1.1. Locality relations in long-distance phonotactics

The cross-linguistic typology of consonant harmony (Hansson, 2010; Rose & Walker, 2004) reveals a dichotomy with respect to locality relations. Attested harmony patterns are either strictly transvocalic, affecting only those consonant pairs that fall within an adjacent-syllable window (…CV.C…), or else unbounded, holding at any distance within the relevant domain, even when other consonants intervene. The sibilant harmony found in several languages of the Omotic family (a branch of Afro-Asiatic), spoken in southern Ethiopia, illustrates the distinction. Transvocalic locality is exemplified by Koorete (a.k.a. Koyra; North Omotic). As shown in (1), the causative suffix /-us/, which surfaces faithfully in (1a), harmonizes with a preceding [–anterior] sibilant when a single vowel intervenes, as in (1b). However, harmony is not enforced across greater distances (i.e., when another consonant intervenes), as seen in (1c).

In the related language Aari (South Omotic), by contrast, this sibilant harmony holds in beyond-transvocalic contexts as well. This is seen in the examples in (2), which involve the perfective suffix /-s/ (cf. /ɡiʔ-s-it/ → [ɡiʔ.sit] ‘I hit,’ /ɡiʔ-er-s-it/ → [ɡi.ʔer.sit] ‘I was hit’; Hayward, 1988, p. 290).

There are two competing interpretations of the locality criterion that defines transvocalic harmony. A common characterization is in terms of syllable adjacency: Harmony applies when the consonants are tautosyllabic or in adjacent syllables, but not further apart (Bennett, 2015a, 2015b; Odden, 1994; Piggott, 1996; Rose & Walker, 2004; Walker, 2000; cf. also Suzuki, 1998; Wayment, 2009). An alternative criterion, implicit in the term ‘transvocalic’ used here, is consonant-tier adjacency: Harmony applies if and only if no other consonant intervenes (Hansson, 2010; Jurgec, 2011; McMullin, 2016; McMullin & Hansson, 2015). Clear-cut evidence bearing on the choice between these interpretations is hard to come by (though consonant-tier adjacency seems slightly better supported; Hansson, 2010, pp. 176–178). We will largely side-step this issue in what follows. For example, our stimuli all consist of strings of CV syllables; all consonant-tier-adjacent C…C sequences are thus also syllable-adjacent and vice versa.

1.2. Explaining the typology

The transvocalic versus unbounded dichotomy amounts to an implicational universal for long-distance assimilation. As the schematic overview in Table 1 shows, interaction at beyond-transvocalic distances entails interaction in transvocalic contexts, but not vice versa. Thus, consonant harmony never follows a strictly beyond-transvocalic locality pattern, whereby consonants in …CV.C… contexts are allowed to disagree in the feature in question while assimilation is enforced for more distant consonant pairs.

Explanations for this typological gap can potentially be sought in two distinct domains. The absence of strictly beyond-transvocalic harmony could be due to such patterns being diachronically inaccessible by the processes through which non-adjacent assimilatory interactions develop; we consider this hypothesis in Section 1.2.1. Alternatively (or in addition), such sound patterns may be synchronically disfavoured or impossible. That is, with respect to the hypothesis space or heuristics available to learners, such patterns are either entirely inaccessible or vulnerable to reinterpretation (imperfect learning) and thus inherently unstable. The hypothesized learning bias in question could be either categorical or probabilistic, and either domain-specific (perhaps to phonological patterns specifically; see Heinz & Idsardi, 2011; Lai, 2012) or attributable to some domain-general aspect of human pattern recognition. In Sections 1.2.2 and 1.2.3 we discuss two different approaches to synchronic explanations along these lines.

1.3. Harmony and dissimilation: Diverging predictions

The preceding discussion has revolved entirely around non-adjacent consonant assimilation (consonant harmony). One question raised in recent research is to what extent the same generalizations hold for non-adjacent dissimilation. For various segment types, both kinds of interaction are attested. For example, liquid harmony is found in Bukusu (Bantu; Hansson, 2010; Odden, 1994), where one of its manifestations is the applicative suffix /-il/ surfacing as [-ir] when the preceding stem contains [r] (e.g., /-rum-il-a/ → [-rumira] ‘send for,’ cf. /-xam-il-a/ → [-xamila] ‘milk for’). Liquid dissimilation is robustly attested as well, for instance in Georgian (Kartvelian; Fallon, 1993; Odden, 1994), where the ethnonym suffix /-uri/ surfaces as [-uli] when the preceding stem contains [r] (e.g., /aprikʼ-uri/ → [aprikʼuli] ‘African,’ cf. /dan-uri/ → [danuri] ‘Danish’). In the Bukusu case, [r…l] sequences are avoided in favour of [r…r], while the exact opposite is true for Georgian.

With respect to the characterization of long-distance phonotactic dependencies as TSL₂ stringsets, harmony and dissimilation are equivalent as regards locality relations. A TSL₂ grammar specifying R = {*rr, *ll} and T = {l, r} amounts to an unbounded dissimilation pattern, banning [r…r] and [l…l] co-occurrences at any distance, regardless of what intervenes. If the tier is defined as including not just liquids but all consonants, T = {l, r, p, t, k, …}, a transvocalic dissimilation pattern is generated: [rVr] and [lVl] sequences are banned, whereas [r…r] and [l…l] are permitted as long as at least one consonant intervenes (e.g., grammatical [ri.pa.ru] versus ungrammatical *[pa.ri.ru]). To translate these TSL₂ grammars into ones generating unbounded and transvocalic liquid harmony, respectively, one need only define the set of prohibited bigrams as R = {*rl, *lr} (cf. the sibilant harmony example described earlier).

In the ABC model, however, where dissimilation is interpreted as ‘correspondence avoidance’ (Bennett, 2015b), the equivalence between harmony and dissimilation breaks down. As harmony (agreement in [F] between correspondents) presupposes a correspondence relation between the two consonants, any restrictions on its application are expected to confine harmony to contexts where correspondence is favoured. Dissimilation, on the other hand, emerges in ABC as a means of evading the need for establishing a correspondence relation in the first place; reducing the similarity of the two consonants renders them free to not stand in correspondence. Dissimilation is therefore expected to gravitate toward contexts where correspondence is disfavoured (or where it is not explicitly required). One corollary of this viewpoint is that various mismatches are predicted to hold between the typology of harmony and that of dissimilation. Bennett (2015b) provides cross-linguistic support for several such mismatch predictions, particularly relating to structural factors (e.g., harmony tending to occur within domains, dissimilation across domain edges).

One such asymmetry between harmony and dissimilation is expected to arise with respect to locality relations. For consonant harmony, strictly transvocalic locality can arise in ABC by either of two mechanisms, as noted in Section 1.2.2. First, a high-ranked constraint demanding correspondence regardless of distance (for the relevant consonant types) may be overridden by even higher-ranked PROXIMITY/CC-SYLLADJ, which explicitly bans correspondence except for …CV.C… consonant pairs. Alternatively, the high-ranked constraint that demands correspondence may itself be one that singles out …CVC… pairs specifically. Either way, the result will be the same: The only consonant pairs subject to harmony (i.e., featural agreement under correspondence) are transvocalic ones. In the case of dissimilatory interactions, these same formal mechanisms that give rise to transvocalic harmony can instead have the effect of limiting dissimilation to consonant pairs beyond the correspondence-favouring …CV.C… window. In this way, the ABC model can generate a strictly beyond-transvocalic locality pattern for dissimilation.

In (3)–(4) we show how this results from the correspondence-delimiting constraint PROXIMITY/CC-SYLLADJ. The constraint CORR-LIQUID[αlateral] demands correspondence for pairs of co-occuring identical liquids ([r…r], [l…l]), regardless of distance, while lower-ranked CORR-LIQUID does the same for all liquid pairs (also [r…l], [l…r]); co-indexation denotes correspondence. IDENT-IO[±lateral] penalizes changes to [±lateral] input specifications.

Furthermore, a transvocalic pair of liquids that already disagrees for [±lateral] will surface faithfully, as shown in (5); the winning candidate does not violate any of the four constraints. The ranking shown here therefore results in a language where the co-occurrence of liquids is unrestricted in transvocalic contexts, while liquid dissimilation is enforced at beyond-transvocalic distances.

Strictly beyond-transvocalic dissimilation can likewise result from the other formal mechanism that generates strictly transvocalic harmony: Special locality-restricted versions of CORR constraints, such as CORR-LIQ_CVC or CORR-LIQ[αlat]_CVC alongside their unrestricted counterparts shown in the above tableaux. To see how this is true, we must first introduce another, independently motivated aspect of the ABC constraint repertoire. In order to derive truly unbounded dissimilation in ABC (e.g., affecting [r…r] and [l…l] regardless of distance or other contextual factors), some constraint or constraints must exist whose effect is to ban correspondence in all contexts. For example, this can result from the combined forces of CC-EDGE(σ), which requires correspondents to be tautosyllabic, and CC-SROLE, which requires correspondents to occupy matching syllable constituents (onset…onset, coda…coda); for justification for both of these, see Bennett (2015b). Since these demands are mutually contradictory (at least for non-adjacent consonants), the net result of enforcing both of these constraints is that no two consonants can ever stand in correspondence. In the tableaux below, we will use the shorthand label CC-NOCORR to represent such an across-the-board ban on correspondence.

