2.1 Toy grammar illustration

While additive cumulativity of the weights of simplex constraints in HG captures standard ganging phenomena well, superadditive phenomena that exhibit independent effects that are not merely additive reveals the shortcoming of additive cumulativity. The toy language in (7) illustrates this case; HG without conjunction makes inaccurate predictions, but HG with conjunction makes accurate predictions, as shown later in this section. Weights for grammars in this section were obtained using hand-weighting and the MaxEnt Grammar Tool (Hayes et al. Reference Hayes, Wilson and George2009).

1. (7)

   

In this toy language, candidates that violate only ℂ2 or ℂ3 are observed in (a, b) to be more frequent than candidates that violate ℂ1, occurring at a 60:40 ratio. A candidate that violates both ℂ2 and ℂ3 is much less frequent than a candidate that violates only ℂ1, occurring at a 10:90 ratio, as in (c). Finally, as shown in (d), a candidate that does not violate any constraint is better than one that violates ℂ1, occurring at a 70:30 ratio.

Given this input pattern, the constraints are weighted evenly, because the grammar is attempting to reconcile two competing forces. First, the individual weights of ℂ2 and ℂ3 must be low enough to allow candidates that violate ℂ2 and ℂ3 individually to win in (7a, b). But at the same time, the sum of ℂ2 and ℂ3 must be high enough to overtake ℂ1 by a large enough margin to achieve the extreme 10:90 gang effect in (c). The weight of ℂ1 is kept in check by the pair in (d).Footnote ³ Under this schema, regular additive ganging cannot capture the extreme effect shown in (c), where the candidate violating both ℂ2 and ℂ3 loses by a large margin. The predictions of the grammar, given in the rightmost column, are wrong, because the grammar does not have enough degrees of freedom to capture the independent slope variances of the gang effect vs. the simplex effect.

Decreasing the weights of ℂ2 and ℂ3 to try to accurately map the surface frequencies for inputs 1 and 2 in the toy language results in a greater inaccuracy in predicting the extreme gang effect. This adjustment is shown in (8).

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The decrease in the weights of ℂ2 and ℂ3 now allows for an accurate prediction in (a, b), where the winning candidates only violate ℂ2 or ℂ3 independently. However, the additive effect of ℂ2 and ℂ3 is also now insufficiently large to capture the gang effect in (c). The grammar, which relies on standard additive cumulativity, mispredicts a ratio of 49:51 between the loser and winner, whereas the observed surface pattern is 10:90. The grammar does achieve a gang effect, such that the sum of ℂ2 and ℂ3 is greater than ℂ1, but crucially, the grammar cannot achieve enough of a gang effect.

Weighted constraint conjunction allows the grammar to specify a separate, independent weight, above and beyond the additive effect of ℂ2 and ℂ3. A grammar with weighted constraint conjunction is given in (9).

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The conjoined constraint ℂ2&ℂ3 shoulders the burden here of the superadditive gang effect: it adds a weight to the harmony score of the losing candidate in (c), making the loser's harmony score greater than the sum of the weights of ℂ2 and ℂ3. This allows ℂ2 and ℂ3 to be properly weighted below ℂ1, generating the correct output candidates for (a) and (b). Furthermore, because ℂ1 is now correctly weighted above ℂ2 and ℂ3, the correct surface output of (d) can also be modelled, where a candidate that does not violate any constraint is more likely to defeat a candidate that violates ℂ1 than is a candidate that violates ℂ2 or ℂ3, as in (a, b).

Thus cases that will necessitate weighted constraint conjunction will be instances of extreme, superadditive gang effects in quantitative data. The following section turns to one such case, from Dioula d'Odienné, which shows cumulative effects that necessitate weighted constraint conjunction.

3.1 Predicting Type 1 vs. Type 2 items

Whether an item is a Type 1 or Type 2 noun is, to an extent, predictable. In his original description of the Dioula tone system, Braconnier (Reference Braconnier1982) points to three factors that contribute to distinguishing between Type 1 and Type 2: (i) the sonority of the final intervocalic consonant, (ii) whether the place of articulation of the final two vowels is identical and (iii) the nasality or orality of the final vowel and final intervocalic consonant. Crucially, here I show quantitatively that the more similar the segments in the word-final VCV# sequence are along these independent planes of similarity, the more the noun is to be Type 2. That is, the more similarity in the final two syllables, the more likely they will be in the same tonal domain (i.e. Type 2).

