2.1 Fricative devoicing

Child-specific devoicing neutralises the voiced–voiceless distinction in obstruents; for example, the mispronunciation of zęby /zɛmbɨ/ ‘teeth’ as [sɛmpɨ] (Łobacz Reference Łobacz1996: 180). In standard textbooks on speech development, this is one of numerous processes occurring in normally developing Polish-speaking children in the third year of life (e.g. Kaczmarek Reference Kaczmarek1988). According to Sołtys-Chmielowicz (Reference Sołtys-Chmielowicz1998: 136), an impressionistic cross-sectional study based on a sample of more than 1000 children aged 3–7, fewer than 2% of children still devoice obstruents at the age of 3. The process is gradient: the refinement of the earlier acquired surface contrast continues after the age of 3 (Łobacz Reference Łobacz1996: 180–181).

There is some indication in the literature that the acquisition of the surface voicing contrast in fricatives may be later than in stops. According to Kaczmarek (Reference Kaczmarek1988), the voiceless prepalatal /ɕ/ is the only fricative in the repertoire of 2-year-olds. Its voiced counterpart /ʑ/ emerges at the age of 3. Alveolar and postalveolar (retroflex) fricatives, both voiceless and voiced, are acquired at ages 4 and 5 respectively.Footnote ³ In comparison, the acquisition of voicing in stops is complete for labials and alveolars by the age of 2; it is delayed till the age of 3 only for velars. Bryndal (Reference Bryndal2015), in her study of acquisition of the inventory of Polish consonants in children aged 3–7, based on 200 subjects tested with a standardised production questionnaire (cf. Krajna Reference Krajna2008), reports non-normative pronunciations of the voiced sibilants in 3- and 4-year-olds. At age 3, none of the three consonants – alveolar /z/, prepalatal /ʑ/ and postalveolar /ʐ/ – reaches the criterion of 90% of normative realisations; at age 4, only /ʑ/ does (2015: 100–101).Footnote ⁴ The acquisition of voicing proceeds faster in stops: for /b/ and /d/ it is complete at age 3; for /g/ at age 4.

Fricative devoicing is also present in J's longitudinal data, which is analysed in the present study. A gradual development towards adult-like surface voicing in fricatives takes place throughout the period of development under investigation; the production of the voicing contrast in stops is predominantly accurate throughout the reported stages. A detailed description of the corpus of the child's speech samples follows in §3.1.

The above asymmetry in the child-specific devoicing pattern is paralleled by cross-linguistic patterns in adult speech. According to Ladefoged & Maddieson (Reference Ladefoged and Maddieson1996: 176–178), most of the world's languages with voiceless fricatives do not have voiced counterparts. The phonetic grounding for this asymmetry is twofold. Voicing is associated with strong low-frequency energy, which may mask the high-frequency frication noise, which has lower amplitude. The vocal tract settings are such that the impedance of airflow at the glottis increases the difficulty of creating turbulence at the point of articulatory constriction. Although many languages (about 32%) have no voicing contrast in either plosives or fricatives, about 33.4% of languages surveyed in Maddieson (Reference Maddieson, Dryer and Haspelmath2013) have a voicing contrast in plosives but not in fricatives; languages with the opposite pattern, i.e. a voicing contrast in fricatives but not in plosives, are relatively rare – 6.7% of the total.

2.2 Voice Assimilation

The output of the across-the-board process of Voice Assimilation in adult Polish is transparent – uniformly voiceless or voiced obstruent clusters. The process has been widely documented in the standard descriptive sources (e.g. Kowalik et al. Reference Kowalik, Grzegorczykowa, Laskowski and Wróbel1998: 91, Ostaszewska & Tambor Reference Ostaszewska and Tambor2012: 87–89), and extensively analysed within the generative phonological tradition (e.g. Gussmann Reference Gussmann2007, Rubach Reference Rubach2008). The automatic character of the process and its categoricity are widely recognised to be a source of difficulty for Polish learners of English (e.g. Sobkowiak Reference Sobkowiak1996: 50–56). Voice Assimilation effects are also present in loanwords and foreign words (e.g. football /fudbɔl/, Macbeth /magbɛt/; Gussmann Reference Gussmann2007: 291).

Voice Assimilation results in voiced–voiceless alternations both inside words (3a) and across word boundaries (3b). It also has static phonotactic effects (3c). As can be inferred from (a) and (b), Voice Assimilation is regressive: the last obstruent in the cluster determines the surface voicing of the whole cluster. The regressive character of Voice Assimilation is also seen in (3d), where the process interacts with Final Devoicing. Some words, e.g. już in (b), do not have realisations which do not involve Voice Assimilation or Final Devoicing.

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Final Devoicing affects obstruents at the end of words; however, when the word-final obstruent precedes a voiced obstruent at the beginning of the following word, the effects of Final Devoicing are masked by Voice Assimilation; cf. ko/d/ dostępu in (3b). In consequence, Final Devoicing can be inferred only on the basis of voicelessness of obstruents at the end of utterances, or in word-final position when the next word starts with a sonorant.Footnote ⁵ Although the child's data show distributions corroborating the presence of Final Devoicing in her grammar, I ignore this process in further discussion, because it cannot interact with child-specific fricative devoicing. An OT ranking for this process can be acquired as a part of the ranking established for Voice Assimilation, as sketched below.

