Modelling the developmental path in the acquisition of /l/-final irregular plurals by Portuguese children

ABSTRACT

Previous empirical research has shown that Portuguese children aged 4;0 to 6;0 are sensitive to the quality of stem-final vowels when acquiring the irregular plural forms of /l/-final words (acquisition order: plurals of /al, ɛl, ɔl, ul/ > plurals of /il/). This study presents a formal account of this emergent pattern. We first construct an initial-state grammar that arguably instantiates Portuguese children’s grammatical knowledge at the onset of morpho-phonological acquisition. We then simulate morpho-phonological learning using two constraint-based models (Stochastic Optimality Theory and Noisy Harmonic Grammar) and their associated Gradual Learning Algorithm. The results of our learning simulations corroborate the experimental data, showing that the plural form of /il/-final words takes longer to master than that of other /l/-final words. In addition to replicating the empirical results, the learning simulation reveals two important implications. First, the raw frequency of individual forms cannot account for the attested patterns, suggesting that the input frequency is mediated by some principles of grammatical learning. Second, Harmonic Grammar outperforms Optimality Theory in simulating real acquisition data. Through the additive constraint interaction, Harmonic-Grammar learning is more restrictive, avoiding the generation of unobserved data in the course of morpho-phonological acquisition.

1. Introduction

Although phonology and morphology are two independent grammatical components, many morphological processes involve phonotactically motivated alternations (e.g., in English, the voicing assimilation of inflectional affixes correlates with a ban on mismatched obstruent voicing in word-final clusters), and phonological computation may likewise be morphologically sensitive (e.g., the English stress-shift triggered by the suffix <-ity> but not by <-ness>; Kiparsky 1979). The interwoven relationship between morphology and phonology has long been addressed in formal linguistic analysis (Troubetzkoy 1929), yet little is known about how this interaction is acquired during childhood (Hayes 2004, Fikkert & Freitas 2006, Kerkhoff 2007, Zamuner et al. 2012; see Tessier 2016 for an overview).

Several studies on European Portuguese (EP) suggest that the interplay between phonology and morphology is intricate in the process of child language acquisition. On the one hand, a phonological structure may be acquired earlier in a morphologically complex environment than in a mono-morphemic one. For instance, while Portuguese children typically master a segment in the word-internal stressed position before one in the word-final unstressed position (Freitas 1997), a reverse acquisition order (word-final unstressed > word-medial stressed) has been observed for the coda fricative /s/¹ (Freitas et al. 2001), and the vowels /a, o, e/ (Freitas 2007). Freitas attributed these two acquisition puzzles to the morphological function that these segments often play in the word-final position, where /s/ serves as the plural morpheme while the vowels /a, o, e/ act as instances of a class marker. Furthermore, Fikkert & Freitas (2006) demonstrated that even at an early stage of development, children may use the allomorphic variation between stressed and unstressed EP stem vowels as a cue to probe their underlying representations.

On the other hand, applying morphology may hinder children from producing a phonological structure that they have acquired in a simplex form. For instance, both nasal diphthongs (around 1;10, Freitas et al. 2010) and the complex rhyme VGC_fricative² (around 2;10, Correia 2004) are available in Portuguese children’s phonological system at early stages; however, when these two phonological structures arise from a morphological process, children tend to have difficulty producing them. Nasal diphthongs in nominal plurals (e.g., [mɐ̃ʁʃ], “hands”) present challenges for children up to the age of 7;0 (Ramalho & Freitas 2012). Similarly, children aged 6;0 cannot consistently produce VGC_fricative structures in morphologically derived contexts (e.g., [ɐ.ni.ˈmajʃ], “plural form of animal”) (Freitas & Afonso 2017).

Some studies have even indicated that morphological learning may interact with phonological acquisition in a more nuanced manner. Freitas & Afonso (2017) provided evidence for this through an experimental study on the acquisition of irregular plural morphology by Portuguese children aged 4;0 to 6;0. The study showed that children do not treat all words ending in /l/ equally, but instead they demonstrate sensitivity to the quality of stem-final vowels. The plurals of words ending in stressed /il/ are produced less accurately than those of words ending in other final stressed vowels (/al/, /ɛl/, /ɔl/, and /ul/). This acquisition order as a function of stem-final vowels cannot be attributed solely to morphological complexity, as this would have the same effect on all words ending in /l/. In this study, we present a formal account for this stem-final vowel effect, simulating the acquisition of the morpho-phonological alternations under examination with both ranked constraints in Optimality Theory (OT, Prince & Smolensky 1993) and weighted constraints in Harmonic Grammar (HG, Legendre et al. 1990).

The present paper is structured as follows. In section 2, we describe the irregular pluralization of /l/-final words in EP and outline previous formal accounts. Section 3 summarizes the findings reported by Freitas & Afonso (2017). After a brief introduction to the theoretical frameworks and the constraint set used in our simulations in section 4, we construct in a stepwise fashion the phonological grammar that arguably instantiates the onset of morpho-phonological acquisition in question in section 5. In section 6, we present the procedure and the results of our simulations. In section 7, we compare the OT and HG simulations, address the issue of input frequency, and discuss the disparities between our simulations and the empirical acquisition data. Section 8 presents our conclusions.

2. Irregular pluralization of /l/-final words in EP

The regular EP plural morphology is quite straightforward. It only requires the plural marker /-s/ to be concatenated to the singular form, without any changes to the stem, as shown in (1) (data here from Morales-Front & Holt 1997). For a more detailed description and analysis of EP plural morphology, the reader is referred to Morales-Front & Holt (1997), Mateus & Andrade (2000), and Mateus (2003), among others.

Table

(1) Regular plural morphology in EP
Singular form	IPA	Plural form	IPA	Gloss
casa	[ˈka.zɐ]	casas	[ˈka.zɐʃ]	‘house(s)’
coisa	[ˈkoj.zɐ]	coisas	[ˈkoj.zɐʃ]	‘thing(s)’
pau	[ˈpaw]	paus	[ˈpawʃ]	‘stick(s)’
rei	[ˈʀɐj]	reis	[ˈʀɐjʃ]	‘king(s)’

In the case of /l/-final words, the application of plural morphology implies different phonological changes depending on the stem-final vowels. After pluralization, the lateral never appears at the surface level. A glide [j] emerges only when the singular form ends in stressed /al/, /ɛl/, /ɔl/, or /ul/, as in (2a), but not when the singular contains a final /il/, as in (2b).

Table

(2a) Pluralization of EP words ending in stressed /al/, /ɛl/, /ɔl/, and /ul/
Singular form	IPA	Plural form	IPA	Gloss
animal	[ɐ.ni.ˈmaɫ]	animais	[ɐ.ni.ˈmajʃ]	‘animal(s)’
papel	[pɐ.ˈpɛɫ]	papéis	[pɐ.ˈpɛjʃ]	‘paper(s)’
caracol	[kɐ.ɾɐ.ˈkɔɫ]	caracóis	[kɐ.ɾɐ.ˈkɔjʃ]	‘snail(s)’
azul	[ɐ.ˈzuɫ]	azuis	[ɐ.ˈzujʃ]	‘blue(s)’

Table

(2b) Pluralization of EP words ending in stressed /il/
Singular form	IPA	Plural form	IPA	Gloss
funil	[fu.ˈniɫ]	funis	[fu.ˈniʃ]	‘filler(s)’
barril	[bɐ.ˈʀiɫ]	barris	[bɐ.ˈʀiʃ]	‘barrel(s)’

Several attempts have been made in the literature to account for such an irregular pluralization of /l/-final words in EP. Mateus (1975) and Andrade (1977) proposed that the surface glide [j] stems from an underlying theme vowel /e/ (as in /animal + e/ “animal”). In their analysis, pluralization first triggers a morphologically sensitive rule that deletes the intervocalic lateral (/animal + e + s/ → /animaes/), after which the theme vowel /e/ in the resulting intermediate form (/animaes/) undergoes gliding due to hiatus avoidance. As for the singular ending in a stressed /il/, after the elision of /l/, the theme vowel /e/ is raised to [i] and then fuses with the identical preceding vowel, for example, /funil + e + s/ → /funi + e + s / → /funi + i + s/ → [fu.ˈniʃ]. However, this account faces at least two problems. First, as noted by Lipski (1973), except for the pluralization of /l/-final words, the intervocalic /l/-deletion rule does not apply at all in the synchronic EP grammar, as evidenced by the preservation of the stem-final lateral in other morphological contexts (e.g., papel ~ papelão “paper ~ cardboard”). Second, such an analysis requires an extrinsic rule ordering (omission of intervocalic lateral > gliding of /e/), which has been criticized for lacking independent evidence (Morales-Font & Holt 1997).

