Documentation

Linglib.Theories.Phonology.ItemSpecificity.RepresentationStrength

Representation Strength (Gradient Symbolic Representations) #

@cite{smolensky-goldrick-2016} @cite{pierrehumbert-2001} @cite{todd-pierrehumbert-hay-2019}

The "frequency lives in the lexicon, not the grammar" theory: the grammar's weights are fixed; what varies with frequency is the activation strength of the underlying representation. A high-frequency item has a more robust UR; a low-frequency item has a weakened UR more susceptible to deletion / repair / coalescence.

Architectural backbone: S-G 2016 GSR #

The gradient-activity machinery follows @cite{smolensky-goldrick-2016}. A symbol's activity is a real-valued degree of presence — the /t/ at the end of petit in their analysis is 0.5·t, half-active. Surface realization is decided by a Harmonic Grammar in which faithfulness constraints (DEP, MAX) are scaled by activity, with the surfacing condition for an intervocalic consonant being χ > θ ≈ 0.73 where χ is the total activity at that position (p. 20). Compounds combine constituent activities additively via coalescence under MAX as a positive constraint (their χ = λ + τ for the W₁-final + W₂-initial blend in petit ami; p. 17 tableau, p. 20).

Frequency channel: Pierrehumbert / TPH #

S-G treat activity values as lexical constants ((λ, τ, ζ, ν) ≈ (0.5, 0.3, 0.3, 0.3) at p. 15) but invite a usage-based hybrid in their §3.6: "Such frequency dependence motivates certain 'usage-' or 'construction-'based accounts of liaison; a formalization of a kind of usage-based account will in fact be blended into the proposed account." This file formalizes that hybrid: activity is a function of token log-frequency, following @cite{pierrehumbert-2001}'s "resting activation level" primitive (TSL 45 p. 141: "each exemplar has an associated strength — which may be viewed as a resting activation level"). Direction-of-effect predictions follow @cite{todd-pierrehumbert-hay-2019}: high-frequency items are more robustly recognized under acoustic ambiguity, with consequences that depend on whether the category moves toward or away from a competitor.

The schema, contrasted with ScaledWeights #

TheoryWhat scales with log-frequencyWhere scaling lives
ScaledWeightsConstraint weightGrammar
RepresentationStrengthUnderlying-form activationLexicon

Both can produce the same surface gradient on simplex datasets, but they diverge on:

The Breiss-Katsuda-Kawahara N2-frequency effect #

@cite{breiss-katsuda-kawahara-2026} report that high N2 token frequency in Japanese compounds blocks nasalisation (preserves the boundary). They themselves analyze this with MaxEnt + Lexical Conservatism (@cite{steriade-2000}); the GSR + frequency-derived activity hybrid here is one of several siblings consistent with the pattern. Under this hybrid: high N2 frequency → high N2-initial activity → activity threshold for faithful boundary preservation (χ > θ) more easily met → less nasalisation. The prediction follows from S-G's optimality condition without further machinery; cf. Phenomena/Phonology/Studies/BreissKatsudaKawahara2026.lean for the discrimination across siblings.

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def Phonology.ItemSpecificity.RepStrength.activation {α : Type} [HasTokenFreq α] (sigmoid : ℝ → ℝ) (a : α) :
ℝ

The activation strength of an item: a sigmoid-like function of its log-frequency, bounded in [0, 1]. We model it abstractly here as a parameter; concrete instances pick a specific shape (logistic, cumulative, etc.).

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    noncomputable def Phonology.ItemSpecificity.RepStrength.logistic (μ x : ℝ) :
    ℝ

    The standard logistic sigmoid 1 / (1 + exp (-(x - μ))).

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      def Phonology.ItemSpecificity.RepStrength.compoundActivation {α : Type} [HasTokenFreq α] (sigmoid : ℝ → ℝ) (combine : ℝ → ℝ → ℝ) (n1 n2 : α) :
      ℝ

      The activation of a compound inherits from its constituents, rather than being looked up from the compound's own frequency. The inherit function (typically min or product) determines how constituent activations combine.

      This is the key architectural commitment: in RepresentationStrength, a high-frequency constituent boosts the compound's activation regardless of the compound's own frequency. Contrast with ScaledWeights, where constraint weights see only the candidate's own log-frequency.

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        def Phonology.ItemSpecificity.RepStrength.addCombine (a b : ℝ) :
        ℝ

        The additive combine rule matching @cite{smolensky-goldrick-2016}'s coalescence: surface activity at the W₁-W₂ boundary is the sum of the constituents' contributions (their χ = λ + τ, p. 17, p. 20), arising from MAX as a positive constraint that rewards faithfulness scaled by underlying activity.

        This is the canonical combine rule for a GSR-grounded compound architecture; new consumers should prefer it over minCombine.

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          def Phonology.ItemSpecificity.RepStrength.minCombine (a b : ℝ) :
          ℝ

          The minimum combine rule — a non-S-G alternative. Predicts that compound activity is bounded above by the weakest constituent; closer in spirit to a Bybee-style storage account where compound accessibility cannot exceed the rarer constituent's accessibility. Useful as a separation contrast against addCombine; not licensed by S-G's coalescence mechanism.

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