From a purely distributional perspective, the contrastive [±voiced] specification is found only in obstruents in presonorant position; elsewhere, it is context-dependent. This simple distribution pattern can be learned at early stages, before children have access to the morphological structure of words and internalise underlying representations of morphemes, which is easily modelled using OT formalism. It can be illustrated using the pure phonotactic learner proposed in Hayes (Reference Hayes, Kager, Pater and Zonneveld2004).Footnote ⁶ The algorithm is based on a stringent constraint installation procedure which, among other things, favours markedness and specificity. Consistent voice agreement in the input will result in installing Agree_obstr[±vcd] in the top stratum. Because of the surface contrast in voicing in presonorant contexts, the next level in the hierarchy will be occupied by a faithfulness constraint. The ‘favour specificity’ requirement will ensure that it is a positional constraint, Ident_preson[±vcd]. Because of the surface contrast in voicing in presonorant contexts, the next level in the hierarchy will be occupied by a faithfulness constraint (assuming the meta-ranking Markedness ⪢ Faithfulness). Note here that standard analyses of regressive Voice Assimilation effects in OT assume either string-based (i.e. presonorant) positional faithfulness or syllable-based (i.e. onset) positional faithfulness (see the discussion in Rubach Reference Rubach2008). Assuming a string-based version of positional faithfulness makes the acquisition of Voice Assimilation independent of prior knowledge of syllable structure. The next two layers will be occupied by the general markedness constraint *Obstr[+vcd] and the general faithfulness constraint Ident[±vcd]; the latter constraint ends up lowest in the hierarchy, as it is favoured neither by markedness nor by specificity. The ranking obtained via this procedure is given in (4). The subhierarchy Ident_preson[±vcd] ⪢ *Obstr[+vcd] ⪢ Ident[±vcd] automatically generates Final Devoicing.Footnote ⁷

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In addition to the patterns described above, Polish also has a process of Progressive Devoicing, which an anonymous reviewer raises as a potential issue. However, its highly restricted scope (Gussmann Reference Gussmann2007: 308), together with various other characteristics, make it inconspicuous from the perspective of a child. The process involves only two fricatives, /ʂ/ and /f/, whose voicelessness is caused by the preceding voiceless obstruent, as evidenced by the alternations exemplified in (5a). To account for the /r/ ~ /ʂ/ alternation, an idiosyncratic /r/ to /ʐ/ change is also needed, as exemplified in (5b), which operates independently of Progressive Devoicing.

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At the surface level, these processes do not apply consistently; consider for example /wɔtrɨ/ łotr-y ‘rascal (nom.pl)’ or /dɔtrɛ/ dotrę ‘I will reach’, in which /r/ is followed respectively by /ɨ/ and /ɛ/, and the examples in (5a.i), in which /ʂ/ occurs in exactly the same contexts. It can further be observed that the /r/ ~ /ʐ/ and /r/ ~ /ʂ/ alternations are often accompanied by /ɛ/ ~ zero alternations inside root morphemes (cf. docierać – dotrze in (5a.i) and brać – bierze in (5b)); likewise, the /ɛ/ ~ zero alternation is seen in szewek – szwy in (5a.ii). In such cases, pre-school children have been reported to have difficulty in perceiving the regularity of the connection between different forms of the same morpheme (Łukaszewicz Reference Łukaszewicz2006b: 16–17). Unlike Voice Assimilation, which can be internalised via pure phonotactic learning, Progressive Devoicing requires access to the morphological structure of words. However, words showing these alternations might not be readily available to the child. Not a single instance of the forms łotrzyk, dotrze and krze in (5a.i), which Gussmann (Reference Gussmann2007: 307) adduces as illustration of surface /ʂ/ alternating with /r/, is found in the Polish child-directed speech corpus (Haman et al. Reference Haman, Etenkowski, Łuniewska, Szwabe, Dąbrowska, Szreder and Łaziński2011). (The same problem is found with the form szewek in (5a.ii), and other examples showing the surface /v/ as an alternant of /f/ in Gussmann Reference Gussmann2007.) I conclude that Progressive Devoicing is unlikely to have any relevance at the stages of acquisition reported in this paper.

Another point raised by the anonymous reviewer is the need for acoustic data from adult Polish which would allow us to assess the categoricity of Voice Assimilation and the potential presence of incomplete neutralisation (phonetic gradience of the output reflecting the underlying specification of the target segment; see e.g. Braver Reference Braver2019 for a discussion of the rich literature on the topic). Below I consider acoustic data from adult Polish, which illustrate the effect of Voice Assimilation on the acoustic parameter most reliably connected with the expression of the voicing contrast in adult Polish, voicing percentage (cf. ‘the voicing ratio’ in Strycharczuk Reference Strycharczuk2012: 659).Footnote ⁸ The measurements were conducted using Praat's Voice Report function (Boersma & Weenink Reference Boersma and Weenink2017), with pitch ranges adjusted for adult male vs. female speakers (in all other respects, the measurement details were the same as for the child's data in §3.1). The data come from high-quality recordings conducted for the purpose of a previous acoustic study investigating phonetic correlates of word stress in Polish (Łukaszewicz & Rozborski Reference Łukaszewicz and Rozborski2008). In that study, children's speech was compared with that of their parents. Speech samples from three adult speakers (one of whom was J's caregiver) were collected in structured interviews. Because of the high level of acquaintance between the experimenter and the participants, the recorded samples (lasting about 90 minutes in total) reflect casual spontaneous speech in informal settings. This sort of speech can be expected to be least prone to incomplete neutralisation, whose extent is known to vary, depending on the communicative context (Port & Crawford Reference Port and Crawford1989; see also the discussion in Gafos Reference Gafos, Goldstein, Whalen and Best2006: 54–55). The inclusion of the caregiver's data ensures a good approximation of the input to which the child was exposed.

