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Readings： French schwa and gradient cumulativity

My thoughts

My Questions

1. What’s a singleton?
   • 在这篇文章的语境里，singleton 指“单辅音”（single consonant）。对比的是“cluster”（辅音连缀/辅音簇）。 - 比如： - C_ 表示 schwa 前面只有一个辅音（singleton）。 - CC_ 表示 schwa 前面有两个辅音（一个辅音簇，如 /st/、/kt/ 等）。
2. Sublinear, linear and superlinear 这个是什么意思？
   • 这三个词描述“当两个约束共同起作用时，累加到概率上的总效果”与“每个约束单独起作用时的效果”之间的对比关系。
   • 设想有两个有利于出现 schwa 的约束 A 和 B：
     • A 单独出现时把概率提高了 ΔA；
     • B 单独出现时把概率提高了 ΔB；
     • A 和 B 同时出现时把概率提高了 ΔA|B（也可看成从只存在 B 的环境再加上 A 后的增量，或反之）。 - 定义： - 次线性（sublinear）：ΔA|B < ΔA（在另一约束存在的“累加”情境里，A 的边际效果更弱）。
   • 含义：当 B 已经把概率推高到 0.40 后，再加 A 的“边际提升”变小了（从 0.30 变 0.20）。这就是“次线性”。 - 线性（linear）：ΔA|B = ΔA（边际效果与单独时相同，等加成）。
   • 含义：A 的效果不因 B 的存在而改变，恰好“等加成”。 - 超线性（superlinear）：ΔA|B > ΔA（两个约束一起时，边际效果更强，产生“1+1>2”的增强）。
   • 含义：在 B 已经存在时，再加 A 的“边际提升”反而更大（从 0.15 变 0.30），两个约束一起“相互增强”，产生 1+1>2 的感觉，这就是“超线性”。 - 有两个有利因素/约束：A（比如“避免辅音三连”）和 B（比如“避免重音冲突”）。
   • 我们比较三种情境下的“概率提升量”（增量）：
     • 只有 A 起作用时，概率从基线 P0 提升到 P(A)；增量 ΔA = P(A) − P0
     • 只有 B 起作用时，概率从 P0 提升到 P(B)；增量 ΔB = P(B) − P0

     • A 和 B 一起起作用时，概率从 P0 提升到 P(A+B)；但为了看“在已有 B 的情况下再加 A 的边际效果”，我们比较 P(A+B) 与 P(B) 的差：ΔA

       B = P(A+B) − P(B)

     • 这三个量里，关键比较的是“ΔA

       B”和“ΔA”。也就是：当 B 已经在起作用时，再加 A 的效果，是否比 A 单独起作用时更弱/一样/更强。

• 论文的发现与结论：
  • Stochastic OT 只能产生“次线性”的累加性（两个约束一起时边际增益变弱）。
  • Noisy HG 和 MaxEnt 能覆盖更广：可以表达次线性、线性乃至超线性。Smith & Pater (2020).pdf

1. cumulative是什么意思？
   • cumulative 在这里指“累加性的、可叠加的”互动。具体到音系学，就是多个约束（比如“避免辅音三连”、 “避免重音冲突”）同时相关时，它们对输出形式（这里是 schwa 是否出现）的影响会“合起来”起作用，而不是彼此互不相干或被简单遮蔽。
   • 文中说的“gradient cumulativity（梯度式累加性）”强调这种累加是“概率大小上的差异”，不是全或无的范畴式对立；并且累加的强度可以是次线性、线性或超线性的。
2. 2 × 2 × 2 factorial design, what’s the eight conditions?
   • *CCC 2 contex
   • deletion and ethpesis
   • stress 2 contex

