The consonant inventory of Aymara contains fifteen stops, exhibiting a three-way laryngeal contrast between plain (voiceless unaspirated), ejective and aspirate at five places of articulation. The full inventory is shown in Table 1 (MacEachern 1997; Hardman 2001).

The distribution of ejective and aspirate stops is restricted within morphemes, both inside suffixes and in roots, which we exemplify in (3). As shown in (3a), ejectives and aspirates may appear in either initial or medial position of roots, which are primarily CV(C)CV. Both ejectives and aspirates are rare in roots with an initial plain stop, however (see (3b)). Pairs of heterorganic ejectives are also rare, though forms with identical ejectives are attested (see (3c)). Other combinations of ejectives and aspirates are attested (see (3d)), though see Section 4.4.4 below for further details. Examples are from de Lucca (1987), and these and other patterns are also discussed in detail in MacEachern (1997) and Bennett (2013).

The three laryngeal combinations that are underattested inside morphemes – plain-ejective, plain-aspirate and ejective-ejective – are attested in words. These combinations arise when suffixes with an ejective or aspirate consonant combine with roots with a plain or ejective stop. Some examples are given in (4), from work with a native speaker consultant in El Alto, Bolivia. These examples involve three verbal suffixes, [-t’a] ‘about to’, [-ʧ’uki] ‘carefully, continuously’, and [-tʰapi] ‘finish’. All three of these suffixes trigger syncope (deletion of the root-final vowel).2

There are several exceptions to the restrictions on tautomorphemic ejective-ejective, plain-ejective and plain-aspirate combinations. These are given in (5). Some of these exceptions occur at the level of a trigram on the linear string – important for our model – while others occur across more intervening material and would be noticeable only when looking at a nonlocal projection. Additionally, there are four combinations of plain-ejective and two combinations of plain-aspirate that occur in clusters. These are reported by our consultant to be monomorphemic forms, though Hardman claims that root-internal stop codas are not found. The morphological structure of these forms is thus in question. An additional observation is that several of the exceptions end in the sequence [t’a], just like the productive suffix.

To assess the statistical evidence available to an inductive learner trying to acquire these restrictions, we looked at the observed combinations of all three series of stops. Our data set is a morphologically segmented word list. This list was compiled by taking an unsegmented web corpus of 88,728 forms, collected from 438 webpages by An Crúbadán (http://crubadan.org/).3 The corpus was converted to lowercase and cleaned to remove numbers, non-alphanumeric characters and English or Spanish forms. Forms were also removed if they contained stray apostrophes, hyphens or other typos that we couldn’t interpet. This list was then crossed with a list of 1846 roots, derived from the de Lucca (1987) dictionary with the help of a native speaker consultant, and a list of 50 suffixes and their allomorphs from the Hardman (2001) grammar and the de Lucca (1987) dictionary. There were 46,164 forms that were divisible into a known root and known suffixes, and these forms comprise the corpus we use in this paper.

Before looking at combinations of stops directly, we report on the distribution of stops by position in our word corpus, comparing the number of each class of stops in root initial position, root medial position, and in a suffix. Table 2 gives the raw counts on the left (e.g., there are 9,113 plain stops which are in root-initial position in the word corpus), as well as the probability of a stop from the given class in the given position (e.g., 20% of our 1,846 roots begin with a plain stop, 7% with an aspirate and 7% with an ejective, the remaining 66% of roots begin with a vowel, fricative or sonorant consonant). These numbers show that plain stops are frequent in both roots and suffixes, while aspirates and ejectives are both much more frequent in roots than in suffixes.

Tables 3 and 4 report the observed counts for stop combinations in tautomorphemic and hetermorphemic strings, on both the baseline and a nonlocal projection containing only stops. For tautomorphemic sequences, we looked at the cooccurrence of nonadjacent stops in a baseline trigram – that is, of a C₁XC₂ string, where X can be any segment except the morpheme boundary symbol – and for adjacent bigrams on the stop projection. For hetermorphemic sequences, we looked at trigrams where the medial gram was the morpheme boundary on the baseline and the stop projection. In all tables, “ejective-ejective” refers to counts made over only non-identical combinations; all other combinations represent both identical (where applicable) and non-identical combinations.

