Frisch, Pierrehumbert & Broe (2004) [FPB04]

Similarity avoidance and the OCP argues that OCP-Place in Arabic verbal roots is gradient: constraint strength is a quantitative function of the similarity of the homorganic pair, where similarity is the natural-classes metric of eq. (7) — shared natural classes over shared plus non-shared, restricted to classes defined by a place feature. A larger, more contrastive region of the inventory generates more natural classes, so any coronal pair is automatically less similar than a comparable labial pair, deriving the strong-coronal vs weak-dorsal/guttural asymmetry that categorical class-based analyses ([McC86], [McC94], [Pad95]) must stipulate.

def FrischPierrehumbertBroe2004.similarity {α : Type u_1} [DecidableEq α] (xs : List (Finset α)) (x y : α) : ℚ

The natural-classes similarity metric of eq. (7) (p. 198): shared natural classes over shared plus non-shared.

Equations

• FrischPierrehumbertBroe2004.similarity xs x y = ↑(List.countP (fun (s : Finset α) => decide (x ∈ s ∧ y ∈ s)) xs) / ↑(List.countP (fun (s : Finset α) => decide (x ∈ s ∨ y ∈ s)) xs)

Labial natural classes (p. 199)

def FrischPierrehumbertBroe2004.labialClasses_fm : List (Finset Arabic.ModernStandard.Consonant)

Labial natural classes for the /f, m/ computation (p. 199): the 2 shared classes, then the 7 non-shared.

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def FrischPierrehumbertBroe2004.labialClasses_bf : List (Finset Arabic.ModernStandard.Consonant)

Labial natural classes for the /b, f/ computation (p. 199): the 3 shared classes, then the 5 non-shared, verbatim from the paper. The {b, w} entry is a substitution typo for {f} — matrix (8) generates {f} and cannot generate {b, w} (derived_bf_classes) — leaving the count, and the reported 3/8, correct.

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theorem FrischPierrehumbertBroe2004.similarity_f_m : similarity labialClasses_fm Arabic.ModernStandard.Consonant.f Arabic.ModernStandard.Consonant.m = 2 / 9

Worked example, p. 199.

theorem FrischPierrehumbertBroe2004.similarity_b_f : similarity labialClasses_bf Arabic.ModernStandard.Consonant.b Arabic.ModernStandard.Consonant.f = 3 / 8

Worked example, p. 199.

Deriving the classes from feature matrix (8)

labialContext records the labial columns of matrix (8), p. 201: [cons], [son], [cont], [voice] on all four labials, [nasal] on the stops only (trivial underspecification). [Bro93a]'s construction over this context regenerates the /f, m/ enumeration exactly and vindicates the reported 3/8 for /b, f/ despite the printed list's typo.

def FrischPierrehumbertBroe2004.labialContext : List (Finset Arabic.ModernStandard.Consonant)

The labial columns of feature matrix (8) (p. 201): the extents on {b, f, m, w} of the feature values the matrix specifies.

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def FrischPierrehumbertBroe2004.derivedLabialClasses : List (Finset Arabic.ModernStandard.Consonant)

The natural classes matrix (8) generates for the labial subinventory.

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theorem FrischPierrehumbertBroe2004.derived_fm_classes : (List.filter (fun (s : Finset Arabic.ModernStandard.Consonant) => decide (Arabic.ModernStandard.Consonant.f ∈ s ∨ Arabic.ModernStandard.Consonant.m ∈ s)) derivedLabialClasses).toFinset = labialClasses_fm.toFinset

The derived classes relevant to /f, m/ are exactly the paper's enumeration.

theorem FrischPierrehumbertBroe2004.derived_bf_classes : (List.filter (fun (s : Finset Arabic.ModernStandard.Consonant) => decide (Arabic.ModernStandard.Consonant.b ∈ s ∨ Arabic.ModernStandard.Consonant.f ∈ s)) derivedLabialClasses).toFinset = insert {Arabic.ModernStandard.Consonant.f} (labialClasses_bf.toFinset.erase {Arabic.ModernStandard.Consonant.b, Arabic.ModernStandard.Consonant.w})

The derived classes relevant to /b, f/ are the paper's enumeration with {f} in place of {b, w}: no description over matrix (8) has extent {b, w}.

theorem FrischPierrehumbertBroe2004.similarity_derived_f_m : similarity derivedLabialClasses Arabic.ModernStandard.Consonant.f Arabic.ModernStandard.Consonant.m = 2 / 9

similarity(/f, m/) = 2/9, derived from matrix (8) rather than the printed enumeration.

theorem FrischPierrehumbertBroe2004.similarity_derived_b_f : similarity derivedLabialClasses Arabic.ModernStandard.Consonant.b Arabic.ModernStandard.Consonant.f = 3 / 8

similarity(/b, f/) = 3/8, derived from matrix (8): the paper's reported value is correct despite its list typo.