Instead of assuming that the surface [j] is derived from a theme vowel, Morales-Font & Holt (1997) postulated that [j] is the surface realization of /l/ as a result of nuclearization (Colman 1983, Girelli 1988). In their proposal, the plural suffix /s/ moves into the syllable coda position that was previously occupied by the lateral, while /l/ is pushed into the syllable nucleus and becomes [j]³. They argued that the nuclearization of /l/ is driven by the EP phonotactics⁴, which disfavors a branching coda. For words ending with stressed /il/, the underlying lateral also undergoes gliding due to nuclearization, and then the resulting intermediate form [ijʃ] is regulated by the Obligatory Contour Principle (OCP), surfacing as [iʃ]⁵.

Subsequent studies have converged on the idea that [j] is derived from an underlying /l/ and that /il/-final words are subject to the OCP effect in pluralization (Mateus 2003, Silva 2020). They hereby differ from Morales-Font & Holt’s (1997) proposal with respect to the formalization of morpho-phonological computation. In the rule-based approach (e.g., Mateus 2003, Silva 2020), it is assumed that the pluralization of words ending with /il/ requires more phonological rules than the remaining /l/-final words. Namely, for words ending with stressed /al/, /ɛl/, /ɔl/, or /ul/, the pluralization triggers only one phonological rule (i.e., /l/ → [j]), whereas for /il/-final words two (ordered) rules are applied, namely first /l/-gliding and then the OCP, which merges the [j] and the stem-final vowel [i] as a monophthong. There is no such difference in the OT analysis of Morales-Font & Holt (1997) since all possible surface forms are evaluated at once by the phonological grammar (constraint ranking). As shown in the next section, Freitas & Afonso (2017) argued that the acquisition data nicely support the rule-based analysis: The need for more phonological rules in derivation causes a delay in the acquisition of the plural forms of words ending in /il/ compared to the plurals of other /l/-final words.

3. Acquisition data

This section summarizes and discusses the findings of Freitas & Afonso (2017), who analyzed both longitudinal spontaneous and experimental cross-sectional production data to explore how Portuguese children acquire /l/-final irregular plurals.

The first piece of evidence presented by Freitas & Afonso comes from the longitudinal spontaneous speech corpus collected by Freitas (1997). The Freitas corpus consists of production data from 7 children acquiring EP (Lisbon variety) as their first language. Each child was assessed longitudinally for at least one year in a naturalistic design. The onset age of the data collection varied between children from 0;10 to 2;7. Each session lasted between 30 and 60 minutes. In the Freitas corpus, only 32 tokens⁶ related to the plurals of /l/-final words (token frequency: 0.16%⁷; 32 out of 18,654 words) were found. Among these 32 tokens, an intriguing pattern was observed: 4 of the 7 toddlers already produced the target plural forms accurately more than half of the time around the age of 2;0, and one child (Luís) even achieved target-like production 80% of the time. At first sight, these spontaneous data seem to suggest that the plurals of EP /l/-final words are acquired at a very early stage. However, there is good reason to be skeptical, as another type of irregular plurals (words ending in nasal diphthongs) is not mastered by Portuguese children until the age of 7;0 (Ramalho & Freitas 2012).

Apart from searching the Freitas corpus, Freitas & Afonso also conducted an elicited production task with 75 older children (Group 1: 20 children aged 4;0 to 5;0, Group 2: 55 children aged 5;0 to 6;0). During the task, each child was first presented with a picture of a single item (e.g., caracol “a snail”) and was then asked to name it; then, a new picture with several instantiations of the same item was shown to elicit the plural form (e.g., caracóis “snails”). The test stimuli consisted of 17 /l/-final EP words, covering all five stem-final vowels (/a, ɛ, i, ɔ, u/). 17 distractors requiring regular pluralization were also included (e.g., bola ~ bolas, “ball(s)”). All 34 items were selected from the Freitas corpus to ensure that they were present in the children’s lexicon by the age of 4;0. The experimental results showed that, in stark contrast to the regular plurals, which are normally available by the age of 2;0 - 2;5 (Freitas 1997), the /l/-final irregular plurals cannot be produced in a target-like manner by Portuguese children until the age of 6;0 (Group 1: M = 0.25; Group 2: M = 0.68), as shown in Figure 1.

Figure 1. Global production accuracy of plurals by 75 Portuguese children in the study by Freitas & Afonso (2017).

These 75 children, despite being much older, were outperformed by their younger peers from the Freitas corpus, suggesting that Portuguese children may exhibit a U-shaped development in their acquisition of /l/-final plurals. U-shaped development, which refers to an initial period of correct use followed by a regression before final mastery, has long been observed in the literature (Cazden 1968). According to MacWhinney’s (1978) Dialectic model, children begin their morpho-phonological exploration by memorizing the adult form that they encounter in the input as an unanalyzed phonological whole (e.g., [ɐ.ni.ˈmajʃ] = as the plural of animal) and reproducing it as a holistic element that superficially resembles the adult form. The Dialectic model also predicts that the use of unanalyzed wholes will be replaced by grammatical combination, for example, the application of regular morphology to an irregular base, also known as over-regularization, leads to a regression in the production accuracy.

This second stage of U-shaped development, over-regularization, which is presumably responsible for the decline in production accuracy, is clearly evident in Portuguese children’s repair strategies. After searching the Freitas corpus, we found that certain toddlers, when unable to produce the target irregular plurals, applied the regular morphology to /l/-final words, adding the plural suffix [ʃ] to the singular form (4 tokens, 25% of all nontarget-like forms), as shown in (3). More strikingly, it was observed that they occasionally employed a highly marked structure, namely a branching coda [ɫʃ], which violates EP phonotactics. Although a schwa may be inserted to break up the consonant cluster, presumably due to phonotactic well-formedness, we believe that the production of both [ɫʃ] and [lɨʃ] instantiates over-regularization.

Table

(3) Portuguese toddlers’ repair of the plurals of /l/-final words (longitudinal data from Freitas 1997)
Target sing. form	Target pl. form	Child pl. form	Age	Gloss
[ka.ʃɨ.kɔɫ]/[kaʃ.ˈkɔɫ]	[kaʃɨ.ˈkɔjʃ]/[kaʃ.ˈkɔjʃ]	[kaʃ.ˈkɔɫʃ]	Luís (2;9.21)	‘scarves’
[pĩ.ˈsɛɫ]	[pĩ.ˈsɛjʃ]/ [pĩ.ˈsɐjʃ]	[pĩ.ˈsɛlɨʃ]	Laura (2;4.30)	‘brushes’
[kɐ.ɾɐ.ˈkɔɫ]	[kɐ.ɾɐ.ˈkɔjʃ]	[kɐ.ɾɐ.ˈkɔlɨʃ]	Pedro (3;5.18)	‘snails’
[kɐ.ɾɐ.ˈkɔɫ]	[kɐ.ɾɐ.ˈkɔjʃ]	[kɐ.ɾɐ.ˈkɔɫʃ]	Pedro (3;5.18)	‘snails’

As further revealed by the experimental data in Freitas & Afonso (2017), over-regularization (e.g., /animal + s/ → [ɐ.ni.ˈmalɨʃ]) remained the most common repair strategy by Portuguese children until the age of 6;0 (86% of all nontarget-like forms). Taken together, both spontaneous and experimental production data point to the same developmental trajectory: The initial target-like irregular plurals are unanalyzed wholes, which are later replaced by over-regularization, a strategy recurrently employed by Portuguese children before mastering target morpho-phonological alternations. The long-standing use of over-regularized forms in the acquisition of irregular morphology has been observed cross-linguistically. It may begin as early as two years of age (Marcus et al. 1992) and continue until school age (Moe et al. 1982, Schaner-Wolles 1989, Veit 1986).