Table 3 shows that, within a baseline trigram, the restricted combinations are unattested tautomorphemically. Ejective-ejective combinations are also nearly unattested in heteromorphemic contexts, but plain-aspirate and plain-ejective combinations are more frequent across a morpheme boundary. The bottom portion of the table shows that other combinations of stops are either frequent in both heteromorphemic and tautomorphemic contexts, or are more frequent tautomorphemically than heteromorphemically (note that while ejectives and aspirates may occur in suffixes, the numbers here reflect the rarity of such suffixes in the corpus as a whole). As noted in Section 2.1 above, vowels in both roots and suffixes syncopate via affixation, creating consonant clusters across morpheme boundaries. This is crucial to the distinction between tautomorphemic and heteromorphemic contexts in Table 3. The plain-aspirate, plain-ejective and ejective-ejective combinations that occur in a heteromorphemic trigram are actually adjacent in the linear string, since the morpheme boundary symbol constitutes the medial gram in the trigram. If Aymara did not have syncope, heteromorphemic combinations would only be noticeable in a tetragram, compare actual [qaq+tʰapi+ɲa] ‘to finish scratching’ to hypothetical [qaqa+tʰapi+ɲa]. We will return to this point below.

Table 4 gives the counts on a stop projection. Here, both [p’at’] and [p’ant’] would count as ejective-ejective combinations. Plain-aspirate and plain-ejective combinations are still much more frequent in hetermorphemic than tautomorphemic contexts, though there are substantially more exceptions tautomorphemically than are observable in a baseline trigram. Ejective-ejective combinations are again rare in both contexts. The bottom portion of the table shows that most combinations (plain-plain, aspirate-plain, ejective-plain) are well attested in both morphological contexts, while aspirate-ejective combinations are somewhat rare in both contexts. Aspirate-aspirate and ejective-aspirate combinations are both more frequent in tautomorphemic combinations, again due to the general rarity of aspirates in suffixes.

The numbers here show that the restrictions on plain-ejective and plain-aspirate combinations in descriptive grammars are supported in counts over a word corpus, at both the baseline trigram level and on a nonlocal projection including only stops. The restrictions on ejective-ejective combinations are more difficult to assess, due to the rarity of heteromorphemic combinations (though our consultant work shows that these are possible, if not frequent in the corpus). The other six combinations of stops are reported to be licit in all morphological contexts, and this appears to be essentially true in our corpus as well, though certain combinations are unattested or nearly unattested (see Section 4.4.4 for discussion of other restricted combinations).

The experiments presented in the next section look at how native speakers of Aymara treat forms with plain-ejective and ejective-ejective combinations, in both heteromorphemic and tautomorphemic contexts. We then present the results of our computational model in Section 4, showing how the counts presented above are reflected in an inductive phonotactic grammar.

Experiment 1 presents Aymara speakers with simple disyllabic forms, which could be interpreted as pseudo-nouns, that contain plain-ejective or ejective-ejective combinations. Speakers’ errors in repeating such forms are evaluated to assess whether speakers have internalized phonotactic restrictions on these combinations.

The goal of Experiment 2 is to test whether speakers’ errors on ejective-ejective and plain-ejective combinations are influenced by morphological structure, and whether these restrictions hold across more than a single intervening vowel. Experiment 2 presents participants with the same kinds of phonotactically illegal structures – ejective-ejective and plain-ejective pairs – as in Experiment 1, but in Experiment 2 the nonce words are pseudo-verbs ending in the infinitval suffix [-ɲa]. The experiment compares performance on stimuli where the illegal ejective in C₂ must be interpreted as part of the root vs. stimuli where the illegal ejective in C₂ may be interpreted as part of a productive suffix [t’a]. Additionally, the pseudo-roots in Experiment 2 all contain a coda consonant, so the interacting consonants are separated by a VC sequence as opposed to the single V in Experiment 1.

