Frequency Effects in the L2 Acquisition of French Liaison: A Corpus-Based Adaptation of Gradient Symbolic Representations

2.1. Phonological Accounts of French Liaison

French liaison is an external sandhi process in which a liaison consonant (LC) is pronounced at the boundary between two words (W1 and W2) if W2 is vowel-initial (e.g., les animaux ‘the animals’ [lezanimo]) or glide-initial. ² The LC is silent elsewhere, especially in isolation (e.g., les ‘the [PL]’ [le]) or in preconsonantal position (e.g., les lacs ‘the lakes’ [lelak]). Just like stable final consonants in the so-called enchaînement consonantique, LCs are subject to resyllabification: the (liaison) consonant forms the onset of a syllable with the initial vowel of W2 as nucleus (e.g., [le.za.ni.mo]), reflecting the preference in French for open syllables and consonant–vowel (CV) sequences.Apart from these usual cases, there are several special liaison contexts: first, some W1s, especially the numerals six ‘six’, huit ‘eight’ and dix ‘ten’, behave ‘normally’ in liaison contexts (e.g., six animaux ‘six animals’ [si.za.ni.mo]) and before consonants (e.g., six lacs ‘six lakes’ [si.lak]) but are realised with a (potentially different) final consonant in isolation ([sis], [ɥit] [dis]). Second, W1s with final nasal vowels (e.g., mon [mɔ͂] ‘my’ or bon ‘good’ [bɔ͂]) form liaison with the nasal or denasalised vowel + [n] depending on the item, as in mon ami ‘my friend’ [mɔ͂.na.mi] or bon ami ‘good friend’ [bɔ.na.mi] (see, e.g., Tranel, 1981). Moreover, certain W2 items that are phonetically vowel-initial nevertheless block liaison (as well as other phonological processes such as elision). These correspond to the class of h aspiré words, such as héros ‘hero’ [eʁo] and hauteur ‘height’ [otœʁ].

The three most frequently occurring LCs are /z/, /n/, and /t/ (Durand et al., 2011; Mallet, 2008). Although the LC is realised as the onset of the first syllable of W2, the nature of the final consonant of W1 determines the form that the liaison will take: a W1 ending in < n > (usually a sign for a final nasal vowel) will generate a liaison in /n/; if it ends in < t > or < d >, the liaison will be in /t/, and W1s in < s, z, x > generate a liaison in /z/.

Liaison occurs only under specific phonological and syntactic conditions, typically when the two words are syntactically close and belong to the same prosodic group (De Jong, 1994). Its production is also highly variable, depending on the morphosyntactic properties of W1 and W2. Several classifications have been proposed to account for this variability, from the classical (essentially prescriptive) description by Delattre (1947) to more recent descriptive and corpus-based approaches, such as those of Durand and Lyche (2008) and Durand et al. (2011), based on the Phonologie du français contemporain (PFC) corpus, which will be used in what follows. These studies distinguish three main categories of liaison: categorical, variable, and impossible. Categorical liaison is systematically produced and occurs in four contexts: between a determiner and a noun or adjective, between a proclitic and a verb or proclitic, between a verb or an enclitic and an enclitic, and in a few fixed expressions. Impossible liaison contexts, where liaison is never realised, include sequences following the conjunction et ‘and’ and singular nouns. Variable liaison occurs in all other contexts, especially after verbs or adverbs, and is influenced by several factors, such as diachronic evolution (Dugua & Baude, 2017; Laks, 2014); diamesic variation, for example, between spontaneous speech and reading (Dugua et al., 2022; Durand et al., 2011); diastratic variation (Gadet, 2003); diatopic variation (Côté, 2017); and communicative context (Mallet, 2008).

Phonological and lexical properties also affect liaison realisation. For instance, liaison occurs more often after monosyllabic words than after polysyllabic words (Laks, 2009; Mallet, 2008). Other phonological factors have been proposed, such as the length of the following sequence (Morin & Kaye, 1982) or whether the LC is part of a consonant cluster (Delattre, 1955), but they have not been confirmed on a large scale. Lexical frequency plays a central role: more frequent W1s are correlated with higher rates of liaison realisation (De Jong, 1994; Fougeron et al., 2001). Moreover, the frequency of the W1–W2 combination, or cofrequency, also seems to be relevant. For instance, Durand et al. (2011, p. 116) show that liaison realisation is more important in the sequence grand honneur ‘great honour’ than in grand émoi ‘great agitation’, where the former is much more frequent than the latter. Similarly, Fougeron et al. (2001) report that higher liaison realisation is correlated with higher W1 frequency, lower W2 frequency, and a greater frequency of resyllabification with W2.

Liaison has been analysed within most major phonological frameworks, as each has sought to explain its complex alternation patterns. A central point of debate concerns the status of LCs, which challenge phonological theory because their behaviour lies between that of word-final and word-initial consonants. Some approaches modelled LCs as final consonants of W1, reflecting their historical origin. For example, in Schane’s (1967) classical generative analysis, a truncation rule deletes word-final consonants except when followed by a vowel-initial word; that is, liaison is the context in which the rule is not applied. In autosegmental theory (e.g., Encrevé, 1988) and Government Phonology and its extension, the CVCV model (see, e.g., Scheer, 2009), LCs are treated as floating elements (at the end of W1) that surface when they associate with skeletal material by filling a following empty onset position.

Analyses treating LCs as final consonants of W1 have been challenged on several grounds, particularly by cases where the LC surfaces as the onset of a W2 that is separated from W1 by prosodic boundaries, hesitation markers or other lexical material (see, e.g., Côté, 2011). Such data can be accounted for by the less common analysis of LCs as W2-initial, proposed by Ternes (1977), who interprets liaison as a form of initial consonant mutation. This view is consistent with exemplar approaches, according to which multiple lexical variants of the same item are stored in the lexicon; for instance, the lexical entry for ami ‘friend’ contains /ami/, but also /zami/, /nami/, and /tami/ (see Côté, 2011). The preceding word selects the appropriate variant. Moreover, it naturally captures the fact that in French-based Creole languages, LCs were often reanalysed as word-initial consonants (e.g., Mauritian Creole zanimo [zanimo] ‘animal’ < French les/ des/ ces animaux ‘DEF.PL/ INDF.PL/ DEM.PL animals’, see, e.g., Henri & Bonami, 2019).

Several usage-based analyses interpret LCs as components of (partially) lexicalised constructions (e.g., Bybee, 2001, 2005). In this view, liaison is not an insertion or deletion process but results from stored constructions, that is, multiword units comprising W1, the LC, and W2. High-frequency sequences containing an LC are more entrenched and therefore more likely to be realised, whereas in low-frequency combinations the general nonliaison pattern prevails more readily.

Finally, liaison has been analysed within several constraint-based frameworks, which enable more fine-grained modelling through the interaction of constraint families such as faithfulness (preserving input–output correspondence), markedness (favouring unmarked structures), and alignment constraints (regulating boundary correspondence). Accounts in classical Optimality Theory (OT, Prince & Smolensky, 2002/1993) treat liaison as a strategy to avoid hiatus (e.g., Féry, 2004; Tranel, 1996, 2000). The LCs are viewed as latent consonants without taking a position on the nature of their lexical source (Tranel, 2000). Yet this relatively limited explanation of hiatus avoidance is challenged, among others, by the existence of the so-called enchaînement vocalique in French, which describes a regular nonresolution of hiatus, even in frequent W1–W2 combinations (e.g., tu as ‘you have [2SG]’ [ty.a]).