The voicing measurements were conducted for 488 obstruents (244 clusters). The results for each of two obstruents forming a cluster (O₁O₂) are plotted as coordinates in Fig. 2. The axes represent the voicing percentage; the experimental data points can have values between 0 and 100 on either of the axes. O₁ is the target of the process (the horizontal axis), O₂ is in presonorant position – it acts as the Voice Assimilation trigger (the vertical axis). S, Z, T and D denote voiceless fricatives, voiced fricatives, voiceless stops and voiced stops respectively. As expected, the data points corresponding to the predicted voiceless Voice Assimilation outputs (145 cluster tokens), e.g. ⫽ZT⫽ and ⫽ST⫽ → /ST/, consistently accumulate in the lower left quarter of the diagram, attracted by the (0,0) point. The predicted voiced Voice Assimilation outputs (99 cluster tokens), e.g. ⫽SD⫽ and ⫽ZD⫽ → /ZD/, consistently accumulate in the upper right quarter; there are 40 data points at exactly (100,100). These cannot be read directly from the diagram, because its axes are continuous. In 70 out of 99 such clusters, O₁ (the target of Voice Assimilation) is 100% voiced. The data are divided depending on the underlying specifications of the obstruents in the cluster. The two sets, target voiceless and target voiced, do not overlap.

Figure 2 Voice Assimilation effects (voicing percentages) in adult Polish O₁O₂ clusters, depending on the underlying voicing of O₁ and O₂. Voiceless outputs are grey; voiced outputs black. The two sets do not overlap.

There are no incomplete neutralisation effects in target voiceless clusters. In target voiced clusters, the incomplete neutralisation effect on O₁ (the Voice Assimilation target) is marginal: there are only four data points in which the underlyingly voiceless O₁ has voicing below 80%. These include two clear instances of the suspension of Voice Assimilation, when the underlyingly voiceless O₁ retains its voicelessness instead of assimilating to the underlyingly voiced O₂ (cf. the two data points near the upper left corner). In general, underlying ⫽ZT⫽, ⫽DS⫽ clusters do not tend to have more voicing in O₁ than underlying ⫽ST⫽, ⫽TS⫽ clusters, i.e. they are not shifted towards the lower right quarter of the diagram; underlying ⫽SD⫽, ⫽TZ⫽ clusters do not show less voicing in O₁ than underlying ⫽ZD⫽, ⫽DZ⫽ clusters, i.e. they are not shifted towards the upper left quarter of the diagram. For the vast majority of tokens, then, the assimilation effect in O₁ is complete. In target voiced clusters, a small ‘gravity’ effect can additionally be observed: O₂ (the trigger of assimilation) is sometimes less voiced than the preceding target. We can ascribe this to the biomechanical difficulty of sustaining voicing throughout the entire cluster.Footnote ⁹

2.3 Hypotheses

The two alternative hypotheses of the interaction between the child's immature production skills and adult phonology can be stated as follows. Will the child-specific fricative devoicing and Voice Assimilation interact in accordance with the ‘feedforward’ scenario, producing an output that is illicit from the point of view of the global adult phonotactics predicted by Voice Assimilation, as in (6a)? Or will it reflect the reversed scenario, with the application of Voice Assimilation conditioned by the child-specific pattern, as in (b)? I assume that the voicing distinction is neutralised only in fricatives; stops in contrastive presonorant contexts are realised in the adult-like manner. As shown earlier in Fig. 2 (§2.2), both ⫽SD⫽ and ⫽ZD⫽ surface as /ZD/ in adult Polish, and both ⫽TZ⫽ and ⫽DZ⫽ surface as /DZ/, so we ignore the differences in underlying representations, which are irrelevant from the point of view of the expected interactions in the child's speech. Also, it is important to note that the hypothesised ‘reversed’ scenario in (6b) is not possible in adult Polish (incomplete neutralisation can potentially affect the target of the neutralising process in adult speech, but never the trigger).

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The diagrams in Fig. 3a schematise the hypothesised position of the child's data points corresponding to target /ZD/ and /DZ/ clusters in the above scenarios. Target voiceless clusters (/ST/, /TS/) are expected to consistently appear in the lower left part of Fig. 3b. In the ‘feedforward’ scenario in Fig. 3a.i, target /ZD/ clusters are expected to occupy the upper left quarter of the diagram, and target /DZ/ clusters the lower right quarter. In the ‘reversed’ scenario in Fig. 3a.ii, realisations of target /ZD/ and /DZ/ are not expected to differ: both are anticipated to appear in the lower left region of the diagram, like the /ST/ and /ST/ clusters in Fig. 3b. The diagrams on the lefthand side of Fig. 3a are based on the assumption that the effects are absolute; those on the righthand side (more plausibly) admit variability of these effects, i.e. some realisations of /DZ/ and /ZD/ may be adult-like. In the latter case, some data points will end up in the upper right quarter, both in Fig. 3a.i and Fig. 3a.ii; cf. the diagram for the adult data in Fig. 2. (For such data, we additionally expect some biomechanically motivated ‘gravity’ effect, such as the one reported for parental speech in §2.2 above.)