With these ingredients in place, the tableaux in (6)–(7) show how, given the right constraint ranking, liquid dissimilation will be enforced in beyond-transvocalic contexts while leaving transvocalic liquid-liquid pairs unaffected. Tableau (8) further shows that transvocalic pairs are unrestricted; they surface faithfully regardless of whether they agree (6) or disagree (8) in the input.¹

In sum, we have a set of diverging predictions as regards the possibility, and hence learnability, of different locality patterns in non-adjacent consonant assimilation and dissimilation, summarized in Table 2. It should be noted that the robustly attested pattern of strictly transvocalic dissimilation can only be generated in ABC if the constraint set includes locality-restricted CORR constraints such as CORR-LIQ[αlat]_CVC or CORR-LIQ_CVC; Bennett (2015b) argues for the inclusion of such constraints (alongside the PROXIMITY/CC-SYLLADJ constraint) for precisely this reason. It is important to note, however, that regardless of whether the ABC constraint repertoire includes both of these constraint types or just one of the two, strictly beyond-transvocalic dissimilation is generated as part of the factorial typology, and is hence predicted to be part of the learner’s hypothesis space.

From the point of view of language-theoretic complexity, if long-distance phonotactic dependencies are indeed confined to the TSL₂ region, we expect strictly beyond-transvocalic locality to be impossible for assimilation and dissimilation alike. We note that the same prediction holds true for non-correspondence-based generative analyses of dissimilation in terms of segment co-occurrence constraints (such as the Obligatory Contour Principle, e.g., OCP[αlateral]), where the transvocalic versus unbounded distinction is encoded as a parameter on the relevant autosegmental rule (Odden, 1994; Piggott, 1996) or markedness constraint (Pulleyblank, 2003; Suzuki, 1998). Alternatively, if non-TSL₂ patterns are within the reach of human phonotactic learners (contra Lai, 2015), then we still have no reason to expect assimilation and dissimilation patterns to differ. In Table 2, this hypothesis is labelled L(T)T, referring to the more complex Locally Testable or Locally Threshold Testable regions of the subregular hierarchy, within which strictly beyond-transvocalic dependencies would fall. Finally, the ABC model predicts an asymmetry: Strictly beyond-transvocalic locality should be learnable for dissimilation but not assimilation.

1.4. Testing the hypotheses: Learning experiments

In evaluating the predictions laid out in Table 2, the cross-linguistic record is of limited help. The only reported case that approximates a strictly beyond-transvocalic pattern is liquid dissimilation in Sundanese (Bennett, 2015a; Cohn, 1992). Here the plural infix /-ar-/ surfaces as [-al-] in the presence of another [r] (9a), but not in the transvocalic configurations of (9b)–(9c).

However, Sundanese is not entirely persuasive as evidence of strictly beyond-transvocalic dissimilation, due to a number of complicating factors. For one thing, dissimilation does apply within the transvocalic window, provided that the two liquids occupy different syllable constituents (onset versus coda), as in (9d). Secondly, within the transvocalic window there is also liquid assimilation under some conditions (9e) but not others (9f). Although Bennett (2015a) presents a complete analysis in ABC terms, other interpretations are possible. For example, Stanton (2018) argues, based on quantitative lexical data, that the full pattern reflects an essentially unbounded dissimilation of [r…r] sequences which is overridden by a pattern of aggressive reduplication (Zuraw, 2002) holding for pairs of adjacent syllables (especially word-initial σ₁σ₂ pairs).

We must therefore turn to other methods to investigate whether strictly beyond-transvocalic locality is learnable and whether dissimilation and assimilation differ in this respect. In situations like these, where the cross-linguistic typology is either too sparse to distinguish systematic gaps from statistical accidents or inherently ambiguous as to the source of those gaps (see Section 1.2.1), artificial grammar learning (AGL) experiments are a particularly helpful tool (Culbertson, 2012; Finley, 2017; Moreton & Pater, 2012a, 2012b).

Certain aspects of the question have been investigated using AGL methods. For sibilant harmony over the (English) [s]-[ʃ] distinction, Finley (2011, 2012) found that English adults seem to interpret interaction in beyond-transvocalic contexts as evidence of an unbounded dependency. When exposed to stems of the shape SVCV, CVSVCV, or SVCVCV (where S is [s] or [ʃ]), together with affixed forms containing a (harmonic) suffix element [-su]∼[-ʃu], learners generalized the dependence of suffix choice on stem sibilant to unfamiliar trigger-target distances, and crucially to transvocalic configurations like (CV)CVSV-Su. McMullin and Hansson (2014) replicated this finding in a design where two distinct suffixes ([-si], [-ʃu]) each triggered a harmony alternation in the preceding stem. Again, participants exposed to evidence of assimilation in beyond-transvocalic CVSVCV-SV contexts extended this to transvocalic CVCVSV-SV. By contrast, both studies found that participants exposed only to items with harmony in transvocalic (CV)CVSV-SV contexts did not generalize this to beyond-transvocalic distances. This fits the predictions of both the TSL₂ hypothesis and the ABC model, but does not distinguish between the two.

Below we report on four AGL experiments that extend this line of inquiry in two important ways. Firstly, we examine learning of co-occurrence patterns at differing locality levels for assimilation and dissimilation alike, allowing comparison of the two. This makes sibilant-sibilant interactions like those featured in previous studies less suitable; whereas sibilant harmony is by far the most frequently attested type of long-distance consonant assimilation (Hansson, 2010), analogous dissimilations (e.g., /ʃ…ʃ/ → [s…ʃ]) are virtually unattested. In the interest of cross-linguistic plausibility, we avoid this typological asymmetry (and any substantive biases that might contribute to it) by using co-occurrence patterns involving liquids [l, r]. As illustrated above (Section 1.3), liquid harmony and liquid dissimilation are both robustly attested.

Secondly, we go beyond previous studies in manipulating the richness of the training data that participants encounter. In Experiments 1 and 2, which target assimilation and dissimilation, respectively, we follow Finley (2011, 2012) in adopting a ‘poverty of stimulus’ paradigm (Finley & Badecker, 2009; Wilson, 2006). Certain critical configurations are withheld from the exposure data, and the test phase examines whether learners generalize the observed pattern to those novel contexts, e.g., from beyond-transvocalic to transvocalic distance. In Experiments 3 and 4, we adopt a ‘rich-stimulus’ paradigm, which directly probes the learnability of the sound pattern under dispute, such as a strictly beyond-transvocalic dependency (cf. Lai, 2015 on first-last harmony, White, 2014 on saltational alternations). Here the exposure data are augmented so as to make their full range compatible only with the pattern at stake (e.g., containing evidence not only of interaction in beyond-transvocalic contexts but also of non-interaction in transvocalic contexts). If learners fail to learn the target pattern even when provided with such ‘rich’ learning data, or if they go against the data by inferring a simpler pattern (e.g., an unbounded dependency), then this is especially persuasive evidence of a strong learning bias against the type of pattern in question.

The remainder of this paper proceeds as follows. In Section 2, we describe aspects of the methodology that apply to all four experiments. Section 3 then summarizes the training conditions for each of the experiments and lays out the full range of predictions under the ABC and TSL₂ hypotheses. The results of Experiments 1 and 2 are presented in Sections 4 and 5. After motivating the use of a ‘rich-stimulus’ training regime in Section 6, the results of Experiments 3 and 4 are presented in Sections 7 and 8. Finally, Section 9 discusses the overall findings and evaluates the competing hypotheses, and Section 10 concludes.

3.1. Experiments 1 and 2

The purpose of Experiments 1 and 2 is to ensure that we can extend previous findings about sibilant harmony (Finley, 2011, 2012; McMullin & Hansson, 2014) to liquids, and to thereby verify that the current design is adequate for a direct comparison of long-distance consonant agreement and disagreement. Experiments 1 and 2 therefore use a ‘poverty of stimulus’ training paradigm, in which the training phase for each group includes direct evidence of the target dependency either at the Medium-range distance or the Short-range distance. However, participants are given no information as to the behaviour of co-occurring liquids at any other distance level. Throughout the remainder of this paper, we use the following labels to refer to each of these experimental groups: M-Harm (for Medium-range harmony), S-Harm (Short-range harmony), M-Diss (Medium-range dissimilation), and S-Diss (Short-range dissimilation). A summary of the types of items that were presented to and withheld from each group during training is presented in Table 5.

Experiment 1 predictions Based on the findings of Finley (2011, 2012), the M-Harm group is predicted to learn an unbounded pattern that applies at each of the three testing distances: Short-range, Medium-range, and Long-range, whereas the S-Harm group is predicted to restrict the pattern to Short-range contexts (even though their training data is in principle consistent with an unbounded pattern as well). These results would be compatible with both the ABC and TSL₂ hypotheses as laid out in Section 1.3.

Experiment 2 predictions In the ABC model, the same constraints that help generate strictly transvocalic harmony will, given certain ranking configurations, generate strictly beyond-transvocalic dissimilation, as illustrated in (3)–(5) and (6)–(8) above. By the ABC hypothesis, a strictly beyond-transvocalic dependency is therefore within reach of the learner, but only if the pattern is a dissimilatory one. If learners are reluctant to generalize to contexts not encountered in training (unless such generalization is the only option available in the hypothesis space), then we should expect the M-Diss group to infer that the pattern is strictly confined to beyond-transvocalic contexts (Medium- and Long-range test items, but not Short-range ones). If M-Diss participants instead infer an unbounded dependency (generalizing to Short- and Long-range items), then such a result could only be reconciled with the ABC hypothesis through additional assumptions about how learners traverse the space of available grammars, e.g., through built-in priors on the relative ranking/weighting of constraints (we take up this question in Section 9.2.1 below).

