[Sto26b]: Systemic Constraints in MaxEnt Grammars

[Sto26b]

Replication of [Sto26b] "A Method to Evaluate Systemic Constraints in Probabilistic Grammars" (Linguistic Inquiry 57(1)).

Key idea

Standard MaxEnt grammars evaluate each input→output mapping independently. Storme shows how to incorporate systemic constraints — constraints that evaluate sets of mappings jointly. The running example is *HOMOPHONY: a constraint penalizing output tuples where distinct inputs receive identical surface forms.

Persian hiatus resolution

The case study is Persian vowel hiatus between /æ/ and /ɑ/. Classical faithfulness (MAX, IDENT) and markedness (*VV, DEP) constraints predict symmetric treatment of /æ.ɑ/ vs. /ɑ.æ/ — deletion of V1 vs. V2 should be equally probable in both. But Storme argues that *HOMOPHONY breaks this symmetry: the grammar prefers output tuples where distinct inputs produce distinct surface forms.

Formalization

We instantiate the MaxEnt machinery — a classical constraint vector classicalCon with weight vector classicalW, plus the systemic *HOMOPHONY constraint — over the Persian hiatus domain from Farsi.Phonology, and verify the key prediction: homophony avoidance breaks the symmetry.

Constraints

Following standard OT/MaxEnt constraint families:

• MAX (faithfulness): penalizes deletion (1 violation per deleted segment)
• DEP (faithfulness): penalizes epenthesis (1 violation per inserted segment)
• *VV (markedness): penalizes vowel hiatus in the output
• IDENT (faithfulness): penalizes feature change (coalescence)
• *HOMOPHONY (systemic): penalizes identical outputs for distinct inputs

@[implicit_reducible]

instance Storme2026.instFintypeHiatusInput : Fintype Farsi.Phonology.HiatusInput

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@[implicit_reducible]

instance Storme2026.instFintypeHiatusOutput : Fintype Farsi.Phonology.HiatusOutput

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def Storme2026.maxConstraint : Constraints.Constraint Farsi.Phonology.HiatusCandidate

MAX: penalizes deletion. 1 violation for deleteV1 or deleteV2, 0 otherwise.

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def Storme2026.depConstraint : Constraints.Constraint Farsi.Phonology.HiatusCandidate

DEP: penalizes epenthesis. 1 violation for epenthesis, 0 otherwise.

Equations

• Storme2026.depConstraint = Constraints.Constraint.binary fun (x : Farsi.Phonology.HiatusCandidate) => match x with | (fst, o) => (o == Farsi.Phonology.HiatusOutput.epenthesis) = true

def Storme2026.starVV : Constraints.Constraint Farsi.Phonology.HiatusCandidate

*VV: markedness constraint penalizing vowel hiatus. 1 violation for faithful (hiatus preserved), 0 for all repairs.

Equations

• Storme2026.starVV = Constraints.Constraint.binary fun (x : Farsi.Phonology.HiatusCandidate) => match x with | (fst, o) => (o == Farsi.Phonology.HiatusOutput.faithful) = true

def Storme2026.identConstraint : Constraints.Constraint Farsi.Phonology.HiatusCandidate

IDENT: penalizes coalescence (feature change). 1 violation for coalescence, 0 otherwise.

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• Storme2026.identConstraint = Constraints.Constraint.binary fun (x : Farsi.Phonology.HiatusCandidate) => match x with | (fst, o) => (o == Farsi.Phonology.HiatusOutput.coalescence) = true

def Storme2026.classicalCon : Constraints.CON Farsi.Phonology.HiatusCandidate 4

The classical constraint set for Persian hiatus, as a CON (constraint vector). Index order: 0 = MAX, 1 = DEP, 2 = *VV, 3 = IDENT.

Equations

• Storme2026.classicalCon = ![Storme2026.maxConstraint, Storme2026.depConstraint, Storme2026.starVV, Storme2026.identConstraint]

def Storme2026.classicalW : Fin 4 → ℝ

MaxEnt weights for classicalCon (MAX = 2, DEP = 1, *VV = 3, IDENT = 2).

Note: weights are illustrative (chosen to make epenthesis the classical winner), not fitted to Storme's experimental data. The qualitative predictions (symmetry, symmetry-breaking) hold for any positive weights since they depend on constraint structure, not specific weight values.

