Documentation

Linglib.Phenomena.PhonologicalAlternation.Studies.Storme2026

@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:

source
instance Phenomena.PhonologicalAlternation.Studies.Storme2026.instFintypeHiatusInput :
Fintype Fragments.Farsi.Phonology.HiatusInput
Equations
  • One or more equations did not get rendered due to their size.
source
instance Phenomena.PhonologicalAlternation.Studies.Storme2026.instFintypeHiatusOutput :
Fintype Fragments.Farsi.Phonology.HiatusOutput
Equations
  • One or more equations did not get rendered due to their size.
source
def Phenomena.PhonologicalAlternation.Studies.Storme2026.maxConstraint :
Theories.Phonology.HarmonicGrammar.WeightedConstraint Fragments.Farsi.Phonology.HiatusCandidate

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

Equations
  • One or more equations did not get rendered due to their size.
Instances For
    source
    def Phenomena.PhonologicalAlternation.Studies.Storme2026.depConstraint :
    Theories.Phonology.HarmonicGrammar.WeightedConstraint Fragments.Farsi.Phonology.HiatusCandidate

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

    Equations
    • One or more equations did not get rendered due to their size.
    Instances For
      source
      def Phenomena.PhonologicalAlternation.Studies.Storme2026.starVV :
      Theories.Phonology.HarmonicGrammar.WeightedConstraint Fragments.Farsi.Phonology.HiatusCandidate

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

      Equations
      • One or more equations did not get rendered due to their size.
      Instances For
        source
        def Phenomena.PhonologicalAlternation.Studies.Storme2026.identConstraint :
        Theories.Phonology.HarmonicGrammar.WeightedConstraint Fragments.Farsi.Phonology.HiatusCandidate

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

        Equations
        • One or more equations did not get rendered due to their size.
        Instances For
          source
          def Phenomena.PhonologicalAlternation.Studies.Storme2026.classicalConstraints :
          List (Theories.Phonology.HarmonicGrammar.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
          • One or more equations did not get rendered due to their size.
          Instances For
            source
            theorem Phenomena.PhonologicalAlternation.Studies.Storme2026.classical_symmetry_deleteV1 :
            Theories.Phonology.HarmonicGrammar.harmonyScore classicalConstraints (Fragments.Farsi.Phonology.HiatusInput.ae_ah, Fragments.Farsi.Phonology.HiatusOutput.deleteV1) = Theories.Phonology.HarmonicGrammar.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.

            source
            theorem Phenomena.PhonologicalAlternation.Studies.Storme2026.classical_symmetry_deleteV2 :
            Theories.Phonology.HarmonicGrammar.harmonyScore classicalConstraints (Fragments.Farsi.Phonology.HiatusInput.ae_ah, Fragments.Farsi.Phonology.HiatusOutput.deleteV2) = Theories.Phonology.HarmonicGrammar.harmonyScore classicalConstraints (Fragments.Farsi.Phonology.HiatusInput.ah_ae, Fragments.Farsi.Phonology.HiatusOutput.deleteV2)
            source
            theorem Phenomena.PhonologicalAlternation.Studies.Storme2026.score_deleteV1 :
            Theories.Phonology.HarmonicGrammar.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).

            source
            theorem Phenomena.PhonologicalAlternation.Studies.Storme2026.score_epenthesis :
            Theories.Phonology.HarmonicGrammar.harmonyScore classicalConstraints (Fragments.Farsi.Phonology.HiatusInput.ae_ah, Fragments.Farsi.Phonology.HiatusOutput.epenthesis) = -1
            source
            theorem Phenomena.PhonologicalAlternation.Studies.Storme2026.score_faithful :
            Theories.Phonology.HarmonicGrammar.harmonyScore classicalConstraints (Fragments.Farsi.Phonology.HiatusInput.ae_ah, Fragments.Farsi.Phonology.HiatusOutput.faithful) = -3
            source
            theorem Phenomena.PhonologicalAlternation.Studies.Storme2026.score_coalescence :
            Theories.Phonology.HarmonicGrammar.harmonyScore classicalConstraints (Fragments.Farsi.Phonology.HiatusInput.ae_ah, Fragments.Farsi.Phonology.HiatusOutput.coalescence) = -2
            source
            def Phenomena.PhonologicalAlternation.Studies.Storme2026.inputsIndexed :
            Fin 4 → Fragments.Farsi.Phonology.HiatusInput

            The four hiatus inputs indexed by Fin 4.

            Equations
            Instances For
              source
              def Phenomena.PhonologicalAlternation.Studies.Storme2026.starHomophony :
              Theories.Phonology.HarmonicGrammar.SystemicConstraint 4 Fragments.Farsi.Phonology.HiatusOutput

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

              Equations
              Instances For
                source
                def Phenomena.PhonologicalAlternation.Studies.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.

                Equations
                • One or more equations did not get rendered due to their size.
                Instances For
                  source
                  def Phenomena.PhonologicalAlternation.Studies.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.

                  Equations
                  Instances For
                    source
                    def Phenomena.PhonologicalAlternation.Studies.Storme2026.homophonousTuple :
                    Fin 4 → Fragments.Farsi.Phonology.HiatusOutput

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

                    Equations
                    Instances For
                      source
                      theorem Phenomena.PhonologicalAlternation.Studies.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
                      source
                      theorem Phenomena.PhonologicalAlternation.Studies.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.

                      source
                      theorem Phenomena.PhonologicalAlternation.Studies.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.

                      source
                      theorem Phenomena.PhonologicalAlternation.Studies.Storme2026.classical_cannot_distinguish :
                      Theories.Phonology.HarmonicGrammar.harmonyScore classicalConstraints (Fragments.Farsi.Phonology.HiatusInput.ae_ah, Fragments.Farsi.Phonology.HiatusOutput.deleteV1) = Theories.Phonology.HarmonicGrammar.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.)

                      source
                      theorem Phenomena.PhonologicalAlternation.Studies.Storme2026.joint_not_separable :
                      ∃ (f : Fin 4 → Fragments.Farsi.Phonology.HiatusOutput) (g : Fin 4 → Fragments.Farsi.Phonology.HiatusOutput), f ⟨1, ⋯⟩ = g ⟨1, ⋯⟩ ∧ persianJointScore f - Theories.Phonology.HarmonicGrammar.harmonyScore classicalConstraints (inputsIndexed ⟨1, ⋯⟩, f ⟨1, ⋯⟩) ≠ persianJointScore g - Theories.Phonology.HarmonicGrammar.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).