly choose the preferred medium.

It is likely however that the diglossia which is temporary (though long-lasting) in the above model will in the real world be preserved by a further mechanism. Used exclusively for low-influence, high-topic purposes, the development of a specialised

vocabulary and literature would tend to stabilize it in that role. This might be modelled by a local tendency for a suitably modified force h1 to maintain a constant value after a given number of iterations. Such a “ratchet” could certainly be built into the current model, though once diglossia is established, it is enough to terminate the run.

As a test of the random nature of the model, C was set to zero, and a series of 100 matrices of size 1010, 2020 and 5050 cells were generated. The 50th iterations of each matrix were divided into those which had become mostly white, and those which had become mostly black. Finally the mostly white matrices were superimposed into a single composite matrix, and similarly the mostly black matrices. In all cases equal numbers of wholly black and wholly white superimpositions were found.

Results of an Application of the Model to 20th-Century Wales

The following information pertaining to late 20th-Century Wales is from Euromosaic (1994), using a sample of 293 people. It is to be regretted that no more recent comparable survey exists. Jones (2015) supports the Euromosaic findings in very broad terms; although Welsh is likely to be more strongly associated with Public Bodies, and with the Welsh Assembly as a substitute for the Welsh Office, Welsh has been entirely divorced from Central Government.

The data of the Euromosaic table entitled “Perceived Interest of Different Bodies in Welsh” is organized in Table 1 (below) along the dimension of increasing influence (j). The “Not Available” and “In-migrants” categories of the original Euromosaic table were excluded, and the three “Average” interest columns split between those of “High” and “Low”. The table “Use of Welsh by Context in the Community” was used to inform the organization of the above categories.

The dimension of decreasing height of topic (i) is similarly used to organize the data of the Euromosaic table “Language of Education”. This is here assumed to stand in close relation to height of topic, as is typically the case in genuinely diglossic communities. The “Eng+Welsh” category was split between the two languages, and the categories “Not Applicable” and “Other” (languages) were excluded. The recategorized data are as given in Table 2.

BODY USE OF WELSH

Central Government 10%

Welsh Office 29%

Public Bodies 44%

Local Authority 58%

Church 79%

Friends 85%

Family 88%

Self 92%

Table 1 Occupations/Social Institutions of Increasing Influence and their perceived Use of Welsh

EDUCATIONAL CATEGORY USE OF WELSH

Primary 60%

Secondary 39%

Higher 34%

Further 25%

Table 2 Low to High Topics of Welsh

It cannot be said that these criteria are exhaustive, nor that they extrapolate well to the rest of Welsh society. In particular, the Government are unlikely to have the choice of using Welsh or not. Further, the Euromosaic table “Community Participation and Language Use” is suggestive of greater importance of Welsh with respect to more inclusive activities on the local level. Nevertheless, following Fishman (1972) and Cooper (1967) they are suggestive of a diglossia in which Welsh is mostly used for low-influence, low topic roles.

The above data was amalgamated by superimposing the two tables, taking the average value of Welsh use according to the superimposition, and generating this average number of black cells in the corresponding area of an automaton (Figure 6).

Figure 6. A simple linguistic model of modern Wales, consisting of language domains along the axes of height of topic and influence of associated social role. There is a marginal domination of Welsh over English (50.3% Welsh).

There has been no attempt to partition the society such that e.g. the far left vertical cells sum to the 10% Welsh usage perceived of the Welsh Office, intersecting with the top two rows as summing to the 60% which represents perceived primary-school usage of Welsh. This might have been possible had there been more certain data, and would have entailed an approximation to a continuous social surface. However, interpretation as to what this amalgamation would mean would be no less difficult. It could not be said that, for instance, the top left cell would represent the proportion of Government officials educated in Welsh at primary school. Hence the present approach, in which that cell represents no more than a perception that the height of topic associated with Primary School education and the activity of Central Government amount to a perceived combined use of Welsh of 35%.

A series of 100 initial automata of sizes 1616, 3232 and 6464 cells were generated, such that the proportion of black (Welsh) to white (English) squares were appropriate in areas corresponding to those above. Within each area, the distribution of black to white squares was otherwise random. Where there are more cells per division, there are more subdivisions to the social domain, so more activity is possible on a smaller scale. Ultimately the subdivision may be to the number of persons

occupying a social role. Therefore, it is assumed that the larger automata are more accurate models than the smaller.

The influence factor C was allowed to be non-zero due to the pervasive influence of English; due to the difficulty of calibration, it was permitted to range over 0.0001, 0.001, 0.01, 0.1 for twenty-five automata of each size. Social temperature Ts was kept at 0.1, as justified above. As with the above test, the 50th iterations of each matrix were divided into those which had become mostly white, and those which had become mostly black. Finally the mostly white matrices were superimposed into a single composite matrix, and similarly the mostly black matrices. An example of the code employed is given in the appendix.