Equations

• Storme2026.classicalW = ![2, 1, 3, 2]

theorem Storme2026.classical_symmetry_deleteV1 : Constraints.harmonyScore classicalCon classicalW (Farsi.Phonology.HiatusInput.ae_ah, Farsi.Phonology.HiatusOutput.deleteV1) = Constraints.harmonyScore classicalCon classicalW (Farsi.Phonology.HiatusInput.ah_ae, Farsi.Phonology.HiatusOutput.deleteV1)

Harmony scores are symmetric across mirror-image inputs: H(ae.ah, deleteV1) = H(ah.ae, deleteV1) under classical constraints alone, because classical constraints only look at output structure.

theorem Storme2026.classical_symmetry_deleteV2 : Constraints.harmonyScore classicalCon classicalW (Farsi.Phonology.HiatusInput.ae_ah, Farsi.Phonology.HiatusOutput.deleteV2) = Constraints.harmonyScore classicalCon classicalW (Farsi.Phonology.HiatusInput.ah_ae, Farsi.Phonology.HiatusOutput.deleteV2)

theorem Storme2026.epenthesis_beats_deletion : Constraints.harmonyScore classicalCon classicalW (Farsi.Phonology.HiatusInput.ae_ah, Farsi.Phonology.HiatusOutput.deleteV1) < Constraints.harmonyScore classicalCon classicalW (Farsi.Phonology.HiatusInput.ae_ah, Farsi.Phonology.HiatusOutput.epenthesis)

The classical grammar ranks the repairs epenthesis ≻ {deletion, coalescence} ≻ faithful (weights chosen so epenthesis is the classical winner). The harmony magnitudes are weight artifacts; the ranking is the prediction.

theorem Storme2026.coalescence_beats_faithful : Constraints.harmonyScore classicalCon classicalW (Farsi.Phonology.HiatusInput.ae_ah, Farsi.Phonology.HiatusOutput.faithful) < Constraints.harmonyScore classicalCon classicalW (Farsi.Phonology.HiatusInput.ae_ah, Farsi.Phonology.HiatusOutput.coalescence)

theorem Storme2026.epenthesis_beats_faithful : Constraints.harmonyScore classicalCon classicalW (Farsi.Phonology.HiatusInput.ae_ah, Farsi.Phonology.HiatusOutput.faithful) < Constraints.harmonyScore classicalCon classicalW (Farsi.Phonology.HiatusInput.ae_ah, Farsi.Phonology.HiatusOutput.epenthesis)

def Storme2026.inputsIndexed : Fin 4 → Farsi.Phonology.HiatusInput

The four hiatus inputs indexed by Fin 4.

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• Storme2026.inputsIndexed ⟨0, isLt⟩ = Farsi.Phonology.HiatusInput.ae_ae
• Storme2026.inputsIndexed ⟨1, isLt⟩ = Farsi.Phonology.HiatusInput.ae_ah
• Storme2026.inputsIndexed ⟨2, isLt⟩ = Farsi.Phonology.HiatusInput.ah_ae
• Storme2026.inputsIndexed ⟨3, isLt⟩ = Farsi.Phonology.HiatusInput.ah_ah

def Storme2026.starHomophony : HarmonicGrammar.SystemicConstraint 4 Farsi.Phonology.HiatusOutput

*HOMOPHONY for the Persian hiatus paradigm: penalizes output tuples where distinct inputs produce identical outputs.

Equations

• Storme2026.starHomophony = HarmonicGrammar.homophonyAvoidance

noncomputable def Storme2026.persianJointScore (f : Fin 4 → Farsi.Phonology.HiatusOutput) : ℝ

Joint harmony score for a complete output tuple, combining classical per-mapping scores with the systemic *HOMOPHONY penalty.

Equations

• Storme2026.persianJointScore f = HarmonicGrammar.jointHarmonyScore Storme2026.inputsIndexed Storme2026.classicalCon Storme2026.classicalW ![1] ![Storme2026.starHomophony] f

def Storme2026.diverseTuple : Fin 4 → Farsi.Phonology.HiatusOutput

An output tuple where the mirror-image inputs /æ.ɑ/ and /ɑ.æ/ use different repair strategies (deleteV1 vs deleteV2). Both still surface as [ɑ], so homophony remains at those positions — but the tuple has fewer *HOMOPHONY violations overall than the uniform-strategy tuple.

Equations

• Storme2026.diverseTuple ⟨0, isLt⟩ = Farsi.Phonology.HiatusOutput.deleteV1
• Storme2026.diverseTuple ⟨1, isLt⟩ = Farsi.Phonology.HiatusOutput.deleteV1
• Storme2026.diverseTuple ⟨2, isLt⟩ = Farsi.Phonology.HiatusOutput.deleteV2
• Storme2026.diverseTuple ⟨3, isLt⟩ = Farsi.Phonology.HiatusOutput.deleteV1

def Storme2026.homophonousTuple : Fin 4 → Farsi.Phonology.HiatusOutput

A homophonous tuple: both mirror-image inputs use the same strategy.