Summary

• Explore the interaction of two phonological factors that condition schwa-zero alternations in French
  1. Schwa is more likely after two consonants than a singleton
     • What’s a singleton?
     • from online: a consonant that is not geminated; it’s a single consonant sound, as opposed to a geminate, which is a long consonant. The primary difference is typically duration, with geminates being significantly longer than their singleton counterparts
  2. schwa is more likely between stressed syllables than elsewhere
• Method: new data from judgment study
  • Finding:
    • Both factors play a role in schwa epenthesis (insert a schwa) and deletion (delete a schwa)
    • Both factors interact cumulatively
      • strong effect together than individually
    • Cumulatively, interaction is better modeled with weighted than with ranked constraints
  • Stochastic OT can model cumulative interactions, but only sublinear ones
    • Downside: the effect of a constraint is weaker in the cumulative context than on its own
  • Weighted constraint models (MaxEnt and Noisy HG)
  • Can model the full range of cumulativity
    • Sublinear, linear, and superlinear
    • 这个是什么意思？
• Stochastic OT is hampered by the fact that the data displays superlinear cuulativity
• Noisy HG and MaxEnt fare better on this dataset = MaxEnt has the best fit

Aim of this paper

• focus on the interaction of two phonological factors that affect the probability of schwa deletion and epenthesis
  1. the cluster factor (whether a singleton consonant or a consonant culture precedes the schwa, *CCC)
     • Deletion is less likely, and epenthesis more likely, when schwa is preceded by a cluster
     • Schwa deletion only applies when a single consonant precedes
       • Dell’s rule abstracts away from the fact that schwa deletion can also apply when a cluster precedes
       • Deletion almost certainly applies in these examples with a lower probability
         • Examples of deletion in the CC_ context in devrait and me
           • [ʒak dvʁɛ paʁtiʁ] Jacques devrait partir. ‘Jacques should leave.’
           • [ʒak m dwa dœ laʁʒɑ̃] Jacques me doit de l’argent. ‘Jacques owes me money.’
         • Schwa epenthesis only applies after a morpheme that ends in a cluster
  2. The stress factor: (Position of the schwa in the phrase (conditioning the probability of both deletion and epenthesis) *Clash )
     • When schwa is followed by a stressed monosyllabic word (its presence avoids a stress clash)
       • Deletion is less likely
       • Epenthesis is more likely
       • Example: film is more likely to be followed by a schwa in un film russe [ɛ̃ ˈfilmœ ˈʁys] than un film danois [ɛ̃ˈfilmœ daˈnwa]
     • Both empirical and theoretical challenges
       • Empirical: data on the relative frequency of outcomes are harder to collect than data on categorical differences
         • native speakers’ ituation is important but they do not provide a fine-grained data to evaluate and compare probabilisitic models
       • theoretical: many phonological frameworks have no way to express the greater probability of the schwa-less realizaiton in 1a,b than 2a,b and why particular contexts favor schwa
         • The standard SPE framework adopted by Dell 1985 allows rules to apply categorically or optionally, but not with some specified probability
         • The variable Rules model can do conditioning factors and separate deletion and epenthesis rules, but this will not make strong predicitons, seems to be no reason that a preceding cluster could not increase the probability of deletion and decrease the probability of epenthesis, the opposite of observed facts
         • Constraint-based models can address both of these theoretical challenges
           • allow a single factor, or constraint to play a role across multiple processes
           • two constrinats (interact cumulatively = schwa is more likely to be realized in contexts which favored by both constraints):
             1. *Clash = assign one violation for every two adjacent stressed syllables, effect = schwa is more likely to be realized in between two stressed syllables compared to one stressed and one unstressed syllables
             2. *CCC = assign one violation for every sequence of three consonants, effect = schwa is more likely to be realized in VCC_C than VC_C

• These differences in predictions and compare 3 constraint-based models of variation in detail
  • Stochastic OT = produces some cumulativity in variable patterns, only sublinear cumulativity, the effect of a constraint is waker in the cumulative context than on its own
  • Noisy Harmonic Grammar = cumulative constraint interaction is a major predictions, weighted constraints produce gang effect,. and probabilistic variants of HG
  • Maximum Entropy Grammar = produce gradient cumulativity
    • All three caputered the fact the schwa realization has the highest probability in CC_stressed syllable
    • but three different models permit differnt patterns of relative probability across the cells
• 不知道什么意思？
• 关键点在“累加性”（cumulativity）：多个约束一起作用时，对概率的提升是“次线性”“线性”还是“超线性”。作者指出：Stochastic OT 只能产生“次线性”的累加效应；而 Noisy HG 与 MaxEnt（加权约束模型）可以产生从次线性到线性再到超线性的更广范围的累加效应。

