Breiss, Katsuda and Kawahara 2026: modelling frequency-conditioned paradigm uniformity

In conservative Japanese dialects a compound whose second member (N2) begins with /g/ realises it as [ŋ], but only optionally when the N2 is also a free word, which begins with [g]. Nasalisation is then more likely the more frequent the compound and less likely the more frequent the free N2. Breiss, Katsuda and Kawahara derive both effects in a MaxEnt grammar under Breiss's Voting Bases model, where a faithfulness violation counts in proportion to the resting activation of the listed form it is assessed against. The free N2 is listed with [g] and an existing compound with [ŋ], so the two effects take opposite signs.

The fitted grammar gives the markedness constraint against medial [g] no weight, against Itô and Mester's markedness-driven account; activation raises faithfulness, where Coetzee and Kawahara let frequency lower it; and the sign of each effect follows from the listed velars, where the earlier output–output account of the same authors stipulated it.

Main definitions

• Item: a compound's listed bases, an optional free N2 and an optional whole compound.
• con, violations, votingBases: the constraints (6), the violation vectors scaled by an item's activations, and the MaxEnt distribution of the item's realisations.
• pNasal: the probability of nasalisation.
• table1: the fitted weights.

Main results

• pNasal_eq_sigmoid: the log-odds of nasalisation are the markedness weight plus each base's weighted activation, signed by its listed velar.
• existing_strictAnti_n2, existing_strictMono_compound, novel_strictAnti_n2: the frequency effects of Figures 2 and 3.
• logit_pNasal_existing: the N2 effect on the log-odds is the same in existing and novel compounds.
• existing_lt_bound: a free N2 lowers nasalisation below that of a bound N2.
• table1_novel_lt_half: under the fitted weights a novel compound nasalises less than half the time.
• tableau7, tableau8, tableau9: the schematic tableaux.

Implementation notes

• A candidate is the realisation of the compound-medial velar, and identity in [nasal] to a base is one violation when the two velars differ.
• The tableaux display activation-scaled violations and the prose speaks of scaled weights; harmony_violations_eq shows the two agree. Scaling the violations keeps the weights the grammar's, shared across items, so the log-likelihood theory of HarmonicGrammar.maxEnt applies to the fit of Table 1.
• Tableau (8b) prints 0.91 for harmonies 2.7 and 0.5, which give σ(2.2) < 0.905 (pNasal_tableau8b_lt).
• The caption of Table 1 describes weights fitted to existing and novel compounds combined, while §5.2 evaluates them as the grammar fitted to existing compounds; table1_novel_lt_half holds on either reading.
• The model gives a bound-N2 compound a probability below 1, so the exceptionless nasalisation of (5) is not derived; the modelled experiment has no bound N2s.

References

• [BKK26a]
• [BKK26b]
• [Bre24]
• [BKK21]
• [IM97]
• [IM03]
• [CK13a]

inductive BreissKatsudaKawahara2026.Velar : Type

The compound-medial velar, oral [g] or nasal [ŋ].

• oral : Velar
• nasal : Velar

@[instance_reducible]

instance BreissKatsudaKawahara2026.instDecidableEqVelar : DecidableEq Velar

Equations

• BreissKatsudaKawahara2026.instDecidableEqVelar x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance BreissKatsudaKawahara2026.instFintypeVelar : Fintype Velar

Equations

• One or more equations did not get rendered due to their size.

@[instance_reducible]

instance BreissKatsudaKawahara2026.instMeasurableSpaceVelar : MeasurableSpace Velar

Equations

• BreissKatsudaKawahara2026.instMeasurableSpaceVelar = ⊤

instance BreissKatsudaKawahara2026.instMeasurableSingletonClassVelar : MeasurableSingletonClass Velar

structure BreissKatsudaKawahara2026.Base : Type

A listed allomorph, given by its medial velar and its resting activation.

• velar : Velar
• activation : ℝ

structure BreissKatsudaKawahara2026.Item : Type

A compound's listed bases (§4.1), the free-standing N2 if the N2 is free and the whole compound if it is an existing word.

• n2 : Option Base
• compound : Option Base

def BreissKatsudaKawahara2026.freeN2 (a : ℝ) : Base

A free N2 is listed with the [g] it shows word-initially.

Equations

• BreissKatsudaKawahara2026.freeN2 a = { velar := BreissKatsudaKawahara2026.Velar.oral, activation := a }

def BreissKatsudaKawahara2026.storedCompound (b : ℝ) : Base

An existing compound is listed with its medial velar nasalised.