The different sizes of automata constitute different, though related, models of the same society, i.e. the smaller do not constitute linear approximations or coarse grainings of the larger. For instance, in the case of the 1616 automata, neglecting the 4 connections made by each cell to its neighbours, the column of highest j makes 16 times more connections than the first column. In the case of the 6464 automaton, this factor is 64. Hence, the highest j-ranked cells of the latter automaton make many more connections than those of the former, and the cells on its 16th rank- representing a social influence much lower than those of its 64th- make as many connections as the highest j-rank cells of the 1616.

The time step represented by an iteration length can be assumed to be independent of the size of the automaton. This assumption is supported by the surprising consistency of times to complete coverage by one or other colour. Where C=0.001 the average numbers of iteration for 1616, 3232 and 6464 automata are 233, 267 and 230.

An opinion of pij =0 does not persist for many iterations, since a zero opinion is easily tipped and cannot propagate. Bahr and Passerini make no provisions for signifying cells of zero opinion, and these are not evident in any of the final-iteration automata examined. Where a low-iteration or intermediary condition requires examination, there are three possible strategies: the cell may remain undecided, in which case grey might be used; the undecided cells are distributed randomly between the two languages; or, where the A language is strongly dominant (as in the case of English) the undecided cells are taken to support this. The authors suggest the last strategy to be most valuable, in that passivity is effectively support for the dominant language (see e.g. Wyburn, 2018).

The superimposed distributions are displayed in Figure 7. Black squares indicate negative numbers, and the use of Welsh, white positive numbers and the use of English.

Figure 7. Possible future Welsh/English diglossia. Black cells indicate those domains in which Welsh is habitually used, white those in which English is used.

In some 70% of cases Welsh becomes the dominant tongue, in a diglossia established

with English as the language of low influence and Welsh as the language of high. No such clear division exists between high and low topic, but there is a marginal preference for Welsh to be low topic.

Nevertheless, in around 30% of cases, English becomes dominant. In these cases, no diglossia is established, English becoming the universal language of both high and low topic, as it is in England.

The Parameters of Differential Iteration and Connectivity

Bahr and Passerini (1998b) employ a mean field analysis to predict the behaviour of the basic model. This approach is not here permissible, in that the two principle criteria, that cell value and significance should be independent of position, and that all cells should be connected (Bahr and Passerini cite Plischke & Bergersen, 1989, in this regard) are explicitly contradicted. Moreover, the average opinion obscures the proportion of runs that lead to diglossia. However, Bahr and Passerini thoroughly examine the parameters h (social force) and T (social temperature), which were introduced in their paper and adopted here.

The present model introduces the parameters of differential iteration, modelling difference in height of language topic, and differential connectivity, modelling influence of language domain. In the present model, iteration is achieved mechanically by making the loop controlling iteration rate dependent on row number i. The same effect can be achieved in a way more conducive to analysis, by the modified equations (3), (4).

PA to B=( 1iq )

e−hA (n A )

T s (∑NB

−pij

∑N

|pij| )1T s

ehA ( nA)

T s ( ∑NA

pij

∑N

|pij|)1T s

+e−h A (n A )

Ts (∑N B

−pij

∑N

|pij| )1Ts

(3)

PB to A=( 1iq )

ehA ( nA)

T s ( ∑NA

pij

∑N

|pij|)1T s

ehA (n A)

T s ( ∑N A

pij

∑N

|pij|)1T s

+e−h A( n A)

T s (∑N B

−pij

∑N

|p ij| )1Ts

(4)

Here iq is the qth power of the i-coordinate of the cell. Given a sufficiently long run, permitting 0<q≤2 allows control over differential iteration of the cells. Therefore q is the differential iteration parameter. Connectivity is achieved by adding a number “trunc(j/c)” of connections to each cell in the jth column. This is the integer part of the ratio of j and c, and hence j/c is the differential connectivity parameter.

If a 3232 automaton is run over 50 runs each of 1024 iterations with neither differential iteration nor connection (that is, if the original Bahr and Passerini model, with the amended force h, is applied to the scenario of Figure 6), the result is never found to be diglossia. Rather, the final iteration divides between 18 all-white (unilingual English) and 32 all-black (bilingual Welsh) scenarios (a long-term average opinion of +0.64). This is presumably to the near equality of black and white cells in the initial distribution. The number 32 = 25 has been chosen to allow results to be easily compared between the amended and the original iteration protocols.

Introducing differential iteration, while differential connectivity remains supressed, naturally delays a final outcome. It also favours the unilingual English scenario as an outcome. This is presumably because the initial English-language domination of the high-topic half of the automaton persists longer with increasing q.

