@cite{storme-2026}: Systemic Constraints in MaxEnt Grammars

@cite{storme-2026}

Replication of @cite{storme-2026} "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 MaxEntGrammar with the Persian hiatus domain from Fragments.Farsi.Phonology, define the classical and systemic constraints, 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 Fragments.Farsi.Phonology.HiatusInput

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@[implicit_reducible]

instance Storme2026.instFintypeHiatusOutput : Fintype Fragments.Farsi.Phonology.HiatusOutput

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def Storme2026.maxConstraint : Core.Constraint.WeightedConstraint Fragments.Farsi.Phonology.HiatusCandidate

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

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def Storme2026.depConstraint : Core.Constraint.WeightedConstraint Fragments.Farsi.Phonology.HiatusCandidate

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

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def Storme2026.starVV : Core.Constraint.WeightedConstraint Fragments.Farsi.Phonology.HiatusCandidate

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

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def Storme2026.identConstraint : Core.Constraint.WeightedConstraint Fragments.Farsi.Phonology.HiatusCandidate

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

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def Storme2026.classicalConstraints : List (Core.Constraint.WeightedConstraint Fragments.Farsi.Phonology.HiatusCandidate)

The classical constraint set for Persian hiatus.

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.classicalConstraints = [Storme2026.maxConstraint, Storme2026.depConstraint, Storme2026.starVV, Storme2026.identConstraint]

theorem Storme2026.classical_symmetry_deleteV1 : Core.Constraint.harmonyScore classicalConstraints (Fragments.Farsi.Phonology.HiatusInput.ae_ah, Fragments.Farsi.Phonology.HiatusOutput.deleteV1) = Core.Constraint.harmonyScore classicalConstraints (Fragments.Farsi.Phonology.HiatusInput.ah_ae, Fragments.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 : Core.Constraint.harmonyScore classicalConstraints (Fragments.Farsi.Phonology.HiatusInput.ae_ah, Fragments.Farsi.Phonology.HiatusOutput.deleteV2) = Core.Constraint.harmonyScore classicalConstraints (Fragments.Farsi.Phonology.HiatusInput.ah_ae, Fragments.Farsi.Phonology.HiatusOutput.deleteV2)

theorem Storme2026.score_deleteV1 : Core.Constraint.harmonyScore classicalConstraints (Fragments.Farsi.Phonology.HiatusInput.ae_ah, Fragments.Farsi.Phonology.HiatusOutput.deleteV1) = -2

Concrete score verification: deletion costs -2 (MAX weight 2, 1 violation), epenthesis costs -1 (DEP weight 1), faithful costs -3 (*VV weight 3), coalescence costs -2 (IDENT weight 2).

theorem Storme2026.score_epenthesis : Core.Constraint.harmonyScore classicalConstraints (Fragments.Farsi.Phonology.HiatusInput.ae_ah, Fragments.Farsi.Phonology.HiatusOutput.epenthesis) = -1

theorem Storme2026.score_faithful : Core.Constraint.harmonyScore classicalConstraints (Fragments.Farsi.Phonology.HiatusInput.ae_ah, Fragments.Farsi.Phonology.HiatusOutput.faithful) = -3

theorem Storme2026.score_coalescence : Core.Constraint.harmonyScore classicalConstraints (Fragments.Farsi.Phonology.HiatusInput.ae_ah, Fragments.Farsi.Phonology.HiatusOutput.coalescence) = -2

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

The four hiatus inputs indexed by Fin 4.

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

def Storme2026.starHomophony : Core.Constraint.SystemicConstraint 4 Fragments.Farsi.Phonology.HiatusOutput

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

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• Storme2026.starHomophony = Core.Constraint.homophonyAvoidance 1

def Storme2026.persianJointScore (f : Fin 4 → Fragments.Farsi.Phonology.HiatusOutput) : ℚ

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

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• Storme2026.persianJointScore f = Core.Constraint.jointHarmonyScore Storme2026.inputsIndexed Storme2026.classicalConstraints [Storme2026.starHomophony] f

def Storme2026.diverseTuple : Fin 4 → Fragments.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.

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

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

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

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

theorem Storme2026.homophony_violation_counts : starHomophony.eval homophonousTuple = 6 ∧ starHomophony.eval 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.eval homophonousTuple ≥ starHomophony.eval 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 : Core.Constraint.harmonyScore classicalConstraints (Fragments.Farsi.Phonology.HiatusInput.ae_ah, Fragments.Farsi.Phonology.HiatusOutput.deleteV1) = Core.Constraint.harmonyScore classicalConstraints (Fragments.Farsi.Phonology.HiatusInput.ah_ae, Fragments.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 → Fragments.Farsi.Phonology.HiatusOutput) (g : Fin 4 → Fragments.Farsi.Phonology.HiatusOutput), f ⟨1, ⋯⟩ = g ⟨1, ⋯⟩ ∧ persianJointScore f - Core.Constraint.harmonyScore classicalConstraints (inputsIndexed ⟨1, ⋯⟩, f ⟨1, ⋯⟩) ≠ persianJointScore g - Core.Constraint.harmonyScore classicalConstraints (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 : Fragments.Farsi.Phonology.HiatusInput) : Core.Constraint.ConstraintSystem Fragments.Farsi.Phonology.HiatusOutput ℝ

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

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