The results of the S-Diss group do not bear on our comparison of the TSL₂ and ABC hypotheses. However, as noted in Section 1.3, the ABC literature employs two different types of locality-based constraints: those that penalize correspondence in non-adjacent syllables (PROXIMITY/CC-SYLLADJ) and those that enforce correspondence only in CVC contexts (e.g., CORR-LIQ_CVC). If the S-Diss group does not generalize to longer distances, this could be taken as support for the inclusion of the latter constraint type, as these are required to generate a strictly transvocalic dissimilation pattern within the ABC model.

By contrast, there is no substantive difference between TSL₂ characterizations of agreement (harmony) and disagreement (dissimilation). From this perspective, participants in the M-Diss and S-Diss conditions are predicted to behave the same way as those in the M-Harm and S-Harm conditions of Experiment 1, respectively: The M-Diss group should infer an unbounded pattern, generalizing to Short- and Long-range contexts, and the S-Diss group should learn a strictly transvocalic pattern, failing to generalize beyond the Short-range context.

3.2. Experiments 3 and 4

The purpose of Experiments 3 and 4 is to determine whether learners can detect strictly beyond-transvocalic dependencies when their training exhibits unambiguous evidence of such a pattern. To achieve this, we implement a ‘rich stimulus’ paradigm, augmenting the training data with direct evidence that the target pattern is not enforced at all distances.

The four new experimental groups are labelled M-Harm-S-Faith, S-Harm-M-Faith, M-Diss-S-Faith, and S-Diss-M-Faith. These labels follow the conventions established for previous experiments: ‘S’ indicates Short-range (CVCVLV stems), ‘M’ indicates Medium-range (CVLVCV stems), ‘Harm’ indicates that the target pattern at that distance was harmony, and ‘Diss’ indicates a target pattern of dissimilation. To further differentiate these groups from those in Experiments 1 and 2, the distance that showed overt evidence of non-alternation (faithful preservation of the stem liquid in the suffixed forms) is indicated by ‘S-Faith’ or ‘M-Faith’ at the end of each group’s label. A summary of the types of training items included for each of the training groups is given in Table 6.

Experiment 3 predictions Both the ABC and TSL₂ accounts of consonant harmony allow for transvocalic and unbounded locality, but neither can generate a strictly beyond-transvocalic assimilation pattern. As such, they do not differ with respect to their predictions about Experiment 3. Specifically, the training data for the S-Harm-M-Faith group is consistent with strictly transvocalic harmony, and is therefore expected to be learned as-is. However, since the training data for the M-Harm-S-Faith group is not compatible with unbounded or strictly transvocalic locality, both the TSL₂ and ABC hypotheses predict that participants will deviate from the training pattern during the test phase. This expected bias against strictly beyond-transvocalic locality could manifest itself in one of two ways. First, participants could force a pattern of liquid harmony into the TSL₂/ABC hypothesis space by learning an unbounded pattern of liquid harmony (over-generalizing into Short-range contexts). Alternatively, if the pattern presented in training is simply too complex for participants to detect any systematic dependency, they should behave indistinguishably from the Control group, either selecting the faithful (non-alternation) response option throughout the test phase, for all three distances, or selecting items at random that illustrate liquid alternations.

Experiment 4 predictions Note that the training data in the M-Diss-S-Faith condition adheres to a pattern of strictly beyond-transvocalic dissimilation. Moreover, since disagreement is not enforced in transvocalic contexts, the training pattern cannot be interpreted as an unbounded dependency. The ABC and TSL₂ hypotheses therefore differ from each other in their predictions for Experiment 4.

Since strictly beyond-transvocalic dissimilation can be generated in the ABC model (see Section 1.3), participants in the M-Diss-S-Faith condition are predicted to be able to learn the pattern as presented in training. Conversely, all dependencies with strictly beyond-transvocalic locality (agreement and disagreement) fall outside of the TSL₂ region, suggesting that they should be inaccessible to learners. The TSL₂ hypothesis therefore predicts that participants in the M-Diss-S-Faith group will fail to learn the pattern exhibited in training, either over-generalizing the dependency to Short-range testing items or learning no discernible pattern whatsoever (as is predicted under both hypotheses for the M-Harm-S-Faith group in Experiment 3; see above).

4.1. Participants

Fifty-five participants (42 female, 13 male, mean age 24.4) took part in Experiment 1. The first 48 of these were distributed across the three training conditions described below, 16 in each group. The remaining seven were recruited in order to obtain a minimum number of ‘successful learners’ in each experimental group, as described in Section 4.3.2.

4.2. Training conditions

Participants were assigned to one of three groups that differed in the type of verb stems encountered in training: two experimental groups (M-Harm and S-Harm), and a Control group. Recall that the training phase consisted of 192 triplets (a trisyllabic verb stem, followed by its two suffixed forms with [-li] and [-ɹu]).

For the experimental groups, half of the stems were distractors containing no liquids (e.g., [dutebi], [dutebi-ɹu], [dutebi-li]). The remaining half of the training stems were of the form CVLVCV for the M-Harm group (where L represents a liquid [l] or [ɹ]) but CVCVLV for the S-Harm group. Participants were exposed to a liquid harmony either across an intervening VCV sequence (M-Harm group; e.g., [pilede], [pilede-li], [piɹede-ɹu]) or across a single vowel (S-Harm group; e.g., [gutoɹo], [gutolo-li], [gutoɹo-ɹu]). For each of these stems, one of the two suffixes triggered an alternation resulting in harmony, whereas the other suffixed form obeyed harmony by morpheme concatenation alone. Training triplets were divided into two blocks, and randomized within block for each participant. Examples of training items for the experimental groups are provided in Table 7.

Participants in the Control group completed the same amount of training, but were not exposed to any stems containing liquids, and therefore saw no evidence for or against liquid harmony. Instead, in each block they encountered the full set of 96 triplets with non-liquid CVCVCV stems (and were thus exposed to each triplet twice). The Control group serves as a baseline for comparison in the statistical analyses, to help eliminate confounding effects of the experimental design, stimuli, or L1 influence (e.g., from English lexical statistics).

4.3. Results and analysis

We present two statistical analyses of the Experiment 1 results. There is precedent for both approaches, and each has advantages and drawbacks. First, in Section 4.3.1, we analyze the responses of a predetermined number of participants (the first 16 in each group), without regard for whether these individuals showed evidence of learning the patterns they were exposed to. This method, common in AGL experiments using a poverty-of-stimulus paradigm (e.g., Finley, 2011, 2012; Finley & Badecker, 2009; Lai, 2015; Wilson, 2006), ensures that the data for each group are sampled from the same population. However, it somewhat obscures the question of interest: whether learners generalize from the types of contexts they encountered in training to novel contexts (e.g., for the M-Harm group, from Medium-range CVLVCV-LV to Short-range CVCVLV-LV). The very notion of ‘generalization’ presupposes that the exposure pattern has been learned in the first place, but this is not a given. Rather, individual performance often tends toward a bimodal distribution between successful learners and non-learners, instead of clustering around the group mean.

To address this problem, non-learners can be filtered out based on some threshold criterion for successful learning. Some studies (e.g., White, 2014) include a screening phase, such that only those who surpass the threshold continue to the testing phase; participants may be cycled repeatedly through the training and screening phases until they meet the criterion (within some time limit). While this method has the benefit of ensuring that each participant is adequately trained, different participants receive different amounts of training. This introduces a potential confound, e.g., if participants in one condition require (and hence receive) more training than those in another condition. Furthermore, as the intermittent screening phases involve 2AFC pairings of a ‘correct’ and an ‘incorrect’ form, they inevitably expose participants to ungrammatical data prior to the testing phase.

We instead examine the testing-phase responses from each participant post hoc, and classify as successful learners those who reliably instantiated a pattern of liquid harmony at the distance at which L…L sequences were present in their training data (Short-range for S-Harm, Medium-range for M-Harm). This approach allows us to investigate group-level effects among participants who all received the exact same amount of training. A predetermined number of successful learners (12 in each experimental group) was set as a target, and additional participants were recruited until that target was reached. In Section 4.3.2 we analyze the data from the first 12 successful learners in each experimental group, as compared against the first 12 participants in the Control group. This method ensures that all participants received an equal amount of training exposure; however, we cannot be certain that the results from each group truly represent the same population. We discuss these issues in Section 4.4.

4.3.1. Analysis of first 16 participants

A mixed-effects logistic regression model was fitted to the responses from the first 16 participants in each group of Experiment 1 (48 participants; 36 female, 12 male, mean age 23), using the glmer function in the lme4 package (Bates, Mächler, Bolker, & Walker, 2015) of R (R Core Team, 2017).³ The categorical dependent variable is whether, on a particular trial, the participant chose the response item that exhibited liquid harmony. The fixed-effects component, summarized in Table 8, includes as main effects the between-participants variable of Group (S-Harm and M-Harm, compared to Control) and the within-participants variable of trigger-target Distance in the test item (Medium- and Long-range, compared to Short-range), as well as an interaction between Group and Distance. Also included are two nuisance variables that significantly improved model fit: Harmony Second (whether the form with harmony was presented as the second of the two 2AFC items) and Harmony Faithful (whether the form with harmony involved mere morpheme concatenation, as opposed to an alternation in the stem liquid). The structure of the random-effects component was determined by the procedure recommended in Bates, Kliegl, Vasishth, and Baayen (2015), and included: random by-item and by-participant intercepts; random by-participant slopes for Harmony Second, Harmony Faithful, and Long-range distance; and random by-participant terms for the interactions between Long-range distance and each of Harmony Second and Harmony Faithful (as well as some correlation parameters). Full details are provided in the Appendix.