A more recent probabilistic OT approach by Storme (2024) assumes that W1s triggering liaison have multiple stored allomorphs, including forms with and without an LC (unlike words with stable final consonants, which have only a single underlying representation). The variable realisation of liaison is thus attributed not to a special status of the LC itself but to paradigm uniformity constraints that favour connected-speech variants resembling citation forms (which typically lack the consonant), thereby disfavouring LC attachment both to the end of W1 and the beginning of W2. However, the main limitation of this account lies in its restricted capacity to explain the factors that influence liaison frequency.

As none of the existing phonological models fully captures liaison in its full complexity and the status of LCs remains unresolved, the following section introduces a more recent analysis that integrates several of these approaches and appears particularly well-suited for modelling the learner data discussed in this study.

2.2. GHG and the Theoretical Modelling of Liaison

The GHG, introduced by Smolensky and Goldrick (2016), represents a major advance in constraint-based phonology. Building on OT and Harmonic Grammar, GHG replaces strict constraint domination with weighted constraints, allowing the overall harmony of a candidate to be computed as a weighted sum of its constraint violations. In combination with GSRs, this framework permits the modelling of gradient and variable linguistic phenomena, including French liaison, which served as the framework’s testing ground. Smolensky and Goldrick (2016) argue that long-standing theoretical debates, such as whether LCs belong to W1 or W2, arise from the assumption that mental representations can only contain one out of two competing structures. Instead, they propose that blends are possible. In the case of liaison, this means that LCs constitute a blend of both competing forms.

The GHG retains the traditional OT assumption that constraints are universal, inherently contradictory, and violable, with languages differing only in their weighting. The familiar constraint families remain, and core constraints such as Max (‘no deletion’) and Dep (‘no epenthesis’) continue to play central roles. In liaison analysis, Onset (‘syllables have an onset’) typically replaces the constraint against hiatus used in earlier accounts. Beyond replacing strict domination with weighted evaluation, GHG introduces several innovations that allow for a more fine-grained modelling of liaison than previous constraint-based frameworks.

A key innovation in Smolensky and Goldrick’s (2016) framework is the introduction of GSRs, which encode partial activity of phonological elements. In this view, the mental representation, called input, consists of a sequence of symbolic elements, each associated with an activity level between 0 and 1. Fully ‘fixed’ segments have an activity of 1, whereas partially present elements, such as LCs, bear lower values. Only fully activated elements (activity = 1) surface in the output. Crucially, the degree of violation of faithfulness constraints depends on these activity levels; for instance, less active segments incur smaller penalties when deleted. The model also permits representations in which one position in the input can be occupied by several weakly activated elements, capturing cases where multiple potential segments are partially present in the mental representation. Consecutive weak elements in the input can also merge into a single element in the output.

With this approach to gradience, GHG differs from precursor models such as Stochastic OT, including the Gradient Learning Algorithm (Boersma, 1997; Boersma & Hayes, 2001), where gradience is located in stochastically evaluated constraint rankings (i.e., constraints are assigned numerical values along a continuous scale rather than discrete ranks).

Applied to liaison, the model assumes that the LC is weakly represented both as the final segment of W1 (e.g., petit ‘small’ /pətit_0.5 ³ /) and as the initial segment of W2 (e.g., ami ‘friend’ /t_0.3ami/). ⁴ Although W1 always contains only one weak consonant at its right edge, the onset of W2 carries weak activations for all possible LCs (e.g., /{t_0.3 + z_0.3 + n_0.3}ami/). When the combined activation of two corresponding weak elements exceeds a given threshold, the LC surfaces; when either contribution is too weak, liaison fails to surface (as in petit copain or petit héros). This blended representation captures the long-standing ambiguity of liaison, its apparent association with both words, without postulating multiple allomorphs or arbitrary selection mechanisms.

The analysis of petit ami ‘little friend, boyfriend’, the main example in Smolensky and Goldrick (2016), is presented in Table 1.

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Table 1.

Analysis of Liaison in petit ami by Smolensky and Goldrick (2016, p. 17) (Our Adaptation of the Presentation).

Weights:		−10	2	1	−0.9	−0.7
/pətit1/ /{t2, z3, n4}ami/t1 = 0.5; t2, z3, n4 = 0.3		Dep	Max	Align-L	Onset	Unif	H
a	pə.ti.a.mi				1−0.9		−0.9
b	pə.ti.t12a.mi	1 − (t1 + t2)−2.0	t1 + t2 1.6	11		1−0.7	−0.1
c	pə.ti.t1a.mi	1 − t1−5	t1 1				−4

The optimal candidate is highlighted in bold.

Candidate (b) with merged t₁ and t₂ violates Unif (‘no fusion of segments’) and Dep, whose violation is attenuated by the partial activation levels of the consonants. However, its rewards from Max and Align-L (‘alignment of segments at the left edge of a morpheme with the left edge of a syllable’) render it the most harmonious candidate.

One of the key advantages of this account lies in its theoretical parsimony: it provides a single mechanism that explains both core and peripheral patterns of liaison. Smolensky and Goldrick (2016) successfully model 14 specific input–output mappings, covering regular cases as well as exceptional or gradient forms that earlier frameworks could not handle uniformly. The notion of weak consonants with variable activation levels also enables a principled distinction between fixed consonants, typical LCs, and exceptional cases (such as the /t/ in huit) without invoking ad hoc rules. Moreover, because the constraint weights in GHG are manually derived, the analysis retains a degree of interpretability and transparency often lacking in computationally fitted models.

Smolensky and Goldrick (2016) also outline a preliminary account of how this blended grammar could be acquired. They propose that children initially store LCs exclusively as W2-initial and gradually move to the adult blended state (where the LC has partial activation in both W1 and W2) through an error-driven process that incrementally adjusts activity levels. Over repeated cycles of error and correction, LC activity is progressively shifted from the onset of W2 to the coda of W1.

Up to this point, only the phonological component of their proposal has been considered. However, the authors additionally integrate prosodic and frequency-based factors to create ‘a blend of the prosodically mediated, syntactically constrained approach of De Jong (1990), and a frequency-based approach (based on the work of Ågren and Bybee)’ (Smolensky & Goldrick, 2016, p. 27).

With respect to prosody, they assume gradient strengths of prosodic boundaries: liaison is more likely if W1 and W2 are separated by a lower-level prosodic boundary in the prosodic hierarchy. This is formalised by scaling the constraint *Cross (‘no coalescence across prosodic boundaries’) according to the boundary’s strength, so that stronger boundaries impose stronger penalties. ⁵ By incorporating gradient usage frequency and prosodic cohesion, the model captures the widely noted tendency for liaison to be favoured in frequent, tightly integrated word pairs, in line with corpus-based findings on spoken French.