Together, Experiments 1 and 2 show that Aymara speakers have learned morphologically sensitive restrictions on ejective-ejective and plain-ejective combinations. Comparison between the two experiments supports a role for both morphological structure and phonetic category in the place of articulation effect found in both studies.

Error rates on forms with dental ejectives were lower than error rates on forms with non-dental ejectives in both experiments. This effect is at least partly an effect of phonetic category: dental ejectives are much more common as the second consonant in a plain-ejective or ejective-ejective sequence than other places of articulation (97% of such combinations in our corpus have [t’] as C₂), because of the frequency of the suffix [-t’a]. We can thus conclude that speakers are sensitive to the different distribution of [t’] compared to other ejectives.

To see if there is also a contribution of morphological structure, we need to consider the increased accuracy on dental ejective forms between Experiments 1 and 2. In Experiment 1, all forms, including those with dental ejectives, have a monomorphemic structure. Some of the dental ejective stimuli in Experiment 1 cannot be decomposed into suffixes (e.g., [qat’i] – there is no suffix [t’i]). Stimuli with [t’a], such as [kat’a] and [nut’a], are unlikely to be analyzed as root-suffix because the suffix [t’a] always attaches to bases that are at least CVC (e.g., [taw+t’a+ɲa] ‘about to row’).5 The other reason for doubting that the dental ejective stimuli in Experiment 1 are morphologically decomposed is that [t’a] is an aspectual suffix that is usually followed by tense and person marking. There are only six words in the corpus where [-t’a] is the last and only suffix, and none of them have the shape CVt’a.

On the other hand, in Experiment 2, forms with a dental ejective have a polymorphemic structure, since they contain the two verbal suffixes [t’a] and [ɲa]. To test for an effect of morphological structure, an additional, post-hoc statistical model was run. Responses to ejective-ejective and plain-ejective stimuli from the two experiments were pooled and a binomial linear mixed model was fitted. The dependent variable was accuracy, and the independent variables were experiment (Experiment 1 or Experiment 2) and place (dental or not). There was a random intercept for participant and a random by-participant slope for place (a model with a random slope for experiment failed to converge). The model found a significant interaction between place and experiment (β = 1.14, SE = 0.28, t = 4.15, p < 0.0001), revealing that the effect of place differs between the two experiments. The difference between accuracy on dental and non-dental forms is larger in Experiment 2 (74% vs. 35%, a 39 point difference), where stimuli have a polymorphemic structure, than in Experiment 1 (52% vs. 37%, a 15 point difference) where stimuli have a monomorphemic structure. While the experiments were not originally designed to be compared in this way, the raw differences in accuracy and the high significance level in the statistical test are supportive of an effect of morphological structure above and beyond place of articulation.

The types of errors between the two experiments also differ, and further show the importance of morphological structure to repetition accuracy. While place of articulation errors are quite rare in Experiment 1, these errors are very frequent in Experiment 2. Errors that map a labial ejective to a dental make up just 4.5% of errors in Experiment 1 but 73% in Experiment 2. Participants’ responses in Experiment 2 are thus tracking the polymorphemic, pseudo-verb structure of the stimuli, skewing responses to create a phonotactically legal form by changing place of articulation and thus morphological structure.

In sum, the experiments here provide behavioral evidence that the laryngeal restrictions on ejectives and their interaction with morphology are part of speakers’ synchronic grammars.

There are several parameters that affect the learner’s ability to find generalizations. First, the segmental features determine whether the learner can group the segments into the right natural classes. The learner is sensitive to the size of the classes, as well as their overall number (see Hayes and Wilson 2008, Gouskova and Gallagher to appear). Since we were primarily interested in laryngeal restrictions, we selected a feature set that uses mostly privative features (specifically, [plain], [cg] and [sg]). Hayes and Wilson’s learner favors constraints whose natural classes mention as few features as possible, so privative features allow certain classes to be picked out more easily; in the Aymara case, this means that plain stops can be picked out as [+plain] instead of as [–cg, –sg]. See Section 4.3 for the full list.