Table 8

Fixed-effects component of the logit mixed model for Experiment 1 (N = 4518; log-likelihood = –2184.2).

Coefficient	Estimate	SE	Pr (>|z|)
Intercept	0.43987	0.12132	0.0003
Harmony Second	–0.51084	0.14367	0.0004
Harmony Faithful	2.61950	0.30531	<0.0001
Medium-range	0.08113	0.18412	0.6595
Long-range	–0.16051	0.25510	0.5292
S-Harm	1.30893	0.38182	0.0006
M-Harm	1.25831	0.38503	0.0011
Medium-range × S-Harm	–1.00769	0.23162	<0.0001
Long-range × S-Harm	–0.95505	0.34226	0.0053
Medium-range × M-Harm	0.28217	0.23734	0.2345
Long-range × M-Harm	–0.38150	0.34577	0.2699

To reduce collinearity, all predictor variables were centered. As a result, the Intercept term reflects the grand mean across all groups, distances, etc.; overall, the odds for choosing the 2AFC option with harmony (matching liquids) were thus e^0.43987 = 1.55 to 1, translating into a probability of 1.55/(1 + 1.55) = 0.61. For each predictor, the coefficient estimate indicates the difference in log-odds between the two factor levels being compared. For instance, the odds of selecting the harmony option when that choice is presented second are only e^–0.51084 = 0.60 times those of doing so where harmony is presented first. The large estimate for Harmony Faithful indicates that participants are much more likely to select harmony on trials where that option involves a faithful stem liquid (e.g., [tuluge … tuɹuge-li, tuluge-li]) than where it involves an alternation (e.g., [pidole … pidoɹe-ɹu, pidole-ɹu]); the odds ratio is e^2.6195 = 13.73.

Because the model includes interaction terms for (the non-reference levels of) Group and Distance, the main-effect estimates for these predictors reflect the difference relative to the reference level of the factor in question (Control for Group, Short-range for Distance). For example, the non-significant effects for Medium- and Long-range show that the Control group is not reliably more or less likely to choose harmony on test trials involving these distances, as compared to trials with Short-range items. The significant main effects of S-Harm and M-Harm show that each experimental group is more likely than the Control group to choose harmony at the Short-range distance. The significant interactions of Medium- and Long-range distance with the S-Harm group show (with negative estimates) that this group applies harmony less often outside of the Short-range window than within it. By contrast, neither of the interactions of Medium- and Long-range distances with this group approach significance, suggesting that the M-Harm group treats all three distances equally. A likelihood ratio test shows that inclusion of the Group × Distance interaction contributes significantly to the model fit (χ²(4) = 37.02, p < 0.0001).⁴

However, a comparison between distances within each experimental group does not answer the question of primary interest. Rather, we wish to determine, for each of the three distances, whether or not a given experimental group is significantly more likely than the Control group to choose harmony. This information can be obtained from the estimates for S-Harm and M-Harm in Table 8, but only for the baseline Short-range distance. We therefore re-fitted the same model, but with each of Medium-range and Long-range serving as the reference level for Distance, allowing for a direct comparison of groups at those distances as well. The combined results are presented in Table 9. Each cell shows an odds ratio (OR); for example, the (estimated) odds that a S-Harm group member would choose the harmony-obeying option for a Short-range test item (CVCVLV-LV) are more than three times (3.70) those of a Control participant doing so. The odds ratios in Table 9 were obtained by exponentiating the coefficient estimates for S-Harm and M-Harm in the relevant version of the model. Thus, for example, the OR values in the Short-range column correspond to the estimates reported in Table 8 (e^1.30893 = 3.70, e^1.25831 = 3.52).

Table 9

Experiment 1: Odds ratios for choosing harmony at each testing distance, comparing experimental groups to Control group (first 16 participants in each). Cells corresponding to contexts encountered in training are shaded; significance is indicated with boldface.

	Type of test item (trigger-target distance)
	Short-range (CVCVLV-LV)	Medium-range (CVLVCV-LV)	Long-range (LVCVCV-LV)
M-Harm versus Control	3.52	4.67	2.40
	p = 0.001	p < 0.001	p < 0.001
S-Harm versus Control	3.70	1.35	1.42
	p < 0.001	p = 0.430	p = 0.106

The Experiment 1 results are displayed visually in Figure 1, which depicts the mean proportion of harmony responses for each group at each testing distance, as well as the proportions for individual participants. The plot and odds-ratio table both illustrate that the experimental groups showed evidence of learning the harmony pattern at the trigger-target distance encountered in training (dashed-line boxes in Figure 1, shaded cells in Table 9). Compared to the Control group, the M-Harm and S-Harm groups displayed significantly greater odds (4.67 and 3.70 times, respectively) of choosing the harmony response on test items of the relevant type.

Figure 1

Proportions of harmony responses across conditions (groups) and test-item type (distance) for Experiment 1. Coloured dots show individual participant performance, with group means indicated by an open circle with error bars (showing standard error); significant between-group differences are extracted from the logit mixed model. Performance on contexts encountered in training is shown with dashed-line boxes.

As for generalizing that pattern to novel contexts, the M-Harm group chose harmony significantly more often than the Control group at the unfamiliar Short-range (OR = 3.52) and Long-range (OR = 2.40) distances. This indicates that the M-Harm group tends to interpret the pattern as an unbounded liquid harmony, generalizing from an observed assimilation in Medium-range CVLVCV-LV contexts both inwards to Short-range CVCVLV-LV and outwards to Long-range LVCVCV-LV contexts. By contrast, the S-Harm group does not generalize the pattern of liquid harmony from the Short-range distance to Medium-range (OR = 1.35, p = 0.430) or Long-range (OR = 1.42, p = 0.106) contexts.

4.4. Discussion of Experiment 1

To summarize, both analyses provide evidence that participants who are exposed to harmony in Medium-range contexts (spanning an intervening VCV sequence) tend to learn an unbounded dependency, generalizing both inwards to Short-range and outwards to Long-range contexts. By contrast, those exposed to consonant harmony in Short-range contexts (where no consonant intervenes) tend to interpret this conservatively: as a harmony requirement that targets only transvocalic consonant pairs. These findings conform to the typological dichotomy described in Section 1.1 above. They also replicate the findings of Finley (2011, 2012) and McMullin and Hansson (2014, Experiment 1), extending these to a different class of segments: liquids [l, ɹ] rather than sibilants [s, ʃ].

With respect to the differing predictions summarized in Section 3, these results are consistent with the TSL₂ hypothesis as well as the ABC hypothesis. Both incorporate the implicational universal that an assimilatory dependency at some beyond-transvocalic distance necessarily entails that the dependency is unbounded, holding at shorter and longer distances as well. Under the ABC and TSL₂ hypotheses, the training data encountered by the M-Harm group should be unambiguous evidence of unbounded harmony, while those seen by the S-Harm group are inherently ambiguous, equally compatible with a transvocalic or an unbounded dependency. Under the hypothesis that non-TSL₂ sound patterns in the Locally Testable (LT) or Locally Threshold Testable (LTT) regions are also within the range of human learners, the results of Experiment 1 (and those of the previous studies cited above) are unexpected. By the L(T)T hypothesis, the M-Harm training data should be no less ambiguous than the S-Harm data: Just as the latter could represent either a strictly-transvocalic pattern or an unbounded one, so could the former be evidence of either a strictly-beyond-transvocalic or an unbounded dependency. The fact that the S-Harm group tends to interpret their ambiguous training input conservatively, favouring a strictly-transvocalic interpretation, makes it all the more surprising that the M-Harm group behaves in the exact opposite way, favouring the less conservative unbounded interpretation over a strictly-beyond-transvocalic one. In summary, Experiment 1 provides strong support for the ABC and TSL₂ hypotheses over the L(T)T hypothesis, but does not distinguish between the former two.

Two final points are worth making regarding finer-grained aspects of the results. First, looking at the successful M-Harm learners, Table 10 shows a dramatic difference in the effect size for generalizing harmony to Short-range (OR = 12.89) versus Long-range (OR = 4.37) contexts. (The same trend, though more subtle, is visible in Table 9). This is likely due to an independent reluctance to extend a phonological alternation into the highly salient word-initial position involved in the Long-range LVCVCV-LV test items (cf. Becker, Nevins, & Levine, 2012; the same effect was found in Finley, 2012). Recall in this connection that all stimuli were produced with initial-syllable prominence, contributing further to the salience of the target liquids in these test items.

Secondly, individual participants sometimes deviate from their group trend; this is particularly notable for the S-Harm condition. As mentioned above, the odds ratio estimates in Tables 9 and 10 for the S-Harm versus Control comparisons on Medium- and Long-range test items suggest that some S-Harm participants are extending harmony to these longer distances, indicating that they inferred an unbounded pattern. Closer examination supports this interpretation: Three of the 12 successful S-Harm learners surpassed the threshold criterion on Medium-range items as well. None were above threshold on Long-range items, however, presumably due to the aforementioned general bias against extending alternation to a word-initial consonant. These observations are consistent with the hypothesis (see Section 1.2.1) that unbounded consonant harmony emerges diachronically as a secondary development through overgeneralization of an originally transvocalic pattern (Dolbey & Hansson, 1999).

