@cite{zuraw-hayes-2017}: Intersecting Constraint Families #
@cite{zuraw-hayes-2017}
@cite{zuraw-hayes-2017} "Intersecting Constraint Families: An Argument for Harmonic Grammar" (Language 93(3): 497–546).
Main claims #
When phonological variation is governed by two independent families of constraints, the data exhibits across-the-board effects with floor and ceiling compression — a family of sigmoid curves.
This pattern is naturally predicted by Harmonic Grammar (MaxEnt and Noisy HG) because constraint effects are additive.
Decision-tree models fail because their multiplicative decomposition produces "claws" (monotonically increasing differentiation), not sigmoid families (§2.6, §3.10).
Stochastic OT fails because ranking paradoxes prevent fitting structured constraint sets to the observed pattern (§2.6, §3.8).
Formalization #
This file proves the decision-tree failure theorem and connects the
empirical Tagalog data to the MaxEnt prediction via the constraint
independence machinery from Separability.lean. The deeper proof
that MaxEnt's success follows from harmony separability is developed
by @cite{magri-2025} — see Magri2025.lean.
The formalization uses the 2×2 sub-case of the Tagalog data
(maŋ-other/paŋ-res × /b//k/) from Fragments.Tagalog.Phonology,
which suffices for the mathematical theorem. The full paper uses a
6×6 grid (6 prefixes × 6 consonants).
Case studies #
- Tagalog nasal substitution (§2): nearly synergistic families (both markedness and prefix UNIFORMITY constraints mostly penalize substitution; only *NC̥ favors it on the consonant side)
- French liaison/elision (§3): synergistic families (word2 ALIGN + word1 USE both favor non-alignment)
- Hungarian vowel harmony (§4): mixed families (stem vowel AGREE + final consonant BILABIAL/SIBILANT/etc.)
Across-the-board consistency: one dimension's effect has the same sign regardless of the other dimension's value. Formally: the product of row-wise rate differences across columns is positive (same sign).
Equations
Instances For
Tagalog nasal substitution rates exhibit across-the-board consistency: maŋ- prefixes have higher substitution than paŋ- prefixes for both voiced (/b/) and voiceless (/k/) stem-initial consonants.
Decision-tree models predict monotonic differentiation (§2.6):
In a multiplicative model P(x,y) = g(x) · h(y), the difference
between two h-values is proportional to g:
g(x) · h(y₂) - g(x) · h(y₁) = g(x) · (h(y₂) - h(y₁))
So at the floor (g → 0), all h-differences vanish, and at the ceiling (g → 1), h-differences are maximal. Differences grow monotonically from floor to ceiling.
This is the "claws" pattern: pinching at one end only. In contrast, MaxEnt predicts humped differentiation: sigmoid families compressed at both extremes — the observed pattern.
In a multiplicative model, the ratio of differences across two g-values exactly equals the ratio of those g-values. Cross-multiplied form (avoids division):
Decision-tree ceiling bound: in a multiplicative model with both factors in [0,1], the product is bounded above by both factors.
This is the mathematical core of why decision trees produce "claws"
instead of sigmoid families: probabilities can never exceed either
component probability. At the floor (g → 0), all products vanish
regardless of h — explaining the pinch at one end. But at the
ceiling (g → 1), differences are preserved — so there is NO
compression at the top. MaxEnt, by contrast, compresses at BOTH
extremes via the sigmoid function 1/(1 + eⁿ).
Both HG variants beat both ranking models for Tagalog (Table 7). This is the paper's core claim: constraint weighting consistently outperforms constraint ranking.
Both HG variants beat both ranking models for French (Table 17).
MaxEnt predicts HZ's generalization for Tagalog nasal substitution:
for any weight assignment w : Fin 6 → ℝ, the MaxEnt logit rates
satisfy the constant-difference identity.
LR(/maŋb/) − LR(/maŋk/) = LR(/paŋb/) − LR(/paŋk/)
The proof instantiates me_predicts_hz (Separability.lean) with the
Tagalog violation differences and their independence (from the
Tagalog fragment).
The constant logit-rate difference equals −w₂ + w₃ + w₄
for both rows, regardless of weights. This follows from the
insensitivity structure of the six constraints: markedness
constraints (C₁–C₄) are insensitive to prefix, so their
contributions cancel in the row difference, while faithfulness
constraints (C₅–C₆) are insensitive to stem consonant, so they
cancel in the column difference.
The HZ identity verified concretely: both row-differences are equal.
NHG produces consistent ordering (@cite{zuraw-hayes-2017} §2.5,
Figure 8): when the harmony scores satisfy ConstantLogitDiff, NHG
probabilities Φ(d(x)/σ) exhibit across-the-board consistency.
Composes constantLogitDiff_mono_consistent (CLD + strict monotonicity
⟹ consistent ordering) with normalCDF_strictMono. Since
x ↦ Φ(x/σ) is strictly monotone for σ > 0, the result follows.
This is the formal version of Z&H's argument that NHG produces sigmoid
families (not claws) because the normal CDF compresses at both
extremes.
NHG predicts consistent ordering for Tagalog nasal substitution: for any weight assignment and noise level, the NHG probabilities of nasal substitution exhibit across-the-board consistency whenever the mang- and pang- prefixes have different logit rates for b-initial stems.
End-to-end chain: Tagalog violation differences (fragment) →
violDiff_independence → maxent_predicts_hz_tagalog (CLD) →
nhg_consistent_ordering (CDF monotonicity) → consistent ordering.