Furthermore, for variable liaison contexts, the probability that the LC surfaces increases with the frequency of the W1–W2 sequence. The input representation for W1–W2 is modelled as a weighted average of the concatenation of the lexical entries for W1 and W2 and a collocation-level lexical entry (W1–W2). The proportion contributed by the collocation entry depends on its activity, which in turn reflects its frequency. Smolensky and Goldrick (2016) distinguish high-, medium-, and low-frequency contexts and assign each a different boundary-activation value. As possible extensions, they also suggest including the ‘non-productive liaison consonants: p, ɡ/k, ʁ, l’ ⁶ (Smolensky & Goldrick, 2016, p. 31) as well as the ‘variation of gradient activity levels across lexical items’ (Smolensky & Goldrick, 2016, p. 31). Both aspects will be examined in more detail in Section 5. A further planned extension is the inclusion of ‘W₁-final nasal vowels’ (Smolensky & Goldrick, 2016, p. 31) (see the discussion in Section 6).

Nonetheless, several challenges remain. The numerical interaction of weighted constraints introduces a level of conceptual complexity absent in OT’s strict hierarchy. Another limitation is the potential risk of overfitting: with multiple adjustable parameters (constraint weights and activation levels), the model could, in principle, accommodate a wide range of data patterns. Smolensky and Goldrick (2016) explicitly address this issue by imposing scalar restrictions on constraint weights and by presenting their acquisition account as a preliminary sketch requiring further refinement. They also deferred the formal derivation of certain liaison phenomena, especially peripheral and debated cases, to future work.

Subsequent work by Smolensky et al. (2020) introduced a learning algorithm called Error-Driven Gradient Activation Readjustment (EDGAR), developed to acquire the complex and often variable patterns of French liaison. The EDGAR model successfully learnt both the constraint weights and the underlying gradient activation levels required for accurate liaison prediction, demonstrating the model’s potential for data-driven acquisition. However, the constraint sets and the learned activity values and weights differ substantially from those manually specified in the analysis presented above, which complicates direct comparison and reduces interpretability.

Despite its limitations, GHG (and its computational implementation within Gradient Symbolic Computation) represents a highly promising theoretical development. Through its gradient representations and quantitative constraint interactions, it bridges the gap between symbolic phonology, probabilistic, and usage-based approaches. Its application to French liaison provides a unified account of a notoriously complex phenomenon and demonstrates how a continuous grammar can capture gradience and variation that categorical frameworks leave unexplained. These properties make GHG a particularly suitable foundation for the modelling of liaison acquisition, which will be addressed in the following sections.

2.3. Empirical Studies on Liaison Acquisition

A substantial body of empirical research has examined the acquisition of French liaison in both L1 and L2 contexts, consistently showing that liaison poses significant challenges for both groups of learners. At the prereading stage, French-speaking children lack stable lexical representations and, by extension, any awareness of orthographic consonants. This makes it difficult for them to develop a productive system for liaison, whose realisation partly depends on silent word-final consonants (Dugua, 2023). The auditory input they receive places them in a contradictory situation: on one hand, syllabification encourages children to associate the LC with W2; on the other hand, its phonetic realisation is determined by W1. Early L1 acquisition of liaison is thus shaped by these opposing tendencies (see, e.g., Harnois-Delpiano et al., 2012).

Most production studies focus on children aged two ⁷ to six, a period critical for liaison acquisition. During this time, children gradually progress from producing holistic or unanalysed utterances to mastering liaison contexts, a process that is typically not complete by age six. Across the numerous empirical studies, three main types of errors in children’s liaison production have been identified, with the first two being most frequent:

Regarding L2 learners, numerous studies over the past 25 years have examined learners from diverse L1 backgrounds, notably within the project Interphonologie du français contemporain (IPFC, Racine et al., 2012). Longitudinal and cross-sectional studies consistently show that learners improve their use of liaison over time (e.g., Andreassen & Lyche, 2015; Pustka et al., 2022). Advanced learners, particularly university students, tend to produce a high proportion of correct liaisons (Mastromonaco, 1999; Pustka, 2015). For L2 learners, three main types of errors have similarly been identified, only partly corresponding to those observed in L1 children:

In L2 acquisition studies, several factors have been proposed to influence liaison acquisition beyond proficiency and years of French study. These include learner variables such as experience abroad (e.g., Howard, 2013) and type of instruction (e.g., Shoemaker & Wauquier, 2019), as well as lexical and phonological properties. Crucially, the frequency of W1 and W2 and the cofrequency of W1 and W2 have been shown to affect liaison acquisition (De Moras, 2011; Pustka et al., 2022).

Overall, learners’ L1 seems to have only a marginal influence on quantitative results. Nevertheless, learners whose L1 is a Germanic language may face specific challenges, including German learners, which will be the focus of investigation in this article. In German (and other Germanic languages), syllabification typically follows word and morpheme boundaries, and sandhi phenomena are rare. This contrasts with French, where resyllabification and sandhi processes, such as liaison, are common (cf. Wauquier, 2009). Recent large-scale investigations by Pustka and colleagues with Austrian learners (Pustka et al., 2021, 2022) confirm these challenges.

Building on these findings, the present study systematically examines the role of frequency effects in liaison acquisition among German learners. The focus is on the relative importance of W1 and W2 frequency, W1–W2 cofrequency, and the cooccurrence rates of specific LCs with particular words in the same morphosyntactic context. This allows us to identify which factors most strongly facilitate or hinder accurate liaison production. Moreover, more fine-grained analyses of error types, including different phonetic variants, are needed. As noted above, comparisons between L1 and L2 acquisition reveal both similarities and differences in error patterns and developmental trajectories. Despite substantial empirical evidence, there is still no consensus regarding the theoretical modelling of the acquisition process. Several theoretical approaches are discussed in the following subsection.

2.4. Theoretical Modelling of Liaison Acquisition

Research on the acquisition of French liaison has been interpreted primarily within two theoretical traditions for a long time: constructionist or usage-based models, which view learning as driven by exposure and lexical frequency, and an autosegmental model, which posits the development of abstract phonological representations and accounts for liaison through floating segments. More recently, these perspectives have been complemented by a proceduralisation model and by an account formulated within GHG. The following sections outline these four approaches and discuss their explanatory potential for liaison acquisition.

Within the autosegmental framework, Wauquier (2009), building on the analysis of liaison by Encrevé (1988), proposes a three-stage model of categorical liaison acquisition in L1 learners. In the first stage, observed before age two, children treat W1–W2 combinations as holistic units in which all segments and syllabic positions are aligned one-to-one with the available skeletal slots, regardless of lexical boundaries. In the second stage, they refine the segmental structure according to the maximal onset principle, interpreting the LC as the onset of W2; however, its precise identity may fluctuate with input frequency or phonological harmony. The third stage, driven by morphological bootstrapping, involves recognising inflected or derived forms of W1, allowing children to reinterpret the LC as a word-final autosegment (Clements & Keyser, 1983). The LC thus attains a floating status, unattached to both the segmental and syllabic positions of W1 but capable of linking to the onset of a following vowel-initial W2. ⁹ Yet, as noted by Dugua (2006), this account faces a limitation: the LC is not always identical to the consonant occurring in morphologically related forms. For example, the adjective grand ‘big (M)’ yields the LC /t/ but surfaces as [d] in grande ‘big (F)’. This morphological inconsistency undermines the proposed bootstrapping mechanism as a general explanation for liaison acquisition.