Introduction

• Dell (1985) found in Parisian French that both deletion of underlying schwa and epenthesis are involved in producing the surface distribution
  • Schwa deletion and epenthesis
    • /dœvrɛ/ → [dvrɛ] Tu devrais partir. ‘You should go.’ (Schwa deletion)
      • œ -> 0 between consonants, produced with a surface schwa
    • /mœe/ → [m] Tu me dois de l’argent. ‘You owe me money.’ (Schwa deletion)
      • œ -> 0 between nasal and vowel, produced with a surface schwa
    • /film/ → [filmœ] un film danois ‘a Danish film’ (Schwa epenthesis)
      • 0 -> œ after nasal at word final position, produced without a surface schwa
  • In some words, schwa never alternates with zero, even in a phonological environment where deletion is usually likely
  • both variations are possible in a neutral rate and register

Schwa epenthesis and deletion

• 

• There are 3 enviroments where schwa alternates with 0
  • clitic boundaries = deletion
  • word boundaries = epenthetic
  • morpheme-internally = deletion
• assume schwas are found morpheme-internally or at clitic boundaries
• assume schwas in clitics are underlying
  • account for generlizaiton that schwa is realized more often in clitics than at word boundaries
• this asymmetry betwee schwas at clitic boundaries and word boundartied follows from the faithfulness constraints MAX and DEP
  • Max prefers schwa to be realized when it is underlying
  • Dep prefers schwa to be absent when it would need to be inserted

The cluster factor

• For both underlying and epenthetic schwa
  • schwa is realized more often after two or more consonants than after a singleton consonant
  • schwa is generally realized more often when it’s underlying (at a clitic boundary) than when it’s epenthetic (at a word boundary)

The stress factor

• schwa as more likely to be realized in the penultimate syllable
  • for both word boundaries and clitic boundaries

• our experimental data show an effect of the number of following syllables even after a single consonant.

Method

• use experimental data on French schwa
• 27 participants after exclusion, age 21 - 48, mean age = 35.11
• 2 × 2 × 2 factorial design, with 8 conditions.
  • c_schwa
  • cc-schwa
  • schaw-stressed
  • schaw-unstressed
  • schaw at the clitic boundary
  • schaw at the word boundary
• Judgments from multiple native speaker on acceptability of realized schwa across contexts
• the realization of schwa is conditioned by segmental context and rhythmic context, which we analyze using the constraints *CCC and *Clash.
• The experiment employed a forced choice paradigm, in which participants were asked to imagine that they were speaking with a friend, and choose between two variants of a phrase: one with a pronounced schwa and one without the schwa.
• In addition to choosing between schwa and no schwa, participants indicated their confidence in the answer as certainement or probablement. Probablement and certainement responses are pooled in statistical analyses of the data, which model the probability of schwa realization
• contorlled for segmental and prosodic context as much as possible
• position of schwa was manipulated by using different tenses of verbs
  • each participant saw every name and verb lexeme only once
• 6 items per condition, 24 for deletion and 24 for epenthesis, and 30 fillers

3.3 Results

• mixed effect logistic regression
  • DV = probability of schwa, which is expressed as log-odds
  • maximal random effect structure
  • random intercepts for subject and item
  • random slope by subject for all of the fixed effects = include the interaction term

  • Schwa ~ Ep/Del + Stress * Seg + (1

    Item) + (1 + Ep/Del + Stress * Seg

    Subject)

  • the inclusion of random slopes mean that some speakers can be exceptionally sensitive or insensitive to phonological context
  • categorical varibales are sum-coded