The other parameters have an effect on the number of constraints induced, the length of segmental strings they scope over, and how closely the learner fits the grammar to the data. We were generous in the number of constraints we allowed the learner to discover, since this learner stops when it cannot identify any constraints that pass the selection criterion. The length of constraint strings on the segmental projection ranged from 1 (as in *[+cg]) to 3 (as in *[–syllabic][–syllabic][–syllabic], “no CCC clusters”); the length of constraint strings on higher projections ranged from 2 to 3. In addition to these two fairly simple parameters, there are several parameters that affect the fit of the model in various ways.

The first parameter is gain (Della Pietra et al. 1997; Wilson and Gallagher 2018; Gouskova and Gallagher to appear), which replaces the O/E threshold criterion in the version of the learner described in Hayes and Wilson (2008). The gain of a constraint is proportional to the reduction in the Kullback-Leibler divergence between the current grammar and the grammar with C added, and the weights of all the other constraints unchanged. Put differently, gain is higher when the probability distributions in the learning data are closer to those generated by the grammar if the constraint were added. A constraint can only be added if its gain exceeds the threshold; the higher the gain, the harder it is to add new constraints. We have found moreover that gain can be lower when the training data sets are small and each datum is relatively informative, but larger data sets yield more sensible grammars when gain is higher.

The second parameter we manipulated is gamma. This parameter affects how the objective function of the learner is calculated each time a constraint is added – it scales the harmony score relative to the negative log probability, with the effect of increasing the impact of constraint violations by individual candidates. Increasing gamma makes it less likely that constraints with very low weights will be learned (NB: both Della Pietra et al. 1997 and Wilson and Gallagher 2018 use ɣ to refer to the gain threshold; this is distinct from the gamma parameter). There are additional parameters that can be manipulated, such as the Laplace regularizer λ, whose function is to penalize constraints with large weights (Wilson and Gallagher 2018: 615). We set λ to a small constant 0.00001.

The corpus we used was based on the Aymara wordlist on the An Crúbadán project website (http://crubadan.org), described in Section 2. We created two versions of the corpus: an unsegmented list of phonological words (transcribed on the basis of the transparent orthography), and a segmented list with morpheme boundaries. Recall that we only used those words in the An Crúbadán list that contained the roots that also occur in the de Lucca (1987) dictionary. Since morpheme boundaries add to the overall length of each string, we had to filter the segmented word list to exclude words above a certain length.7 This left us with 46,164 words each in the segmented and unsegmented lists.

The feature set we used for all simulations is shown in Table 11. In addition to the phonemes of Aymara, the feature set contains the morpheme boundary “+”, the word boundary, and a special “copy” segment X, which has just one privative feature, [+copy]. As explained in Gouskova and Gallagher (to appear), this copy notation is necessary because the learner does not implement algebraic notation in its constraint language (Berent et al. 2012). Thus, to allow the learner to distinguish between the allowed identical pairs of ejectives and the disallowed non-identical ones (recall Section 2.1), we transcribe words such as [t’ant’a] as [t’anXa].

To further establish the role of morphological information in the success of our model, we ran learning simulations on the same word corpus, but with morpheme boundary markers removed. Recall from Table 3 that there are many instances of plain-aspirate, plain-ejective and ejective-ejective combinations in the corpus as a whole, but they are mostly found across a morpheme boundary. We tested whether these sequences were frequent enough to obscure the restrictions if morpheme boundaries are not represented. To start, we look at the number of plain-ejective, plain-aspirate and ejective-ejective combinations that occur in a baseline trigram in the unparsed data set in Table 19; for comparison, we repeat the numbers for tautomorphemic and heterormorphemic sequences in the parsed data set from Table 3. For all three restricted combinations, there are exceptions in the unparsed data set, and plain-aspirate combinations in particular are quite frequent. It is worth noting that the heteromorphemic trigram combinations in the parsed data set are actually bigrams in the unparsed data set (for example, [ati+p+t’a+ɲ] in the parsed data set appears as [atipt’aɲ] in the unparsed data set), so these forms do not introduce exceptions at the trigram level in the unparsed data set. Instead, many of the exceptions that we see in the unparsed data are actually tetra- or penta-grams in the parsed data set that appear as trigrams once morpheme boundaries are removed. For example, the form [huk’ampst’aɲ] appears in the unparsed data set but is [huk’a+m+p+s+t’a+ɲ] in the parsed data set.