Equations

• BreissKatsudaKawahara2026.storedCompound b = { velar := BreissKatsudaKawahara2026.Velar.nasal, activation := b }

def BreissKatsudaKawahara2026.novel (a : ℝ) : Item

A novel compound with a free N2.

Equations

• BreissKatsudaKawahara2026.novel a = { n2 := some (BreissKatsudaKawahara2026.freeN2 a), compound := none }

def BreissKatsudaKawahara2026.existing (a b : ℝ) : Item

An existing compound with a free N2.

Equations

• BreissKatsudaKawahara2026.existing a b = { n2 := some (BreissKatsudaKawahara2026.freeN2 a), compound := some (BreissKatsudaKawahara2026.storedCompound b) }

def BreissKatsudaKawahara2026.bound (b : ℝ) : Item

An existing compound with a bound N2.

Equations

• BreissKatsudaKawahara2026.bound b = { n2 := none, compound := some (BreissKatsudaKawahara2026.storedCompound b) }

Constraints, scaled violations, and the MaxEnt probability

def BreissKatsudaKawahara2026.idNasal : Option Base → OptimalityTheory.Constraint Velar

Identity in [nasal] to an optional listed base (6b, c) is violated when the candidate's velar differs from the base's, and never without a base.

Equations

• BreissKatsudaKawahara2026.idNasal (some β) = OptimalityTheory.Constraint.binary fun (x : BreissKatsudaKawahara2026.Velar) => x ≠ β.velar
• BreissKatsudaKawahara2026.idNasal none = 0

def BreissKatsudaKawahara2026.con (it : Item) : OptimalityTheory.ConstraintSet Velar (Fin 3)

The constraint set (6), *INTERNAL-[g] and identity in [nasal] to the N2 and to the compound.

Equations

• One or more equations did not get rendered due to their size.

def BreissKatsudaKawahara2026.activation : Option Base → ℝ

The resting activation of an optional base, zero without one.

Equations

• BreissKatsudaKawahara2026.activation (some β) = β.activation
• BreissKatsudaKawahara2026.activation none = 0

def BreissKatsudaKawahara2026.activations (it : Item) : Fin 3 → ℝ

The activations by which an item scales the violations of (6) (§4.2), each faithfulness constraint's by the activation of its base and markedness's not at all.

Equations

• BreissKatsudaKawahara2026.activations it = ![1, BreissKatsudaKawahara2026.activation it.n2, BreissKatsudaKawahara2026.activation it.compound]

def BreissKatsudaKawahara2026.violations (it : Item) (v : Velar) : Fin 3 → ℝ

The violation vector of a candidate realisation of an item, its violations of (6) scaled by the item's activations, as the tableaux display them.

Equations

• BreissKatsudaKawahara2026.violations it v = BreissKatsudaKawahara2026.activations it * (BreissKatsudaKawahara2026.con it).violations v

structure BreissKatsudaKawahara2026.Weights : Type

The weights of the three constraints.

• markedness : ℝ
• n2 : ℝ
• compound : ℝ

def BreissKatsudaKawahara2026.Weights.vec (w : Weights) : Fin 3 → ℝ

The weights as a vector indexed like con.

Equations

• w.vec = ![w.markedness, w.n2, w.compound]

noncomputable def BreissKatsudaKawahara2026.votingBases (w : Weights) (it : Item) : MeasureTheory.Measure Velar

The Voting Bases grammar's MaxEnt distribution of an item's realisations.

Equations

• BreissKatsudaKawahara2026.votingBases w it = HarmonicGrammar.maxEnt (BreissKatsudaKawahara2026.violations it) w.vec

noncomputable def BreissKatsudaKawahara2026.pNasal (w : Weights) (it : Item) : ℝ

The probability that the compound nasalises.

Equations

• BreissKatsudaKawahara2026.pNasal w it = (BreissKatsudaKawahara2026.votingBases w it).real {BreissKatsudaKawahara2026.Velar.nasal}

theorem BreissKatsudaKawahara2026.harmony_apply (w : Weights) (it : Item) (v : Velar) : HarmonicGrammar.harmony (violations it) w.vec v = -(w.markedness * ↑(OptimalityTheory.Constraint.binary (fun (x : Velar) => x = Velar.oral) v) + w.n2 * (activation it.n2 * ↑(idNasal it.n2 v)) + w.compound * (activation it.compound * ↑(idNasal it.compound v)))

The harmony of a realisation weighs each faithfulness violation by its base's activation.

theorem BreissKatsudaKawahara2026.harmony_violations_eq (w : Weights) (it : Item) : HarmonicGrammar.harmony (violations it) w.vec = HarmonicGrammar.harmonyScore (con it) (w.vec * activations it)

Scaling the violations by the activations is scaling the weights by them (§4.2, §6.3).

def BreissKatsudaKawahara2026.pull : Option Base → ℝ

A base's pull toward nasalisation is its activation, signed by its listed velar.