However, diglossia is not established while differential connectivity remains supressed. Introducing differential connectivity, while suppressing differential iteration, leads to divergent end values including those pseudo-equilibria here associated with diglossia. This suggests that influence, rather than height of topic, is the critical factor in determining development towards diglossia in Wales.

Figure 8 plots the averages over the 50 runs at an iteration differential of i1.5, over connectivities of j/8 to j/2.

Figure 8. Averages from 50 3232 automata for q = 1.5 over the following values of j/c:j/2, no diglossia, a long-term average opinion of +0.550 determined by the preponderance of all-white end states.j/4, no diglossia, average opinion +0.189.j/6, 21 instances of diglossia out of 50 final states, average opinion +0.022. j/8, 24 instances of diglossia out of 50 final states, average opinion 0.266.

This pattern is evident for 1 < q < 2. Below an iteration differential of q=1, the relation between Welsh diglossia and increasing levels of differential iteration is more sensitive to initial conditions. While j/8 and above is consistently associated with

Welsh-dominant diglossia, the progression of favourability to Welsh for j/2 to j/6 is not predictable. The combination of low q and j/c values perhaps makes the model more sensitive to random activity (in generating the initial automata, and in progressing these).

Given a low enough T (<1), there will certainly emerge some stable areas, but these need not exhibit the axial polarisation typical of a diglossia. This is particularly evident in the smaller automata (<3030) wherein individual cells more strongly influence the outcome.

If there is an initial strong domination of one language over the other, total coverage will proceed too quickly to allow a pseudo-equilibrium. This possibility is mitigated if the non-dominant language is strong in the high-influence region of the automaton. Only a rule of thumb is possible while the model cannot be analytically described, but a cell in row n has the integer part of n/c more connections than one in row 1, and is this much more influential. It can be seen that a relatively low differential connectivity is required for Welsh to overcome the dominance of English at the high-influence end of the initial cell distribution.

A similar advantage is possessed by cells in the low-iteration region, where resistance to change gives marginally greater strength of opinion. That is, the same opinion has more than one opportunity (iteration) to make itself known without itself changing. In the present case, this has proved to be not so decisive a factor- early consolidation of the high-iteration areas of the automaton by Welsh proves decisive. That is, the values of q employed make the initial advantage of English in the low-iteration cells very fragile.

A high enough value of q, however (>2), does indeed encourage polarisation along the high/low topic axis rather than the high/low influence axis. It may be that the choice of educational language, required of the material available, is not as strongly representative of the high/low polarity as is necessary, and criteria employing a greater range of q values (literature, perhaps) would be preferable.

Conclusion

The model may be of use wherever a language is in revival, in determining how a diglossia between it and the dominant language may come about. It may also be used to identify those domains most effectively targeted by language planning measures.

Domains currently employing the minority language may be consolidated by awarding these greater influence, equivalent to affording their corresponding cells greater connectivity. This may be accomplished by the following measures.1. Inculcation of the minority language as proper to that domain, popularly and in formal education.2. The creation in, or dedication to the minority language of, communication channels

serving the domain and its interests.3. Where the domain is distinguished by academic or other qualifications, a minimum standard of proficiency in the minority language.

Similarly, the domain may by rendered more resistant to change, corresponding to a lower iteration rate, by the following.4. The production of minority language media, and the translation of majority-language media, appropriate to the domain. 5. The composition and tuition of domain-specific vocabularies in the minority language.

Those domains which are predicted to be likely to fall to the minority language (i.e. those whose corresponding cells are noted to become black in the majority of cases), may be targeted by:6. Specific intervention from an appropriate authority, formalizing the minority language as that of the domain concerned. Such intervention is possible only on those domains “higher”” than that of domestic use.  Once such usage has been established, strategies 1-5 may be employed.

It has been assumed that the force (h) encouraging either language is an environmental factor comprised of its proportional use. However, the effectiveness of this is mediated by perception. Ambient use of the minority language in public places, and increased use in popular broadcasts and entertainment, may permit a modification to the force more along the lines of Bahr and Passerini’s original. Referring to the illustrative example here presented (the waiting room), if the television in the room is using Welsh, it will encourage the use of Welsh.

It is under conditions of bilingualism without diglossia that language death may occur (Fishman 1967). Individual choice, if made consistently, comes to determine the occupation norms of the language. Diglossia may be established, resulting in language maintenance. If however one language achieves currency in more domains, or in more important domains than the other, this may displace the other language, leading to language death.

The language which establishes an early lead as the dominant language of high influence (connectivity) and low topic (high iteration frequency) forces the alternative language to roles of low influence and high topic, and ultimately assumes these roles also. Any language which is employed for no social roles is unlikely to be passed onto the next generation. Therefore the model would, under the ranges of values adopted for modern Wales, seem to indicate that the “common touch” is the prerequisite for survival. The form of the diglossia is thus an indicator of the popularity and breadth of application of a language, an important determinant in the language’s long-term future.

