@cite{westerbeek-koolen-maes-2015}

@cite{dale-reiter-1995}

Stored Object Knowledge and the Production of Referring Expressions: The Case of Color Typicality. Frontiers in Psychology 6, 935.

Core Argument

Speakers mention color in referring expressions more often when an object's color is atypical (blue banana) than when it is typical (yellow banana), even when color is not necessary for unique identification. This cannot be explained by physical salience (they control for this computationally via Erdem & Erdem 2013) — it arises from a contrast with stored object knowledge: the perceived color deviates from the prototypical color stored in long-term memory.

The effect is proportional to degree of atypicality (continuous, not binary) and is moderated by color diagnosticity: it is strongest for high color-diagnostic objects (simple shapes like apples, where color is a primary identifying feature) and weaker for low color-diagnostic objects (complex shapes like lobsters, where shape alone identifies the object).

Experiments

Experiment 1 (N = 43, 40 analysed after 3 excluded for audio issues): Simple objects varying in color typicality. Speakers describe target objects in 6-object visual scenes where color is never necessary for disambiguation. Strong main effect of color typicality on color mention rate. Pearson r(40) = −0.86 between typicality score and proportion of color descriptions.

Experiment 2 (N = 62 participants in 31 speaker–addressee pairs): 2 × 2 design crossing color typicality (typical / atypical) with shape diagnosticity (high / low, i.e., complex / simple shapes). Significant main effect of atypicality, and significant interaction: the typicality effect is larger for low shape-diagnostic (= high color-diagnostic, simple shape) objects.

Architecture

The meaning function is constructed from two components:

1. typicalityφ — shared structure of the reference game: a 2 × 2 matrix parameterised by a single value, the bare noun's effectiveness for the target. All other cells are fixed.

2. bareEffectiveness — maps color typicality (from a Prototype) to recognition effectiveness: how well the bare noun identifies a target whose color has this typicality. Linear in typicality; even a completely atypical object has nonzero effectiveness (category membership is not zero, just degraded).

The composition typicalityφ ∘ bareEffectiveness ∘ proto.typicality produces meaning functions grounded in stored knowledge by construction.

Verified Predictions

1. S1(withColor) is strictly decreasing in bare noun effectiveness (monotonicity theorem, general)
2. Atypical objects receive more color mention than typical objects
3. Color mention is preferred in both conditions, but more strongly for atypical
4. The color preference gap is larger for atypical than typical
5. Color diagnosticity interaction: the typicality effect is larger for high color-diagnostic (simple shape) objects
6. RSA prediction: color modifier preferred for atypical target
7. Meaning functions are derived from Prototype via bareEffectiveness
8. IA produces identical output for typical and atypical targets (preference order is static)

inductive WesterbeekKoolenMaes2015.World : Type

Objects in a referential display. The target is always of a unique type (e.g., the only banana among non-bananas), so color is never required for disambiguation. The question is whether the speaker voluntarily mentions the target's color.

• target : World

  Target object (e.g., the banana)

• distractor : World

  Distractor (different category, e.g., an apple)

@[implicit_reducible]

instance WesterbeekKoolenMaes2015.instDecidableEqWorld : DecidableEq World

Equations

• WesterbeekKoolenMaes2015.instDecidableEqWorld x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def WesterbeekKoolenMaes2015.instReprWorld.repr : World → ℕ → Std.Format

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

@[implicit_reducible]

instance WesterbeekKoolenMaes2015.instReprWorld : Repr World

Equations

• WesterbeekKoolenMaes2015.instReprWorld = { reprPrec := WesterbeekKoolenMaes2015.instReprWorld.repr }

@[implicit_reducible]

instance WesterbeekKoolenMaes2015.instInhabitedWorld : Inhabited World

Equations

• WesterbeekKoolenMaes2015.instInhabitedWorld = { default := WesterbeekKoolenMaes2015.instInhabitedWorld.default }

def WesterbeekKoolenMaes2015.instInhabitedWorld.default : World

Equations

• WesterbeekKoolenMaes2015.instInhabitedWorld.default = WesterbeekKoolenMaes2015.World.target

@[implicit_reducible]

instance WesterbeekKoolenMaes2015.instFintypeWorld : Fintype World

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inductive WesterbeekKoolenMaes2015.Utterance : Type

Referring expressions. The speaker can use a bare noun ("the banana") or add a color modifier ("the yellow/blue banana").