Equations

• Storme2026.homophonousTuple ⟨0, isLt⟩ = Farsi.Phonology.HiatusOutput.deleteV1
• Storme2026.homophonousTuple ⟨1, isLt⟩ = Farsi.Phonology.HiatusOutput.deleteV1
• Storme2026.homophonousTuple ⟨2, isLt⟩ = Farsi.Phonology.HiatusOutput.deleteV1
• Storme2026.homophonousTuple ⟨3, isLt⟩ = Farsi.Phonology.HiatusOutput.deleteV1

theorem Storme2026.homophony_violation_counts : starHomophony homophonousTuple = 6 ∧ starHomophony diverseTuple = 3

Concrete violation counts:

• homophonousTuple: all 4 positions use deleteV1 → C(4,2) = 6 collisions
• diverseTuple: 3 positions use deleteV1, 1 uses deleteV2 → 3 collisions

theorem Storme2026.homophony_penalizes_uniform : starHomophony homophonousTuple ≥ starHomophony diverseTuple

*HOMOPHONY incurs more violations on the homophonous tuple than on the diverse tuple. This is the core mechanism by which systemic constraints break symmetry.

theorem Storme2026.diverse_higher_joint_score : persianJointScore diverseTuple ≥ persianJointScore homophonousTuple

The diverse tuple has at least as high joint harmony as the homophonous tuple, because it incurs fewer *HOMOPHONY violations while having the same classical constraint violations.

theorem Storme2026.classical_cannot_distinguish : Constraints.harmonyScore classicalCon classicalW (Farsi.Phonology.HiatusInput.ae_ah, Farsi.Phonology.HiatusOutput.deleteV1) = Constraints.harmonyScore classicalCon classicalW (Farsi.Phonology.HiatusInput.ah_ae, Farsi.Phonology.HiatusOutput.deleteV1)

Classical constraints alone assign the same score to deleteV1 for /æ.ɑ/ and /ɑ.æ/ — the grammar cannot distinguish mirror-image inputs at the individual mapping level. (Restated from §2 for contrast with the non-separability result below.)

theorem Storme2026.joint_not_separable : ∃ (f : Fin 4 → Farsi.Phonology.HiatusOutput) (g : Fin 4 → Farsi.Phonology.HiatusOutput), f ⟨1, ⋯⟩ = g ⟨1, ⋯⟩ ∧ persianJointScore f - Constraints.harmonyScore classicalCon classicalW (inputsIndexed ⟨1, ⋯⟩, f ⟨1, ⋯⟩) ≠ persianJointScore g - Constraints.harmonyScore classicalCon classicalW (inputsIndexed ⟨1, ⋯⟩, g ⟨1, ⋯⟩)

Under *HOMOPHONY, the joint distribution over all four mappings is not a product of independent per-mapping distributions. The marginal probability of a specific output for /æ.ɑ/ depends on what outputs are chosen for the other inputs — this is the core mechanism by which systemic constraints influence individual mapping probabilities.

This theorem verifies that the joint score is not additively separable (i.e., there exist tuples f, g agreeing on position 1 but differing in joint score minus the classical score at position 1).

At each input, the classical MaxEnt model is a ConstraintSystem HiatusOutput ℝ: candidates = Finset.univ, score = harmony, decoder = softmaxDecoder 1. This is the same ConstraintSystem API used by HayesWilson2008.onsetSystem — different domain, identical scaffolding.

The systemic-constraint (*HOMOPHONY) story in §§3–6 sits above this per-input view: it couples the per-input distributions into a joint distribution on Fin 4 → HiatusOutput. With zero systemic weight, the joint factorises and each marginal equals the per-input predict.

noncomputable def Storme2026.stormeSystem (i : Farsi.Phonology.HiatusInput) : Core.Optimization.ConstraintSystem Farsi.Phonology.HiatusOutput ℝ

The classical MaxEnt distribution at input i, packaged as a generic ConstraintSystem. Score = harmonyScore classicalCon classicalW (i, ·), decoder = softmaxDecoder 1.

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theorem Storme2026.stormeSystem_epenthesis_gt_deleteV1 : (stormeSystem Farsi.Phonology.HiatusInput.ae_ah).predict Farsi.Phonology.HiatusOutput.deleteV1 < (stormeSystem Farsi.Phonology.HiatusInput.ae_ah).predict Farsi.Phonology.HiatusOutput.epenthesis

For input /æ.ɑ/, the system predicts a higher MaxEnt probability for epenthesis (cost −1) than for deleteV1 (cost −2). This is a comparison of actual softmax probabilities (numerator / partition function over all 5 outputs), not just exponentiated harmonies.

theorem Storme2026.stormeSystem_isProb (i : Farsi.Phonology.HiatusInput) : ∑ o : Farsi.Phonology.HiatusOutput, (stormeSystem i).predict o = 1

The classical Persian system at /æ.ɑ/ is a probability distribution over HiatusOutput. Follows from the generic softmaxDecoder_isProb.