Nouns with more sonorous final intervocalic consonants are more likely to be Type 2 nouns, with the domain of tone change being the final two syllables: e.g. /mèlì/ → [mélí] in (10b). This sonority effect scales quantitatively with increasing sonority (Shih Reference Shih2013), as illustrated in Fig. 1. Thus, as the sonority of the final intervocalic consonant becomes more similar to the sonority of its surrounding vowels, the tonal domain for the definite alternation is more likely to include the entire VCV# sequence (i.e. the final two syllables). This is especially true when comparing the nasals ([m n ɲ ŋ]) with the liquids ([l r]) and with [g], which behaves phonotactically like a sonorant and often surfaces as a voiced fricative [ɣ] (Braconnier Reference Braconnier1983). Braconnier (Reference Braconnier1983: 50) notes that [j] behaves phonotactically like a fricative, which corresponds to its behaviour here: nouns with [j] in final intervocalic consonant position are more likely to be Type 1. There are too few tokens of [w] (two Type 1 occurrences and one Type 2) to draw any definite conclusion about its behaviour. Obstruent consonants are the least similar in sonority to the flanking vowels, and so items with obstruent final intervocalic consonants are the least likely to be Type 2: there are only four exceptions, out of 294 total final intervocalic obstruents.

Figure 1

Final intervocalic consonants by Type. Width of bars are scaled by proportion of data.

Nasal similarity also contributes to distinguishing Type 1 vs. Type 2 items, as shown in Fig. 2. Nouns in which the final vowel and intervocalic consonant are both nasal or both oral are more likely to be Type 2 than nouns in which the final vowel and intervocalic consonant disagree in nasality or orality: /sànã̀/ → [sánã́] in (10b).

Figure 2

Nasality agreement by Type. Width of bars are scaled by proportion of data.

The third predictor of Type 1 vs. Type 2 items noted by Braconnier (Reference Braconnier1982) is long-distance featural identity between vowels, shown in Fig. 3. If the final two vowels are featurally identical (i.e. they agree in height, rounding and backness), then the likelihood of being Type 2 increases: e.g. /bìlì/ → [bílí] in (10b).

Figure 3

Vowel-identity agreement by Type. Width of bars are scaled by proportion of data.

These similarity preconditions on definite tone behaviour gang up in a cumulative similarity interaction. The more similarity that is exhibited amongst the members of the word-final VCV# sequence in terms of sonority, nasality and vowel identity, the more likely it is that the noun will be Type 2, with the tonal domain extending over the final two syllables: e.g. /tùrù/ → [túrú] in (10b). This ganging effect is illustrated in Fig. 4: the rightmost bar, which has the largest proportion of Type 2 items, represents nouns that agree in vowel identity, have a sonorant intervocalic final consonant and agree in either nasality or orality. The leftmost bar, in contrast, has the largest proportion of Type 1 items, and represents nouns that share no similarity across the VCV# sequence.Footnote ⁵ The bars in between represent items that only share similarity along a single dimension: from left to right, in vowel identity but not in nasality or sonority, in nasality but not in vowel identity or sonority, or in sonority but not in vowel identity or nasality.

Figure 4

Final intervocalic consonants by Type. Width of bars are scaled by proportion of data.

The ganging up effect demonstrated by the Dioula nouns amounts to a superadditive phenomenon, wherein the more similar the vowels and consonant of the final two syllables are, the more likely they are to belong in the same tonal domain. The ganging effect is made up of three (somewhat) independent similarity conditions. They are independent because each contributes individual effects on Type 1 vs. Type 2 likelihood: for example, as is evident in Fig. 4, agreement in nasality/orality is much more likely to predict Type 2 behaviour than agreement in vowel identity alone. We will see this borne out in the constraint-weighting results in §4.

3.2 An OT approach to Dioula tone harmony

The Dioula tone pattern is formalised using Agreement by Correspondence (ABC) theory, adapted from the analysis in Shih (Reference Shih2013). While ABC was originally developed for long-distance consonant-harmony phenomena (Walker Reference Walker2000, Hansson Reference Hansson2001, Reference Hansson2010, Rose & Walker Reference Rose and Walker2004), it has since been extended to other segmental and tonal interactions, including vowel harmony (e.g. Sasa Reference Sasa2009, Walker Reference Walker2009, Reference Walker2014, Rhodes Reference Rhodes2012), dissimilation (e.g. Bennett Reference Bennett2013, Reference Bennett2015a, Reference Bennettb), local harmonies (e.g. Wayment Reference Wayment2009, Inkelas & Shih Reference Inkelas and Shih2014, Lionnet Reference Lionnet, Kingston, Moore-Cantwell, Pater and Staubs2014) and tone (e.g. Shih & Inkelas Reference Shih, Inkelas, Kingston, Moore-Cantwell, Pater and Staubs2014, Inkelas & Shih Reference Inkelas and Shih2016). ABC builds on the core insight that segments that are similar and proximal are more likely to interact (e.g. Kaun Reference Kaun1995, Burzio Reference Burzio and Boucher2002, Zuraw Reference Zuraw2002, Frisch et al. Reference Frisch, Pierrehumbert and Broe2004). As such, ABC is particularly well suited for modelling parasitic patterns such as the similarity-based tone domains in the Dioula definiteness alternation, in which phonological similarity and proximity beget phonological interactions via correspondence domains. I focus here on the similarity bases that determine the tonal domains for the definite/indefinite alternation. For further details of the Dioula analysis that do not immediately pertain to the similarity issue, see Shih (Reference Shih2013).

