The basic reproductive ratio as a link between acquisition and change in phonotactics

Language acquisition and change are thought to be causally connected. We demonstrate a 1 method for quantifying the strength of this connection in terms of the ‘basic reproductive 2 ratio’ of linguistic constituents. It represents a standardized measure of reproductive success, 3 which can be derived both from diachronic and from acquisition data. By analyzing English 4 data, we show that the results of both types of derivation correlate, so that phonotactic 5 acquisition indeed predicts phonotactic change, and vice versa . After drawing that general 6 conclusion, we discuss the role of utterance frequency and show that the latter only exhibits 7 destabilizing effects on late acquired items, which belong to phonotactic periphery. We 8 conclude that – at least in the evolution of English phonotactics – acquisition serves 9 conservation, while innovation is more likely to occur in adult speech and affects items that 10 are less entrenched but comparably frequent. 11 and morphological productivity.


Introduction 21
Languages are systems of mental instructions that are shared by their speakers. They are 22 instantiated in the mind-brains of many individuals and transmitted across generations 23 through communicative interaction and language acquisition. For a constituent of linguistic 24 knowledge to be successfully transmitted across generations, it needs to be used and 25 expressed by adult speakers in such a way that new generations can acquire it successfully. 26 Thus, the history of language constituents depends on language use and language acquisition 27 and is likely to reflect constraints on both of them. This paper focusses on the relation 28 between history and acquisition. 29 That language acquisition is crucial for language history is trivially true and generally 30 acknowledged (Briscoe, 2008;Smith & Kirby, 2008). After all, constituents that are not 31 acquired cannot survive. However, the matter is both more complex and more interesting 32 than that. On the one hand, there is considerable disagreement about how much language 33 acquisition contributes to linguistic change, and on the other hand, some correlations between 34 acquisition and diachronic stability appear to be quite specific. For instance, Monaghan 35 7 centrally in epidemiological research due to its straightforward properties: whenever it holds 123 for a population (e.g. a subpopulation of infected individuals) that , that population 124 increases in size and spreads. 125 In our model, entails that the population of users approaches a stable 126 equilibrium , so that . If, on the other hand, , the 127 fraction of users approaches 0. The linguistic item vanishes. 128 represents a standardized measure of reproductive success that reflects the 129 diachronic stability of linguistic items. Its greatest asset is that it can be derived from 130 different types of data and that all derived estimates are situated on the same scale. Thus, 131 estimates derived from different data types can be compared directly and without further 132 transformation. In our paper, we exploit this for comparing the derived from diachronic 133 frequency data to the derived from language-acquisition data. We show that such a 134 comparison yields interesting perspectives on the relation between age of acquisition and 135 historical stability. 136

Estimating reproductive success from diachronic growth 137
The model of linguistic spread outlined in the previous section can be reformulated in terms 138 of a logistic equation (Hethcote, 1989 In order to estimate the intrinsic growth rates of final CC clusters, we use logistic 145 growth rates obtained from diachronic frequency data as a proxy (see also the discussion 146 in section 5). For that purpose, we determine a trajectory of normalized token frequencies f 147 from 1150 to 2012 for each word-final CC cluster. The token frequencies were retrieved from 148 various historical and contemporary language databases and corpora (see Table 1, which also 149 indicates who carried out the phonological interpretation We chose 1150 to 2012 as our observation period because word final CC clusters 161 were rare before (i.e. in Old English). The vast majority of them was only first produced by 162 schwa loss in final syllables, which started roughly at this time (Minkova, 1991). Note that 163 although the phonological process of schwa loss affected word final sequences quite 164 uniformly in the early Middle English period, the different cluster types it produced 165 developed relatively independently of each other after schwa loss was completed (in the 15 th 166 century). This reflects the post-medieval influx of loans ending in CC clusters as well as 167 phonological processes other than schwa lossfor instance final devoicingthat produced 168 new clusters. For most of the observation period the dynamics of the individual cluster types 169 can thus be considered as relatively independent from each other. 170 The derived trajectories were normalized to the unit interval with respect to their 171 maximum values, and subsequently fit to a logistic model given by 172 , where was set at the middle of the observation period. Non-linear 173 least-squares regression was used to estimate for each cluster. The quality of this estimate 174 depends on the actual shape of the empirical trajectory. Since the model presupposes 175 (positively or negatively) unidirectional development, estimates can be unreliable for 176 clusters who show (inverse) U-shaped developments. Therefore, we also computed 177 Spearman's Rho ( ) for each cluster. We excluded clusters for which scored below the 178 threshold of 0.1, to rule out clearly non-monotonous developments. 3 This also eliminated 179 clusters that occurred only sporadically in a few periods. Finally, we did not consider final 180 10 cluster types that are absent in Present Day English such as /mb/ in limb because there are no 181 data on the age at which they are acquired. Thus, a total of 58 final CC types entered our 182 analysis ( Table A1 in  show a non-trivial monotonous development (decreasing or increasing). The graphs were 190 selected in order to represent a large variety of diachronic patterns. In some cases (e.g. /sk/, 191 /ts/, /sk/) trajectories fit the logistic pattern remarkably well. In other cases (e.g. /rn/, /fs/, /sp/) 192 they don't. Some clusters feature extremely low frequencies in early periods. should increase or decrease the diachronic stability of CC clusters. 233 5 This operationalization of AoA is most compatible with the underlying population dynamical model. We found that the exact operationalization of AoA is crucial to the comparison of the two derived estimates. AoA ratings for clusters that are derived from the AoAs of all words containing it get implausibly high because some of those words are inevitably acquired extremely late and unlikely to play any role in the acquisition of a cluster.