• bare : Utterance

  "the banana" — head noun only

• withColor : Utterance

  "the [color] banana" — color modifier + head noun

@[implicit_reducible]

instance WesterbeekKoolenMaes2015.instDecidableEqUtterance : DecidableEq Utterance

Equations

• WesterbeekKoolenMaes2015.instDecidableEqUtterance x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance WesterbeekKoolenMaes2015.instReprUtterance : Repr Utterance

Equations

• WesterbeekKoolenMaes2015.instReprUtterance = { reprPrec := WesterbeekKoolenMaes2015.instReprUtterance.repr }

def WesterbeekKoolenMaes2015.instReprUtterance.repr : Utterance → ℕ → Std.Format

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

def WesterbeekKoolenMaes2015.instInhabitedUtterance.default : Utterance

Equations

• WesterbeekKoolenMaes2015.instInhabitedUtterance.default = WesterbeekKoolenMaes2015.Utterance.bare

@[implicit_reducible]

instance WesterbeekKoolenMaes2015.instInhabitedUtterance : Inhabited Utterance

Equations

• WesterbeekKoolenMaes2015.instInhabitedUtterance = { default := WesterbeekKoolenMaes2015.instInhabitedUtterance.default }

@[implicit_reducible]

instance WesterbeekKoolenMaes2015.instFintypeUtterance : Fintype Utterance

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def WesterbeekKoolenMaes2015.typicalityφ (bareTarget : ℚ) : Utterance → World → ℚ

Meaning function parameterised by the bare noun's effectiveness for the target. This is the single parameter that varies across typicality conditions — all other cells are fixed:

| Utterance | target     | distractor |
|-----------|------------|------------|
| bare      | bareTarget | 1/10       |
| withColor | 19/20      | 1/20       |

The distractor values are fixed because distractor identity does not depend on the target's color. The withColor values are fixed because mentioning the actual color resolves any recognition mismatch regardless of typicality.

The key architectural point: all condition-specific meaning functions are instances of this single constructor. The condition structure is explicit in the parameterisation rather than hidden in four separate case-by-case definitions.

Equations

• WesterbeekKoolenMaes2015.typicalityφ bareTarget WesterbeekKoolenMaes2015.Utterance.bare WesterbeekKoolenMaes2015.World.target = bareTarget
• WesterbeekKoolenMaes2015.typicalityφ bareTarget WesterbeekKoolenMaes2015.Utterance.bare WesterbeekKoolenMaes2015.World.distractor = 1 / 10
• WesterbeekKoolenMaes2015.typicalityφ bareTarget WesterbeekKoolenMaes2015.Utterance.withColor WesterbeekKoolenMaes2015.World.target = 19 / 20
• WesterbeekKoolenMaes2015.typicalityφ bareTarget WesterbeekKoolenMaes2015.Utterance.withColor WesterbeekKoolenMaes2015.World.distractor = 1 / 20

inductive WesterbeekKoolenMaes2015.BananaColor : Type

Colors of the target banana.

• yellow : BananaColor
• blue : BananaColor

@[implicit_reducible]

instance WesterbeekKoolenMaes2015.instDecidableEqBananaColor : DecidableEq BananaColor

Equations

• WesterbeekKoolenMaes2015.instDecidableEqBananaColor x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance WesterbeekKoolenMaes2015.instReprBananaColor : Repr BananaColor

Equations

• WesterbeekKoolenMaes2015.instReprBananaColor = { reprPrec := WesterbeekKoolenMaes2015.instReprBananaColor.repr }

def WesterbeekKoolenMaes2015.instReprBananaColor.repr : BananaColor → ℕ → Std.Format

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def WesterbeekKoolenMaes2015.bananaPrototype : Core.Prototype BananaColor

Stored knowledge: yellow is the prototypical banana color. These values encode the contrast that drives color mention — a blue banana violates stored expectations.

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theorem WesterbeekKoolenMaes2015.typical_more_prototypical : bananaPrototype.MoreTypical BananaColor.yellow BananaColor.blue

Yellow (typical) is more prototypical than blue (atypical).

def WesterbeekKoolenMaes2015.bareEffectiveness (colorTypicality : ℚ) : ℚ

Maps stored color typicality to bare noun recognition effectiveness. Linear: bareEffectiveness(t) = 9/20 + t/2.