Constructionist and usage-based approaches, primarily developed based on L1 acquisition, conceptualise liaison learning as emerging from exposure to concrete linguistic sequences rather than from abstract phonological rules. In the constructionist model proposed by Chevrot et al. (2009, 2013), children initially store multiword sequences, with or without liaison, as single lexical units. Over time, they reorganise these into constructions by identifying recurrent patterns associated with specific lexical items, such as determiners. For example, exposure to sequences like les animaux, les lacs, and les héros, children form a general schema les + X, where X represents a variable slot that may host any noun, regardless of whether it begins with a consonant, a vowel, or an LC. At a subsequent stage, these schemas become more specific, encoding which noun variants occur after particular determiners (e.g., les + zX for /z/ liaison). Later, learners integrate such specific constructions into broader generalisations; for instance, the constructions les + zX, des + zX, and mes + zX converge into an overarching association between plural determiners and liaison [z] (Dugua, 2023; cf. Harnois-Delpiano et al., 2019).

Usage frequency is central throughout, both at the level of individual sequences and entire constructions. Nonetheless, this model faces two main limitations. First, lacking phonological constraints, it cannot account for the restriction of liaison to vowel-initial words. Second, its focus on categorical liaison in determiner-noun contexts limits its applicability to the full range of liaison phenomena (cf. Wauquier, 2009). These limitations underscore the need for a model that integrates representational and frequency-based components within a unified framework.

The proceduralisation model developed by Harnois-Delpiano et al. (2019) conceptualises the acquisition of liaison in L2 learners as qualitatively distinct from L1 development. Drawing on Anderson’s (1983) theory of skill acquisition, it assumes that declarative knowledge is gradually transformed into procedural knowledge, allowing automatic production in appropriate contexts. L2 learners are first able to judge the correctness of liaison in speech, a competence that generally precedes accurate production and thus reveals the mediating role of awareness. At the initial stage, learners possess declarative knowledge of orthographic forms and their presumed grapho-phonemic correspondences, alongside proceduralised literacy skills such as automated reading and writing. Their ability to evaluate liaison, sometimes expressed through characteristic teaching gestures, is interpreted as evidence of conscious awareness of the phenomenon. In later stages, learners begin to produce first categorical and then variable liaisons, reflecting the gradual automatisation of production. Ultimately, sociolinguistic knowledge is integrated, enabling context-appropriate realisation of variable liaison.

However, several aspects of this account remain uncertain. The assumption that knowledge of standard spelling implies an understanding of grapho-phonemic relations is empirically weak, and the use of gestures does not necessarily demonstrate explicit declarative knowledge. Furthermore, as the model relies on a single empirical study in which Stage 1 capacities were assumed to be already in place for all participants, it cannot clarify either the developmental sequence or the factors influencing early L2 acquisition.

Within constraint-based frameworks, language acquisition is typically modelled as a process in which learners infer constraint weights from observable output, guided by universal constraints that are gradually reweighted over time. In their simulation of liaison acquisition, Tessier and Jesney (2021) build directly on Smolensky and Goldrick’s (2016) analysis in the GHG framework (see Section 2.2 and Table 1). They model a GSR-based learner and compare the resulting error patterns with those observed in French-speaking children. In the simulation, constraint weights are assumed to be fixed in advance; learning occurs through the adjustment of segmental activation rather than constraint weights. Liaison errors thus serve as cues to reweight the activation of relevant segments. The W1–W2 pairs are randomly sampled from a small lexicon without considering cofrequency.

The model performs well: from any starting point, activation updates following erroneous outputs eventually yield target-like patterns: liaison with the target LC in liaison contexts and its absence elsewhere. During the learning process, the simulation reproduces the two major child error types, liaison substitution, and omission. More specifically, substitution errors arise in the simulation when an inappropriate LC is initially overactivated in W2 (e.g., ours ‘bear’ stored as /nuʁs/). Through error-driven updates, this activation decreases, and substitution errors eventually disappear. Omission errors occur in all runs, either as an initial state (when W2 is stored without LC), which persists until enough activation of the correct LC accumulates across both W1 and W2, or as a transitional phase following substitution (when the activation of the incorrect LC decreases but the correct one has not yet reached threshold activation).

Despite its overall success, the simulation diverges from attested child data in several ways. It fails to generate another attested child error associated with liaison, the substitution of W2 onsets with LCs (e.g., les nuages ‘the clouds’ realised as [le.z.u.aʒ] instead of [le.ny.aʒ]), while producing unattested ones, such as ‘excess liaison’ patterns, for example, LCs before consonant-initial words (e.g., petit prof ‘small professor’ realised as [pœ.tit.pʁɔf])) or the realisation of two LCs before a vowel-initial W2 (e.g., petit ours ‘small bear’ realised as [pœ.tit.nuʁs]). Furthermore, the model’s final state lacks generalisation to novel lexical items, contradicting native-speaker intuitions and their pilot data with adult learners on nonce words.

Tessier and Jesney (2021) conclude that as the GHG framework and especially GSRs successfully capture key aspects of developmental liaison patterns, further refinement, potentially involving morphological and lexical frequency factors, is required to achieve closer alignment with empirical acquisition data. The present article takes up this challenge, extending the model to L2 learner data and exploring how such refinements can account for the patterns observed in experimental results.

4.1. Realisation Types

After transcribing the W1–W2 contexts, 504 relevant instances were retained. Individual participants produced between 4 and 10 analysable contexts (M = 7.8, SD = 1.3). These realisations were subsequently classified into seven distinct categories:

All categories were attested in multiple speakers. In addition, an ‘Other’ category was defined to include rare or unique realisations, such as metathesis ([wit le.ze.fɑ̃] for huit éléphants), insertion of non-LCs in the onset of W2 ([tʁwa hø.ʁɔs] for trois euros), or deletion of the word-initial vowel ([tʁwa kɔl] for trois écoles). The following overall distribution was found (Figure 1):

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Figure 1.

Overall distribution of liaison realisation types.

Target-like liaison realisations with resyllabification accounted for 26% of all instances. The most frequent error types were glottal stop insertion, both without (38.1%) and with LC realisation (23.6%). The /l/-insertion was observed less frequently, whereas other realisation types were rare.

Examining the effect of years of French instruction reveals changes in the distribution of realisation types (Figure 2).

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Figure 2.

Distribution of liaison realisation types across 5 years of learning.

The proportion of target-like liaison with resyllabification increased with years of instruction (7.6% in the first, 17.2% in the third, and 52.6% in the fifth year). In contrast, glottal stop insertion, with and without the LC, decreased over time (combined: 81.1% in the first, 65.0% in the third, and 34.5% in the fifth year). Other realisation types remained relatively stable. Contrary to expectations from previous studies, liaison had not yet been fully acquired by the average learner after 5 years of French instruction.

Furthermore, to undertake a more detailed examination of individual speakers, they were grouped according to the realisation types they demonstrated at least once (Figure 3):

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Figure 3.