We then turned to examine whether the learner found placeholder trigram constraints when trained on the unparsed corpus (we call this the “Induced Unparsed Model”, in contrast to the model we presented in Section 4.4.3).8 We tested a range of gain and gamma combinations, and compared the placeholder trigrams found in the baseline model when trained on the parsed and unparsed data sets. All runs of the learner were asked to find a maximum of 200 constraints. We report on a representative sample of the numerous combinations we tried in Table 20. Morpheme boundary trigram constraints corresponding to at least two of the three restricted combinations are found for the parsed data under almost all settings, and settings with a higher gain or gamma allow the model to detect all three restrictions in the baseline grammar. For the unparsed data, placeholder trigram constraints are only found with a very low gamma of 1. Even with gamma this low, the Induced Unparsed model only finds a single placeholder trigram on ejectives; this model never finds a placeholder trigram corresponding to the plain-aspirate restriction and may miss the ejective-ejective restriction as well.

One of the grammars built after training on the unparsed corpus is investigated in more detail in Figure 5 and Table 21. This grammar’s gain is 500, and gamma = 1. The baseline grammar constraint *[+plain]-any_seg-[+cg] motivates the [–son, –cont] projection, on which all restrictions could be correctly stated. But the final grammar achieves a poor fit to the experimental data, as shown by the nearly vertical lines in Figure 5. The correlations for Experiment 1 are τ = 0.22 and ρ = 0.35; the correlations for Experiment 2 are τ = 0.43 and ρ = 0.60.

The Induced Unparsed Model’s lack of success at representing phonologically meaningful underattestations in the unparsed data becomes clear when we look at the constraints it posits on the stop projection (see Table 21). Their low weights are due in part to the gamma setting; almost all of them are violated frequently by Aymara words. While these constraints penalize some restricted combinations, they don’t capture the full extent of the restrictions nor are their weights high enough to distinguish restricted from unrestricted structures.

With this low gamma setting, the Induced Unparsed Model does not succeed in distinguishing meaningful underattestations in the data, and instead learns many, low weighted constraints with numerous exceptions. Our more succesful Induced Parsed Model in Section 4.4.3 has higher gamma and gain (in addition to access to morheme boundaries), which makes the Induced Parsed Model more selective and a better fit to the phonological distinctions supported by traditional phonological analysis and experimental work with native speakers. Given these same settings (400 gain, 150 gamma), the Induced Unparsed Model’s grammar fails to include any placeholder trigram constraints from which it could posit nonlocal projections.

We also considered whether it was possible to capture phonological distinctions on the stop projection with unparsed data, when the learner was given a higher gain and gamma and we supplied the stop projection manually. This is the Manual Unparsed Model. As shown in Figure 6, this model’s grammar achieves a better fit to Aymara speakers’ performance in the repetition experiments than the grammar in Figure 5. But there are interesting differences in the details.

The Manual Unparsed Model, trained on unparsed data, captures the distinction between dental and labial ejectives via constraints on everything but dentals (in Table 22, (c), (e)–(g), (k)–(m)). But this model fails to make a distinction between control stimuli such as [lap’a] (100% correct) and [p’it’a] (67% correct) – they all receive a harmony score of –6. This is because this grammar does not include a general constraint against ejectives in second position on the stop projection. Constraint (c) on the ejective position is too specific. This grammar is overfitting, learning overly specific constraints to accommodate the exceptions in the data.