Equations

• BreissKatsudaKawahara2026.pull (some { velar := BreissKatsudaKawahara2026.Velar.oral, activation := a }) = -a
• BreissKatsudaKawahara2026.pull (some { velar := BreissKatsudaKawahara2026.Velar.nasal, activation := a }) = a
• BreissKatsudaKawahara2026.pull none = 0

@[simp]

theorem BreissKatsudaKawahara2026.pull_freeN2 (a : ℝ) : pull (some (freeN2 a)) = -a

@[simp]

theorem BreissKatsudaKawahara2026.pull_storedCompound (b : ℝ) : pull (some (storedCompound b)) = b

@[simp]

theorem BreissKatsudaKawahara2026.pull_none : pull none = 0

theorem BreissKatsudaKawahara2026.pNasal_eq_sigmoid (w : Weights) (it : Item) : pNasal w it = (w.markedness + w.n2 * pull it.n2 + w.compound * pull it.compound).sigmoid

The probability of nasalisation is the sigmoid of the markedness weight plus the weighted pulls of the two bases, so each frequency effect takes the sign of its base's listed velar.

The frequency effects

theorem BreissKatsudaKawahara2026.pNasal_existing (w : Weights) (a b : ℝ) : pNasal w (existing a b) = (w.markedness - w.n2 * a + w.compound * b).sigmoid

An existing compound with a free N2 nasalises with probability σ(w_M − w_N2·a + w_C·b).

theorem BreissKatsudaKawahara2026.pNasal_novel (w : Weights) (a : ℝ) : pNasal w (novel a) = (w.markedness - w.n2 * a).sigmoid

A novel compound has no whole-compound base.

theorem BreissKatsudaKawahara2026.pNasal_bound (w : Weights) (b : ℝ) : pNasal w (bound b) = (w.markedness + w.compound * b).sigmoid

A bound N2 (5) has no free base.

theorem BreissKatsudaKawahara2026.existing_strictAnti_n2 (w : Weights) (hw : 0 < w.n2) (b : ℝ) : StrictAnti fun (a : ℝ) => pNasal w (existing a b)

The more active the free N2, the less an existing compound nasalises (Figure 2, right).

theorem BreissKatsudaKawahara2026.existing_strictMono_compound (w : Weights) (hw : 0 < w.compound) (a : ℝ) : StrictMono fun (b : ℝ) => pNasal w (existing a b)

The more active the stored compound, the more it nasalises (Figure 2, left).

theorem BreissKatsudaKawahara2026.novel_strictAnti_n2 (w : Weights) (hw : 0 < w.n2) : StrictAnti fun (a : ℝ) => pNasal w (novel a)

The N2 effect persists, with the same sign, in novel compounds (Figure 3).

theorem BreissKatsudaKawahara2026.logit_pNasal_existing (w : Weights) (a b : ℝ) : (pNasal w (existing a b)).logit = (pNasal w (novel a)).logit + w.compound * b

The N2 effect on the log-odds of nasalisation is the same in an existing compound as in a novel one, the stored compound adding a constant, so the frequency effect is uniform across existing and novel compounds (§3.2).

theorem BreissKatsudaKawahara2026.novel_lt_existing (w : Weights) (hw : 0 < w.compound) (a : ℝ) {b : ℝ} (hb : 0 < b) : pNasal w (novel a) < pNasal w (existing a b)

A stored compound's pull adds nasalisation at any N2 activation.

theorem BreissKatsudaKawahara2026.existing_lt_bound (w : Weights) (hw : 0 < w.n2) {a : ℝ} (ha : 0 < a) (b : ℝ) : pNasal w (existing a b) < pNasal w (bound b)

Only a free N2 is listed with [g], so a free N2 lowers nasalisation below that of a bound N2 under the same compound, (2)–(4) against (5).

theorem BreissKatsudaKawahara2026.bound_half_le (w : Weights) (hM : 0 ≤ w.markedness) (hC : 0 ≤ w.compound) {b : ℝ} (hb : 0 ≤ b) : 2⁻¹ ≤ pNasal w (bound b)

A bound-N2 compound nasalises at least half the time under nonnegative weights, whatever its frequency.

theorem BreissKatsudaKawahara2026.zero_markedness_iff (w : Weights) (hM : w.markedness = 0) (a b : ℝ) : 2⁻¹ < pNasal w (existing a b) ↔ w.n2 * a < w.compound * b

With *INTERNAL-[g] at zero weight nasalisation is the majority outcome exactly when the compound base's weighted activation exceeds the N2 base's (§5.1, §6.2).