Groups of learners based on their realisation types.

The two largest groups, comprising 15 and 14 speakers, consist of learners who produced no target-like liaison realisations, likely reflecting the initial stage of liaison acquisition. ²³ Only one participant, a very small minority, produced exclusively target-like realisations with resyllabification, suggesting liaison is fully mastered, at least in the context and words examined. Among most remaining learners, target-like realisations cooccur with glottal stop insertions (with and without LC), indicating intermediate stages in which competing strategies are still active. The less frequent realisation types generally occur in learners who also show multiple other types, suggesting that they belong to a transitional and internally inconsistent interlanguage.

Finally, when comparing the distribution of realisation types across W1–W2 combinations, item-specific differences are obvious (Figure 4):

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Figure 4.

Distribution of realisation types by W1–W2 combination.

Target-like liaison realisations with resyllabification were relatively consistent across W1–W2 combinations (from 18.5% for un avion to 35.9% for trois euros). Glottal stop insertion, with or without the LC, appeared to be item-dependent, with LC realisation predominating in six avions, huit éléphants and dix élèves. /l/-insertion was most frequent in *un école (33.3%) and trois écoles (43.4%), suggesting a strong influence of W2. Concerning the less frequent realisation types, reduplication is most common in dix élèves (8.2%) and enchaînement vocalique in *un école (3.7%).

Statistical analyses of these factors are presented in Section 4.3, preceded by the results of the frequency analysis, which may serve as additional predictors.

4.2. Corpus-Based Analysis of Potential Frequency Predictors

The frequency analysis of the liaison contexts under investigation in ESLO showed the following results (Table 2).

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Table 2.

Frequency Variables for W1–W2 Combinations in ESLO.

W1–W2	W1 frequency(total)	W2 frequency(total)	W1–W2cofrequency(total)	W1LCcooccurrence (rate)	LCW2cooccurrence(rate)
deux éléphants	8724	20	0	0.26	0.10
deux ordinateurs	8724	273	2	0.26	0.10
dix élèves	1774	1457	2	0.46	0.69
huit éléphants	1391	20	0	0.56	0.05
six avions	1540	100	0	0.45	0.14
trois écoles	4205	5102	0	0.29	0.13
trois euros	4205	454	18	0.29	0.32
un avion	58336	100	13	0.14	0.24
*un école	58336	5102	0	0.14	0.08

Out of the items used in the experiment, the indefinite article un is unsurprisingly by far the most frequent W1, and école and élève are the most frequent W2s. Most W1–W2 combinations have a low cofrequency, with several pairs even having a W1–W2 cofrequency of 0. Only the cofrequency of trois euros is somewhat higher. The occurrence rates of W1s with LC are clearly distributed between words only rarely triggering liaison (un), and those where the consonant is often realised, which are, as expected, the more special liaison cases (six, dix, and huit) where the LC is also realised in isolation.

The LCW2 cooccurrence rates differ considerably between W1 and W2 pairs. A majority of the rates are located somewhere between 5% ([t] before éléphant[s]) and 14% ([z] before avion[s]). The [n] before avion(s) and [z] before euros occur more frequently, whereas [z] before élève(s) is dominant with 69%. Taking a more detailed view on the occurrence rate of different consonants at the onset of W2, the analysis showed some interesting trends:

Particularly striking is the high frequency of /z/ before élève(s), /n/ before éléphant(s), and /l/ before école(s). In general, LCs play a central role in this onset position, but other consonants, especially [l], should not be neglected. When we combine this with the results presented in Figure 3, we find that /l/-insertion occurs only where /l/ cofrequency is high, mainly with école(s) and to a smaller degree with ordinateur(s) and avion(s). Moreover, ‘double LC’ realisations occur where W1LC and LCW2 cooccurrence rates are high. To verify the statistical validity of these observations and identify further correlations, the results of statistical tests are presented in the following section.

4.3. Variables Affecting Target-Like Realisations of Liaison

First, the influence of several variables on the rate of target-like liaison realisation with resyllabification was tested. A generalised linear mixed-effects model (GLMM) was fitted with ‘resyllabification’ as the binary dependent variable (presence vs. absence of target-like liaison with resyllabification) and a random intercept for speaker. Fixed effects included participant-level predictors (year and school ²⁴ ) and the frequency variables presented above (W1 frequency, W2 frequency, W1–W2 cofrequency, W1LC cooccurrence rate, and LCW2 cooccurrence rate), which were scaled for better comparison.

The mixed-effects logistic regression revealed several robust predictors of resyllabification. Speakers from School B showed a markedly higher likelihood of producing target-like liaison than those from School A (β = 1.8, p = .006). Target-like liaison realisation also increased significantly across year groups (β = 1.09, p < .001), indicating substantial developmental progression. Two frequency predictors for W1 were negatively associated with resyllabification: higher W1 frequency (β = −1.00, p < .001) and higher W1LC cooccurrence rate (β = −0.74, p = .003), both reduced the odds of producing target-like liaison. In contrast, W2 frequency had a positive effect (β = 0.50, p = .018). Neither the LCW2 cooccurrence rate nor the W1–W2 cofrequency reached significance.

The estimated variance of the random intercept for speakers was 3.96 (SD = 1.99), showing considerable interspeaker variability. The adjusted ICC of 0.55 confirmed that speaker differences accounted for a large share of the variance. Model diagnostics indicated no multicollinearity, no overdispersion, and acceptable residual behaviour. The marginal R² suggested that fixed effects explained 39% of the variance, whereas the conditional R² of 72% highlighted the substantial contribution of random effects.

Subsequently, the influence of the same predictors on the different realisation types was investigated. As the complex models did not converge (see Section 3), a smaller dataset with only the most common realisation types had to be used. A Bayesian multinomial logistic regression model was fitted using the brms package, with the three most frequent realisation types ‘resyllabification’, ‘glottal stop insertion’ and ‘LC and glottal stop insertion’ as the outcome variable. The predictors were equivalent to those used for the binary model: school, year, and lexical frequency measures as predictors and random intercepts for speakers. Four chains with 4,000 iterations each (2,000 warm-up) were run, using a categorical (link = ‘logit’) family. Sampling was conducted with the No-U-Turn Sampler (NUTS) and the control parameter max_treedepth = 15. The model showed excellent convergence (Rhat = 1.00 for all parameters). Pareto-k diagnostics indicated stability (99.5% < 0.7; 0.5% > 0.7), suggesting reliable out-of-sample predictive performance.

The model confirmed the trends observed in the binary model: both realisation types with glottal stop insertion were less frequent among speakers from School B (without LC: β = −2.23, with LC: β = −1.64) and decreased with more years of French instruction (without LC: β = −1.32, with LC: β = −0.97). Moreover, substantial between-speaker variation was again found (without LC: SD = 2.48, with LC: SD = 1.53). Regarding frequency effects, higher W2 frequency decreased the probability of glottal stop insertion with LC (β = −23.60) and, to a lesser extent, without LC (β = −0.64). An important difference is found concerning the W1LC rate: a higher W1LC rate leads to lower probabilities of glottal stop insertion without LC (β = −2.65), but to a much higher probability of glottal stop insertion with LC realisation (β = 5.15). This supports our hypothesis that the final LC occurs more frequently in learners’ realisations, the more frequently the consonant is realised in this position. For glottal stop insertion with LC, further significant predictors emerged: its probability increased with higher W1 frequency (β = 10.11) and LCW2 rate (β = 8.80), but decreased with higher W1–W2 cofrequency (β = −10.88).