When it comes to Experiment 2, however, the Manual Unparsed Model’s fit to behavioral data is comparable to the model we reported in Section 4.4.3 (although the differences between “good” and “bad” forms are smaller in Experiment 2 – the opposite of the parsed grammar). The reasons for this have to do with the abundance of dental ejectives in Aymara suffixes; by positing place-specific constraints against non-dental ejectives in 2nd position on the stop projection, the model manages to approximate the same generalizations.

For a quantitative comparison, Table 23 summarizes the non-parametric correlations between the harmony scores each model assigns to the experimental stimuli and the averaged accuracy in the repetition experiments with Aymara speakers. Model (a), which uses parsed data both for inducing the projections and for the final grammar has the best correlations with Experiment 1, and the best correlations overall. Model (b), which includes no morphological information at all, achieves the lowest correlations across the board. The third model, (c), does worse on the first experiment and slightly better on the second experiment, but its overall correlations with behavioral data are not as good as the model in (a).

Zooming out from laryngeal cooccurrence restrictions, the Manual Unparsed Model is not quite right in other ways. Aymara morphemes obey an exceptionless constraint against CCC clusters (0 of them in the corpus), but such clusters are created by syncope at morpheme boundaries (recall Section 2.1). The right constraint to capture this would be *[–syll][–syll][–syll]. The Induced Parsed Model (a) contains such a constraint, and gives it a high weight of 14.506. The Manual Unparsed Model cannot motivate such a constraint – there are 3322 words in the Aymara corpus that have such clusters. What this model does instead is posit many specific constraints, sometimes with rather low weights, against various CCC clusters that it sees few examples of – e.g., *[–son, –cont][+labial][–cont], with a weight of 6.778. This is just one example of a morpheme structure constraint that a morphology-agnostic learner cannot capture.

The experimental and modeling work show that the distinction between tautomorphemic and heteromorphemic stop-ejective sequences is likely both a direct effect of morphology and also an effect of place of articulation. Participants in Experiment 1 made slightly fewer errors on stimuli with dental ejectives than with other ejectives, reflecting the frequency of dental ejectives in non-initial position. This effect was exaggerated in Experiment 2, where the structure of nonce words favored a polymorphemic parse. Similarly, the constraints in the grammar are sensitive to both morphological structure and place of articulation. The model includes constraints on tautomorphemic but not hetermorphemic laryngeal combinations and it also includes more specific constraints on individual segments; due to their presence in suffixes, [t’] and [ʧ’] are more frequent than other ejectives in non-initial position, and the constraints in the grammar reflect this asymmetry.

One element of the experimental results not directly captured by the model is that while participants were more accurate on dental ejectives than labial ejectives in Experiment 2, they still made more errors on dental ejective forms than on filler items. In contrast, our model predicts forms like [k’ast’aɲa] to be fully grammatical. Errors on dental ejectives likely reflect the two available morphological parses for these forms. Since these were nonce words that weren’t associated with any meaning, speakers did not know for certain that [t’a] in these forms was the verbal suffix. There are also roots in the language that end in [t’a]. Errors in the repetition task are predicted if participants occasionaly parse a form like [k’ast’aɲa] as [k’ast’a+ɲa], while accurate repetitions are expected if the [k’as+t’a+ɲa] parse is hypothesized.

Our modeling simulations showed that while some pieces of the restrictions can be detected in an unsegmented corpus, morpheme boundaries are necessary to fully capture the patterns. This means that infants and children learning Aymara cannot have a complete phonotactic grammar until they have learned enough morphology to segment the speech stream. The prediction is that the trajectory of phonotactic awareness of laryngeal restrictions should be different in Aymara learners than in learners of a language where laryngeal restrictions are categorical and hold at the word level, like Quechua.