Figure 3 (a) Hypothesised position of the target voiced clusters: (i) feedforward; (ii) reversed. (b) Hypothesised position of the target voiceless clusters (adult-like).

3.1 The data and acoustic measurements

The corpus of child data used for acoustic analyses consists of 4 hours 40 minutes of audio recordings of naturalistic speech of a normally developing Polish speaking child, interacting with the experimenter during play sessions. The child (J) is female, aged 2;3–2;8 at the time of the recordings and raised in a monolingual environment in which standard Polish (the Warsaw standard) is spoken. The recordings were divided into seven sessions, conveniently covering the span of 21 weeks during which a shift from a complete neutralisation of voicing in fricatives towards a nearly adult-like surface voicing was observed.

The corpus contains rich audio material of quality sufficient to attempt acoustic analyses.Footnote ¹⁰ However, the spontaneous and naturalistic character of the data poses a challenge to acoustic measurement, and requires a rigorous data preselection procedure. The data were rejected for analysis if they were distorted by background sounds (the experimenter's voice, clattering toys, etc.), or if they had low amplitude caused by the child moving away from the microphone. Tokens were also rejected if the obstruent occurred within a larger span of ceased phonation, which sometimes happened towards the end of an utterance.

Some data had to be rejected during segmentation because of interference from certain reduction and substitution patterns. As typical of development between ages 2 and 3, the child's data did not show variability only on the voiced–voiceless dimension. Two phenomena are relevant here: the occasional realisation of fricatives as affricates (e.g. /z/ → [ʥ] in język ‘tongue (nom. sg)’ /jε̃w̃zɨk/ [jɛɲʥik]), and erroneous omission or replacement of sounds. As far as the former issue is concerned, analyses both including and discarding such segments were performed, but this did not have a crucial effect on the outcome (see §3.2 for details). Omission of a sound, resulting in reducing a cluster to a singleton, excluded the token from analysis (e.g. the word gdzie /gʥɛ/ ‘where’ was sometimes rendered as [ʥɛ]). Reduction of obstruent clusters was infrequent at all stages. I also discarded tokens in which an obstruent was replaced by a sonorant or vice versa, e.g. zaśpiewać /zaɕpjɛvaʨ/ [naɕpjɛvaʨ] ‘to sing’, piesek /pjɛsɛk/ [pɕɛsɛk] ‘dog (dim)’ (2;2.30). Such examples occurred only sporadically – with the notable exception of the labiodental /v/, which was replaced by [w] in all contexts. Simultaneously, target /f/ was predominantly rendered as [x], which means that labiodental fricatives were generally absent from J's speech. The focus of the study is thus on clusters containing coronal fricatives. The remaining 1248 obstruent tokens were segmented in Praat; three types of phonetic segments were distinguished: closure, frication and ‘burst’ (positive VOT), as in Fig. 4.

Figure 4 Example of obstruent segmentation: (to) jest drugi (mikrofon) /jεz druɟi/ ‘(it) is another (microphone)’ (J 2;5.10).

As is well-known, phonological contrast in voicing can be expressed in terms of various parameters, depending on the language. The most reliable cue for voicing in adult Polish is the voicing ratio, i.e. glottal pulsing duration divided by closure duration in the case of stops, and by frication duration in the case of fricatives (Strycharczuk Reference Strycharczuk2012: 659). Measurements of a related parameter, voicing percentage (i.e. the percentage of closure/frication during which voicing occurs), were carried out using Praat's Voice Report function, with a pitch range of 200–450 Hz, cross-correlation analysis method and default advanced pitch settings.Footnote ¹¹ The measurements were performed in the sound editor window on the part of the sound corresponding to the earlier annotated closure or frication portion of an obstruent. This allowed for visual inspection of pulses. The outcome of the voicing measurements in fricatives in presonorant position is exemplified in Fig. 5. All three spectrograms represent the target voiced fricative /ʐ/, produced with a different degree of voicing depending on the token: (a) duża /duʐa/ ‘big (fem)’, (b) kucharza /kuxaʐa/ ‘cook (gen.sg)’, (c) dużo /duʐɔ/ ‘a lot’. For stops, I also measured the portion of the speech signal corresponding to the release burst plus any subsequent aspiration, i.e. the positive VOT (see Appendix A3).

Figure 5 Example of voicing measurements in target voiced fricatives, using Praat's Voice Report function: (a) duża /duʐa/ ‘big (fem.nom.sg)’ (J 2;2.30; 23% voicing); (b) kucharza /kuxaʐa/ ‘cook (gen.sg)’ (J 2;5.10; 60% voicing); (c) dużo /duʐɔ/ ‘a lot’ (J 2;7.19; 100% voicing).

Voicing in fricatives and stops was measured independently in singletons in contrastive (presonorant) non-assimilatory contexts and in obstruent clusters (Voice Assimilation contexts); the measurement data are available in the online supplementary materials.Footnote ¹² Most target voiced obstruent clusters come from external sandhi contexts, and consist of a sequence of a fricative and a stop, as exemplified in (7a). Among these are tokens containing the word jest ‘is’, in which the final t was omitted (as in casual adult speech). There were also a number of tokens in which the fricative was preceded by a stop (7b), or where either word-internal sandhi (c) or static phonotactics (d) were involved. These were also included, because they were expected to exhibit the same voiced ~ voiceless variation pattern. In the analyses of clusters, I excluded some tokens with fricative–fricative sequences (e) and stop–stop sequences (f), as the lack of clear transitions in such sequences often made their acoustic segmentation infeasible. Both (e) and (f) were rare. Such sequences cannot provide a compelling argument for the ‘feedforward’ vs. ‘reversed’ interaction between child-specific fricative devoicing and Voice Assimilation (although we do expect more target-appropriate voicing in target [+voiced] stop–stop sequences than in target [+voiced] fricative–fricative sequences, because of the presence of fricative devoicing but no stop devoicing in the child's system).