• When t = 9/10 (typical): bareEffectiveness = 9/10. The bare noun is highly effective — the listener imagines a yellow banana.
• When t = 1/10 (atypical): bareEffectiveness = 1/2. The bare noun is degraded — the listener imagines yellow, but the object is blue.
• The intercept 9/20 represents base category identification: even a maximally atypical color doesn't reduce effectiveness below 9/20 (it IS still a banana).
• The slope 1/2 captures how much typicality boosts recognition.

Equations

• WesterbeekKoolenMaes2015.bareEffectiveness colorTypicality = 9 / 20 + colorTypicality / 2

theorem WesterbeekKoolenMaes2015.bareEff_typical : bareEffectiveness (9 / 10) = 9 / 10

theorem WesterbeekKoolenMaes2015.bareEff_atypical : bareEffectiveness (1 / 10) = 1 / 2

theorem WesterbeekKoolenMaes2015.bareEffectiveness_strictMono (t₁ t₂ : ℚ) (h : t₁ < t₂) : bareEffectiveness t₁ < bareEffectiveness t₂

bareEffectiveness is strictly increasing: higher color typicality → higher bare noun effectiveness. This is the first link in the chain Prototype → bareEffectiveness → typicalityφ → S1, and it ensures that typicality ordering is preserved through the transform.

def WesterbeekKoolenMaes2015.φ_typical : Utterance → World → ℚ

Meaning function for the typical color condition (e.g., yellow banana). Derived from the Prototype via bareEffectiveness: typicalityφ (bareEffectiveness (bananaPrototype.typicality .yellow)) = typicalityφ (9/10).

φ(bare, target) = 9/10: the bare noun is highly effective because the listener's stored representation matches reality.

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def WesterbeekKoolenMaes2015.φ_atypical : Utterance → World → ℚ

Meaning function for the atypical color condition (e.g., blue banana). Derived from the Prototype via bareEffectiveness: typicalityφ (bareEffectiveness (bananaPrototype.typicality .blue)) = typicalityφ (1/2).

φ(bare, target) = 1/2: the bare noun is degraded because the listener imagines yellow but the object is blue.

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theorem WesterbeekKoolenMaes2015.φ_typical_grounded : φ_typical Utterance.bare World.target = bananaPrototype.typicality BananaColor.yellow

Sanity check: φ_typical(bare, target) coincides numerically with the Prototype's typicality value. This is a consequence of the linear parameters (intercept = 9/20, slope = 1/2) chosen for bareEffectiveness, not a deep structural necessity.

theorem WesterbeekKoolenMaes2015.shared_structure : φ_typical Utterance.bare World.distractor = φ_atypical Utterance.bare World.distractor ∧ φ_typical Utterance.withColor World.target = φ_atypical Utterance.withColor World.target ∧ φ_typical Utterance.withColor World.distractor = φ_atypical Utterance.withColor World.distractor

Grounding: the typical and atypical meaning functions differ only in the Prototype input — all other structure is shared.

def WesterbeekKoolenMaes2015.L0_q (φ : Utterance → World → ℚ) (u : Utterance) (w : World) : ℚ

L0 posterior: φ(u,w) / Σ_{w'} φ(u,w').

Equations

• WesterbeekKoolenMaes2015.L0_q φ u w = φ u w / (φ u WesterbeekKoolenMaes2015.World.target + φ u WesterbeekKoolenMaes2015.World.distractor)

def WesterbeekKoolenMaes2015.S1_q (φ : Utterance → World → ℚ) (u : Utterance) (w : World) : ℚ

S1 probability: L0(w|u) / Σ_{u'} L0(w|u'). With α = 1 and cost = 0 (No-Brevity), S1 is proportional to L0.

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

S1(withColor | target) is strictly decreasing in the bare noun's effectiveness t. Algebraically, S1 = (190t+19)/(390t+19), whose derivative is −3800/(390t+19)² < 0. Intuitively: a more effective bare noun makes color mention less necessary, so S1 allocates less probability to it.

All point predictions in §7–§9 are instances of this monotonicity.

def WesterbeekKoolenMaes2015.S1_colorMention (t : ℚ) : ℚ

S1(withColor | target) as a closed-form function of bare noun effectiveness. Equivalent to S1_q (typicalityφ t) .withColor .target when denominators are non-zero (i.e., t > 0).