Even though languages do not mark every morpheme boundary phonotactically, speakers of languages such as Finnish and Dutch have been shown to use phonotactic knowledge to segment speech in experimental settings (Suomi et al. 1997; McQueen 1998). The model we presented makes a simplifying assumption that at some point, the learner examines a fully parsed corpus with boundaries. A more realistic approach would use phonotactics to deduce where boundaries are located (as in the StaGe model of Adriaans and Kager 2010). StaGe uses bigram probabilities to posit word boundaries. Aymara would lend itself to such an approach: ejectives and aspirates are most common root-/word-initially (recall Table 2), so the distribution of plain-ejective bigrams would be a clue to boundaries even for a learner that does not yet have detailed morphological segmentation information. We leave for future work an implementation of a more complete model that deduces both where morpheme/word boundaries are and whether they lead to nonlocal projections.

In our baseline grammar for Aymara, the nonlocal restrictions on stop combinations are reflected in morpheme boundary trigrams like *[+plain][–mb][+sg, –continuant]. The pattern of syncope in Aymara is crucial to these constraints being found, because syncope allows stops to appear adjacent to one another across an intervening morpheme boundary.9 Without syncope, a hypothetical form like /lipa+t’a/ would be realized as [lipa+t’a] (as opposed to [lip+t’a]), with the interacting stops only appearing in a tetragram; syncope creates a consonant-morpheme_boundary-consonant trigram. In a language without syncope, the restrictions on stop combinations may still be observable in a baseline trigram, but they would be reflected in a placeholder trigram (in which the medial gram is ‘any segment’, e.g., *[+plain][][+sg, –continuant]) as opposed to in a morpheme boundary trigram. Under our proposal about how nonlocal projections are induced, both placeholder trigrams and morpheme boundary trigrams trigger the learner to add a nonlocal projection to their search space of constraints, so syncope should not be crucial to the learning of a nonlocal restriction.

A learner does not know in advance whether including morpheme boundaries on to projections will lead to better generalizations. We showed in Gouskova and Gallagher (to appear) that in languages like Quechua, for example, it is possible to discover the nonlocal interactions between stops from phonological words alone, and presumably Quechua learners acquire this knowledge before they are morphologically aware, since the restrictions are categorical within words. It is important in Quechua that the nonlocal projection include all stops but not include morpheme boundaries, since the relevant restrictions hold both within morphemes and across morpheme boundaries.

We hypothesize that in both Quechua-type languages and Aymara-type languages, learning starts on unparsed words, and if any projections are discovered, they include word but not morpheme boundaries. When words are morphologically segmented, the phonotactic grammar is reassessed, in case any generalizations were missed in the unparsed grammar. In a language like Aymara, we showed that the nonlocal laryngeal restrictions are not noticeable from the unparsed data, so a learner of Aymara would not be able to learn these restrictions until they had acquired some morphological structure, at which point nonlocal projections with the morpheme boundary symbol and constraints on this projection would be added to the grammar. In a language like Quechua, where similar laryngeal restrictions are not morphologically sensitive, the laryngeal restrictions should be learned earlier and represented on a nonlocal projection that does not include morpheme boundaries. When the Quechua learner returns to phonotactic learning with morphological information, the learner should not uncover any new placeholder trigrams on stops, since the distribution of stop combinations is already fully accounted for on the nonlocal stop projection without the morpheme boundary symbol. We verified that this is in fact how things work for Quechua. We first trained a baseline model on an unparsed corpus (about 10k words, described in Gouskova and Gallagher to appear), from which the leaner built a nonlocal stop projection. We then trained a model with this projection on the parsed data, and indeed, the learner found constraints on the nonlocal stop projection but did not learn any new placeholder trigrams on the default projection that would motivate adding the morpheme boundary symbol to the stop projection. We leave it to future work to examine in more detail how morphologically insensitive generalizations can be incorporated into a later stage of learning where morphological structure is represented.