4 Presentation of modeling results

• compare the ability of three models of variation to fit our experimental data

• Stochastic OT
  • schwa realization is optimal iff a schwa-preferring constraint is ranked above NoSchwa
    • probabilistic variation of OT.
    • Each constraint is given a real numbered ranking value, when grammar is used to evaluate a candidate set, the ranking values are converted to an ordinal OT ranking
    • varation happens becuase the constraint value has a real number added to it and is sampled from a Gaussian distribution centered on 0 (resampled from each constraint)
    • predicts greater probability in a gang effect context
    • two differences between the patterns of gradient cumulativity between OT and MaxEnt
      1. if two violations in the cumulative case come from a single constraint, then OT will not show an increase in probability when there are two violations being avoided instead of just one
      2. there are patterns of strong cumulativitiy that cannot be represented by OT, but can be represented by MaxEnt
         • the weakness is that its cumulativity is always sublinear
           • Definiation: contribnution of a constraint = the difference between the probability of that outcvome in a context in which the constraint applies compared to minimally different context in which it is irrevelant
           • *CCC is the probability of schwa in C minus the probability in A, and *CLASH is the probability in B minus the probability in A

           • ΔCon1

             Con2 is the difference between the cumulative context and the non-cumulative context for Con2

           • Whether a case of cumulativity is sublinear, linear, or superlinear depends on how the cumulative contribution of a constraint compares with its independent contribution. If the cumulative contribution is less than the independent contribution, the pattern is sublinear. If the cumulative contribution is greater than the independent contribution, the pattern is superlinear. If the two contributions are equal, the pattern is linear.
           • OT. In Stochastic OT, ranking values are sampled from a normal distribution over the unbounded space of possible ranking values, so no ranking ever has zero probability. Therefore, cumulativity will always be sublinear in Stochastic OT.

4.1.3 Sublinearity through superlinearity in MaxEnt and Noisy HG

• MaxEnt
• In MaxEnt the probability of a candidate is proportional to the exponential of the weighted sum of violations. In terms of the difference vectors, the probability of the realized schwa is en/(1 + en), where e is the base of the natural logarithm (approximately, e = 2.71828) and n is the weighted sum of difference scores.
  • Harmony difference between 2 candidates is the log-odds of candidate a
  • This is superlinear because the cumulative contributions of *Clash and *CCC (0.46) are greater their independent contributions (0.22).
• HG
  • optimal candidate is the one whose weighted sum of the contraint scores which is harmoney is the highest
  • This gang effect, or cumulative constraint interaction, cannot be modeled in standard deterministic OT: no ranking of these constraints will produce an output schwa in only the top row; it will always be accompanied by an optimal output schwa in one of the middle rows.
  • Similar to OT, except the values of the constraints are used in a weighted constriant evaluation of the candidate set
  • superlinear cumulativity
  • but they are not identical

• Finally, Noisy HG and MaxEnt can display superlinear cumulativity in probability differences, as shown in Table 12 in which NoSchwa is given a higher value than *CCC and *Clash (again 2 vs. 1). In MaxEnt, we get predictable superlinearity when the result of adding a single constraint is probability less than 0.50

  

4.2 Models fitted to French data

Predictions and directions for future work

• A richer model would take into account factors that we controlled for and mentioned in passing, such as the sonority profile of the consononant cluster, the number of preceding syllables, h-aspiré, and individual differences between speakers.

Conclusion

• modeled two phonological factors that condition french schwa alternations
  • consonant cluster
  • penultimate syllable or elsewhere = stress
• This is been identified in the liteature for French schwa, but no one looked at their interactions in the probability space
• using data from a judgement study, this study showed that both factors play a role in schwa epenthesis and deletion
• including in contexts where the stress factor has previously been described as having no effect
• so procied the cumulative interaction as sub trhough super-linear, showing that OT is limited to sublinear cumulativeity
• since superlinearity is attested in the experimental data, OT fared less well in fitting the data than the weighted constraint probabilistic models such as HG and MaxEnt
• MaxEnt did the best fit to the data