@cite{kursat-degen-2021}

@cite{degen-etal-2020} @cite{waldon-degen-2021} @cite{engelhardt-etal-2006}

Perceptual difficulty differences predict asymmetry in redundant modification with color and material adjectives. Proceedings of the Linguistic Society of America 6(1): 676-688, 2021.

Core Argument

Material adjectives are harder to perceive than color adjectives (Exps 1, 3), and color adjectives are used redundantly more often than material adjectives (Exp 2). This anti-correlation between perceptual difficulty and redundant-use rate supports a noise-based RSA account (@cite{waldon-degen-2021}) where the noise parameter reflects perceptual difficulty of property verification.

Experiments

• Exp 1 (§2, N = 105: 120 recruited, 15 excluded): Perceptual difficulty norms. Participants verified whether an adjective applied to an object. Material adjectives produced higher error rates (β = 0.48) and slower RTs (β = 5.44).
• Exp 2 (§3, N ≈ 95: 100 recruited, 5 excluded): Redundant modifier production. Speakers described objects in contexts where one property was sufficient. Color was used redundantly more than material (β = 2.32).
• Exp 3 (§4, N = 376: 400 recruited, 24 excluded): Perceptual difficulty measured with Exp 2 displays. Material remained harder (error β = 0.96, RT β = 0.24).

Verified Data

All regression coefficients verified against paper text (§2.3, §3.3, §4.3).

Derivational Chain

The cs-RSA model (@cite{degen-etal-2020}) explains redundant modification via noisy perception. The derivation proceeds in four steps:

1. Model structure: The cs-RSA meaning function φ decomposes into independent per-feature noise channels (proven in DegenEtAl2020: φ_product_of_experts).

2. Parameterization: Each noise channel has match/mismatch parameters that determine its discrimination (noise gap). The cs-RSA model's default color params (0.99/0.01) match the RSA.Noise module's (proven below: csrsa_params_match_noise).

3. Ordering prediction: The noise gap determines how much signal a modifier provides. Color's gap (0.98) exceeds material's gap (0.40), so color modifiers provide more discriminative signal to the L0 listener (proven: color_gap_exceeds_material).

4. Empirical confirmation: The predicted ordering (color more redundant than material) matches Exp 2 (β = 2.32, p < .0001), and the perceptual difficulty ordering that grounds the noise parameters is confirmed by Exps 1 and 3.

Note: the material noise parameters in RSA.Noise (0.70/0.30) are hypothetical, not derived from this paper. This paper establishes the ordering (color easier than material), not the specific channel parameters. The full S1 redundancy prediction requires the incremental model of @cite{waldon-degen-2021}.

@[reducible, inline]

abbrev Phenomena.Reference.Studies.KursatDegen2021.PropertyType : Type

Property types tested across experiments — re-exported from Core.PropertyDomain for local use.

Equations

• Phenomena.Reference.Studies.KursatDegen2021.PropertyType = Core.PropertyDomain

structure Phenomena.Reference.Studies.KursatDegen2021.RegressionResult : Type

A regression result from one of the paper's mixed-effects models. Sign convention varies by experiment: in Exps 1/3, positive β means material > color (harder); in Exp 2, positive β means color > material (more redundant). See individual def docstrings for interpretation.

• beta : Float

  Fixed-effect coefficient

• se : Float

  Standard error

• tStat : Option Float

  t-statistic (when reported)

• significant : Bool

  All reported effects are p < .0001

def Phenomena.Reference.Studies.KursatDegen2021.instReprRegressionResult.repr : RegressionResult → ℕ → Std.Format

Equations

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

instance Phenomena.Reference.Studies.KursatDegen2021.instReprRegressionResult : Repr RegressionResult

Equations

• Phenomena.Reference.Studies.KursatDegen2021.instReprRegressionResult = { reprPrec := Phenomena.Reference.Studies.KursatDegen2021.instReprRegressionResult.repr }

def Phenomena.Reference.Studies.KursatDegen2021.exp1_error : RegressionResult

Material → higher error rates (§2.3: β = 0.48, SE = 0.12, p < .0001). Log odds of incorrect response.

Equations

• Phenomena.Reference.Studies.KursatDegen2021.exp1_error = { beta := 0.48, se := 0.12, significant := true }

def Phenomena.Reference.Studies.KursatDegen2021.exp1_rt : RegressionResult

Material → slower RTs (§2.3: β = 5.44, SE = 4.74, t = 11.49, p < .0001). RT difference in ms (material − color). NOTE: The paper's SE and t are inconsistent (5.44/4.74 ≈ 1.15 ≠ 11.49); likely SE = 0.474 (giving 5.44/0.474 ≈ 11.48). Values as printed.

Equations

• Phenomena.Reference.Studies.KursatDegen2021.exp1_rt = { beta := 5.44, se := 4.74, tStat := some 11.49, significant := true }

def Phenomena.Reference.Studies.KursatDegen2021.exp2_redundancy : RegressionResult

Color used redundantly more than material (§3.3: β = 2.32, SE = 0.64, p < .0001). Log odds of including the redundant modifier.