In order to investigate that issue, our study takes frequency into consideration as an 234 additional factor. Since cluster-specific utterance frequencies fluctuate during the observation 235 period, we first extracted per million normalized token frequencies for all cluster types in 236 every single period of 50 years. In addition, we computed average token frequencies for each 237 cluster type across all 18 periods, denoted as in order to obtain a more compact 238 summary measure (see Table A1 in the appendix).  The direct comparison of the two estimates of (model 1a, Fig. 2)  Model 2 (Fig. 3a, right) reveals that the relationship between and , 285 established in model 1, is much tighter for frequent clusters (e.g. /ns/ as in hence vs. /st/ as in 286 best) than for infrequent ones, where it is approximately constant (/rp/ as in harp vs. /lk/ as in 287 milk; interaction term: , , ). Another way of looking at Fig. 3a  288 is this: in the phonotactic core inventory (i.e. among early acquired clusters), frequency does 289 not affect diachronic stability, while in the phonotactic periphery (among late acquired 290 clusters), frequency reduces it significantly (Fig 3a, left). 291 In model 3a (Fig. 3b ). Frequent CC clusters are acquired significantly earlier than rare ones. Model 4a 297 ( Fig. 3c) shows that frequency and were inversely related in the beginning of the 298 observation period but not during more recent periods. The relationship between frequency 299 and (model 4b, Fig. 3c) was slightly negative in the early part of the observation period 300 but evolved towards a strongly positive interaction later on (interaction term: , 301 , ). 302 303 8 Model 3a was additionally fit to all clusters with ('core' items) and ('periphery' items), respectively, in order to make the effect of frequency more clearly visible. Core items: smooth term at , , ( , explained deviance). Periphery items: significantly decreasing smooth term at , , ( , explained deviance). The correlation between the estimates derived from acquisition data and diachronic 327 evidence supports the widely shared view that age of acquisition and diachronic stability are 328 causally linked. Concurring with Monaghan (2014), our study suggests that what is acquired 329 early is diachronically more stable (and vice versa). Interestingly, however, the tightness of 330 this relationship increases with the frequency of CC clusters. This means that frequent 331 9 Defined as the expected number of learners that acquire an item from a single user. clusters are not simply acquired before rare ones, but that the historical stability of a cluster 332 can be more confidently predicted from the age at which it is acquired when that cluster is 333 frequent. Among rare clusters the correlation is not as tight. At the same time, these results 334 show that late acquired items from the phonotactic periphery suffer most from frequency 335 driven effects such as assimilation, reduction, or deletion. In that respect, they differ strongly 336 from early acquiredand highly entrenchedcore items. Thus, the notion than utterance 337 frequency reduces historical stability still applies (e.g. via erosion in adult speech; Bybee, 338 2007), but we have demonstrated it to be restricted to the periphery. 339 The correlation between frequency and R 0 estimated from AoA is not surprising. It 340 reflects the way in which the (linguistic version of the) basic reproductive ratio is derived. 341 According to Nowak (2000), find that their stability is positively correlated with frequency. The divergence between their 365 result and ours is noteworthy. We suspect that it reflects that the frequencies employed in 366 Lieberman et al. (2007) were derived from contemporary data (CELEX) rather than 367 historically layered sources: in the slice representing most recent periods in Figure 2b (right), 368 a negative interaction between stability and frequency is not visible either. We think that 369 averaged frequencies, which cover the entire observation period, provide a more robust 370 picture. 10 371 Alternatively, there might be fundamental differences between phonotactics and the 372 lexical domain. In the sublexical domain, the destabilizing effect of frequency might be 373 stronger than in the lexical domain, because for the recognition of lexical items listeners can 374 rely on the syntactic, semantic and pragmatic context, and may therefore recognize them even 375 in phonetically reduced forms (Ernestus, 2014). In this regard, cluster perception is supported 376 at best by morphological cues and benefits much less from linguistic redundancy. Therefore, 377 weakly entrenched phonotactic items may be more vulnerable to the destabilizing effects of 378 frequency than weakly entrenched lexical items. 