Although some problems remain and some things couldn’t be tested because of the small number of realisations of some types, a few key points can be noted: frequency effects on liaison realisation of L2 learners are confirmed, and they not only affect whether (target-like) liaison is realised or not, but also which concrete liaison realisation is more probable. Based on these results, the theoretical account will be presented in the following section.

5.1. Adjustments to the Model by Smolensky and Goldrick (2016)

Although the original model successfully derives target-like liaison, it fails to generate several of the learner realisation types. A prominent example is /l/-insertion: this candidate violates the same constraints as the insertion of an LC, yet the latter always incurs lower penalties because the LC has some activation, whereas /l/ has none and therefore receives the full Dep and Max penalties. As a result, /l/-insertion is systematically disfavoured and can never emerge as the most harmonic candidate. As these realisation types occur in learner speech (and some of them with nonnegligible frequency), the model must be able to derive them as most harmonic under certain conditions to capture the empirically observed patterns.

In addition, the considerable item-specific variation observed in the empirical data requires not only different tableaux (as mentioned above) but also a principled mechanism for distinguishing items within the model. Smolensky and Goldrick (2016) already noted this need but did not formalise a solution. As the statistical analysis revealed no item-level random effects and instead showed effects of frequency variables, these variables should also constitute the basis of item differentiation in the theoretical account. Nevertheless, the question remains as to which frequency measures should be taken into account. The original model provides a formalisation only for W1–W2 cofrequency (see Section 2.2). Because all item pairs in the present study exhibit low cofrequency values and nearly no statistical effect was detected, this component was not implemented here. It is nevertheless expected to play a crucial role in future work involving other morphosyntactic contexts, such as frequent pronoun–verb sequences. The integration of the frequency of W1 and W2 into the modelling of a single item combination also did not appear to be effective. Following Tessier and Jesney (2021), we assume that activation adjustments occur more frequently for high-frequency words, implying that their activity levels may approach target values earlier than those of less frequent items. Word frequency is thus more indicative of the progression in the learning process from one tableau to the next, which cannot be encoded directly in a single tableau.

A further extension concerns the activation values assigned to consonants. Smolensky and Goldrick (2016) already suggested that different consonants, and possibly different lexical items, may require distinct activity levels. In their original analysis, however, all consonants share the same activation value across all items, which makes it impossible to capture the item-specific patterns observed in the empirical data. Drawing on the statistical results, particularly the effects of W1LC and LCW2 cooccurrence rates, the present model assumes that the activity of GSRs directly reflects their empirical frequency in the relevant position. Thus, activity is neither manually set based on the optimal candidate (as in Smolensky & Goldrick, 2016) nor automatically computed (as in Smolensky et al., 2020; Tessier & Jesney, 2021), but instead derived from corpus-based occurrence rates. Accordingly, based on the results of the frequency analysis presented in Section 4.2, the underlying representation of trois is analysed as /tʁwaz_0.29/ and that of école as /{z_0.13, n_0.08, t_0.01}ekɔl/, reflecting the relative frequency with which each consonant occurs in the respective positions. As these rates differ across items (compare école above to élèves /{z_0.69, n_0.08, t_0.03}elɛv/), item-specific differences in realisation patterns can naturally be accounted for. This approach is supported by the results of the statistical analysis, which demonstrated the effects of the W1LC and LCW2 rates, and aligns with usage-based perspectives such as exemplar theory, according to which mental representations are shaped by the distributional properties of the input. This adaptation builds on earlier constraint-based models, most notably Stochastic OT (Boersma & Hayes, 2001), in which empirically derived frequency measures have been integrated into OT analyses (see, e.g., Gabriel & Meisenburg, 2009, for an application on French h aspiré words). The key difference, however, lies in the locus of frequency integration: whereas Stochastic OT incorporates frequency through gradient (stochastically evaluated) constraint ranking, the approach proposed here integrates frequency directly into the activity of symbolic representations, rather than into the weighting of constraints.

A related issue concerns the status of non-LCs in the same position. As briefly mentioned by Smolensky and Goldrick (2016) in a possible extension, a comprehensive model of liaison must distinguish LCs from other consonants. This is all the more crucial in child acquisition, where orthography cannot serve as a cue. The error pattern with /l/-insertion observed here suggests that not only LCs but also other consonants that appear in the onset of vowel-initial words have some degree of underlying activity. Extending the representation of école accordingly yields/{z_0.13, n_0.08, t_0.01, l_0.72, d_0.04, ʁ_0.01}ekɔl/ (see Table 3 for the occurrence rates). As French adults presumably show reduced activation for non-LCs, ²⁸ L2 learners may first assign higher activation to these segments. Without such activation, candidates involving /l/-insertion could not emerge as optimal under the constraints used in the liaison analysis by Smolensky and Goldrick (2016).

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Table 3.

Cooccurrence Rate of Selected Consonants With W2 Items in ESLO.

W2	/z/	/n/	/t/	/l/	/ʁ/	/d/
avion(s)	0.14	0.24	0.02	0.34	0.10	0.12
école(s)	0.13	0.08	0.01	0.72	0.01	0.04
éléphant(s)	0.10	0.55	0.05	0.10	0	0.05
élève(s)	0.69	0.08	0.03	0.06	0.01	0.10
euro(s)	0.32	0.08	0.23	0.09	0.06	0.03
ordinateur(s)	0.10	0.23	0	0.43	0.11	0.07

A further challenge in adapting the original account arises from the deterministic nature of most tableaux. Smolensky and Goldrick (2016) assume that, for any input, a single candidate is selected as most harmonic. This is sufficient for a purely theoretical analysis, where demonstrating that all competing candidates receive lower harmony values is adequate. However, once empirical data are considered – particularly L2 data, but also variable liaison in native French – true variation between several candidates must be accounted for. To address this, harmony scores are converted into predicted probabilities following the approach of Zuraw and Hayes (2017) within Maximum Entropy Grammar (MaxEnt). ²⁹ This probabilistic reformulation preserves the relative harmony relations between candidates by allowing them to surface with different probabilities. It thus provides a natural way to integrate structured variation into the analysis. Moreover, it makes it possible to directly compare the model’s predictions to the empirical distribution of realisation types observed in the learner data.

Finally, the constraint set must be adapted to account for the influence of a German L1. German resolves onsetless syllables primarily by inserting a glottal stop, ³⁰ which implies a relatively low-ranked Dep.ʔ (‘no glottal stop epenthesis’) in the German grammar (and a high-ranked Onset) (Féry, 2024). Moreover, NoCoda (‘syllables do not have a coda’) needs to be included in the constraint inventory to penalise the closed syllables that are frequent in German. Incorporating these constraints captures transfer patterns that are structurally available in the learners’ L1 and clearly visible in the realisation types with glottal stop insertion and/or final LC realisation. Moreover, in contrast to Smolensky and Goldrick (2016), but following Tessier and Jesney (2021), all constraints are assigned negative weights. This ensures that each violation lowers harmony by a uniform principle and improves the interpretability of the tableaux, especially when comparing weighted contributions across several interacting constraints.