Table IV (p. 203): O/E by similarity, adjacent pairs

def FrischPierrehumbertBroe2004.adjacentPairOE : List (ℚ × ℚ)

The adjacent-pair column of Table IV: pairs of similarity-bin representative (bin midpoint) and observed-over-expected co-occurrence rate. O/E falls from 1.22 at similarity 0 to near zero from similarity 0.4 upward — the gradient pattern no two-valued model can match.

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Gradient vs categorical: no threshold TSL₂ grammar fits Table IV

A TSL₂ grammar forbidding tier-adjacent pairs with similarity ≥ t decides each labial pair by the two-valued step function similarity < t, so as an O/E predictor it realises at most two values — one per side of the threshold. Table IV has more than two distinct O/E levels, so no such model fits it exactly. This is the corpus-free core of FPB's quantitative argument; their own comparison is the R² fit of Table V.

def FrischPierrehumbertBroe2004.categoricalAtThreshold (t c₁ c₂ sim : ℚ) : ℚ

Step-function O/E prediction of a threshold model: c₁ strictly below the threshold t, c₂ at or above it.

Equations

• FrischPierrehumbertBroe2004.categoricalAtThreshold t c₁ c₂ sim = if sim < t then c₁ else c₂

def FrischPierrehumbertBroe2004.thresholdedTSL (xs : List (Finset Arabic.ModernStandard.Consonant)) (t : ℚ) : Subregular.TSLGrammar 2 Arabic.ModernStandard.Consonant

The TSL₂ grammar over Consonant forbidding tier-adjacent labial pairs of similarity at least t — [HRT11]'s forbidden-pair schema instantiated with FPB's metric.

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theorem FrischPierrehumbertBroe2004.thresholdedTSL_lang_isTSL2 (xs : List (Finset Arabic.ModernStandard.Consonant)) (t : ℚ) : Language.IsTierStrictlyLocal 2 (thresholdedTSL xs t).lang

TSL₂ witness: the threshold grammar's stringset is tier-based strictly 2-local.

theorem FrischPierrehumbertBroe2004.thresholdedTSL_lang_isBTSL2 (xs : List (Finset Arabic.ModernStandard.Consonant)) (t : ℚ) : Language.IsBTSL 2 (thresholdedTSL xs t).lang

BTSL₂ corollary: the threshold grammar's stringset is in the multitier closure of strictly local languages, hence consumed by the [Lam26] BTC framework.

theorem FrischPierrehumbertBroe2004.thresholdedTSL_pair_iff (xs : List (Finset Arabic.ModernStandard.Consonant)) (t : ℚ) {x y : Arabic.ModernStandard.Consonant} (hx : x.IsLabial) (hy : y.IsLabial) : [x, y] ∈ (thresholdedTSL xs t).lang ↔ similarity xs x y < t

The threshold grammar accepts a labial pair iff its similarity is strictly below the threshold — the precise sense in which any similarity-threshold TSL₂ grammar collapses to the two-valued categoricalAtThreshold prediction.

theorem FrischPierrehumbertBroe2004.categorical_cannot_fit_adjacentPairOE (t c₁ c₂ : ℚ) : ¬∀ p ∈ adjacentPairOE, categoricalAtThreshold t c₁ c₂ p.1 = p.2

No exact categorical fit to Table IV: for every threshold t and predicted rates c₁, c₂, some Table IV bin is missed. A threshold model realises at most two O/E values, but Table IV contains at least three (1.22, 0.59, 0.06). FPB's own argument is the aggregate R² comparison of Table V (p. 207: categorical 0.70 vs natural classes 0.75), which requires the [Cow79] corpus; the exact-fit impossibility here is its corpus-free core.