Equations

• WesterbeekKoolenMaes2015.S1_colorMention t = (190 * t + 19) / (390 * t + 19)

theorem WesterbeekKoolenMaes2015.S1_q_eq_S1_colorMention (t : ℚ) (h : 0 < t) : S1_q (typicalityφ t) Utterance.withColor World.target = S1_colorMention t

Equivalence: S1_q at (withColor, target) equals the closed-form S1_colorMention for any typicalityφ t meaning function. This ensures the monotonicity theorem applies to all S1_q predictions, not just the specific parameter values checked below.

theorem WesterbeekKoolenMaes2015.S1_closed_typical : S1_q φ_typical Utterance.withColor World.target = S1_colorMention (9 / 10)

S1_colorMention equals S1_q at the specific parameter values used in the model. (Corollaries of S1_q_eq_S1_colorMention.)

theorem WesterbeekKoolenMaes2015.S1_closed_atypical : S1_q φ_atypical Utterance.withColor World.target = S1_colorMention (1 / 2)

theorem WesterbeekKoolenMaes2015.S1_colorMention_decreasing (t₁ t₂ : ℚ) (h₁ : 0 < t₁) (h₁₂ : t₁ < t₂) : S1_colorMention t₁ > S1_colorMention t₂

Monotonicity: S1(withColor | target) is strictly decreasing in bare noun effectiveness. Higher typicality → more effective bare noun → less color mention.

Proof: Cross-multiplying the fraction comparison reduces to 3800 · (t₂ − t₁) > 0, which holds since t₁ < t₂.

theorem WesterbeekKoolenMaes2015.monotonicity_instances : S1_colorMention (1 / 2) > S1_colorMention (7 / 10) ∧ S1_colorMention (7 / 10) > S1_colorMention (9 / 10)

Monotonicity instantiated at the three parameter values used in the model: 1/2 (atypical), 7/10 (low color-diag atypical), 9/10 (typical). The ordering is strict.

theorem WesterbeekKoolenMaes2015.L0_typical_bare : L0_q φ_typical Utterance.bare World.target = 9 / 10

L0(target | bare) = 9/10 in the typical condition.

theorem WesterbeekKoolenMaes2015.L0_atypical_bare : L0_q φ_atypical Utterance.bare World.target = 5 / 6

L0(target | bare) = 5/6 in the atypical condition. Lower than typical (9/10) because stored color expectations create a mismatch.

theorem WesterbeekKoolenMaes2015.L0_withColor_constant : L0_q φ_typical Utterance.withColor World.target = 19 / 20 ∧ L0_q φ_atypical Utterance.withColor World.target = 19 / 20

L0(target | withColor) = 19/20 in both conditions. Color mention resolves any mismatch — this is fixed by construction in typicalityφ.

theorem WesterbeekKoolenMaes2015.bare_better_for_typical : L0_q φ_typical Utterance.bare World.target > L0_q φ_atypical Utterance.bare World.target

The bare noun is more effective for typical than atypical targets.

theorem WesterbeekKoolenMaes2015.atypical_more_color : S1_q φ_atypical Utterance.withColor World.target > S1_q φ_typical Utterance.withColor World.target

Main prediction: S1 allocates more probability to color mention for atypically colored objects than for typically colored objects. Instance of S1_colorMention_decreasing at t₁ = 1/2, t₂ = 9/10.

theorem WesterbeekKoolenMaes2015.typical_color_preferred : S1_q φ_typical Utterance.withColor World.target > S1_q φ_typical Utterance.bare World.target

In the typical condition, color mention is still slightly preferred (S1 > 0.5 for withColor). Even typical colors carry some signal.

theorem WesterbeekKoolenMaes2015.atypical_color_strongly_preferred : S1_q φ_atypical Utterance.withColor World.target > S1_q φ_atypical Utterance.bare World.target

In the atypical condition, color mention is more strongly preferred.

theorem WesterbeekKoolenMaes2015.atypical_gap_larger : S1_q φ_atypical Utterance.withColor World.target - S1_q φ_atypical Utterance.bare World.target > S1_q φ_typical Utterance.withColor World.target - S1_q φ_typical Utterance.bare World.target

The color preference gap is larger in the atypical condition.

noncomputable def WesterbeekKoolenMaes2015.atypicalCfg : RSA.RSAConfig Utterance World

RSA model for the atypical color condition. Demonstrates that the ℚ-valued predictions correspond to a genuine RSAConfig.

Architecture: same as @cite{degen-etal-2020}'s cs-RSA — the only difference is the meaning function, which now varies with color typicality rather than with perceptual noise parameters.

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theorem WesterbeekKoolenMaes2015.rsa_atypical_color_preferred : atypicalCfg.S1 () World.target Utterance.withColor > atypicalCfg.S1 () World.target Utterance.bare

RSA prediction: color modifier is preferred for the atypical target.