Equations

• Phenomena.Reference.Studies.KursatDegen2021.exp2_redundancy = { beta := 2.32, se := 0.64, significant := true }

def Phenomena.Reference.Studies.KursatDegen2021.strongVersionSupported : Bool

The strong version of the perceptual difficulty hypothesis — within-property-type difficulty predicts item-level redundancy — is not supported (§3.3: insufficient speakers for material items).

Equations

• Phenomena.Reference.Studies.KursatDegen2021.strongVersionSupported = false

def Phenomena.Reference.Studies.KursatDegen2021.exp3_error : RegressionResult

Material → higher error rates in context (§4.3: β = 0.96, SE = 0.09, p < .0001).

Equations

• Phenomena.Reference.Studies.KursatDegen2021.exp3_error = { beta := 0.96, se := 9e-2, significant := true }

def Phenomena.Reference.Studies.KursatDegen2021.exp3_rt : RegressionResult

Material → slower RTs in context (§4.3: β = 0.24, SE = 0.018, t = −59.62, p < .0001). Log-transformed RT.

Equations

• Phenomena.Reference.Studies.KursatDegen2021.exp3_rt = { beta := 0.24, se := 18e-3, tStat := some (-59.62), significant := true }

theorem Phenomena.Reference.Studies.KursatDegen2021.material_harder : exp1_error.significant = true ∧ exp1_rt.significant = true ∧ exp3_error.significant = true ∧ exp3_rt.significant = true

Material is harder to perceive than color: both error and RT effects are significant in Exp 1 (isolated properties) and Exp 3 (in context).

theorem Phenomena.Reference.Studies.KursatDegen2021.difficulty_direction : exp1_error.beta > 0 ∧ exp1_rt.beta > 0 ∧ exp3_error.beta > 0 ∧ exp3_rt.beta > 0

All difficulty effects have positive β: material produces more errors and slower RTs than color.

theorem Phenomena.Reference.Studies.KursatDegen2021.color_more_redundant : exp2_redundancy.significant = true ∧ exp2_redundancy.beta > 0

Color is used redundantly more than material: positive β in Exp 2.

theorem Phenomena.Reference.Studies.KursatDegen2021.difficulty_predicts_redundancy : exp1_error.significant = true ∧ exp1_error.beta > 0 ∧ exp3_error.significant = true ∧ exp3_error.beta > 0 ∧ exp2_redundancy.significant = true ∧ exp2_redundancy.beta > 0

The core finding: perceptual difficulty and redundant use are anti-correlated across property types. Material is harder to perceive (positive β in Exps 1, 3) AND less redundantly used (positive β in Exp 2 means color > material). All effects significant.

The cs-RSA model's meaning function φ (@cite{degen-etal-2020}) is a product of independent per-feature noise channels (proven in DegenEtAl2020.φ_product_of_experts). The φ function uses RSA.Noise parameter values by construction — this is structural, not coincidental (proven in DegenEtAl2020.φ_grounded_in_noise).

theorem Phenomena.Reference.Studies.KursatDegen2021.csrsa_params_match_noise : DegenEtAl2020.φ DegenEtAl2020.Utterance.blue DegenEtAl2020.World.smallBlue = RSA.Noise.colorMatch ∧ DegenEtAl2020.φ DegenEtAl2020.Utterance.blue DegenEtAl2020.World.bigRed = RSA.Noise.colorMismatch ∧ DegenEtAl2020.φ DegenEtAl2020.Utterance.small DegenEtAl2020.World.smallBlue = RSA.Noise.sizeMatch ∧ DegenEtAl2020.φ DegenEtAl2020.Utterance.small DegenEtAl2020.World.bigBlue = RSA.Noise.sizeMismatch

The cs-RSA φ function uses RSA.Noise parameters by construction. Re-exported from the study file for local use.

def Phenomena.Reference.Studies.KursatDegen2021.propertyToDiscrimination : PropertyType → ℚ

Map property types to RSA Noise discrimination values (noise gap = onMatch − onMismatch). Larger gap → the feature provides a cleaner signal to the L0 listener via the cs-RSA φ function. Delegates to PropertyDomain.noiseDiscrimination for the three parameterized domains.

Equations

• Phenomena.Reference.Studies.KursatDegen2021.propertyToDiscrimination Core.PropertyDomain.color = RSA.Noise.colorDiscrimination
• Phenomena.Reference.Studies.KursatDegen2021.propertyToDiscrimination Core.PropertyDomain.size = RSA.Noise.sizeDiscrimination
• Phenomena.Reference.Studies.KursatDegen2021.propertyToDiscrimination Core.PropertyDomain.material = RSA.Noise.materialDiscrimination
• Phenomena.Reference.Studies.KursatDegen2021.propertyToDiscrimination x✝ = 0

theorem Phenomena.Reference.Studies.KursatDegen2021.propertyToDiscrimination_canonical : Core.PropertyDomain.color.noiseDiscrimination = some (propertyToDiscrimination Core.PropertyDomain.color) ∧ Core.PropertyDomain.size.noiseDiscrimination = some (propertyToDiscrimination Core.PropertyDomain.size) ∧ Core.PropertyDomain.material.noiseDiscrimination = some (propertyToDiscrimination Core.PropertyDomain.material)