Apart from the protracted struggle with irregular plurals, Freitas & Afonso (2017) observed that, when acquiring the irregular plural morphology, Portuguese children are sensitive to the quality of stem-final vowels, as shown in Figure 2. It was statistically confirmed that the plural forms of words ending in stressed /il/ were less accurately produced than those of other words ending in /l/. The plurals of words ending in /ul/ had the highest degree of accuracy. There was no significant difference between the plurals of words ending in /al/, /ɛl/, and /ɔl/. This study aims to account for what causes a delay in the acquisition of /il/-final plurals compared to other /l/-final plurals. We speculate that the high accuracy in the production of /ul/-final plurals arises from a methodological issue, which is addressed in section 7.3.

Figure 2. Production accuracy of plurals of /l/-final words as a function of stem-final vowels in the study by Freitas & Afonso (2017).

The attested order of acquisition as a function of stem-final vowels was interpreted by Freitas & Afonso (2017) as an instantiation of derivational complexity. Recall that, according to the rule-based approach (Mateus 2003, Silva 2020), the production of the plurals of /il/-final words demands the activation of two phonological rules (/l/-gliding and OCP), whereas the pluralization of other /l/-final words requires only /l/-gliding. Freitas & Afonso argued that the additional rule required for /il/-final plurals may interfere with their acquisition. Although the acquisition data seem to support the rule-based view well, Freitas & Afonso (2017) warned that such an explanation hinges on the assumption that the degree of grammatical (phonological) complexity increases when more phonological rules are involved in the computation. There is still an ongoing debate in the literature about what effective indicators of derivational complexity are (e.g., Pellegrino et al. 2009, Prickett 2019).

In the remainder of this paper, we present an alternative formal account of the attested acquisition order, adopting two constraint-based frameworks, namely Stochastic OT (Boersma 1998) and Noisy HG (Boersma & Pater 2016). In contrast to rule-based approaches, where different rules apply in a serial manner (one rule at a time), all output candidates in constraint-based phonological theories are evaluated simultaneously. Therefore, the controversial derivational complexity does not apply. Our learning simulations show that the stem-final vowel effect reported by Freitas & Afonso (2017) emerges naturally through constraint interaction in the course of morpho-phonological development.

4. Theoretical models and constraint set

The learning simulation was performed with two constraint-based frameworks, both of which assign numerical values to constraints. The main difference between Stochastic OT and Noisy HG lies in their decision-making mechanism. To illustrate, consider a toy grammar consisting of three constraints with constraint values of 70, 50, and 30, respectively, whereby two candidates are competing to be selected as the output of the grammar.

In Stochastic OT, the optimum is selected based on the hierarchical ranking of constraints, which is reflected in their numerical differences. Tableau (4) illustrates that the competition between the two candidates is terminated by Candidate 1’s violation of the highest-ranked Constraint A (with the highest numerical value). The constraint values are shown above the constraint names, and “!” indicates a fatal violation. Lower-ranked constraints B and C, although both penalizing the other candidate, do not play a decisive role in the choice of the optimum. As a result, Candidate 2 wins.

In the case of Noisy HG, the winner is the candidate that obtains the highest harmony, which is the negated weighted sum of its constraint violations (number of violations × constraint value). Since constraint violations are treated as penalties (negative numbers), the highest possible harmony value for a candidate is zero, so the optimal candidate is the one whose harmony is closest to zero. The way the harmony score is calculated determines that the decision-making in Noisy HG is sensitive to cumulativity (the ganging-up effect). In the Noisy HG version of our toy grammar, as shown in Tableau (5), Candidate 1 receives a harmony score of -70 due to its violation of the highest weighted Constraint A, and Candidate 2’s harmony is -80, which is the sum of its violation of Constraint B and Constraint C. In this case, Candidate 1 is the winner, since the two weak constraints gang up to jointly beat a stronger one.

The performance of ranked and weighted constraints in simulating empirical data has been evaluated in several studies. While some have found that Stochastic OT and Noisy HG achieve comparable results (Boersma & Escudero 2008, Hamann et al. 2012), others have shown that HG outperforms OT (e.g., Jesney & Tessier 2008, 2011; Boersma & Pater 2016, Zhou & Hamann 2024). We present our OT and HG simulations in 6.4 and 6.5, respectively, and their differences in section 7.1.

In the current study, the following three types of constraints are employed: input-output-based faithfulness constraints, markedness-related structural constraints, and output-output faithfulness constraints.

The IO-faithfulness constraints (IO-Faith), which have been familiar to phonologists since McCarthy & Prince (1995), aim to ensure a correspondence between input and output. They militate against output candidates that (i) contain a segment that is absent in the input (Dep-IO), (ii) omit a segment that is present in the input (Max-IO), or (iii) have a feature value that is different from the one in the input (Ident-IO).

The structural constraints (Structural) only target output candidates, enforcing the phonotactic restrictions on a particular language. In the case of the two structural constraints relevant to the current study, Syll (following Morales-Front & Holt 1997) expresses the syllable well-formedness in EP (e.g., no branching coda), while *[ij] disfavors the occurrence of the sequence [ij] at the surface level. One may wonder whether *[ij] is interchangeable with a conventional markedness constraint, the Obligatory Contour Principle (OCP), which militates against adjacent identical elements. Although the OCP has been held responsible for the nonoccurrence of [ij] in previous formal accounts (Morales-Front & Holt 1997, Mateus 2003, Silva 2020), and would work equally well in our simulations, we prefer the more specified constraint *[ij]. It has long been recognized that the OCP needs to be relativized in a language-specific way (McCarthy 1986, Odden 1988). An explicitly stated constraint such as *[ij] seems more appropriate than the most general form of the OCP, which prohibits adjacent identical elements without specifying which structural property is being targeted.

A third type of constraint required for our simulations are Output-to-Output Faithfulness constraints (OO-Faith; McCarthy 1998, Benua 2000, Hayes 2004, Tessier 2012). Similar to IO-Faith constraints, OO-Faith constraints assess the correspondence between two forms. However, unlike the IO-Faith constraints, which evaluate the relationship between an input and an output, the OO-Faith constraints hold between two output strings: the output of a morphological base and the output of that base within a morphologically more complex word. OO-Faith requires that the derived forms have the same phonological properties as their morphological bases. This study employs the three OO-Faith constraints from Tessier (2016), namely Dep-Derived, Max-Derived, and Ident-Derived. Analogous to the IO-Faith triplet, Dep-Derived is violated if a segment missing in the base is inserted in the derived form. Max-Derived does the opposite, militating against an output candidate that omits a segment contained in the base. Ident-Derived penalizes any feature change from the base to the morphologically derived form.

5. EP grammar at the onset of learning morpho-phonological alternation

Having introduced the theoretical models and the set of constraints, we now progressively construct a constraint-based grammar that is believed to instantiate the grammatical knowledge of Portuguese children at the onset of the morpho-phonological acquisition in question. We start from a zero state, where neither phonological nor morphological learning has taken place (5.1), and then implement a necessary change in the grammar as a result of early phonotactic learning (5.2). We show that the inferred initial-state grammar produces both the unanalyzed wholes and the over-regularized forms, as observed in children’s early production (5.3).

5.1. The zero state (before any phonological or morphological learning takes place)

Both theoretical learnability studies and empirical research converge on the idea that certain learning biases affect the initial stage of phonological acquisition. These learning biases not only have significant implications for learnability, for example, ensuring restrictive learning by preventing the acquisition of “superset grammars” (see Smolensky 1996 and Jesney & Tessier 2011 for a discussion), but also provide elegant solutions to several intriguing “puzzles” in child language acquisition (e.g., Tessier 2009, 2012).