5.2. GHG Analyses of Learner Realisations

Based on these modifications, we constructed tableaux for each of the nine item combinations, both for an idealised target state (reached by only one participant in the study) and for the two most relevant learner groups identified in the empirical analysis (see Figure 3): the group of 14 learners who produced only glottal stop insertion (with or without an LC) and the group of 15 learners who produced glottal stop insertion as well as /l/-insertion. As presenting all 27 tableaux would exceed the scope of this section, we focus on illustrating the central analytical steps; the full set of tableaux is available on OSF (Gemmeke, 2026).

For each of the three groups, activity levels were first set on the basis of the ESLO frequencies (see Section 5.1) and theoretically motivated hypotheses (e.g., the LC of numerals is assumed to be highly activated in beginner learners). An empirical probability of occurrence was then determined for every realisation type in each W1–W2 combination. Constraint weights – and activity levels where necessary – were subsequently refined iteratively to bring the predicted probabilities as close as possible to the observed distributions. In the idealised target state, resyllabification is expected for all W1–W2 combinations and occurs categorically. In contrast, the two learner groups exhibit variation in several item combinations, which the model aims to capture.

The baseline analysis of the target-like liaison realisation could be modelled as follows (Table 4).

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Table 4.

Model of trois écoles With Target-Like Liaison Realisation (Group ‘Target-Like Realisation’).

Weights:		−15	−15	−15	−10	−10	−0.5	−0.5
/tʁwaz1 //{z2, t3, n4, l5}ekɔl/z1 = 0.29, z2 = 0.13, l5 = 0.072		Onset	NoCoda	Dep.ʔ	Dep.C	Max	Align	Unif	H	P
a	tʁwa.e.kɔl	1−15 31				z1 + z2 + l5 −4.92			−19.92	0
b	tʁwa ʔe.kɔl			1−15		z1 + z2 + l5 −4.92			−19.92	0
c	tʁwa le.kɔl				1 − l5 −9.28	z1 + z2 −4.2			−13.48	0
d	tʁwas ʔe.kɔl		1−15	1−15	1 − z1 −7.1	z2 + l5 −2.02			−39.12	0
e	tʁwas le.kɔl		1−15		(1 − z1) + (1 − l5)−16.38	z2 −1.3			−32.68	0
f	tʁwas ze.kɔl		1−15		(1 − z1) + (1 − z2)−15.8	l5 −0.72			−31.52	0
g	tʁwa.ze.kɔl				1 − (z1 + z2)−5.8	l5 −0.72	1−0.5	1−0.5	−7.52	1

The optimal candidate is highlighted in bold.

We assume that this tableau reflects the target state that L2 learners ultimately acquire. The activation values of the LCs are derived from corpus frequencies, whereas those of the non-LCs are scaled down by a factor of 10 to prevent them from surfacing systematically; this preserves their relative frequencies by keeping their contribution low in the competition. The tableau displays only the activity levels of consonants that appear in at least one candidate (here: z₁, z₂, and l₅), because all others contribute identical penalties across candidates. The constraint weights differ substantially from those used by Smolensky and Goldrick (2016) and by Tessier and Jesney (2021), as their values cannot be maintained once corpus-based activation values and MaxEnt probabilities are introduced. The general logic of Max and Dep.C remains the same: Max penalises activity that fails to surface in the output, and Dep.C penalises inserted material according to the amount of additional activity required to surface. As candidates may contain two LCs, Dep.C can be violated twice, with each violation weighted by (1 – activity).

In this configuration, the liaison candidate (g) is the most harmonic: it incurs only low-weight violations of Align and Unif and receives comparatively small penalties from Max and Dep.C, as only the activity of l₅ is absent in the output and the combination of z₁ and z₂ requires minimal additional activity. The same reasoning applies to the other item combinations for this group.

Now that we have seen that the target variant can be effectively modelled even after all the adjustments presented above, the subsequent description focuses on the learner realisations in the two most important groups.

For the first learner group, which only realised glottal stop insertions (with or without LCs), the item-specific distinction between realisation types can be modelled using the same constraint weights and activation logic (Tables 5 and 6):

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Table 5.

Model of trois écoles With Glottal Stop Insertion (Group ‘Only Glottal Stop Insertion’).

Weights:		−15	−15	−15	−15	−10	−3	−0.5
/tʁwaz1 / /{z2, t3, n4, l5}e.kɔl/z1 = 0.029, z2 = 0.013, l5 = 0.072		Onset	Align	Dep.C	Max	Unif	NoCoda	Dep.ʔ	H	P
a	tʁwa.e.kɔl	1−15			z1 + z2 + l5 −1.71				−16.71	0
b	tʁwa ʔe.kɔl				z1 + z2 + l5 −1.71			1−0.5	−2.21	1
c	tʁwa le.kɔl			1 − l5 −13.92	z1 + z2 −0.63				−14.55	0
d	tʁwas ʔe.kɔl			1 − z1 −14.57	z2 + l5 −1.28		1−3	1−0.5	−19.34	0
e	tʁwas le.kɔl			(1 − z1) + (1 − l5)−28.49	z2 −0.20		1−3		−31.68	0
f	tʁwas ze.kɔl			(1 − z1) +(1 − z2)−29.37	l5 −1.08		1−3		−33.45	0
g	tʁwa.ze.kɔl		1−15	1 − (z1 + z2)−14.37	l5 −1.08				−40.45	0

The optimal candidate is highlighted in bold.

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Table 6.

Model of dix élèves With LC and Glottal Stop Insertion (Group ‘Only Glottal Stop Insertion’).

Weights:		−15	−15	−15	−15	−10	−3	−0.5
/diz1 / /{z2, t3, n4, l5}elεf/z1 = 1, z2 = 0.069, l5 = 0.006		Onset	Align	Dep.C	Max	Unif	NoCoda	Dep.ʔ	H	P
a	di.e.lεf	1−15			z1 + z2 +l5 −16.13	1			−31.13	0
b	di ʔe.lεf				z1 + z2 +l5 −16.13			1−0.5	−16.63	0
c	di le.lεf			1 − l5 −14.91	z1 + z2 −16.04				−30.95	0
d	dis ʔe.lεf			1 − z1 0	z2 + l5 −1.13		1−3	1−0.5	−4.63	1
e	dis le.lεf			(1 − z1) + (1 − l5)−14.91	z2 −1.04		1−3		−18.95	0
f	dis ze.lεf			(1 − z1) + (1 − z2)−13.97	l5 −0.09		1−3		−17.06	0
g	di.ze.lεf		1−15	1− (z1 + z2) 32 0	l5 −0.09	1−10			−25.09	0

The optimal candidate is highlighted in bold.