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4.1 Phonological grammar as a dynamical landscape: basic notions

Dynamical systems can be viewed as attractor landscapes, formally expressed through non-linear mathematics (for an overview see Thelen & Smith Reference Thelen and Smith1994, especially Chapter 3). Attractors are behavioural modes, which the system develops (via self-organisation) as a function of the interaction of its internal components and their responsiveness to external conditions. Dynamical principles, either stated explicitly in terms of differential equations or only alluded to at the level of metaphor, have been widely applied in natural sciences (for a mathematical introduction, with emphasis on applications, see Strogatz Reference Strogatz1994). A wealth of illustrative examples, ranging from modelling the Belousov–Zhabotinsky oscillatory reaction in chemistry to applications in motor and cognitive development research (for example on the ontogeny of treadmill stepping in infants or the puzzle of the Piagetian A-not-B error), can be found in Thelen & Smith (Reference Thelen and Smith1994). Of interest here is the possibility of conceptualising the relationship between phonological organisation and phonetic parameters using these principles. The symbolic-dynamical representations described below are in terms of first-order differential equations, as formalised in Gafos (Reference Gafos, Goldstein, Whalen and Best2006). The mathematical formalism of dynamical systems has been used in various ways to model phonological–phonetic phenomena in adult speech: articulatory gestures defined as invariant mathematical laws (Kelso et al. Reference Kelso, Saltzman and Tuller1986, Browman & Goldstein Reference Browman, Goldstein, Port and van Gelder1995), (in)stability of speech perception (Tuller et al. Reference Tuller, Case, Ding and Scott Kelso1994), incomplete neutralisation (Gafos Reference Gafos, Goldstein, Whalen and Best2006, Gafos & Benus Reference Gafos and Benus2006), shifts in phonetic indices dependent on qualitative differences in syllable parses (Gafos et al. Reference Gafos, Charlow, Shaw and Hoole2014) and perception-induced updates in speech production planning (Roon & Gafos Reference Roon and Gafos2016), to mention just a few examples. For a non-mathematical treatment of the shift in phonological organisation from babbling to first words, espousing the dynamical systems approach, see Vihman et al. (Reference Vihman, DePaolis, Keren-Portnoy, Bavin and Naigles2015).

As illustrated in (8), the behaviour of a dynamical system can be pictured as a particle landing in potential wells. The position of the particle is the system's state x at some time t. The wells reflect the system's preferred modes, the so-called ‘attractors’; the relative depth and steepness of the wells reflects the attractors’ relative stability. The peaks are unstable points (repellers): the system is unlikely to reside in these positions.

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Attractors are the states towards which the system tends, and thus correspond to mean values of parameters in experimental data. How close x is with respect to those target values at time t depends on its initial position and its trajectory, determined by a differential equation relating dx/dt (the instantaneous velocity of x) to the so-called force function referring to the attractor's slope (for clarification see below). It also depends on the amount of noise by which the system is perturbed. (Internal fluctuations are a consequence of interaction of multiple subcomponents of a complex system.) A relatively stable attractor is less prone to change; (8c) is an intuitive illustration. When perturbed to a relevant degree by internal noise or external forces, the particle is likely to visit the deeper well of the attractor to the left instead of dropping back into the shallow well of the attractor to the right. Globally, i.e. when starting from different initial positions, the system is more likely to reside in the former attractor than in the latter. The measure of relative variability around a mean state is a powerful tool in assessing the degree of the attractor's stability. Thus, a major progress offered by the dynamical approach is that it allows us to treat a certain kind of variation in experimental data as information, not noise; cf. the RSDs for target [+voiced] fricatives vs. stops in Fig. 11 above, indicative of unstable vs. stable categories.

Characteristically, dynamical systems produce coherent patterns only within a certain range of values of their control parameters. As the control parameter is scaled continuously, it passes through a critical value at which a qualitative transition in the attractor landscape (so-called bifurcation) occurs. Under different conditions, the components of a complex dynamical system are free to reassemble into different stable modes. It is this ‘soft assembling’ property to which dynamical systems owe their enormous flexibility; it is also why they come with the potential to model developmental data in an insightful way (Thelen & Smith Reference Thelen and Smith1994: 60). One of the primary research goals is then to single out factors as potential control parameters that affect the evolution of a dynamical system.

The geometric characterisation of the dynamical landscape in (8) allows us to express the notion of phonological contrast in an intuitively straightforward way. The attractors reflect the macroscopic observable, i.e. the phonological organisation of the system. Given some continuous physical parameter (e.g. the extent of vocal fold vibration), the absence vs. presence of a binary phonological contrast (e.g. a voiced–voiceless distinction) corresponds to a single-attractor vs. two-attractor system, as in (8a) and (b) respectively. The system's state x at time t corresponds to measurable phonetic output. In the dynamical view of the phonology–phonetics link, phonology is not antecedent to phonetics, as no translation from discrete symbolic to physical continuous is necessary. The continuous (phonetic) and discrete (phonological) properties of speech are inseparable, yet distinguishable. The qualitative and quantitative aspects of the system are both expressed with non-linear mathematics (cf. Browman & Goldstein Reference Browman, Goldstein, Port and van Gelder1995, Gafos Reference Gafos, Goldstein, Whalen and Best2006).