The most well-established learning bias in the OT literature is that Structural constraints must initially outrank IO-Faith constraints⁸ (Demuth 1995, Smolensky 1996, Pater 1997, Boersma & Levelt 2000, Gnanadesikan 2004). The Structural ≫ IO-Faith bias effectively accounts for the observation that children start with unmarked structures (e.g., CV syllables) and, as a function of re-ranking between Structural and IO-Faith constraints, gradually acquire more marked ones (e.g., CVC and CCV) over time (Jakobson 1941/1968, Fikkert 1994, Freitas 1997, Rose 2000).

Another learning bias relevant to the current study concerns the ranking between the two types of output-based constraints: OO-Faith constraints should initially dominate all Structural constraints (McCarthy 1998, Hayes 2004, Tessier 2012). The OO-Faith ≫ Structural bias is particularly revealing when one considers the evidence that children temporarily override the phonological regularities only in derived words, but not in simplex ones. As noted in Kazazis (1969), a child acquiring Greek was able to produce the velar and palatal fricative allophonic alternation (conditioned by the quality of the following vowel) in underived contexts, but she struggled to do so in verb conjugation. Likewise, Bernhardt & Stemberger (1998) observed that an English-learning child (aged between 2;00 and 3;08) consistently flapped /t/ in morphologically simple words (e.g., wa[ɾ]er), yet faithfully realized stops in derived words (si[t]ing from base si[t]). Hayes (2004) interpreted children’s resistance to phonological changes between the base and derived forms as a consequence of Paradigm Uniformity (e.g., Venneman 1972, Burzio 1994). Experimental evidence supporting the preference for OO-faithfulness in early language acquisition was later presented by Tessier (2012). In a Wug-test, twelve English-acquiring children (mean age 4;07) applied Paradigm Uniformity only to the words with morphological bases in their lexicon, but not to those without bases. Children’s preference for Paradigm Uniformity, even in the absence of such a requirement in the target language, may be due to the fact that Paradigm Uniformity simplifies an aspect of the morphology during the acquisition process (Tessier 2016).

Taken together, in accordance with Hayes (2004) and Tessier (2012), we propose that, prior to any phonological and morphological learning, the constraints in the Portuguese children’s phonological grammar should conform to the following ranking/weighting condition: OO-Faith ≫ Structural ≫ IO-Faith.

5.2. After early phonotactic learning, yet before developing morphological knowledge

Several lines of research have converged on the observation that children develop an understanding of the phonotactics of their native language before exploring the morphological relationships between words (Hayes 2004, Tessier 2012, 2016). As Tessier (2016) points out, the compelling evidence for this developmental trajectory comes from studies on infant speech perception (see Saffran et al. 2006 for a review). For instance, by 9 months of age, infants are able to distinguish sound sequences that occur in their native language from those that violate their native phonotactics (e.g., Friederici & Wessels 1993, Jusczyk et al. 1993). The sensitivity to native phonotactics implies that children have rearranged some of the Structural and IO-faithfulness constraints, thereby enabling their phonological grammar to accommodate increasingly marked structures in the target language. Early phonotactic learning can aid in the morpho-phonological acquisition that follows shortly thereafter, since many morpho-phonological alternations essentially resemble a language’s phonotactics. In a longitudinal study of a child acquiring Polish, Łukaszewicz (2006) observed that the child mastered alternations in accordance with Polish’s phonotactics much earlier than those alternations conditioned by particular morphemes.

As previously discussed, it is conceivable that some phonotactic learning shapes the zero-state grammar proposed in 5.1 (OO-faithfulness ≫ Structural ≫ IO-faithfulness) before morpho-phonological learning takes place. Evidence for this can be found in the Freitas corpus. Recall that the child forms for /l/-final irregular plurals can be broadly classified as unanalyzed wholes (no morphological operation) and over-regularized forms (application of regular morphology). The co-occurrence of these forms is clearly instantiated in the production of one child, Pedro (3;5.18), for the same target plural during a single interview session, see (6).

Table

(6) Variants of irregular plurals produced by Pedro (3;5.18) (longitudinal data from Freitas 1997)
Target form	Child form	Age	Gloss
[kɐ.ɾɐ.ˈkɔjʃ]	[kɐ.ɾɐ.ˈkɔjʃ]	Pedro (3;5.18)	‘snails’
[kɐ.ɾɐ.ˈkɔjʃ]	[kɐ.ɾɐ.ˈkɔlɨʃ]	Pedro (3;5.18)	‘snails’
[kɐ.ɾɐ.ˈkɔjʃ]	[kɐ.ɾɐ.ˈkɔɫʃ]	Pedro (3;5.18)	‘snails’

On the basis of these child forms, it is possible to identify a trace left by early phonotactic learning. The use of unanalyzed wholes is not surprising in terms of the well-formedness requirements, given that the target phonological structure VGC_fricative (cais [ˈkajʃ], “dock”) was consistently produced by the Portuguese children at the age of 2;10 (Correia 2004). The early production of the VGC_fricative suggests that some structural constraint against this complex rhyme was already dominated by IO-Faith. Turning to the over-regularized forms, the employment of a branching coda [ɫʃ] (in violation of EP phonotactics; Mateus & Andrade 2000) similarly implies that the structural constraint Syll that penalizes coda clusters must be dominated, at least temporarily, by a constraint disfavoring epenthesis. Otherwise, the other over-regularized form [lɨʃ]⁹, which is consistent with EP phonotactics, would always have been chosen.

There are two constraints that target output candidates with epenthesis: Dep-Derived and Dep-IO. According to the zero-state grammar in 5.1, the OO-Faith constraint Dep-Derived already overrides Syll as part of the learning bias OO-Faith ≫ Structural. However, Dep-Derived may not be the most suitable constraint to rule out a surface form ending in [lɨʃ], as it is uncertain whether the epenthetic vowel occurs within the morphological base, is inserted into the suffix, or has any morphological affiliation at all¹⁰ On the contrary, there is little doubt that the epenthesis violates Dep-IO. The question then arises as to what leads children to move from Syll ≫ Dep-IO to Dep-IO ≫ Syll, resulting in a highly marked structure that is not even allowed by EP phonotactics. Our speculation is that Dep-IO ≫ Syll is driven by the productive vowel reduction and deletion processes in EP, which produce a large number of surface-level coda clusters (Mateus & Andrade 2000, Vigário et al. 2003; e.g., [malɨʃ] → [maɫʃ], “ills (pl.)”; [elɨʃ] → [eɫʃ], “they”; [fiɐ̃bɾɨ] → [fjɐ̃bɾ], “ham”). Children would not know the exact nature of these consonantal sequences at the earliest stage and would parse them faithfully as underlying representations (e.g., /ɫʃ/; the Identity Map Hypothesis; Prince & Tesar 2004, Hayes 2004)¹¹ In this case, the reduced form [ɫʃ] can be regarded as the faithful realization of the input /ɫʃ/, while the adult form [lɨʃ] may be considered as being derived via epenthesis. An example of such an initial misanalysis is given in (7). The form marked “√” is the one that was (mis)analyzed as the faithful output, while the pointing finger “☞” indicates the children’s current production. In order to faithfully produce the underlying input that they have misanalyzed, children would have to adapt their phonological grammar¹² by downgrading Syll ¹³ and upgrading Dep-IO to accommodate these coda clusters (for further evidence on the early production of coda clusters by Portuguese toddlers, see Freitas 2003).

It should be noted that Dep-IO does not dominate Syll once and for all. Portuguese toddlers actually produce both [lɨʃ] and [ɫʃ] as a result of over-regularization. This variation can be more realistically formalized in Stochastic OT and Noisy HG. In particular, in these two probabilistic grammatical models, a random noise value¹⁴ (e.g., transmission/background noise) is temporarily added to each constraint value at each evaluation time. The decision-making can be influenced by the addition of noise, as this can modify the relative ranking or weighting of the constraints, leading to the selection of different winners at different evaluation instances. This variation due to noisy evaluation is illustrated in (8) and (9) in the case of Stochastic OT.