In ABC, surface correspondence relationships are determined by the phonological similarity and proximity of segments, encoded in correspondence constraints. In essence, segments with similar features precondition the domain of phonological interaction such as agreement or dissimilation. Under an ABC approach for Dioula, the tone domains that distinguish Type 1 vs. Type 2 items are correspondence domains over which tone interactions take place. The set of relevant Corr constraints for Dioula nouns are given in (12). The constraints here follow pairwise correspondence and identity assessment (Hansson Reference Hansson2007, Rhodes Reference Rhodes2012).Footnote ⁶

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Correspondence constraints are needed for each independent dimension of similarity involved in determining the tone domain for Type 1 vs. Type 2 items. The constraints in (12a) mandate that segments that are sufficiently similar in sonority to adjacent segments be in a correspondence relationship; given the scalar nature of sonority, a set of scaled correspondence constraints is used here. The constraint in (12b), Corr-[X∷X]σ[±nas], states that segments within a syllable that share nasality or orality are in a correspondence relationship.Footnote ⁷ The constraint in (12c), Corr-VV[F], mandates that consecutive vowels that are featurally identical be in a correspondence relationship. For simplicity in tracking the individual similarity-based correspondence relationships, I assume here that each correspondence constraint (or set of correspondence constraints, in the case of the scalar sonority constraints) projects its own correspondence set, as demonstrated in (13) (for a similar approach, see Hansson Reference Hansson2014 and Walker Reference Walker2014; cf. Bennett Reference Bennett2013 for single correspondence sets in a single word). Correspondence relationships are indicated by subscripts.Footnote ⁸

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While correspondence constraints set up similarity-conditioned correspondence relationships, phonological interactions predicated on these correspondence sets are triggered by Corr-limiter constraints (Bennett Reference Bennett2013). Limiter constraints create unstable surface correspondence relationships, wherein corresponding segments are similar enough to interact but are too uncomfortably similar to stably coexist at a certain distance (see Wayment Reference Wayment2009 and Inkelas & Shih Reference Inkelas and Shih2014 for discussion on instability). Unstable correspondences give rise to repairs by harmony (i.e. attraction, whereby similar segments become more similar) or dissimilation (i.e. repellence, whereby similar segments become sufficiently dissimilar).

The Dioula definiteness alternation can be characterised by mandating that corresponding segments agree in tone, using the limiter constraint in (14).

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Under this analysis, Type 2 behaviour in underlyingly L-toned items means that the H definite morpheme spreads onto both penultimate and final syllables: e.g. L.L → Hx.Hx. If the relevant Corr and limiter constraints outweigh input–output faithfulness to tone, (15), then similar segments will correspond and assimilate in tone, meaning that if the segments are similar enough, the noun will be Type 2.

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Shih (Reference Shih2013) argues that underlyingly H-toned items can also follow the same correspondence-based analysis, even though the surface tonal alternations are different. The contour-tone formation in Type 1 items and lack of H-tone agreement in Type 2 items for underlyingly H-toned nouns stems from antihomophony and OCP avoidances of adjacent, heteromorphemic H tones. For simplicity here, I set aside the surface tonal differences between underlyingly H- and L-toned nouns, and focus on the similarity-based correspondence relationships, because these determine the relevant tonal domains in Type 1 vs. Type 2 items, which cut across any differences in underlying tones.

Hand-weighted HG tableaux are shown below as illustrations of the ABC system for Dioula tone harmony. As a simplification, the suffix H tone is assumed to be realised on the final vowel already: Shih (Reference Shih2013) accomplishes this in the full Dioula system using high-ranked RealiseMorph. In (16), a sonorant final intervocalic consonant (e.g. [l]) facilitates tone agreement between the final vowel and the penultimate vowel by satisfying high-weighted Corr-X∷X{V,R} and the mandated tonal identity between corresponding segments (i.e. Ident-XX[tone]). The relevant Corr constraints that give the necessary precondition for correspondence and tone agreement must outweigh input–output faithfulness in order for the corresponding and tonally agreeing candidate, (16d), to win.

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In (17), a non-sonorant final intervocalic consonant t (e.g. [s]) does not facilitate tone harmony, because it is insufficiently sonorous to incur violations of Corr-X∷X{V,R} when between two vocalic segments. The winner when there is an obstruent, then, is the disharmonic candidate with no correspondence in (17a).