The schematic illustrations of different dynamical systems in (8) are based on polynomials of different degrees.Footnote ¹⁴ These are potential functions, V(x), defined over some order parameters (e.g. the degree of voicing). Their minima correspond to attractors, i.e. stable equilibrium points at which a particle will eventually land (assuming deterministic dynamical systems), starting from different initial positions. Their maxima are repellers, i.e. unstable equilibrium points. Both attractors and repellers are equilibria in the sense that the rate of change is zero at these points, i.e. dV(x)/dx = 0. The number and depth of attractors depend on the degree of the polynomial as well as on the value of its control parameters, which will be discussed below with reference to grammatical V^G(x) and lexical (intentional) potentials V^I(x), as defined in Gafos (Reference Gafos, Goldstein, Whalen and Best2006). Before we turn to exploring the role of these potentials in shaping the attractor landscape in child vs. adult phonologies, let us consider a basic characterisation of stability in mathematical terms.

The essence of symbolic-dynamical representations is that they are assembled in time and are understood as mathematically defined laws. They are defined in terms of differential equations, which have the general form dx/dt = ―dV(x)/dx. (For simplicity, I ignore the noise component which occurs on the righthand side of the equation and introduces indeterminacy in the system's behaviour; see Gafos & Benus Reference Gafos and Benus2006: 908–909.) The lefthand side of the equation is the rate of change of x (with respect to t). The righthand side of the equation corresponds to the force function, which is the negative of the first derivative of the potential function V(x). The mathematical relationship between the potential, V(x), and the force function, F(x) = ―dV(x)/dx, reflects a well-established relationship for oscillating systems in physics. It is the relationship between the potential energy and the restoring force which acts on an oscillating system to bring it back to its equilibrium. The restoring force depends on the degree of displacement – linearly in harmonic, non-linearly in anharmonic systems. Its direction is opposite to the direction of displacement, hence the minus sign in ―dV(x)/dx. First-order dynamical equations, dx/dt = ―dV(x)/dx, express the evolution of the system in time. It is important to note that they have equilibria solutions, not periodic solutions (see Strogatz Reference Strogatz1994: 28). The solution to the equation is a function of x with respect to t, or, to be more precise, an infinite set of functions for an infinitude of different initial conditions (x ₀). The solutions are trajectories which converge towards the stable fixed point(s) and diverge from the unstable fixed point(s).

Convergence towards the stable fixed points (attractors) is illustrated in Fig. 12 on the basis of relatively simple single-attractor dynamics: dx/dt =―dV(x)/dx = ―2x and dx/dt = ―dV(x)/dx = ―½x respectively. As we will see later, single-attractor dynamics generalised as dx/dt = ―dV(x)/dx = ―αx, with α > 0, can be considered to be a dynamical equivalent of lexical representations, because they specify the target value of a phonological feature unambiguously, regardless of the shifts in the system's control parameter α. The solutions to these differential equations, which are x(t) = x ₀e ^―2t and x(t) = x ₀e ^―½t respectively, are plotted in the (iii) panels; in both plots the system's initial states are assumed to take the arbitrarily selected values x ₀ = {―20, ―15, ―10, ―5, 0, 5, 10, 15, 20}.Footnote ¹⁵ We can readily observe how the differences in the attractors’ stability (steepness of the wells) in the (i) panels in Fig. 12 correspond to the differences in slopes (the force functions) in (ii), and ultimately, to the degree of variability in the output data at a given time point in (iii). For example, if we assume that the decision time for assembling representations during the speech planning process is t = 1 (indicated by the arrow), this renders output x_t values in Fig. 12a much less variable relative to Fig. 12b.

Figure 12 (a) A more stable attractor; (b) a less stable attractor.

Differences in stability are one of several expected differences between the adult and child phonological systems, making the latter more liable to change. The systems in Fig. 12 are nevertheless qualitatively the same, i.e. their trajectories converge towards the same stable fixed point.

As proposed in Gafos (Reference Gafos, Goldstein, Whalen and Best2006: 62), a two-way surface phonological contrast (e.g. in voicing) can be represented in terms of the quartic function in Fig. 13, V^G(x) = kx − ½x ² + ¼x ⁴, where k stands for a control parameter. (The quartic degree is the lowest polynomial degree allowing us to have two minima and one maximum.) The dynamics of phonological grammar are defined in terms of the differential equation: G(x) = dx/dt =―dV(x)/dx = ―k + x − x ³. The G(x) formula is referred to as the ‘tilted anharmonic oscillator’ in Gafos (Reference Gafos, Goldstein, Whalen and Best2006: 66), and proposed as a first approximation of the grammar dynamics.Footnote ¹⁶ (See Tuller et al. Reference Tuller, Case, Ding and Scott Kelso1994: 7 for an earlier application of the same formula in the domain of speech perception.) By scaling the parameter k, we obtain a qualitative shift (bifurcation) between a single-attractor system and a two-attractor system, which corresponds to phonological grammars characterised by the absence vs. presence of binary contrast, as in Fig. 13. The bistable organisation emerges only at some critical (―k_c) value of k, and is sustained only within a certain range of k ∈ (―k_c, k_c). The bistable grammatical potential is symmetrical when k = 0: the chances of a particle landing in either of the two attractors is then equal, as in Fig. 13a. I take this symmetrical potential to represent the adult-like voiced–voiceless distinction.