The basic constraint values of Dep-IO and Syll are 70 and 70, respectively, given in brackets. The values after the addition of noise are shown below the basic values, according to which either the candidate with epenthesis (in Tableau 8) or the one without epenthesis (in Tableau 9) wins. Such a change in the winning candidates is only possible when the involved constraints have close values.

It may be worth considering whether early phonotactic learning also leads to changes in the ranking/weighting of the other two IO-Faith constraints (Max-IO and Ident-IO). The variable use of [ɫʃ] and [lɨʃ] in children’s production might suggest that Syll and Dep-IO are outranked/outweighed by Max-IO and Ident-IO in Portuguese children’s phonological grammar. Otherwise, lateral deletion (penalized by Max-IO) and /l/-gliding (penalized by Dep-IO) should also have been used as repair strategies. Although this could well be the case, the empirical data analyzed in this study only tell us that [ɫʃ] and [lɨʃ] are produced during morpho-phonological acquisition, where the forms involving /l/-deletion and segmental change can be excluded by OO-Faith constraints, Max-Derived and Ident-Derived. Therefore, it is not easy to assess the ranking/weighting of Max-IO and Ident-IO, since their potential effects are masked by Max-Derived and Ident-Derived.

In any case, if early phonotactic learning causes Max-IO and Ident-IO to dominate Syll and Dep-IO, one would expect that before mastering other marked structures (e.g., onset clusters, which are penalized by a high ranked or weighed structural constraint *ComplexOnset), when producing morphological simplex words (no OO-Faith constraints are accessed), children should prefer epenthesis over deletion and segmental change as a repair strategy, as a result of *ComplexOnset ≫ Max-IO, Ident-IO ≫ Dep-IO. Freitas (2003) examined the acquisition of onset clusters by Portuguese children and found that cluster reduction (C1C2 → C1) is the most frequent repair strategy when children begin to deal with the target branching onset. Cluster reduction remains the most prevalent repair strategy for complex onset in the production of morphological simplex words by Portuguese children until the age of 2;10 (Freitas 2003:33). Meanwhile, the coda cluster [ɫʃ] has been found in the pluralization of /l/-final words by a child aged 2;9 (Luís; see the Online Supplementary material). Therefore, it is not completely impossible that a Portuguese child reduces onset clusters in a context without morphological processing, while producing coda clusters as a result of over-regularization at the same stage of development. In conclusion, when considering EP phonological acquisition from a holistic perspective, the ranking/weighting condition OO-Faith ≫ Syll, Dep-IO, (*ComplexOnset) ≫ Max-IO, Ident-IO seems theoretically appealing as it accounts for the occurrence of a particular developmental stage in which Portuguese children produce coda clusters for morphologically complex words, but not onset clusters for simplex ones.

In this section, we have argued for a change in children’s phonological grammar as a result of early phonotactic learning. More specifically, based on the early spontaneous production data, we speculate that the constraint value of Dep-IO will become similar to that of Syll.

5.3. Onset of learning EP morpho-phonological alternation

After early phonotactic learning, the constraints in Portuguese children’s phonological grammar conform to the following ranking/weighting condition: {Max-Derived, Ident-Derived}¹⁵ ≫ {Dep-IO, Syll, *[ij]} ≫ {Ident-IO, Max-IO}¹⁶. To explicitly express this relationship between the constraints and to prepare them for later simulation, we assign arbitrary numerical values¹⁷ to all constraints, as in (10).

Before children realize that there is a morphological relationship between the singular and plural forms of a word (both share the same base), OO-Faith is not assessed (vacuously satisfied). Unanalyzed wholes that superficially resemble the target forms are thus produced, as shown in (11). Only when morphological knowledge is developed will OO-Faith constraints be activated, yielding over-regularized forms, as in (12). Tableaux (11) and (12) are illustrations of the OT grammar, followed by the HG analogy in (13) and (14).

The alternative use of unanalyzed wholes and over-regularized forms in Portuguese children’s early production suggests that they may parse a given input in various ways, depending on whether or not they relate it to a morphological base. It has been shown that children’s ability to detect morphological relationships can be conditioned by several linguistic factors, such as semantic transparency, lexical frequency, and phonological and orthographic similarity (see Ku & Anderson 2003 for a review). It should thus come as no surprise that children shift between pre- and post-morphology grammars before they have developed adult-like morphological awareness.

The morpho-phonological learning simulation, which is presented in the next section, takes the two post-morphology grammars in (12) and (14) as the starting point.

6. Simulated learning

In this section, we describe in detail our learning simulations of how Portuguese children acquire the morpho-phonological alternations conditioned by the plural morphology of /l/-final words. We begin with a description of the input data that will be fed to the virtual learners (section 6.1), followed by an illustration of the adopted learning algorithm (section 6.2). After defining the plasticity and noise values in section 6.3, we first present the simulation results of the OT learner in 6.4 and then those of the HG learner in section 6.5.

6.1. Input

The data fed to the virtual learners comprise input pairs consisting of an underlying form (UF) and an adult surface form (SF) (for how input pairs trigger the error-driven learning algorithm adopted in this study, see section 6.2). Recall that the main goal of the current study is to model the acquisition order (/al, ɛl, ɔl, ul/ > /il/) with respect to the irregular plurals of EP /l/-final words (Freitas & Afonso 2017). To assess the effect of the stem-final vowel, we create 5 different types of pairs, namely “/animal + s/ - anima[j]s”, representing the Portuguese words ending with stressed /al/, “/papɛl + s/ - pape[j]s” for /ɛl/, “/kaɾakɔl + s/ - caracó[j]s” for /ɔl/, “/azul + s/ - azu[j]s” for /ul/, and “/funil + s/ - funi[∅]s” for /il/.

For the moment, we set aside the frequency difference between these five types of /l/-final words, so that each type occurs at the same frequency in the virtual learners’ input. For instance, if we fed the learners 100 data, there would be 20 input-output pairs for each of the five stem vowels (20 pairs ending with /al/, 20 with final /ɛl/, 20 with final /ɔl/, 20 with final /ul/, and 20 with final /il/). Of course, such an input frequency distribution is far from realistic. We return to the issue of input frequency in section 7.2.

6.2. Error-driven learning

The learning simulations are implemented in Praat (Boersma & Weenink 2021) by the error-driven Gradual Learning Algorithm (GLA; see Boersma 1998, Boersma & Hayes 2001 for OT; see Boersma & Pater 2016 for HG). The GLA is error-driven because it induces a change in the numerical value of constraints whenever there is a mismatch between the target form and the learner’s output. In other words, each time the simulated learner “hears” an adult form (the SF from an input pair introduced in section 6.1), she will compare it with her own production (which is generated by her current phonological grammar, given the UF from an input pair). If the two forms differ, the GLA will be triggered, so that the constraint value increases for those violated by the wrongly winning candidate and, at the same time, decreases for those that penalize the correct candidate. To illustrate how the GLA works, we return to the initial-state OT grammar in (12), replicated here in (15).

To illustrate, the OT learner is now fed with only one UF-SF pair, namely /animal +s/ - anima[jʃ]. The form marked “√” is the SF (the correct form chosen by the target grammar), while the pointing finger “☞” indicates the learner’s current production (the incorrect optimum selected by the learner grammar). Following Prince (2002), Ws and Ls indicate whether each constraint “prefers” the winner or the loser in the current evaluation. In the example in (15), since the learner forms (anima[lɨʃ] or anima[ɫʃ]) differ from the adult surface form (e.g., anima[jʃ]), the GLA is triggered, increasing (indicated by the leftward arrow) the values of the winner-preferring constraints and simultaneously decreasing (indicated by the rightward arrow) the values of the loser-preferring constraints.

6.3. Plasticity and noise

In addition to the input data and the learning algorithm, there are two other parameters that need to be defined when running simulations in Praat, namely the plasticity and the noise.