In the case of trois écoles, the candidate with glottal stop insertion (b) emerges as the most harmonic. For dix élèves, however, the candidate with LC and glottal stop insertion (d) is predicted as most harmonic, which accurately reflects the empirical distribution. In both items, the same constraint weights are applied, and activity levels are derived from corpus frequencies, with the final consonant in special items (like dix) fully activated, whereas other items retain lower activation values (corpus frequencies divided by 10). This approach demonstrates that, by maintaining consistent constraint weights and adjusting activation values based on corpus data, the model can successfully capture item-specific variation in learner realisations.

For the second group of learners, who produced either glottal stop insertions (with or without LC) or /l/-insertions, the same constraint weights can be applied, as activation values are fully corpus-based (Table 7).

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Table 7.

Model of trois écoles With /l/-Insertion (Group ‘Glottal Stop or /l/-Insertion’).

Weights:		−15	−15	−15	−15	−10	−3	−0.5
/tʁwaz1 / /{z2, t3, n4, l5}e.kɔl/z1 = 0.29, z2 = 0.13, l5 = 0.72		Onset	Align	Dep.C	Max	Unif	NoCoda	Dep.ʔ	H	P
a	tʁwa.e.kɔl	1−15			z1 + z2 + l5 −17.1				−32.1	0
b	tʁwa ʔe.kɔl				z1 + z2 + l5 −17.1			1−0.5	−17.6	0.08
c	tʁwa le.kɔl			1 − l5 −4.2	z1 + z2 −6.3				−10.5	0.92
d	tʁwas ʔe.kɔl			1 − z1 −10.65	z2 + l5 −12.75		1−3	1−0.5	−26.9	0
e	tʁwas le.kɔl			(1 − z1) + (1 − l5)−14.85	z2 −1.95		1−3		−19.8	0
f	tʁwas ze.kɔl			(1 − z1) + (1 − z2)−23.7	l5 −10.8		1−3		−37.5	0
g	tʁwa.ze.kɔl		1−15	1 − (z1 + z2)−8.7	l5 −10.8	1−10			−44.5	0

The optimal candidate is highlighted in bold.

In the case of trois écoles, the candidate with /l/-insertion (c) emerges as the most harmonic, predicting a probability of occurrence of 92%, closely matching the 100% observed in the empirical data. ³³ Only the penalties associated with Max and Dep.C differ from the previous group (Table 6), reflecting the changed activation values. All other constraints and calculations remain identical, demonstrating that corpus-based activation effectively captures the increased occurrence of /l/-insertion. This confirms that the model can represent both dominant realisations and variation within a learner group.

Finally, to illustrate that the model can also perfectly capture items where true variation of realisation types was found in the data, the following tableau shows the analysis of deux ordinateurs, where, again in the group with glottal stop insertion or /l/-insertion, 87% of the realisations in the empirical study were glottal stop insertions (b) and 13% were /l/-insertion (c) (Table 8).

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Table 8.

Model of deux ordinateurs With Glottal Stop Insertion or /l/-Insertion (Group ‘Glottal Stop or /l/-Insertion’).

Weights:		−15	−15	−15	−15	−10	−3	−0.5
/ døz1 / /{z2, t3, n4, l5} ɔʁ.di.na.tœʁ /z1 = 0.26, z2 = 0.1, l5 = 0.43		Onset	Align	Dep.C	Max	Unif	NoCoda	Dep.ʔ	H	P	O 34
a	dø.ɔʁ.di.na.tœʁ	1−15			z1 + z2 + l5 −11.85				−26.85	0	0
b	dø ʔɔʁ.di.na.tœʁ				z1 + z2 + l5 −11.85			1−0.5	−12.35	0.83	0.87
c	dø lɔʁ.di.na.tœʁ			1 − l5 −8.55	z1 + z2 −5.4				−13.95	0.17	0.13
d	døs ʔɔʁ.di.na.tœʁ			1 − z1 −11.1	z2 + l5 −7.95		1−3	1−0.5	−22.55	0	0
e	døs lɔʁ.di.na.tœʁ			(1 − z1) + (1 − l5)−19.65	z2 −1.5		1−3		−24.15	0	0
f	døs zɔʁ.di.na.tœʁ			(1 − z1) + (1 − z2)−24.6	l5 −6.45		1−3		−34.05	0	0
g	dø.zɔʁ.di.na.tœʁ		1−15	1 − (z1 + z2)−9.6	l5 −6.45	1−10			−41.05	0	0

The optimal candidate is highlighted in bold.

The model predicts that 83% of occurrences correspond to glottal stop insertions (candidate b) and 17% to /l/-insertions (candidate c), closely matching the distribution found in the empirical study. Compared with trois écoles, the activation values for z₁ and z₂ are very similar, whereas the activation for l₅ is considerably lower before ordinateur. This difference increases the Dep.C penalty for candidate (c), which reduces its overall harmony and allows variation with glottal stop insertions to occur. Overall, these results demonstrate that the model can also perfectly capture items where true variation of realisation types was found in the data.

5.3. Overview of Constraint Weights and Activity Values

Finally, we provide an overview of the analysis across the three groups. Comparing the constraint weights of the learner groups with those of the target group (Table 9) reveals significant shifts from the presumed beginning to the end of the acquisition process. The most important changes are logically consistent with what is known about syllable structure in German and French: German allows more syllables with codas and frequently inserts glottal stops in onsets, resulting in lower weights for NoCoda and Dep.C compared with French. Furthermore, alignment of morpheme and syllable boundaries and uniformity constraints appear to be more influential, although these tendencies require further confirmation.

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Table 9.

Comparison of Constraint Weights for the Two Learner Groups and the Target Group.

Group(s)	Align	Dep.C	Dep.ʔ	Max	NoCoda	Onset	Unif
Learner groups	−15	−15	−0.5	−15	−3	−15	−10
Target group	−0.5	−10	−15	−10	−15	−15	−0.5

The two learner groups are primarily differentiated by activity levels, which are summarised in the following table (Table 10):

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Table 10.

Comparison of Activity Values for the Two Learner Groups and the Target Group.

Groups	LCs	Special LCs (in six, huit, dix)	Nonliaison consonants (e.g., /l/)
‘only glottal stop insertion’	low values (frequency-based values divided by 10)	fully activated	low values (frequency-based values divided by 10)
‘glottal stop or /l/-insertion’	frequency-based values	high values (frequency-based values multiplied by 1.5)	frequency-based values
‘target-like realisation’	frequency-based values	frequency-based values	low values (frequency-based values divided by 10)

The values are consistent with our hypotheses presented above: in the group with only glottal stop insertions, LCs and non-LCs generally show low activity, whereas special LCs (e.g., in six, huit, and dix) are fully activated. In the group with glottal stop or /l/-insertion, frequency-based activity values are used, with special LCs showing higher activation (multiplied by 1.5), which could represent an emerging development from fully activated consonants to the frequency-based values. The target group displays frequency-based values for LCs and special LCs, while non-LCs remain low. All deviations from frequency-based activity values should be understood as approximations, aimed at representing general trends in learner realisations rather than exact activation for each consonant.

These results suggest that not only the final target-like state but also error patterns observed in learner realisations, can be explained by the occurrence rate of LCs and other consonants in specific positions. Overall, the model demonstrates that the different realisation types produced by L2 learners can be effectively captured within the GHG framework when activity values are at least partially derived from corpus-based frequencies.