Figure 13 Grammatical potentials: V ^G (x) = kx−1/2x ² + 1/4x ⁴.

Expanding Gafos’ hypothesis to the domain of acquisition, there can be a twofold difference between the child's system and the corresponding adult system from the point of view of the grammar dynamics. First, there may be only one attractor in the child's system, resulting in absolute contrast neutralisation, as in Fig. 13b; cf. the behaviour of target voiced fricatives at the initial stage reported in this paper, when they are not distinguished from target voiceless fricatives in J's outputs. (The one-attractor system in (b) can also represent contextual voicing neutralisation in adult speech, e.g. the cross-linguistically common patterns of coda devoicing, as described in Gafos Reference Gafos, Goldstein, Whalen and Best2006.) Second, there may be two attractors in the child's system, but their strength may be unequal, as illustrated in (c). With the latter representation, variable contrast neutralisation ensues; cf. the unstable behaviour of target voiced fricatives at some stages of J's development reported in this paper. The depth of the well of the ‘voiced’ attractor gets bigger as the k parameter is scaled up from the critical value ―k_c towards 0 (to ultimately match the stability of the ‘voiceless’ attractor at k = 0). Thus, given the three different landscapes in Fig. 13, the predicted order of acquisition is: (b) (initial stage), (c) (intermediate stage), (a) (adult-like stage).

With the grammatical potential supplying two competing attractors for binary contrast (voiced–voiceless in presonorant position), we need a mechanism to ensure that the feature is realised unambiguously in lexical items. In Gafos (Reference Gafos, Goldstein, Whalen and Best2006), lexical features are expressed in terms of intention dynamics, I(x) = dx/dt = α(x^REQ − x), supplying attractors at the required voicing values (x^REQ); α is a control parameter and expresses the strength of the intention. Accordingly, in Fig. 14 I assume (arbitrarily) that the values x^REQ = ―1 and x^REQ = 1 correspond to the categories ‘voiced’ and ‘voiceless’ respectively. The intention parameter in the illustration in Fig. 14 is assumed to take the value of α = 1. The ‘voiced’ vs. ‘voiceless’ intentional dynamics are mutually exclusive.

Figure 14 Intentional (lexical) potentials: V ^I(x) = ɑ(1/2x ² − x ^REQ × x).

In what follows, I assume that every output can be viewed as a combined effect of the grammar dynamics and the intention dynamics, as in (9). For simplicity, the weighting of the grammar dynamics will be kept constant (w_G = 1). This idealisation reflects the expected early acquisition of contrasts and phonotactics (recall the discussion in §1 and §2). The relative weighting will be achieved by allowing the control parameter α = w_I of the intention dynamics to assume different values within the range from 0 to 1 (cf. Gafos Reference Gafos, Goldstein, Whalen and Best2006: 70).

1. (9)

   

Let us first consider a straightforward case of a fully symmetrical voiced–voiceless contrast in adult speech. In the dynamical interpretation, grammar predicts a voiced–voiceless alternation in presonorant contexts, which needs to be lexically disambiguated. The control parameter in the grammar dynamics is k = 0 (as in (Fig. 13a) above). The values x^REQ = {―1, 1} correspond to voiced and voiceless categories respectively. If the weights of both the grammar and intention dynamics are set to w_G = w_I = 1, we find unambiguously voiced or voiceless outputs, because the resultant force function ―dV(x)/dx = G(x) + I(x) = ―x ³ + x^REQ assumes the value of zero only at one point: x^REQ = ―1 for the potential in Fig. 15a vs. x^REQ = 1 for the potential in (b). Note that this function does not have a linear term, which precludes the possibility of having two attractors in the corresponding potential function.

Figure 15 Adult surface contrast as a consequence of combining Grammar and Intention. The solid black lines represent the combined potentials.

4.2 Contrast emergence in the dynamical landscape

The relative weighting of the grammar and intention dynamics may be different in the child and the adult. Given the order of acquisition, intention may be expected to be weighted lower in children. If the grammar potential is symmetrical (i.e. k = 0), and the weight of intention is as low as 0.3, monostable combined potentials still result, as depicted in Fig. 16. Such a system will produce target-appropriate voiced and voiceless fricatives, although the distinction between the two categories will be much less stable than in the adult system in Fig. 15 above. Nevertheless, the systems in Figs 15 and 16 are qualitatively the same: they are characterised by the same attractors, x^REQ = {―1, 1}.

Figure 16 Weaker (less stable) surface contrast due to weaker Intention (w _I = ɑ = 0.3).