The local propertyToDiscrimination agrees with the canonical PropertyDomain.noiseDiscrimination for all parameterized domains.

theorem Phenomena.Reference.Studies.KursatDegen2021.color_gap_exceeds_material : RSA.Noise.noiseGap RSA.Noise.colorMatch RSA.Noise.colorMismatch > RSA.Noise.noiseGap RSA.Noise.materialMatch RSA.Noise.materialMismatch

Color's noise gap (0.98) exceeds material's (0.40). In the cs-RSA product-of-experts model, this means a color modifier contributes a stronger discriminative signal to φ than a material modifier: φ_match/φ_mismatch = onMatch/onMismatch, which increases with gap.

theorem Phenomena.Reference.Studies.KursatDegen2021.discrimination_ordering : propertyToDiscrimination Core.PropertyDomain.color > propertyToDiscrimination Core.PropertyDomain.size ∧ propertyToDiscrimination Core.PropertyDomain.size > propertyToDiscrimination Core.PropertyDomain.material

Full discrimination ordering: color > size > material. Each step in this chain means the modifier provides less signal to the L0 listener, so the S1 speaker has less reason to include it redundantly.

theorem Phenomena.Reference.Studies.KursatDegen2021.noise_predicts_redundancy : propertyToDiscrimination Core.PropertyDomain.color > propertyToDiscrimination Core.PropertyDomain.material ∧ exp2_redundancy.significant = true ∧ exp2_redundancy.beta > 0 ∧ exp1_error.significant = true ∧ exp1_error.beta > 0 ∧ exp3_error.significant = true ∧ exp3_error.beta > 0

The predicted ordering (color more redundant than material) matches the observed data: Exp 2 β = 2.32 > 0 (color used redundantly more), and the noise gap ordering (color > material) is grounded by the perceptual difficulty data (Exps 1 and 3).

The full redundancy prediction requires the incremental CI-RSA model (@cite{waldon-degen-2021}), which processes utterances word-by-word. The incremental model's three predictions are verified as theorems in WaldonDegen2021.lean — here we re-export the key result showing that redundant color > redundant size in English, which is the color/size analogue of the color/material asymmetry tested in Exp 2.

theorem Phenomena.Reference.Studies.KursatDegen2021.incremental_model_predicts_color_asymmetry : WaldonDegen2021.englishSS.trajectoryProb () WaldonDegen2021.Referent.smallBlue [WaldonDegen2021.Word.small, WaldonDegen2021.Word.blue, WaldonDegen2021.Word.pin, WaldonDegen2021.Word.stop] > WaldonDegen2021.englishCS.trajectoryProb () WaldonDegen2021.Referent.smallBlue [WaldonDegen2021.Word.small, WaldonDegen2021.Word.blue, WaldonDegen2021.Word.pin, WaldonDegen2021.Word.stop]

The CI-RSA incremental model predicts English speakers use redundant color more than redundant size (Waldon & Degen 2021, Prediction 1). This is the color/size version of the color/material asymmetry observed in Exp 2.

theorem Phenomena.Reference.Studies.KursatDegen2021.converging_production_with_engelhardt : EngelhardtEtAl2006.exp1_target_1ref.modified > 0 ∧ exp2_redundancy.significant = true ∧ exp2_redundancy.beta > 0 ∧ propertyToDiscrimination Core.PropertyDomain.color > propertyToDiscrimination Core.PropertyDomain.material

Both this study and @cite{engelhardt-etal-2006} show that speakers produce unnecessary modifiers. @cite{engelhardt-etal-2006} Exp 1 finds a 31% overall over-description rate; this study's Exp 2 further shows the rate varies by property type (β = 2.32: color > material). The noise model explains this variation: high-discrimination properties (color) provide more signal, so the S1 speaker has more reason to include them even when not strictly necessary.

theorem Phenomena.Reference.Studies.KursatDegen2021.discrimination_predicts_preference : propertyToDiscrimination Core.PropertyDomain.color > propertyToDiscrimination Core.PropertyDomain.size ∧ propertyToDiscrimination Core.PropertyDomain.size > propertyToDiscrimination Core.PropertyDomain.material ∧ exp2_redundancy.significant = true ∧ exp2_redundancy.beta > 0 ∧ DaleReiter1995.BrevityInterpretation.noBrevity.strength = 0

@cite{dale-reiter-1995}'s Incremental Algorithm uses a fixed PreferredAttributes list. This study's Exp 2 data — colour used redundantly more than material (β = 2.32) — suggests the preference ordering should track discrimination: higher-discrimination properties (colour) are preferred over lower ones (material).

The noise discrimination ordering (colour > size > material) from RSA.Noise provides exactly this ranking, connecting D&R's preference-based REG to RSA's noise-based semantics.