5.1. Participants

Participants for two additional training conditions, M-Diss and S-Diss, were recruited and compensated as described above, yielding a total of 40 new participants for Experiment 2 (30 female, 10 male, mean age 24.1). The first 32 of these were distributed evenly across these two conditions, 16 in each. The additional eight were recruited in order to obtain 12 successful learners in each condition, just as in Experiment 1; see Section 5.3.2 for details.

5.2. Training conditions

All training stems were identical to those used in Experiment 1 (see Table 7 above). However, instead of liquid harmony, participants in the experimental groups were exposed to consistent liquid dissimilation in the relevant suffixed forms. The M-Diss group was exposed to 96 triplets that exhibited dissimilation in CVLVCV-LV contexts (e.g., [pilede], [piɹede-li], [pilede-ɹu]), while the S-Diss group saw 96 corresponding triplets with CVCVLV-LV. As before, the training phase for each group also included 96 stems with no liquid. The testing phase in Experiment 2 was identical to that of Experiment 1 (see Table 4 for examples).

5.3. Results and analysis

The Experiment 2 results are analyzed in the same way as those in Experiment 1. As the two experiments were designed and implemented in parallel, we use the response data from the same set of Control participants as in Experiment 1. An analysis of the first 16 participants in each group is presented in Section 5.3.1, followed in Section 5.3.2 by an analysis of the first 12 successful learners obtained in each group.

5.3.1. Analysis of first 16 participants

A mixed-effects logistic regression model was fitted to the responses of the first 16 participants in each condition (48 in total), with choice of the response option with two different liquids as the dependent variable. As with Experiment 1, the fixed-effects component includes the predictors of Group (M-Diss and S-Diss, compared against Control) and Distance (Medium- and Long-range, compared against Short-range), and an interaction between Group and Distance. The model again includes two nuisance variables, Dissimilation Faithful (whether the stem liquid in the option with disagreeing liquids was faithful to that of the bare-stem form) and Dissimilation Second (whether the option with disagreeing liquids was presented as the second 2AFC item). The random-effects structure was selected by the same procedure described for Experiment 1, and consists of random by-item and by-participant intercepts, random by-participant slopes for Dissimilation Second, Dissimilation Faithful, and Distance (Medium- and Long-range), and random by-participant terms for the interactions between Dissimilation Faithful and each of Medium- and Long-range distance (as well as a correlation parameter for these two interaction terms).

A likelihood ratio test shows that the Group × Distance interaction significantly improves model fit (χ²(4) = 79.57, p < 0.0001).⁷ Full details of the model, including coefficient estimates, are provided in the Appendix. Here we instead focus on the more informative odds-ratio comparisons between the experimental and Control groups at each test-item distance. As before, odds ratios are extracted from the model by exponentiating the coefficient estimates for the relevant Group predictor (M-Harm and S-Harm, respectively); to obtain such estimates for Medium-range and Long-range, the same model was re-fitted using each of these as the reference level of the Distance predictor (instead of Short-range). This comparison of each experimental group’s performance against that of the Control group is provided in Table 11. A visual representation of the results from Experiment 2 is provided in Figure 2.

Table 11

Experiment 2: Odds ratios for choosing harmony at each testing distance, comparing experimental groups to Control group (first 16 participants in each). Cells corresponding to contexts encountered in training are shaded; significance is indicated with boldface.

	Type of test item (trigger-target distance)
	Short-range (CVCLV-LV)	Medium-range (CVLVCV-LV)	Long-range (LVCVCV-LV)
M-Diss versus Control	4.19	4.08	1.48
	(p < 0.001)	(p < 0.001)	p = 0.218
S-Diss versus Control	10.08	1.57	0.78
	(p < 0.001)	p = 0.150	p = 0.423

Figure 2

Proportions of disharmony responses across conditions (groups) and test-item type (distance) for Experiment 2. Coloured dots show individual participant performance, with group means indicated by an open circle with error bars (showing standard error); significant between-group differences are extracted from the logit mixed model. Performance on contexts encountered in training is shown with dashed-line boxes.

Both experimental groups show evidence of learning. Overall, the M-Diss group enforces dissimilation in Medium-range contexts; they are more than four times (OR = 4.08) more likely than the Control group to choose dissimilation in CVLVCV-LV test items. Likewise, the S-Diss group is over ten times (OR = 10.08) more likely than the Control group to choose dissimilation in the CVCVLV-LV context that they encountered in training. The M-Diss group seems to generalize this pattern inwards to the Short-range test items (OR = 4.19); for Long-range (LVCVCV-LV) test items, the corresponding effect (OR = 1.48) is too small to reach significance. Finally, the S-Diss group does not display a significant preference for dissimilation in either Medium- or Long-range contexts.

5.4. Discussion of Experiment 2

The goal of Experiment 2 was to replicate Experiment 1 as faithfully as possible, but for dissimilatory as opposed to assimilatory dependencies. The purpose was to evaluate the competing TSL₂ and ABC hypotheses, with which the Experiment 1 results were equally compatible.

The S-Diss group clearly preferred dissimilation in Short-range test items, indicating learning of the exposure pattern, but did not extend this dissimilation preference to either Medium- or Long-range contexts. They thus appear to have inferred a strictly transvocalic dissimilation dependency. This result is in line with the predictions of the TSL₂ hypothesis as well as those of the ABC model. It is also consistent with the cross-linguistic typology of consonant dissimilation; transvocalic dissimilation is relatively common and need not imply dissimilation at greater distances. It is noteworthy that, just like the S-Harm group in Experiment 1, S-Diss participants tended to interpret their inherently ambiguous training data conservatively, preferring a strictly transvocalic interpretation over an unbounded one.

As for the M-Diss condition, both analyses show this group to be significantly more likely to choose disagreeing liquids not only for Medium-range but also Short-range items. This is consistent with the TSL₂ hypothesis, by which interaction in beyond-transvocalic contexts entails interaction in transvocalic contexts as well (i.e., an unbounded dependency). It is somewhat unexpected under the ABC hypothesis, by contrast, as that model predicts strictly beyond-transvocalic dissimilation to be a perfectly possible (and hence learnable) sound pattern. Given that participants in the S-Harm and S-Diss conditions tended to act conservatively, we might similarly have expected the M-Diss group not to generalize the observed Medium-range dependency into Short-range contexts. Finally, both hypotheses treat the two non-syllable-adjacent distances (Medium- and Long-range) as equivalent and thus predict that if dissimilation is enforced at Medium-range the same should hold at Long-range. At first glance, the fact that the first 16 participants in the M-Diss group showed no significant preference for dissimilation on Long-range test items is thus somewhat problematic for both hypotheses. However, the analysis of the 12 successful learners suggests that those who learned dissimilation at the Medium-range distance did indeed tend to generalize outwards to Long-range contexts.

All in all, the Experiment 2 results reveal the same patterns of learning and generalization for dissimilation as Experiment 1 did for harmony. Crucially, the M-Diss group did not interpret dissimilation in Medium-range contexts as indicative of a strictly beyond-transvocalic dependency; this despite the fact that such a locality pattern is predicted by the ABC hypothesis to be possible, and that a strictly beyond-transvocalic (or perhaps strictly Medium-range) interpretation would have parallelled the ‘conservatism’ exhibited by the S-Diss (and S-Harm) group. Rather, the behaviour of the M-Diss group suggests that they tended to infer an unbounded dissimilation pattern, consistent with the predictions of the TSL₂ hypothesis—though obscured in Long-range test items by a general dispreference for alternation in word-initial syllables (just as in Experiment 1). In sum, the same dichotomy between strictly transvocalic and unbounded locality appears to emerge in the learning of dissimilation patterns as for harmony patterns.

7.1. Participants, Stimuli, and Procedure

Thirty-two participants (26 female, 6 male, mean age 23) were recruited and compensated as described for the previous experiments, with 16 assigned to each of the M-Harm-S-Faith and S-Harm-M-Faith conditions described below. The experiment was performed using the same stimulus set and procedures as for Experiments 1 and 2.

7.2. Training Conditions

The two new groups of participants were subjected to the exact same testing phase as in Experiments 1 and 2, but were exposed to a substantively different training phase. Recall that in Experiments 1 and 2, only 50% of the training stems for each of the experimental conditions (S-Harm, M-Harm, S-Diss, M-Diss) contained a liquid in the relevant position. The remaining half were distractors, containing no liquids whatsoever (e.g., [dutebi]), such that all stem consonants remained faithful when the suffixes [-li] or [-ɹu] were attached. In Experiment 3, this other half of the training data was replaced with a different set of stems, which likewise stayed faithful when a suffix was added, but which in this case contained a (non-alternating) liquid [l] or [ɹ]. Depending on whether the alternating stems had their alternating liquid in the final or penultimate syllable, the faithful stems contained a liquid in the other position. In other words, participants were still given explicit evidence that pairs of co-occurring liquids were subject to harmony at one of the two distances (Short- or Medium-range), but they now also saw evidence that liquids at the other distance did not harmonize. As an illustration of the contents of the training phase, Table 13 provides a breakdown of the number and type of stimuli encountered by the S-Harm-M-Faith group.

Note that a pattern of this type is fully compatible with the attested strictly transvocalic variant of consonant harmony. It is thus expected to be learned as-is, especially given the behaviour of the S-Harm group in Experiment 1 (who inferred non-harmony at Medium-range even without overt evidence). Inclusion of a S-Harm-M-Faith condition is nevertheless important for the interpretability of results. Consider the possibility that the M-Harm-S-Faith group behaves indistinguishably from the Control group. We could not necessarily attribute this failure of learning to the fact that the pattern is more complex than the M-Harm pattern of Experiment 1, since there are additional methodological changes. For example, the fact that all training stems now contain liquids, and that only some of these alternate, might introduce additional processing difficulties that make learning more difficult. It is therefore essential to have a baseline of comparison not only in the Control group, but also the S-Harm-M-Faith group, in order to better interpret the results for the M-Harm-S-Faith group.