Patterns such as those of Aymara present two types of subset problem (Baker 1979; Bowerman 1988). First, phonotactic learning in general requires the learner to err on the side of assuming more restrictive grammars and to construct its own negative evidence. In order to posit these more restrictive grammars, an Optimality-Theoretic learner with innate constraints requires a bias to keep faithfulness constraints ranked low; the negative evidence comes from the theory’s Gen component (Hayes 2004; Prince and Tesar 2004). If the learner induces constraints from data instead, it constructs its own negative evidence by comparing the attested data to plausible phonotactic distributions generated at random (Hayes and Wilson 2008). But even such a learner will not notice nonlocal interactions – it must either be given the nonlocal representations a priori (as in Hayes and Wilson’s proposal), or it must be nudged in the direction of looking for nonlocal representations. In our proposal, the learner does a second pass of phonotactic learning in response to generalizations that it may have missed on the first pass, signaled by placeholder trigram constraints. This kind of bias is designed to alert the learner to the need for more restrictive constraints.

Second, if the learner assumes that the sequences allowed in words are also allowed in morphemes, then this morphologically agnostic learner will learn the superset grammar. For a language such as English, the superset learner hears words with [md] clusters and assumes, incorrectly, that such clusters are allowed anywhere, not just at morpheme boundaries. Our experiments suggest that Aymara speakers make the more conservative assumption. We suggest that in order to learn the right level of generalization, an Aymara learner has to revisit phonotactic learning once morphological information is available. We implemented this by supplying morpheme boundaries and using sequences at boundaries as a clue that nonlocal interactions are present.

The alternative we did not discuss is to split the learning data into morphemes, and learn phonotactics over these morphemes. For Aymara, a plausible learning data set that would reveal the right regularities would be a corpus of roots. This is the learning data we used in Gouskova and Gallagher (to appear). The benefit of using roots as learning data is that the learner may use just one simple cue, segmental placeholder trigrams, without attending to morpheme boundaries or reifying them to the representational level of segments with feature values. This would essentially use a sublexicon to learn morpheme-level phonotactics that hold over just a subset of the language’s forms (Gouskova and Becker 2013; Becker and Gouskova 2016). The main reason we did not use a sublexicon model here is that it is not clear how to define phonotactics over bound roots; verbal roots in Aymara are obligatorily suffixed. The application of phonotactic learning to words is straightforward to implement, whether they are embedded in connected speech or taken as a lexicon-like list. On the other hand, bound roots and suffixes are not complete, pronounceable phonological objects, and learning over such entities would be one level of abstraction removed from a realistic learning scenario.

An anonymous reviewer suggests an alternative to using morpheme boundaries: marking segments of the root with a [±root] feature. This would effectively allow the learner to capture the root-internal cooccurrence restriction by including [+root] in the constraint. One problem with this move is computational implementation: the addition of this binary feature would mean a larger set of natural classes (and therefore many more constraints) for the learner to analyze. As for capturing the right level of generalization, this would be fitting for a language like Quechua, where affixes do not have laryngeals at all (motivating *[–root, +cg] and *[–root, +sg]). But in the case of Aymara, the restrictions do hold both inside roots and inside suffixes, so this would probably not be sufficient – Aymara requires something more along the lines of McCarthy’s (1989) planar separation for morphemes, so they dwell on different planes and escape cooccurrence restrictions.

Our approach uses properties of the learning data to detect other properties of the language, and as such, it can be considered to be an example of cue-based learning (Dresher and Kaye 1990; Gibson and Wexler 1994; Dresher 1999). A critique of cue-based learning is that it assumes a lot of learning machinery specific to language (Nazarov and Jarosz 2017). We do believe that the problem of boundary-sensitive phonotactics is a fairly language-specific one, and it is not immediately clear how one would approach it without recognizing morphemes as separate pieces with boundaries.

The logic of inducing nonlocal interactions from trigrams is not strictly phonological: one could argue for a domain-general status of the deduction that if A and B cannot cooccur nearby in a configuration A-X-B, it worthwhile to check whether A and B can cooccur at longer distances. Our learner uses an independent criterion to check whether A and B can interact at all – they must be part of a natural class, as defined by the language’s phonological contrasts and alternation system. This requirement, along with the requirement that constraints in the grammar must receive robust statistical support, minimizes the learner’s ability to notice nonlocal interactions between unrelated segments, which most linguists would describe as accidental.