379 In summary, it appears that linguistic entrenchment is a function of both age of 380 acquisition and frequency rather than just the latter (Ellis, 2012; Schmid, 2016). If we 381 operationalize entrenchment by means of diachronic stability (because of the conserving 382 function of routinization) then our analysis suggests that the relative age at which an items is 383 acquired plays a key role in linguistic entrenchment. One straightforward mechanistic 384 explanation is this: an item that happens to be acquired early has more time for being 385 routinized than an item that is acquired late. Crucially, this holds irrespectively of how 386 frequent an item is. Another mechanism discussed by Monaghan (2014: 533), applies to the 387 lexical domain and involves higher plasticity of the cognitive system at early ages. Lexical 388 items that are acquired early (for whatever reason) are more easily entrenched because the 389 cognitive system is still more flexible. This, then, should also apply to complex processes of 390 cognitive planning, articulation and perception relevant in the sublexical domain (Cholin,  391 Dell, & Levelt, 2011;Levelt & Wheeldon, 1994). 11 392 Finally, the comparison between the reproductive ratios derived from our two data 393 sets, sheds light on the question how much acquisition contributes to language change. To see 394 this, note that the ratios derived from AoA data are considerably larger than the ones derived 395 from diachronic data (Fig. 2, boxplots). While that difference may partly be an artefact of our 396 method 12 , it may also be revealing. Thus, it might plausibly be interpreted as reflecting the 397 different contributions which first-language learners and proficient speakers make to the 398 actuation of linguistic change (Bybee, 2010;Croft, 2000). Since age-of-acquisition data 399 predict greater diachronic stability than is derivable from actual diachronic evidence, this 400 potentially suggests that language use by adults may play a more important role in causing 401 linguistic innovation than language acquisition by new generations of children (Diessel, 402 2012). Of course, further research is still needed to corroborate this suspicion, but the 403 methods we have demonstrated in this paper may help to make the question addressable in 404 quantitative terms. 405 12 To some extent, the difference may reflect the way in which has been estimated, because linguistic tokens and speakers represent two different dimensions in the first place. We suppose our token-frequency based proxy to represent a lower bound for the intrinsic growth rate in the population-dynamical model. This is because the spread of an item in a population of tokens involves both its spread through a population of speakers (i.e. ), and its spread through the linguistic system and the lexicon (Kroch (1989) (2012)). The two dimensions are hard to disentangle on the basis of the limited number of historical texts available. Only quantitative empirical and computational approaches that incorporate both dimensions can shed more light on this issue.
As to , one possible reason why it might be overestimated is that our measure of AoA is based on lexical acquisition. Of course, the first form of a word that a child uses may not be the one containing the relevant cluster, nor will a child's first productions of what is a cluster in the target form always be accurate. Moreover, considering only AoA for estimating neglects the possibility that clusters, once acquired, may disappear again in adult speechnot only through language attrition and articulatory loss (see Seliger and Vago (1991); Ballard, Robin, Woodworth, and Zimba (2001); Torre and Barlow (2009)), but also through natural phonological backgrounding and deletion processes. If the proportion of individuals abandoning a particular cluster is underestimated, this will result in being overestimated.

Outlook 406
Although our case study has been restricted to a very specific set of phonotactic constituents 407 and to a single language, namely English, there is no a priori reason why our approach 408 should not work in other domains (e.g. modeling the spread of single phonemes or words), 409 and for other languages. The two operationalizations of , however, require (a) diachronic 410 data that cover the complete histories of constituents (ideally from the period of their first 411 emergence), as well as (b) corresponding acquisition data. As so often, English enjoys a 412 privileged status in this regard. A large number of historical sources have been digitized, and 413 also research on acquisition has produced a large amount of data. Testing the methods 414 described in this study against other languages is likely to face difficulties, although it would 415 of course be important. At least on the lexical level, however, the prospects are not so bad. 416 For core-vocabulary items in 25 languages a set of AoA ratings has been compiled by 417