7.3. Results and analysis

The response data from the M-Harm-S-Faith and S-Harm-M-Faith groups were subjected to a mixed-effects logistic regression analysis in much the same way as for Experiments 1 and 2, again using the same Control group for a baseline comparison. The structure of the fixed-effects component of the model is identical to that of Experiment 1 (see Section 4.3.1), and the random-effects structure (obtained by the same procedure as described there) is minimally different. Full model specifics are provided in the Appendix; here we present only the odds-ratio comparisons of experimental groups against the Control group, shown in Table 14 (for description of how odds ratios were obtained, see Section 4.3.1). The plot in Figure 3 shows the performance of each group as well as that of individual participants.

Figure 3

Proportions of harmony responses across conditions (groups) and test-item type (distance) for Experiment 3. Coloured dots show individual participant performance, with group means indicated by an open circle with error bars (showing standard error); significant between-group differences are extracted from the logit mixed model. Performance on contexts where harmony alternations were encountered in training is shown with dashed-line boxes.

The M-Harm-S-Faith and S-Harm-M-Faith groups both learned liquid harmony at the trigger-target distance where harmony was present in their training data (shaded cells in Table 14, dashed-line boxes in Figure 3). The relevant estimated odds ratios (relative to the Control group) are 2.50 for M-Harm-S-Faith and the considerably higher 7.42 for S-Harm-M-Faith. The small but significant effect for the M-Harm-S-Faith condition is notable, given that this group was exposed to evidence of non-harmony in Short-range contexts; evidently this was not enough to stop a number of participants from learning the Medium-range dependency. Visual inspection of Figure 3 suggests a roughly bimodal distribution for M-Harm-S-Faith participants on Medium-range items, with seven individuals appearing to apply harmony while the remaining nine cluster in the same range as the Control group.

As a whole, the M-Harm-S-Faith group also chose harmony more often than the Control group in the novel Short-range (OR = 2.22) and Long-range contexts (OR = 1.74). Note that they did so despite explicit evidence in the training data that the liquids in CVCVLV stems do not alternate, and that all possible combinations of liquids ([l…l], [[l…r], [r…l], [r…r]]) are permitted in the Short-range CVCVLV-LV contexts. The small but significant effects for both Short- and Long-range test items show that many M-Harm-S-Faith participants overgeneralized harmony from the Medium-range context to the other distances, interpreting the exposure pattern as an unbounded dependency despite overt evidence to the contrary. Once again, Figure 3 depicts an apparently bimodal distribution among M-Harm-S-Faith participants on these types of test-items: Seven individuals tend toward harmony on Short-range items, and four do so on Long-range ones. Not surprisingly, six of the former seven (and all four of the latter) are indeed among the abovementioned seven M-Harm-S-Faith participants who appear to apply harmony in the (familiar) Medium-range context. (See Section 7.4 below for further discussion of individual results).

A small effect is also present for the S-Harm-M-Faith group on Medium-range items (OR = 1.57). This effect barely reaches statistical significance (p = 0.042) and, judging by Figure 3, it stems from a small number of participants who seem to generalize the liquid harmony from Short-range to Medium-range contexts. Nonetheless, this is slightly surprising in light of the fact that this group was exposed to evidence that stem liquids remain faithful in Medium-range contexts. It appears from Figure 3 that only one S-Harm-M-Faith participant applies harmony in Long-range contexts, and the effect for the group as a whole does not reach significance (OR = 1.50, p = 0.062). The S-Harm-M-Faith results, though peculiar when considering only whether or not an effect reaches statistical significance at the group level, are quite similar to those for the S-Harm group in Experiment 1. In both cases, we see that participants who are exposed to data compatible with strictly transvocalic harmony are very likely to learn that pattern as such, but there is a small tendency (within individuals and, as a result, for their group as a whole) to generalize the dependency to longer distances as well.

In each of Experiments 1 and 2, we pursued a follow-up analysis examining only those participants who had met a criterion for ‘successful learning’ of the pattern embedded in their training data. That procedure cannot be straightforwardly extended to the rich-stimulus training conditions of Experiments 3 and 4. Strictly speaking, truly ‘successful’ learners in the M-Harm-S-Faith and S-Harm-M-Faith conditions would be those showing not only a significant preference for harmony at one distance, but a significant absence of any such preference at the other distance as well.

However, defining and applying such a criterion would not be of interest here. Recall from the predictions outlined in Section 3 that the M-Harm-S-Faith group was expected to fail in learning the pattern embedded in their exposure data (since strictly beyond-transvocalic harmony is both typologically unattested and incompatible with the ABC and TSL₂ hypotheses alike). Such failure could potentially be manifested in several different ways: One participant might detect the Medium-range dependency but erroneously infer an unbounded harmony; another might ignore that dependency and instead generalize the faithful non-alternation from Short-range to other distances; yet another might detect no consistent pattern at all (e.g., responding randomly). It is the first of these patterns that emerges from the aggregated group-level responses, evidently due to a sufficient number of individuals having inferred an unbounded harmony. We emphasize, however, that those participants cannot be considered any more or less ‘successful’ than those who inferred the absence of any harmony dependency (e.g., consistent non-alternation at all distances). Both types of outcome involve accurate learning of the pattern exhibited at one distance level (harmony at Medium-range, or non-alternation at Short-range) combined with overgeneralization of that pattern to the other distance level, in contradiction to the training data. As there was no a priori expectation of one outcome being likelier than the other, a group-level result for M-Harm-S-Faith where harmony was not significantly preferred for any of the three test-item types would have been equally consistent with the predictions of ABC and TSL₂ alike.

For these reasons, we do not single out for special consideration those M-Harm-S-Faith participants who did acquire a pattern involving harmony in Medium-range contexts, nor did we attempt to recruit additional participants who did so for the purpose of further analysis. In the discussion below, we do however highlight certain aspects of the response patterns at the level of individual participants, as these raise some issues of potential interest.

7.4. Discussion of Experiment 3

In all important respects, the results of Experiment 3 agree with those of Experiment 1. Just like the M-Harm group of Experiment 1, the M-Harm-S-Faith group of Experiment 3 tended to learn an unbounded harmony pattern, generalizing the Medium-range harmony observed in the training data to Short- and Long-range contexts as well—this despite the fact that half of the training data directly contradicted such an interpretation (the CVCVLV-LV cases). Our results thus indicate a strong learning bias against strictly beyond-transvocalic harmony as a possible type of sound pattern, in line with the ABC and TSL₂ hypotheses. These findings, with learners inferring a simpler (or otherwise favoured) pattern in spite of overt evidence for a more complex (or disfavoured) pattern, are analogous to those reported by Lai (2012, 2015) for first-last harmony and by White (2014) for saltatory alternations.

It is worth noting that even though the M-Harm-S-Faith group exhibits a significant preference for harmony at all three distances, the effect sizes are considerably smaller than for the M-Harm group of Experiment 1. The odds ratios relative to the Control group range from 2.40 to 4.67 for the M-Harm group (see Table 9), but from 1.74 to 2.50 for the M-Harm-S-Faith group. (Though it is harder to detect, the same difference is evident from the group means for M-Harm and M-Harm-S-Faith in Figures 1 and 3, respectively.) This dampening of the overall effect is not surprising, since it was expected that the (presumed) unlearnable status of the strictly beyond-transvocalic harmony pattern embedded in the M-Harm-S-Faith training data might cause a greater number of participants not to learn any consistent pattern at all, or to generalize the Short-range pattern (faithful non-alternation) to Medium-range contexts.

The results for the S-Harm-M-Faith group likewise resemble those of the S-Harm group in Experiment 1, except in that the former exhibit a significant group-level preference for harmony in Medium-range items, suggesting a tendency to generalize from Short-range to Medium-range contexts (though not to Long-range items). Nevertheless, this effect is very weak and the relevant odds ratios for Medium- and Long-range items are rather similar across the two groups: 1.35 and 1.42, respectively, for S-Harm (see Table 9) and 1.57 and 1.50, respectively, for S-Harm-M-Faith (Table 14).

It is possible that the composition of the training data in the rich-stimulus conditions of Experiment 3 rendered them more confusing than those in the corresponding Experiment 1 conditions. This could explain the smaller effect sizes in the M-Harm-S-Faith condition as compared to the M-Harm condition of Experiment 1. However, there is reason to be skeptical of such an interpretation. For example, in Short-range items, participants in the (ostensibly confusing) S-Harm-M-Faith condition departed even more decisively from the Control group than did the S-Harm group of Experiment 1: The relevant odds-ratio estimates are 7.42 versus 3.70. This suggests that, if anything, participants’ ability to detect the presence of harmony in the Short-range context was aided, rather than undermined, by the addition of training items with liquids in a different position in the stem, and the failure of these liquids to exhibit a harmony alternation. In any case, the close parallel between the results of Experiments 1 and 3 suggests that the rich-stimulus training regime used here is an appropriate design for probing the existence of a learning bias against strictly beyond-transvocalic locality. This makes us more confident in applying the same regime to test whether such a locality pattern is also disfavoured in the case of dissimilation; Experiment 4 investigates this question.