The plasticity values determine the learning rate. For instance, with a plasticity value of 1, a constraint with an initial value of 100 needs 20 violations to fall to a value of 80, while with a plasticity value of 0.1, 200 violations would be needed. Following Boersma & Levelt (2000), we set the plasticity value to 0.1 in order to obtain a more gradual learning curve. In this study, the plasticity value is kept constant by setting the number of plasticities to 1 (see Jesney & Tessier 2011 for the implications of biased plasticity in grammar learning).

Throughout the simulations, we fix the noise standard deviation at 2.0 with a mean of zero (Boersma 1998). This entails that if two constraints are ranked by a distance of about 10 or more, the output is essentially categorical; if the distance is much smaller than 10, there may be variation and optionality in the output, as illustrated in (8) and (9).

6.4. OT learner

For the sake of space, the following tableaux only illustrate how the Portuguese words ending in /al/ and /il/ are produced at each stage of development. Words ending in /ɛl/ /ɔl/ and /ul/ follow the same developmental path as those ending in /al/, since the pluralization of all of them requires /l/ → [j] alternation, which emerges when a certain ranking/weighting condition is met.

As shown in (16), as long as the OT learner (Praat decision strategy: OptimalityTheory) realizes that the singular and plural forms share the same base, OO-Faith constraints are accessed and over-regularized forms are produced. This behavior of the OT learner indicates that, at the beginning of morpho-phonological learning, the stem-final vowel effect is not expected (no difference exists between 16.a and 16.b).

Learning is now triggered by a mismatch between the adult target form (“√”) and the learner forms (“☞”). The GLA will increase the values of the two winner-preferring constraints Dep-IO and Syll, while decreasing the values of the two loser-favoring constraints Max-Derived and Ident-Derived. It is important to note that the constraint value of Ident-Derived decreases faster than that of Max-Derived. This is because Ident-Derived is implicated in the errors induced by 80% of the input (16.a) (i.e., words ending with /al/, /ɛl/, /ɔl/, /ul/), whereas Max-Derived is only targeted by the GLA when the input ends in /il/ (16.b). Such a divergence in input distribution (80% vs. 20%) leads to a difference in terms of a constraint violation frequency between Ident-Derived and Max-Derived. Different frequencies of constraint violation will give rise to a moment when the two winner-preferring constraints dominate Ident-Derived, but not yet Max-Derived.

This crucial intermediate-stage ranking occurs after approximately 300 data have been received. A representative example of constraint values is shown in (17). At this intermediate stage, the OT learner starts to produce target-like plural forms for all /l/-final words (17.a), except those ending in /il/ (17.b), for which the OT learner still needs more time (data). This acquisition order as a function of the stem-final vowel is exactly what was reported by Freitas & Afonso (2017).

Apart from the stem-final vowel effect, another intermediate-stage pattern emerges. As shown in (17.b), the OT learner over-generalizes /l/-gliding in the pluralization of /il/-final words, even without evidence from the target language. Given the ranking condition in (17), the OT learner treats all /l/-final words in the same manner. Regardless of the stem-final vowel, the OT learner produces /l/ → [j] alternation approximately 77% of the time and over-regularized forms with and without epenthesis 15% and 8% of the time, respectively. The numbers to the right of the tableau are the output probabilities given an underlying form; they were computed by running this underlying form through the current grammar 1,000 times.

Learning continues from this stage as the remaining errors in (17.b) trigger the GLA, further decreasing the constraint value of Max-Derived and Max-IO. Meanwhile, the GLA increases the value of the three winner-preferring constraints, namely Ident-Derived, *[ij] and Ident-IO. In fact, only the ranking value of *[ij] increases drastically because the changes in Ident-Derived and Ident-IO are largely cancelled out when the OT learner is fed with input data ending in /al/, /ɛl/ /ɔl/, and /ul/ to ensure /l/ → [j] alternation. As soon as the Structural constraint *[ij] outranks Max-Derived, the OT learner starts to produce target-like plurals for /il/-final words. The constraint value of *[ij] exceeds that of Max-Derived after approximately 1,300 data, as illustrated in (18). Since the distance between {Dep-IO, Syll, *[ij]} and Max-Derived is still very small (less than 10), the evaluation noise 2.0 leads to probabilistic behavior, which means that the OT learner is still not able to produce the target plurals of /il/-final words consistently. At this stage, she only produces the target-like plurals of /il/-final words 55% of the time (forms with /l/-gliding 36% of the time and over-regularized forms 9% of the time), yet she has already achieved high accuracy in pluralizing other /l/-final words, with 90% of forms being target-like, 9% showing lateral deletion and 1% being over-regularized.

The final stable grammar is reached after 5,000 data, as shown in (19). At this stage, all /l/-final plurals are realized in a target-like manner. The OT learning was repeated 50 times and the developmental trajectories were highly consistent.

The simulation using stochastic OT shows that achieving the adult-like grammar requires two crucial constraint re-rankings, specifically {Syll, Dep-IO} ≫ Ident-Derived and *[ij] ≫ Max-Derived. {Syll, Dep-IO} ≫ Ident-Derived emerges earlier than *[ij] ≫ Max-Derived, given that the former, which gives rise to the /l/ → [j] alternation, is prompted by 80% of the input (tokens ending in /al/, /ɛl/ /ɔl/, and /ul/), while the latter, responsible for the realization of the target plurals of /il/-final words, can be triggered by only 20% of the input. Therefore, it is the difference in the constraint violation frequency that underlies the emergent order of acquisition (/al, ɛl, ɔl, ul/ > /il/).

6.5. HG learner

This section presents the learning simulation results of the HG learner (Praat decision strategy: HarmonicGrammar), which behaves similarly to her OT counterpart. The HG learner likewise exhibits the stem-final vowel effect, but she learns in a more restrictive manner than the OT learner. A detailed discussion of the differences between the OT and the HG learning simulations is provided in section 7.1.

At the onset of morpho-phonological acquisition, as shown in (20), the HG learner produces over-regularized forms irrespective of the stem-final vowel.

After receiving approximately 400 data, the HG learner produces /l/ → [j] alternation for /al/-final inputs 92% of the time, using the over-regularized form for the remaining 8% of the time, as shown in (21.a). This learning outcome is achieved when the values of winner-preferring constraints, Syll and Dep-IO, increase enough for each of them to outweigh the summed weights of IDENT-DERIVED and IDENT-IO (both of which militate against /l/ → [j] alternation). The stem-final vowel effect emerges at this point of development as the HG learner produces target-like plurals for words ending in /al/, /ɛl/ /ɔl/, and /ul/ (21.a), but not for those ending in /il/ (21.b). Interestingly, the HG learner at this intermediate stage refrains from over-generalizing /l/-gliding to /il/-final words, in contrast to her OT counterpart. Instead, the HG learner consistently uses the over-regularized forms when producing /il/-final plurals (51% of the time with epenthetic vowel and 49% without it). The over-generalized form (e.g., funi[jʃ]) will never be selected by the HG grammar as the optimum due to the ganging-up effect of three constraints, namely, Ident-Derived, *[ij], and Ident-IO.

Learning continues as the remaining errors in (21.b) trigger the GLA, which further reduces the weight of Max-Derived and Max-IO. The target plurals for /il/-final words can only be produced when the weights of these two Max decrease to a point where they can no longer gang up against either Dep-IO or Syll. This happens after approximately 1,400 data, as demonstrated in (22). Under the weighting condition outlined in (22), the HG learner achieves high levels of accuracy in producing EP irregular plurals, with an accuracy rate of 59% for /il/-final words and 100% for other /l/-final ones. When the HG learner fails to produce the target-like plural, she relies on regular morphology (e.g., funi[lɨʃ] or funi[ɫʃ]).

The adult-like grammar, shown in (23), is attained after approximately 5,000 data. The HG learning was repeated 50 times and the developmental trajectories were highly consistent.

In summary, the stem-final vowel effect likewise emerges in the HG learning simulation because the HG learner is confronted with the same scenario as the OT learner. The crucial weighting condition w{Syll}, w{Dep-IO} > w{Ident-Derived + Ident-IO} for /l/-gliding is implicated in all errors that occur during learning, whereas the weighting condition w{Syll}, w{Dep-IO} > w{Max-Derived + Max-IO} required for the target pluralization of /il/-final words is only implicated in the errors induced by /il/-final tokens (20% of the input).