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The tableau in (18) illustrates the cumulative addition of constraint weights. In this case, sonorant consonants and featurally identical vowels facilitate tone spread, due to the additive effect of two high-weighted Corr constraints, Corr-X∷X{V,R} and Corr-VV[F].

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If the additive effect shown in (18) is implemented as weighted constraint conjunction on top of regular cumulative additivity (19), then there is an increase in the magnitude of harmony-score differences (ΔH) between the winner and the loser candidates. While the ΔH between the harmonic winner, (d), and completely non-harmonic non-corresponding candidate, (a), in the additive approach in (18) is 10, the ΔH in the superadditive approach in (19) is 14, denoting a greater predicted likelihood of tone harmony.

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4.1 Comparing weighted conjunction and additive cumulativity

The necessity of weighted constraint conjunction can be tested using model comparison of grammars with and without conjoined constraints. If a gang effect is merely additive, then a grammar with conjoined constraints should not contribute any additional explanatory power. If, however, a gang effect is superadditive, then a grammar with conjoined constraints will demonstrate improved explanatory power over a grammar without conjunction.

Model-comparison tests for the necessity of constraints have been used sparingly in HG and OT approaches, largely due to the usual assumption that Con provides a constraint set and the grammar's primary concern is constraint weighting or ranking. Under this mode of operation, the assessment and rejection of the viability of a constraint's existence is largely left to arguments on conceptual or phonological grounds: e.g. Occam's Razor and naturalness. There are notable exceptions in the MaxEnt literature, however, that appeal to model comparison for constraint assessment: see e.g. Wilson & Obdeyn (Reference Wilson and Obdeyn2009), using a maximum a posteriori approach, and Hayes et al. (Reference Hayes, Wilson and Shisko2012), using model log-likelihood comparison.

One quantitative approach to assessing competing grammars – i.e. one which allows conjunction and one which does not – is Akaike Information Criterion (AIC)-based model comparison, which allows for the comparison of significant improvements in capturing information between competing grammars. AIC model comparison is an approach founded on the idea that all models (i.e. grammars) are mere approximations of full reality, an ideal for which the true parameters (β) remain unknown (e.g. Kullback & Leibler Reference Kullback and Leibler1951, Burnham & Anderson Reference Burnham and Anderson2002, Reference Burnham and Anderson2004). The aim in AIC model comparison is to reduce the amount of information loss in a candidate grammar: the less information that a candidate grammar loses, the more weight of evidence there is in favour of that particular grammar.

Information criteria measures come in various forms.Footnote ¹⁰ I use second-order AICc, as shown in (22), because it penalises an increasing number of constraints against a sample size, with adjustments for sample size and the number of constraints in the grammar (Burnham & Anderson Reference Burnham and Anderson2002, Reference Burnham and Anderson2004).

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When n/K>40, AICc begins to converge with AIC. Because AICc regularises for sample size, it is the more conservative measure for model comparison in general (Burnham & Anderson Reference Burnham and Anderson2004: 269–270).

Algebraically lower AICc values suggest better model support. As a rule of thumb in comparing candidate models, any AICc difference greater than 10 between two candidate models is considered large. Translated into an evidence ratio E, as in (23), a 10-point difference between two candidate models is equivalent to about a 150:1 odds that the second-best model has essentially no evidential support of being as good as the best candidate model (e.g. Anderson Reference Anderson2008: 89–90).

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AICc is based on the standard likelihood ratio test, which maximises descriptive accuracy given the observed data (see e.g. Hayes et al. Reference Hayes, Wilson and Shisko2012 for its use in phonology). But, unlike a likelihood ratio, AICc penalises a loss of restrictiveness in the grammar that potentially comes with the addition of constraint conjunction.Footnote ¹¹ A further advantage of AICc comparison is that its results for assessing restrictiveness and generalisability have also been shown to asymptotically converge with k-fold cross-validation as sample size increases, while remaining computationally faster than k-fold cross-validation (e.g. Stone Reference Stone1977).

It is crucial to note that the AICc number on its own is not a statistical test of significance or of stand-alone goodness of fit: AICc metrics must be taken as comparison statistics of more than one candidate model and between at least two AICc measures.

4.2 Results of model comparison

Results from AICc model comparison are as follows. A model with constraint conjunctions has an AICc value of 1101, while a model without conjunction has an AICc value of 1083. This 18-point difference between AICc scores indicates a rough 8000:1 odds (E = 8103.08) that there is substantially more support for a grammar with weighted constraint conjunctions. That is, having weighted constraint conjunctions better approximates truth in predicting Type 1 vs. Type 2 items in Dioula, even after penalising for increased model complexity (ΔK=4).