In Wales, the positive outcome for the Welsh language seems to entail a diglossia with Welsh dominant, and having the high-occupation domains for its own.  Given suitably long-term stability under these conditions, it may be possible for Welsh to extend its active domains. Should Welsh come to be used in all domains, intergenerational transmission of English may stall, and Welsh Unilingualism become possible.

The authors would like to thank the anonymous reviewers of this paper, whose suggestions considerably improved both its form and content.

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Appendix 1: Pseudocode for the Cellular Automaton

Let A[k]:= be the kth matrix of type A Let A[k][i,j]:= be the entry in the ith row and jth column of matrix A

A[0]:=an m*m matrix:

p:=number of iterations:

for k from 1 to p do

#new matrix for the kth iterationA[k]:=A[k-1];

h:=value of social force: Ts:=value of social temperature:

f:=define a piecewise function on x, f(0)=m, f(0<x<m+1)=x, f(m+1)=1:

for i from 1 to m dofor j from 1 to n do

sigma_A_neighbours:=0:sigma_B_neighbours:=0:

for a in set {A[k-1][i,j],A[k-1][f(i-1),j],A[k-1][f(i+1),j],A[k-1][i,f(j-1)],A[k-1][i,f(j+1)]}do:

if a>0 then sigma_A_neighbours:=sigma_A_neighbours+a:else if a<0 thensigma_B_neighbours:=sigma_B_neighbours+abs(a):end if:sigma_all_neighbours:=abs(A[k-1][i,j])+abs(A[k-1][f(i-1),j])+abs(A[k-1][f(i+1),j])+abs(A[k-1][i,f(j-1)])+abs(A[k-1][i,f(j+1)]):end do:

P_A_to_B:=evalf(((exp(-h/Ts))*(sigma_B_neighbours/sigma_all_neighbours)^(1/Ts))/(((exp(h/Ts))*(sigma_A_neighbours/sigma_all_neighbours)^(1/Ts))+((exp(-h/Ts))*(sigma_B_neighbours/sigma_all_neighbours)^(1/Ts)))):

P_B_to_A:=evalf(((exp(h/Ts))*(sigma_A_neighbours/sigma_all_neighbours)^(1/Ts))/(((exp(h/Ts))*(sigma_A_neighbours/sigma_all_neighbours)^(1/Ts))+((exp(-h/Ts))*(sigma_B_neighbours/sigma_all_neighbours)^(1/Ts)))):

r:=generate a rational number in the range 0..1:

if A[k-1][i,j]>0 and r<P_A_to_Bthen A[k][i,j]:=-A[k-1][i,j]:else if A[k-1][i,j]<0 and r<P_B_to_Athen A[k][i,j]:=-A[k-1][i,j]:end if:

end do:end do:end do:

Appendix 2: Modifications for the General Diglossia ModelThe set of neighbouring cells is no longer subject to the function f(x) permitting periodicity of the surface. Instead, the set is redefined according to whether i=1 or m, j=1 or n e.g.

if i=1 thenneighbours:={A[k-1][i,j],A[k-1][f(i+1),j],A[k-1][i,f(j-1)],A[k-1][i,f(j+1)]}

The functional modification of h_1 is:h_1:=evalf(C*(nA[k-1]-nB[k-1])/(m*n)):

To achieve selective iteration according to the value of i, the iteration criterion may be mechanically changed for higher i. Equivalently, the probability of change might be reduced as a function of i (assuming the probabilities to be drawn from a uniform distribution). For instance, in 1616 cell automata, this is conveniently done by multiplying the probability by 1/i.

To achieve increased connectivity according to increasing j, additional connections are randomly generated as follows, and added onto the sigma_all_neighbours quantifiers:

for a from 1 to trunc(j/c) dox:=generate a non-zero integer in the range 1..m:y:=generate a non-zero integer in the range 1..m:sigma_all_neighbours=sigma_all_neighbours+abs(A[k-1][x,y]):if A[k-1][x,y]>0 thensigma_A_neighbours:=sigma_A_neighbours+A[k-1][x,y]:else if A[k-1][x,y]<0 thensigma_B_neighbours:=sigma_B_neighbours+abs(A[k-1][x,y]):end if:end do:

Appendix 3: Sample Code for the Initial Wales Matrices

Consider the area in the top left corner designated as having 35% Welsh language involvement, comprising cells of 1 i m/4, 1 j n/8. For this area alone, the code is required to generate black and white cells in the proportion 35:65. This part of the code reads as follows.

A[0]:=an m*m matrix:

for i from 1 to m/4 dofor j from 1 to m/8 dos:=generate a non-zero integer in the range 1..100:if s<36 thenA[0][i,j]:=-1:elseA[0][i,j]:=1:end if:end do:end do: