Harmonic Grammar: Basic Definitions @cite{smolensky-legendre-2006}

The shared foundation for all stochastic Harmonic Grammar variants: Maximum Entropy (MaxEnt), Noisy HG (NHG), and Normal MaxEnt.

A Harmonic Grammar assigns numerical weights to constraints. The harmony score of a candidate is the negated weighted sum of its constraint violations: H(c) = -Σⱼ wⱼ · Cⱼ(c).

Different stochastic variants map harmony scores to probabilities via different mechanisms:

• MaxEnt (@cite{goldwater-johnson-2003}): P(c) ∝ exp(H(c)) (softmax)
• NHG (@cite{boersma-pater-2016}): add Gaussian noise to weights
• Normal MaxEnt (@cite{flemming-2021}): add i.i.d. Gaussian noise to candidates

All three share the WeightedConstraint and harmonyScore definitions.

structure Theories.Phonology.HarmonicGrammar.WeightedConstraint (C : Type) extends Core.OT.NamedConstraint C : Type

A weighted constraint for Harmonic Grammar. Extends NamedConstraint with a rational-valued weight.

• name : String
• family : ConstraintFamily
• eval : C → ℕ
• weight : ℚ

  Constraint weight (higher = more important).

def Theories.Phonology.HarmonicGrammar.harmonyScore {C : Type} (constraints : List (WeightedConstraint C)) (c : C) : ℚ

Harmony score: H(c) = -Σⱼ wⱼ · Cⱼ(c). Negative because violations are penalized.

Equations

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

noncomputable def Theories.Phonology.HarmonicGrammar.harmonyScoreR {C : Type} (constraints : List (WeightedConstraint C)) (c : C) : ℝ

Harmony score as a real number, for interfacing with softmax.

Equations

• Theories.Phonology.HarmonicGrammar.harmonyScoreR constraints c = ↑(Theories.Phonology.HarmonicGrammar.harmonyScore constraints c)