7. Discussion

7.1. Is there a difference between OT and HG learning?

Although the stem-final vowel effect attested by Freitas & Afonso (2017) is replicated in both learning simulations, unlike the OT learner, the HG learner does not over-generalize the /l/ → [j] alternation to words ending in /il/ at the intermediate stage of development; compare (17.b) and (21.b). This divergence between the two virtual learners can be attributed to their decision-making mechanism. Specifically, at the intermediate stage of OT learning (17.b), the three constraints (Ident-Derived, *[ij], and Ident-IO) that penalize the over-generalized form (e.g., funi[jʃ]) are all dominated (not decisive for decision-making). Since the choice between possible outcomes in (Stochastic) OT is determined by strict constraint ranking, the over-generalized form, disfavored by merely three non-decisive constraints, emerges as the winner at this stage of development. In the case of the HG learning (21.b), the three constraints violated by the over-generalized form can gang up to overcome another constraint in the evaluation. Accordingly, an over-generalized form such as funi[jʃ] will never be selected as the optimum, because it will always receive a much lower harmony score than any other candidate.

The question is which of the learning simulation results more closely resembles the empirical data. On the one hand, if Portuguese children over-generalized /l/-gliding when pluralizing words ending in /il/ (e.g., /funil + s/ → funi[jʃ]), then the OT simulation would be superior to the HG one. The absence of such an emergent pattern, on the other hand, would provide strong support for the HG account. Given that the repair strategies for target plurals reported by Freitas & Afonso (2017:87) are not detailed enough¹⁸ for our intended survey, we revisited the production data of 25¹⁹ (out of 75) Portuguese children from Freitas & Afonso’s (2017) study and found 48 productions for /il/-final plurals. These 48 tokens were transcribed again by a trained female native EP listener. The transcription results (available at https://osf.io/q8hz4/) corroborate what was reported in Freitas & Afonso (2017), indicating that when Portuguese children fail to realize the target-like plural of /il/-final words, they predominately apply the regular morphology (e.g., /funil + s/ → funi [lɨʃ]/ funi[ɫʃ]). We did not find any form indicating that children over-generalize /l/-gliding to /il/-final plurals (e.g., /funil + s/ → funi[jʃ]). The reanalysis of the production data in Freitas & Afonso (2017) thus suggests that, compared to the OT modelling, the HG simulation better corroborates the experimental data. The HG learning is more restrictive in the sense that it does not generate unobserved forms ([ij]) at the intermediate stage of development.

Restrictiveness has been one of the main research topics in the learnability literature. The results of our simulations align with various previous studies showing that HG outperforms OT when learning the most restrictive grammar (e.g., Jesney & Tessier 2008, 2011; Boersma & Pater 2016). An illustration can be found in studies modelling the Intermediate Faith stages, where children produce marked structures only in privileged positions (Fikkert 1994, Pater 1997, Rose 2000). In the acquisition of complex onset, French-acquiring children produce this marked structure in the stressed position before the unstressed position (e.g., Rose 2000). In HG, the Intermediate Faith stage can be modelled employing a structural constraint against onset clusters, a general IO-Faith constraint that protects complex onset in all contexts, and a specific IO-Faith constraint that favors complex onset only in the privileged (stressed) position. In Jesney & Tessier (2008, 2011), the Intermediate Faith stage emerged naturally in their HG simulation (learning algorithm: GLA). This is because, as the weights of the two IO-Faith constraints rise during learning (starting from the initial Structural ≫ IO-Faith bias), there will come a moment when the General IO-faithfulness alone cannot outweigh the Structural constraint, but both General and Specific IO-faithfulness are strong enough to defeat markedness. Such an emergent stage, however, poses a serious challenge to the OT modelling adopting the GLA (Tessier 2009) due to the fact that the General IO-Faith is implicated in a superset of errors that Specific IO-Faith is implicated in (the non-production of complex onset in the stressed position violates both General and Specific IO-faithfulness, while the non-production of complex onset in the unstressed position violates only General IO-Faith). In other words, according to the GLA, which is sensitive to the frequency of constraint violations, Specific IO-Faith will be violated less frequently (being promoted more slowly) than General IO-Faith constraints, such that the Intermediate Faith stage, which requires the ranking Specific IO-faithfulness ≫ Structural ≫ General IO-faithfulness, never occurs during OT learning. The results of this study show another way in which the strict domination in OT and the cumulative interaction in HG predict different intermediate stages of phonological acquisition.

In addition to learnability studies, several other research perspectives have also converged on the belief that constraint interaction is additive (Guy 1997, Rose & King 2007, Pater 2009, Breiss 2020, Breiss & Albright 2022), as assumed by HG. For instance, in a speech-error elicitation task performed by speakers of two Ethiopian Semitic languages, Amharic and Chaha, Rose & King (2007) observed that the participants produced more errors when they were presented with stimuli that violated several consonant co-occurrence restrictions than when given items that violated each restriction independently. Similarly, in a recent study adopting a series of artificial grammar learning experiments, Breiss (2020) found that participants judged words that violated two phonotactics of the artificial grammar to be less well-formed than those that only disobeyed one phonotactic restriction. These two cumulative effects cannot be easily dealt with when the constraints are strictly ranked as in OT.

7.2. Role of input frequency

Both the OT and HG learning simulations indicate that the stem-final vowel effect stems from the fact that the ranking/weighting condition responsible for /l/-gliding is learned faster than that required for the target-like realization of /il/-final plurals. The former is promoted by any errors that occur prior to the mastery of /l/-[j] alternation, while the latter is implicated only in the errors induced by /il/-final tokens (20% of the input in the simulation). Thus, the relatively low frequency of /il/-final plurals in the input does indeed cause a delay in their acquisition. This finding echoes the significant volume of experimental studies showing that input statistics play an important role in children’s phonological acquisition and processing (e.g., Juczyck et al. 1993, Zamuner et al. 2005, Ota 2006, Tsurutani 2007).

On the other hand, our modelling reveals that raw input frequency alone may not be a reliable source for learning and should be mediated by some principles of grammatical learning (Tessier 2009). Namely, the learning of phonological grammar is sensitive to differences in the frequency of constraint violations (Boersma 1998, Boersma & Levelt 2000). A search in the P-PAL corpus²⁰ (Soares et al. 2018) shows that in EP, the /l/-final irregular plurals (words and adjectives) with different stem-final vowels have a rather diverse frequency distribution, as shown in Table 1.

Table 1. Frequency distribution of the plurals of EP /l/-final nouns and adjectives as a function of the stem-final vowel.

	/al/	/ɛl/	/ɔl/	/ul/	/il/
Frequency per million words (fpmw)	4148.64	117.64	92.72	28.83	158.15
Log (fpmw)	8.33	4.77	4.53	3.36	5.06

The experimental data in Freitas & Afonso (2017) clearly show that Portuguese children do not simply rely on input frequency. Figure 3 plots the Log(fpmw) against the production accuracy of two groups of Portuguese children who participated in Freitas & Afonso’s study.

Figure 3. Comparison between the log frequency (bars) of EP /l/-final irregular plurals from the P-PAL corpus and the production accuracy (lines) of two groups of children in the study by Freitas & Afonso (2017).

Figure 3 clearly shows that the input frequency does not account for children’s acquisition of /l/-final plurals. The plurals of /il/-final words pose the greatest challenge to Portuguese children (their production accuracy of /il/-final plurals was significantly lower than that of other /l/-final plurals; Freitas & Afonso 2017), but not because they are the least frequent in the input. The most uncommon ones in the input, /ul/-final plurals, are in fact mastered before other /l/-final plurals. Moreover, there is no evidence that children’s production accuracy differs between the plurals of words ending in /al/, /ɛl/, and /ɔl/, despite their distinct distribution in the input. However, the lack of such a difference is nicely accounted for by our simulation, in which the pluralization (/l/-gliding) of words ending in /al/, /ɛl/, and /ɔl/ is yielded by the same ranking/weighting condition.