We close this section with some comments on individual results from the M-Harm-S-Faith group in Experiment 3. Full details and plots of these results are provided in the Appendix. In the preceding section, we noted a bimodal distribution whereby only a subset of the sixteen participants appeared to apply harmony while the others’ responses clustered in the same range as those of the Control group. Application of the same 95% confidence-limit threshold criterion used in Experiments 1 and 2 (see Section 4.3.2) confirms this: Seven out of the 16 M-Harm-S-Faith participants show evidence of applying harmony in the Medium-range context, and all but one of these seven learners generalize the harmony into Short-range contexts, overriding the counterevidence present in the training data. By the same criterion, only four out of the seven show the same generalization to Long-range contexts; this is similar to Experiments 1 and 2, where participants were likewise reluctant to extend an observed alternation to the word-initial position of LVCVCV-LV items. However, this relatively clear-cut picture hides a certain amount of idiosyncratic variation at the individual level.

While we emphasize the dangers inherent in attempting to identify and interpret individual differences in a study of this type, the response patterns of two M-Harm-S-Faith participants in particular deserve highlighting. First, one individual (participant 509) appears to have inferred a strictly transvocalic harmony, in blatant contradiction to the training data for both Short-range and Medium-range contexts. (The proportions of harmony responses are 0.74, 0.56, and 0.47 on Short-, Medium-, and Long-range items, respectively; only the first of these exceeds the 95% confidence threshold.) One tentative interpretation (keeping in mind the above caveat) might be that this learner did detect that the training data contained a locality-restricted harmony pattern, but reanalyzed the distribution of harmony versus non-harmony contexts in a way that shifted the pattern into the available hypothesis space (the TSL₂ region or, equivalently, the factorial typology of the ABC model).

Secondly, another individual (participant 521) displays a pattern of harmony responses across contexts that does appear to be consistent with a genuine strictly beyond-transvocalic harmony. The relevant harmony proportions are 0.43, 0.88, and 0.79 on Short-, Medium-, and Long-range items, respectively; the latter two are both well above the 95% confidence threshold. As noted above and in Section 1.3, such a pattern is incompatible with both the ABC hypothesis and the TSL₂ hypothesis, and the behaviour of this one participant—if it is anything but a statistical fluke—is therefore surprising. It is of course impossible to conclude much from isolated observations like these, since the experiment was not designed to draw post hoc conclusions from individual participants’ deviations from group-level trends. However, we raise them here to shed light on the kinds of underlying individual variation that can at times affect the aggregate group-level results (and their statistical significance). We will return to this issue in the discussion of Experiment 4 in Section 8.4 below, where such individual-level idiosyncracies turn out to pose a bigger problem for interpreting the experimental results.

8.1. Participants, stimuli, and procedure

Thirty-two new participants (21 female, 11 male, mean age 22) were recruited and compensated in the same manner as described for previous experiments, with 16 assigned to each of the M-Diss-S-Faith and S-Diss-M-Faith conditions (described below). We note that an additional 24 participants were subsequently recruited for the M-Diss-S-Faith condition (a follow-up investigation intended to resolve certain ambiguities of interpretation; see Section 8.3 for discussion). Experiment 4 uses the same stimulus set and procedures as in all previous experiments.

8.2. Training conditions

Training items for the two additional experimental conditions were assembled from the stimuli used in the previous experiments, in a manner analogous to Experiment 3 except here the liquid alternation (present in either Short-range or Medium-range items, depending on condition) was one of dissimilation, just as in Experiment 2. In each condition, half of the stems had the shape CVCVLV and the other half CVLVCV (96 stems of each type). In the S-Diss-M-Faith condition, dissimilation was seen in CVCVLV-LV items (Short-range context) while the stem liquids stayed faithful in CVLVCV-LV items (Medium-range context). Conversely, in the M-Diss-S-Faith condition the Medium-range items displayed dissimilation while the Short-range items showed faithful non-alternation. Table 15 provides examples from the training phase of the M-Diss-S-Faith group.

Note that the pattern in Table 15 complies with the strictly beyond-transvocalic variant of dissimilation that is generated within the ABC model (see Section 1.3); it is incompatible with an interpretation that the dependency is unbounded (let alone that it is strictly transvocalic). Recall from the predictions laid out in Section 3 that if the ABC model accurately reflects the hypothesis space available to the phonological learner, such a pattern should be learnable by participants in the M-Diss-S-Faith condition. From the perspective of formal language theory, by contrast, all dependencies with strictly beyond-transvocalic locality fall outside of the TSL₂ region no matter whether they involve agreement or disagreement. The TSL₂ hypothesis therefore predicts that strictly beyond-transvocalic dissimilation should be inaccessible to learners, due to its greater computational complexity. The results for the M-Diss-S-Faith group should thus resemble those for the M-Harm-S-Faith group in Experiment 3: Participants should either treat the dissimilation as unbounded, or fail to apply dissimilation at any trigger-target distance. As noted in Section 7.3 above, we have no particular expectation as to the relative prevalence of the former versus the latter outcome among the group. Consequently, an aggregate result for the M-Diss-S-Faith group involving either a consistent preference for dissimilation across all three distance levels or the consistent absence of such a preference would be equally consistent with the TSL₂ hypothesis.

8.3. Results and analysis

The response data from the two groups, M-Diss-S-Faith and S-Diss-M-Faith, were analyzed in the same manner as for Experiment 3. Once again, we continue to use the same Control group as a baseline for comparison, and fitted the same kind of mixed-effects logistic regression model over the data from these three groups. As with Experiment 3, we omit the full statistical model here (see the Appendix for details) and instead present the odds ratios extracted from coefficient estimates of the relevant predictors. Table 16 compares the group-level performance of each experimental group against the Control group for each test-item distance. Figure 4 plots the proportions of dissimilation responses for each group as well as for individual participants.

Figure 4

Proportions of disharmony responses across conditions (groups) and test-item type (distance) for Experiment 4. Coloured dots show individual participant performance, with group means indicated by an open circle with error bars (showing standard error); significant between-group differences are extracted from the logit mixed model. Performance on contexts where dissimilation alternations were encountered in training is shown with dashed-line boxes.

Both groups selected test items with liquid disharmony more often than the Control group at the trigger-target distance where that dependency was present in their training data (shaded cells in Table 16, dashed-line boxes in Figure 4). As expected, the S-Diss-M-Faith group seems to reliably learn the target pattern, as revealed by the large odds ratio of 7.67 on Short-range test items. In the case of the M-Diss-S-Faith group, there is a comparatively weak effect; the odds ratio (relative to the Control group) on Medium-range items is 1.81. Though this effect is statistically significant (p = 0.009), visual inspection of Figure 4 reveals it to be carried entirely by two or three participants who apply Medium-range dissimilation at varying rates (one of them with 100% consistency); other M-Diss-S-Faith participants are clustered near a 0.50 proportion on Medium-range test items (similar to their S-Diss-M-Faith counterparts). Details on individual results from all 16 M-Diss-S-Faith participants are provided for reference in the Appendix.

As is evident from Table 16 and Figure 4, neither experimental group generalized liquid dissimilation to trigger-target distances other than where it was present in the training data. For the S-Diss-M-Faith group, this pattern of responses is quite robust: As can be seen in Figure 4, the vast majority of participants in this group reliably apply dissimilation in Short-range items but show no preference for dissimilation in Medium-range (or Long-range) items. For the M-Diss-S-Faith group, by contrast, the same is not true. Rather, the statistically significant preference for dissimilation in Medium-range items is carried entirely by just two or three individual participants (who, like the remainder of the group, show no analogous preference for dissimilation on Short- or Long-range test items).

As learners’ responses to the M-Diss-S-Faith training condition are of central interest, in that they bear on the differing predictions of the ABC and TSL₂ hypotheses, we attempted to strengthen the empirical basis for that comparison by recruiting additional participants for the M-Diss-S-Faith condition. The goal was to obtain a minimum of 12 participants who reliably replicated the Medium-range dissimilation featured in the training data (using a 95% confidence-limit threshold criterion). This was analogous to the procedure used in Experiments 1 and 2 for obtaining sufficient numbers of successful learners (see Sections 4.3.2 and 5.3.2); it should nevertheless be kept in mind that in this case, ‘successful learner’ would not be an accurate characterization (see Section 7.3 for discussion on this point). In what follows, we instead refer to them as Medium-range dissimilators.

However, after running a total of 40 native English speakers in this training condition (including the original 16), the number of participants meeting this criterion had only risen to eight, at which point recruitment was terminated due to practical constraints. Figure 5 shows the response patterns of the eight Medium-range dissimilators, including the three from the original set of 16 who had sufficiently high rates of Medium-range dissimilation (participants 705, 712, 723). Of these eight, all of whom show a reliable preference for dissimilation on Medium-range items, five surpass the threshold on Short-range items as well (participants 732, 736, 752, 757, 766). In other words, none of the Medium-range dissimilators in the additional pool of 24 M-Diss-S-Faith recruits turned out to display the non-generalization to Short-range contexts that was seen for those three who had been in the original group of 16 participants.

Figure 5

Experiment 4: Individual results for M-Diss-S-Faith participants classified as Medium-range dissimilators, drawing on an expanded pool of 40 participants (see text for discussion). Performance in the context where dissimilation was encountered in training is highlighted in red. The dotted horizontal line indicates the threshold criterion for reliable application of dissimilation.