The mirror images of the voiced and voiceless categories in Fig. 16 are inadequate from the point of view of the asymmetric behaviour of target voiced and voiceless fricatives for most of the stages reported in this paper. The initial bias towards monostable voiceless realisations and the subsequent emergence of bistability (variation) in the target voiced fricatives (and obstruent clusters containing these segments) must follow from changes in the control parameter k. As the parameter gradually shifts from negative values upward towards zero, which reflects the child's growing capacity to produce voicing in fricatives, the landscape changes from asymmetric to symmetric, passing through a critical point where a qualitative shift from a monostable (single-attractor) to a bistable (two-attractor) system occurs. In Fig. 17, I illustrate predicted successive stages of development of the surface voicing contrast in fricatives, assuming k = {―0.5, ―0.3, ―0.1}. The weighting of the intention dynamics is also assumed to shift, reflecting the expected gradual refinement of lexical representations (hypothetically proportional to the development of production skills): w_I = {0.15, 0.2, 0.25}; see Appendix C for a summary of these formulas. For simplicity, we model development only at three points: contrast neutralisation (Fig. 17a), variable contrast with weak stability (b) and still variable but more stable contrast (c). The changes in the parameter k crucially influence the modes of stability in target voiced fricatives (the (i) panels), but do not have a major effect on target voiceless fricatives (the (ii) panels). For the latter, the pattern is monostable throughout the reported stages. (The scale is the same in all graphs.)

Figure 17 (a) No surface contrast: target voiced fricatives are realised as voiceless. (b) Loss of stability in the child-specific fricative devoicing pattern: targetappropriate and voiceless renditions of target voiced fricatives concur. (c) Target voiced fricatives are realised in a nearly adult-like manner.

Fig. 17a illustrates the stage at which the voicing contrast in fricatives is neutralised completely in the child's speech. Despite differences in intention (cf. the parabolas drawn as dashed lines), target voiced (i) vs. target voiceless (ii) outputs are not rendered contrastively: both are characterised by a single ‘voiceless’ attractor. The grammatical component (k = ―0.5), which supplies a single ‘voiceless’ attractor, is too strong to be outweighed by the contradictory intention (w_I = 0.15) supplying a weaker ‘voiced’ attractor in (a.i). Hence there is no qualitative difference between (a.i) and (a.ii).

Fig. 17b depicts the stage at which target voiced fricatives are produced variably as voiced or voiceless in the child's speech. The parameter k assumes a slightly higher value in comparison with the previous stage (k = ―0.3). The linear combination of the grammar and the ‘voiced’ intention allows a second attractor to emerge in the combined potential (b.i). However, the value of k is still too low to yield a monostable ‘voiced’ pattern. With the same grammar but the opposite, i.e. ‘voiceless’, intention, we get the monostable ‘voiceless’ potential in (b.ii).

In Fig. 17c, a further shift in the parameter k towards 0 (k = ―0.1) produces nearly mirror-image landscape patterns for target voiced (c.i) and voiceless fricatives (c.ii); cf. Fig. 16 above. In (c.i), the voiced attractor is now much more stable; the competing voiceless attractor is almost negligible. Voiceless fricatives, as in previous stages, are represented by a consistently monostable ‘voiceless’ potential.

The dynamical account of gradient contrast neutralisation in child speech relies on a trade-off between grammar and intention specifying contradictory attractors for target voiced segments. The same mechanism has been applied to account for incomplete neutralisation in adult speech (Gafos Reference Gafos, Goldstein, Whalen and Best2006). However, the role of grammar vs. intention is different in these two accounts. In adult speech, the gradient surface effect is caused by an increase in the intention weight (the lexical factor), while grammar is kept constant. In child speech, grammar is affected by the shift in the control parameter k, which reflects development of a phonetic skill.

I now turn to the question of how combining the grammar and intention dynamics predicts Voice Assimilation ‘late repairs’ in target voiced obstruent clusters containing fricatives in the child's system. In regressive Voice Assimilation, the voicing specification of the trigger (O₂) is anticipated in the target of assimilation (O₁). The simplest way in which this effect can be formalised within the dynamical systems approach is in terms of a linear combination of the grammatical potential for the entire cluster (with the k parameter reflecting the current capacity of producing voicing in fricatives) and the lexical (monostable) potential of the trigger (O₂). Assembling voicing for the entire cluster, rather than for each segment separately, mimics the effect of the top-ranked Agree_obstr[±vcd] constraint in OT. On this account, the predicted development of target voiced obstruent clusters fully parallels the development of target voiced fricatives, as illustrated in Fig. 17. This is because the intention dynamics in Fig. 17 correspond to underlying voicing specifications for an obstruent in presonorant position, which is the contrastive context, regardless of whether this obstruent is a singleton or part of a cluster, whether it is a stop or a fricative. For example, in spytać /spɨtaʨ/ ‘to ask (pfv)’ vs. zbadać /zbadaʨ/ ‘to examine (pfv)’ (but also in pytać /pɨtaʨ/ ‘to ask’ vs. badać /badaʨ/ ‘to examine’, as well as in szyć /ʂɨʨ/ ‘to sew’ vs. żyć /ʐɨʨ/ ‘to live’), the voicing contrast in /p/ vs. /b/ (or /ʂ/ vs. /ʐ/) in presonorant position will be the basis for the voiceless vs. voiced intention I(x). Combining it with the grammar dynamics G(x) (whose attractor landscape is influenced by the control parameter k) will eventually determine the surface voicing of the entire cluster. In consequence, the same dynamics pertain to target voiced fricatives in non-assimilatory contrastive contexts and in clusters because in both the intention I(x) is voiced, and the grammar G(x) has the same tilt defined by the current value of the control parameter k (the capacity of combining voicing with frication). (This account ignores the lower voicing values for O₂, which were ascribed to biomechanical effects in §2.2 and §3.2 above.)