The fitted grammar

noncomputable def BreissKatsudaKawahara2026.table1 : Weights

The fitted weights of Table 1.

Equations

• BreissKatsudaKawahara2026.table1 = { markedness := 0, n2 := 7.39, compound := 7.09 }

theorem BreissKatsudaKawahara2026.table1_novel_lt_half (f : ℝ) : pNasal table1 (novel f.sigmoid) < 2⁻¹

Under the fitted weights a novel compound, with an activation σ(f) for its N2's centred log-frequency f (§4.4), nasalises less than half the time; §5.2 reports predicted rates from 0.002 to 0.39 against observed rates up to 0.79.

The schematic tableaux

def BreissKatsudaKawahara2026.w78 : Weights

The weights of tableaux (7) and (8).

Equations

• BreissKatsudaKawahara2026.w78 = { markedness := 2, n2 := 1, compound := 1 }

noncomputable def BreissKatsudaKawahara2026.w9 : Weights

The weights of tableau (9).

Equations

• BreissKatsudaKawahara2026.w9 = { markedness := 0.1, n2 := 1, compound := 2 }

theorem BreissKatsudaKawahara2026.tableau7 : HarmonicGrammar.harmony (violations (novel 0.7)) w78.vec Velar.oral = -2 ∧ HarmonicGrammar.harmony (violations (novel 0.7)) w78.vec Velar.nasal = -0.7 ∧ HarmonicGrammar.harmony (violations (novel 0.3)) w78.vec Velar.oral = -2 ∧ HarmonicGrammar.harmony (violations (novel 0.3)) w78.vec Velar.nasal = -0.3 ∧ pNasal w78 (novel 0.7) = Real.sigmoid 1.3 ∧ pNasal w78 (novel 0.3) = Real.sigmoid 1.7 ∧ pNasal w78 (novel 0.7) < pNasal w78 (novel 0.3)

Tableau (7) evaluates novel compounds whose N2s have activations 0.7 and 0.3, and the more active N2 pulls harder toward [g].

theorem BreissKatsudaKawahara2026.tableau8 : HarmonicGrammar.harmony (violations (existing 0.5 0.7)) w78.vec Velar.oral = -2.7 ∧ HarmonicGrammar.harmony (violations (existing 0.5 0.7)) w78.vec Velar.nasal = -0.5 ∧ HarmonicGrammar.harmony (violations (existing 0.5 0.3)) w78.vec Velar.oral = -2.3 ∧ HarmonicGrammar.harmony (violations (existing 0.5 0.3)) w78.vec Velar.nasal = -0.5 ∧ pNasal w78 (existing 0.5 0.7) = Real.sigmoid 2.2 ∧ pNasal w78 (existing 0.5 0.3) = Real.sigmoid 1.8 ∧ pNasal w78 (existing 0.5 0.3) < pNasal w78 (existing 0.5 0.7) ∧ pNasal w78 (novel 0.5) < pNasal w78 (existing 0.5 0.3)

Tableau (8) evaluates existing compounds with N2 activation 0.5 and compound activations 0.7 and 0.3. The more active compound nasalises more, and either nasalises more than the same N2 without a stored compound.

theorem BreissKatsudaKawahara2026.pNasal_tableau8b_lt : pNasal w78 (existing 0.5 0.7) < 0.905

The probability of (8b) is below 0.905, so it rounds to 0.90, not to the printed 0.91.

theorem BreissKatsudaKawahara2026.tableau9 : HarmonicGrammar.harmony (violations (existing 0.5 0.7)) w9.vec Velar.oral = -1.5 ∧ HarmonicGrammar.harmony (violations (existing 0.5 0.7)) w9.vec Velar.nasal = -0.5 ∧ HarmonicGrammar.harmony (violations (existing 0.5 0.3)) w9.vec Velar.oral = -0.7 ∧ HarmonicGrammar.harmony (violations (existing 0.5 0.3)) w9.vec Velar.nasal = -0.5 ∧ pNasal w9 (existing 0.5 0.7) = Real.sigmoid 1 ∧ pNasal w9 (existing 0.5 0.3) = Real.sigmoid 0.2 ∧ 2⁻¹ < pNasal w9 (existing 0.5 0.3) ∧ pNasal w9 (existing 0.5 0.3) < pNasal w9 (existing 0.5 0.7)

Tableau (9) weakens markedness and strengthens faithfulness to the compound. The nasal candidate is still the majority outcome for either compound of (8), and the compound effect keeps its sign.