7.3. Divergence from the empirical data and an alternative account

Although the learning simulations in this study successfully achieve the goal of replicating the stem-final vowel effect on the acquisition of morpho-phonological alternation reported by Freitas & Afonso (2017), they differ from the empirical data with regard to one aspect. The plurals of /ul/-final words are notably better produced by Portuguese children than the plurals of other /l/-final words. The preference for /ul/-final plurals is, however, not predicted by our modelling, since these plurals were yielded by the same ranking/weighting condition that underlies the pluralization of words ending in /al/, /ɛl/, and /ɔl/. We speculate that children’s production of /ul/-final plurals might be a result of methodological issues. In Freitas & Afonso (2017), the production of irregular plurals was elicited through a picture-naming task. This task required all the test items to be present in the children’s lexicon at the time of testing. Due to this restriction, only one item ending in /ul/ (azul, “blue”) was included. If it were possible to assess the plurals of more words ending in /ul/, this preference might cease to exist.

A reviewer noted that the preference for /ul/-final plurals might offer a clue to an alternative articulatory-based account. As shown in Figure 2, the difference in production accuracy between /il/ and other /l/-final plurals seems to be gradient and somehow correlated with the backness of the stem-final vowel, i.e., the more frontal the vowel is, the less accurate the production of the plural forms: /ul/ > /al/ > /ɔl/ > /ɛl/ > /il/. Although articulatory difficulty may condition the morpho-phonological acquisition in question (e.g., the children sometimes produced a quite simplified/reduced form of the target plural: /animal + s/ → [imáj] by João 2;3 in the Freitas corpus), we deem that the grammatical account presented in this study is superior to the articulatory one for at least two reasons. First, the accuracy of production cannot be reliably predicted by the vowel backness: in EP, the vowel [ɔ] is more back than [a] (Delgado-Martins 1973, Escudero et al. 2009), but /al/-final plurals appear to be acquired before /ɔl/-final ones (this difference needs to be statistically confirmed in future research), as shown in Figure 2. Second, among the plurals of /l/-final words, those ending in /il/ involve the least articulatory difficulty. The pluralization of words ending in /il/ only requires the mastery of CVC_fricative (e.g., [fu.ˈniʃ]), in contrast to the plurals of other /l/-final words, which demand a more complex syllable structure CVGC_fricative (e.g., [ɐ.ni.ˈmajʃ]). Hence, the delay in acquiring the plurals of /il/-final words can hardly be explained on the basis of articulatory difficulty.

8. Conclusion

In this study, we presented a formal account of the stem-final vowel effect reported by Freitas & Afonso (2017) during their investigation of the acquisition of EP irregular plural morphology by children aged 4;0 to 6;0. Our learning simulations, which employed both Scholastic OT and Noisy HG, produced outcomes that mostly resemble the empirical data, demonstrating that the attested acquisition order arises naturally during the course of grammatical learning. Furthermore, upon revisiting previous experimental data, it was suggested that the HG model outperforms the OT one in learning the most restrictive grammar, without yielding unobserved forms at the intermediate stage of development. Restrictiveness is ensured in HG by its sensitivity to cumulative constraint interactions.

Acknowledgments

We would like to thank the Associate Editor Mitsuhiko Ota and two anonymous reviewers for their excellent commentary and suggestions, which helped us to improve the article greatly.

Data Availability Statement

The data transcribed and analyzed during the current study are available in the Open Science Framework repository at https://osf.io/q8hz4/.

Disclosure Statement

The authors declare that they have no conflict of interest.

Additional information

Funding

This work has been funded by the Fundação para a Ciência e a Tecnologia (FCT, Foundation for Science and Technology), grant UIDB/00214/2020.

Notes

¹ Following Mateus & Andrade (2000), in this study we assume that the Portuguese coda fricative (plural morpheme) is an underlying /s/, underspecified for place and voicing; it is realised as [ʃ]/[ʒ] by a coda rule that adds [-anterior] and an assimilation rule for [+/-voicing] assignment.

² V stands for vowel, G for glide, and C for consonant.

³ Schwindt & Wetzels (2016) argued that, instead of nuclearization, /l/-gliding is better analyzed as coda vocalisation (/l/ → [w]), followed by place assimilation with the plural suffix ([wʃ] → [jʃ]). Schwindt & Wetzels deem that lateral nuclearization induced by pluralization is unmotivated in some Portuguese dialects where the underlying /l/ is already vocalized in singular forms. In the current study, we take a neutral position on the underlying force that gives rise to /l/ ~ [j] alternation, as it goes beyond the scope of this article and is irrelevant to our modelling.

⁴ In EP, both branching and non-branching onsets are present, while only singleton codas exist. (Mateus & Andrade 2000).

⁵ According to the phonological analysis of EP by Mateus & Andrade (2000), the glide [j] and the vowel [i] are derived from the same underlying form /i/.

⁶ The transcription of these data is available online via the PhonBank Project. The production by Inês can be found in the CFF corpus (https://phon.talkbank.org/access/Romance/Portuguese/CCF.html; sessions 0;11 to 1;11). The data produced by the other 6 children are available from the Pereira/Freitas Corpus (https://phon.talkbank.org/access/Romance/Portuguese/PereiraFreitas.html). The transcription of these 32 child forms of /l/-final plurals can also be consulted at https://osf.io/q8hz4/.

⁷ The low occurrence of these irregular plurals is also attested in the child-directed speech: type frequency: 0.24%, 18 out of 7563; token frequency: 0.056%, 161 out of 286990 tokens (data from CDS_EP; Frota et al. 2013). CDS_EP is a database of child directed speech for EP, which contains the speech data of eight caregiver-child dyads given to Portuguese children between 0;11 and 3;04.

⁸ Hale & Reiss (1998) argued for the inverse initial ranking IO-Faith ≫ Structural; but see the criticism voiced by Hayes (2004).

⁹ Freitas & Afonso (2017) reported that [lɨʃ], but not [ɫʃ], is the most prevalent repair form employed by Portuguese children aged 4;0 to 6;0. The preference for [lɨʃ] over [ɫʃ] is presumably due to the EP phonotactics.

¹⁰ We are grateful to both anonymous reviewers for their guidance on selecting between the two Dep constraints.

¹¹ Thanks to an anonymous reviewer for pointing out the implication of early phonotactic learning.

¹² The changes in the phonological grammar can be modelled by the Gradual Learning Algorithm, which is presented in detail in section 6.2.

¹³ The structural constraint Syll that militates against branching coda will need to be undominated to reflect EP phonotactics (no branching coda). What matters here is that the child production data observed by Freitas (1997) (see also Freitas 2003) suggest that this structural restriction could be temporarily overridden. Vigário et al. (2003) discussed how Portuguese children may resolve the mismatch between phonology and phonetics over the course of development.

¹⁴ The noise value can be drawn from a normal distribution with a mean of 0 and a standard deviation of 2; see the discussion in section 6.3.

¹⁵ Since there are no observed child forms that violate Dep-Derived, it is not included in the following tableaux for the sake of space.

¹⁶ We consider that there is no need to determine the relative ranking/weighting between Ident-IO, Max-IO. This is because both lateral gliding (violating Ident-IO) and lateral deletion (violating Max-IO) were attested in children’s production by Freitas (1997). Their relative ranking/weighting is not crucial for our simulations.

¹⁷ The constraint scores chosen in this paper are arbitrary, since it is the relative ranking/weighting between constraints that matters. In the case of HG, to calculate the relative constraint values in a given grammar that allows a certain candidate to be selected as the optimum, one may use error vectors; see Boersma & Pater (2016) for a discussion.

¹⁸ In the study by Freitas & Afonso (2017), the repair strategies employed by Portuguese children were classified into 4 categories. No distinction was made based on stem-final vowel.

¹⁹ Among all the experimental data collected by Freitas & Afonso (2017), the audio files of only 25 children are available.

²⁰ P-PAL is a frequency lexicon of European Portuguese based on a corpus of over 227 million words.