8.4. Discussion of Experiment 4

The observed behaviour of the S-Diss-M-Faith group was unsurprising and is compatible with both hypotheses under consideration: They did not generalize dissimilation from Short-range contexts to greater distances, and in this respect behaved similarly to their S-Diss counterparts in Experiment 2. The M-Diss-S-Faith results are less clear-cut. As a group, the initial 16 M-Diss-S-Faith participants appear to have acquired a pattern of dissimilation in Medium-range contexts without generalizing it to other distances. This pattern, though non-TSL₂, could be accommodated under the ABC hypothesis, provided there is some independent account of the general reluctance (seen across all the experiments) to generalize alternations into word-initial position (cf. Becker et al., 2012). In other words, at first glance, the aggregate M-Diss-S-Faith results in Table 16 seem compatible with a characterization that this group learned a strictly beyond-transvocalic dissimilation (the effects of which are suppressed in word-initial position), contrary to the TSL₂ hypothesis.

However, as noted in the preceding section, such a conclusion is premature. The weak-but-significant trend in the M-Diss-S-Faith group toward dissimilation on Medium-range items (and the lack of generalization of this dissimilation to Short-range items) is carried entirely by three individuals. Note that these three could truly be called ‘successful learners,’ in that their responses seem to reflect both aspects of the training data: dissimilation in Medium-range contexts, combined with the absence of dissimilation (i.e., faithful non-alternation) in Short-range contexts. From the point of view of the ABC hypothesis, the small number of individuals who fit this description is quite surprising, especially in the context of other findings in this study. The M-Diss-S-Faith training data were completely consistent with a possible constraint-ranking configuration in the ABC model (in fact, with several such configurations; see Section 1.3 and the Appendix for details). There is thus no obvious reason why the number of such ‘successful learners’ (in this sense, i.e., of a strictly beyond-transvocalic dissimilation) should differ so sharply from what is seen in the S-Diss-M-Faith condition, or in the S-Harm-M-Faith group of Experiment 3, where the vast majority appear to acquire the strictly transvocalic dependency embedded in the training data. A comparison with the M-Harm-S-Faith group of Experiment 3 raises similar questions: That group was exposed to training data that were, by the ABC hypothesis (as well as the TSL₂ hypothesis), incompatible with any analysis available to the learner, and should thus have caused learners more confusion than the M-Diss-S-Faith training data. Yet we see considerably more M-Harm-S-Faith participants successfully replicating the Medium-range dependency than is the case for the M-Diss-S-Faith group. (A comparison of the M-Harm and M-Diss groups of Experiments 1 and 2 gives no support for such an asymmetry; the effect sizes for learning of the Medium-range dependency were quite similar for both groups; see Tables 9 and 11.) We examine the difficulties of capturing these and other asymmetries between training conditions within the ABC model in Section 9.2.1 below.

Secondly, with the effect of interest being driven by such a small number of individuals, the question arises of how reliable that effect is. Recall in this context how the response patterns of individual participants in earlier experiments sometimes deviated from expectations in surprising ways (see Section 7.4). When we attempted to ascertain whether the three M-Diss-S-Faith participants in question were representative of a reliable trend, by expanding the total number of participants in the M-Diss-S-Faith condition from 16 to 40, the number of learners who replicated the Medium-range dissimilation rose from three to eight. All five of these ‘new’ Medium-range dissimilators did generalize the pattern to Short-range contexts. While the number of relevant participants is still too low to draw any decisive conclusions, this suggests that the original group-level results in Table 16 should be taken with a grain of salt. In the expanded set of data, the majority of Medium-range dissimilators (five out of eight) generalized this dependency to Short-range items, in contradiction to the evidence provided in the training data, just as did the majority of learners of the Medium-range harmony (six out of seven) in the M-Harm-S-Faith condition of Experiment 3 (see Section 7.3). Once again, this apparent similarity between the M-Diss-S-Faith and M-Harm-S-Faith conditions is unexpected under the ABC model, given that the training data for the former, but not the latter, were fully consistent with the hypothesis space that is presumed to be available to the learner.

In sum, we do not believe that the results of Experiment 4 should be interpreted as support for the ABC hypothesis over the TSL₂ hypothesis. On the contrary, data from the expanded set of M-Diss-S-Faith participants suggests that Medium-range dissimilators do in fact tend to overgeneralize this into transvocalic contexts (i.e., infer an unbounded rather than strictly beyond-transvocalic dependency). This behaviour is comparable to the results of the M-Harm-S-Faith group of Experiment 3 and is consistent with the predictions of the TSL₂ hypothesis.

9.1. Evidence for learning biases

The results of Experiments 1 and 2 point to a hierarchy in terms of which locality parameters are preferred by learners, both for long-distance consonant assimilation and dissimilation: No dependency > Strictly transvocalic dependency > Unbounded dependency > Strictly beyond-transvocalic dependency. The evidence suggests that this is the order in which a typical learner will consider each of these hypotheses; in the face of ambiguous evidence, learners tend to interpret the data conservatively on the above scale. For example, a training phase which contains no stems with liquids whatsoever is, in principle, compatible with any type of dependency between liquids. The Control group, who were exposed to just such input, showed no preference for harmony or dissimilation at any of the test-item distances. Likewise, the training phases for the S-Harm and S-Diss conditions (in Experiments 1 and 2, respectively) contained no evidence about liquid-liquid co-occurrences at Medium- or Long-range distances, and were thus consistent with either a strictly transvocalic or an unbounded dependency. However, participants in these conditions were reluctant to extend the inferred pattern to distances beyond the …CVC… window, preferring instead to interpret the dependency as strictly transvocalic. Finally, the M-Harm and M-Diss groups (Experiments 1 and 2) were exposed to a Medium-range dependency only, with no evidence bearing on Short- or Long-range distances. While this input was consistent with an interpretation of the dependency as either strictly beyond-transvocalic or unbounded, participants preferred the latter, generalizing to the Short-range and also (though somewhat less reliably) the Long-range distances.

The results of Experiments 3 and 4 suggest that patterns with strictly beyond-transvocalic locality, which appear to be cross-linguistically unattested (for the questionable case of Sundanese dissimilation, see Section 1.4) are indeed extremely difficult for phonological learners to access. Only a small number of participants displayed individual results that seem consistent with such a pattern (one M-Harm-S-Faith participant, three M-Diss-S-Faith participants). However, most had difficulty detecting any Medium-range dependency at all (in contrast to the M-Harm and M-Diss conditions of Experiments 1 and 2, where participants had no trouble picking up on interactions at that distance). Of those who did, the majority over-generalized to Short-range contexts as well, thus deviating from the pattern shown to them in training. This suggests that strictly beyond-transvocalic locality patterns are situated at the outer limit of, if not beyond, the learner’s available hypothesis space and that any viable theory of long-distance dependencies should be reflective of this bias.

9.2. Theoretical implications

9.3. Further issues

The discussion of language-theoretic classes and complexity relations in preceding sections has been framed exclusively in terms of phonotactics and phonotactic learning. This may appear to be at odds with the fact that the experiment itself involved morpho-phonological alternations between forms. Admittedly, there is not yet a deep understanding of the formal and learnability properties of subregular functions that can model string-to-string mappings. However, recent work has defined a number of relevant classes and identified some of their formal properties (e.g., Chandlee, 2014; Chandlee, Eyraud, & Heinz, 2015; Mohri, 1997), including a number of promising results regarding learnability (Chandlee, Eyraud, & Heinz, 2014; Chandlee & Jardine, 2014). Of particular relevance is the finding that subsequential functions (Mohri, 1997) characterize many attested patterns of long-distance consonant dissimilation (Payne, 2014) and consonant harmony (Luo, 2017), while excluding a number of complex pathologies. Further work will determine whether or not the class of TSL functions defined by Chandlee, Heinz, Jardine, and McMullin (2017), which are more restrictive than the subsequential functions, can be learned with a technique similar to that employed by Jardine and McMullin (2017).

As noted in Section 1.1, we have sidestepped the question of whether the dependencies we have described as strictly ‘transvocalic’ (as well as strictly beyond-’transvocalic’) are based on consonant-tier adjacency or syllable adjacency. Our formal-language-theoretic characterizations of such patterns have assumed the former; they are defined over strings of segments, with no reference to syllable constituency.¹⁵

For the ABC model, however, things are more complex: The most elaborated and formally explicit version of the model, that of Bennett (2015b), contains constraints referencing each of these two locality criteria. On the one hand, the CC-Limiter constraint CC-SYLLADJ (equivalent to PROXIMITY in Rose & Walker, 2004) bans correspondence between segments that are separated by an intervening syllable, i.e., are not syllable-adjacent. On the other hand, special CVC versions of the correspondence-inducing CORR-X constraints are posited (following Hansson, 2010), which apply only to consonant pairs that are consonant-tier-adjacent. As discussed in Section 9.2.1, each of these constraint types can give rise to ‘transvocalic’ harmony, as well as to beyond-‘transvocalic’ dissimilation, whereas only the CORR-X_CVC constraint type can generate ‘transvocalic’ dissimilation. A true characterization of the factorial typology of the ABC model as per Bennett (2015b) is thus more nuanced. While harmony can be either strictly consonant-tier-adjacent, strictly syllable-adjacent, or unbounded, dissimilation, can be strictly consonant-tier-adjacent, strictly beyond-consonant-tier-adjacent, strictly beyond-syllable-adjacent or unbounded. Notably, while strictly syllable-adjacent locality is available for harmony, it is predicted to be impossible for dissimilation patterns. It is not clear whether this particular mismatch prediction for harmony versus dissimilation is intentional on the part of Bennett (2015b); we are not aware of any cross-linguistic basis for it.