Experiment 2 crosses color typicality (typical / atypical) with shape diagnosticity (high / low) in a 2 × 2 design.

Terminology mapping (the paper and our code use different axes):

| Paper term              | Our term              | Shape    | Example  |
|-------------------------|-----------------------|----------|----------|
| Low shape diagnosticity | High color diagnostic | simple   | apple    |
| High shape diagnosticity| Low color diagnostic  | complex  | lobster  |

The interaction: the typicality effect is LARGER for high color-diagnostic (simple shape) objects, because color is their primary identifying feature, so deviations from typical color are more disruptive.

The model captures this by varying the bare noun effectiveness at the atypical level: for high color-diagnostic objects, atypical color severely degrades the bare noun (1/2); for low color-diagnostic objects, shape compensates and the bare noun remains fairly effective (7/10).

def WesterbeekKoolenMaes2015.φ_lowColorDiag_atypical : Utterance → World → ℚ

Meaning function for low color-diagnostic (complex shape) objects in the atypical color condition. φ(bare, target) = 7/10: higher than the high color-diagnostic atypical condition (1/2) because the distinctive shape compensates for the color mismatch.

Equations

• WesterbeekKoolenMaes2015.φ_lowColorDiag_atypical = WesterbeekKoolenMaes2015.typicalityφ (7 / 10)

theorem WesterbeekKoolenMaes2015.s1_values : S1_q φ_typical Utterance.withColor World.target = 19 / 37 ∧ S1_q φ_atypical Utterance.withColor World.target = 57 / 107 ∧ S1_q (typicalityφ (9 / 10)) Utterance.withColor World.target = 19 / 37 ∧ S1_q φ_lowColorDiag_atypical Utterance.withColor World.target = 38 / 73

S1 color mention rates for all four conditions of Exp 2:

Condition	bareTarget	S1(withColor)
High color-diag + typical	9/10	19/37
High color-diag + atypical	1/2	57/107
Low color-diag + typical	9/10	19/37
Low color-diag + atypical	7/10	38/73

theorem WesterbeekKoolenMaes2015.diagnosticity_interaction : S1_q φ_atypical Utterance.withColor World.target - S1_q φ_typical Utterance.withColor World.target > S1_q φ_lowColorDiag_atypical Utterance.withColor World.target - S1_q φ_typical Utterance.withColor World.target

Interaction: the typicality effect on color mention is larger for high color-diagnostic (simple shape) objects.

High color-diag gap: 57/107 − 19/37 = 76/3959 Low color-diag gap: 38/73 − 19/37 = 19/2701

This captures the paper's Exp 2 finding: "the effect of typicality on using color in a referring expression was larger for [low shape- diagnostic] objects" — i.e., objects where color is more diagnostic and deviations from typical color are more disruptive.

theorem WesterbeekKoolenMaes2015.all_prefer_color : S1_q φ_typical Utterance.withColor World.target > S1_q φ_typical Utterance.bare World.target ∧ S1_q φ_atypical Utterance.withColor World.target > S1_q φ_atypical Utterance.bare World.target ∧ S1_q (typicalityφ (9 / 10)) Utterance.withColor World.target > S1_q (typicalityφ (9 / 10)) Utterance.bare World.target ∧ S1_q φ_lowColorDiag_atypical Utterance.withColor World.target > S1_q φ_lowColorDiag_atypical Utterance.bare World.target

All four conditions prefer color mention, but at different rates.

structure WesterbeekKoolenMaes2015.RegressionResult : Type

Mixed-effects logistic regression result.

• β : Float
• se : Float
• significant : Bool

def WesterbeekKoolenMaes2015.instReprRegressionResult.repr : RegressionResult → ℕ → Std.Format

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

@[implicit_reducible]

instance WesterbeekKoolenMaes2015.instReprRegressionResult : Repr RegressionResult

Equations

• WesterbeekKoolenMaes2015.instReprRegressionResult = { reprPrec := WesterbeekKoolenMaes2015.instReprRegressionResult.repr }

def WesterbeekKoolenMaes2015.exp1_typicality : RegressionResult

Exp 1 main effect: color typicality predicts color mention. Predictor is typicality (higher = more typical), so β is negative: more typical → less color mention. β = −2.36, SE = 0.25, p < .001. N = 43 participants (40 analysed; 3 excluded for audio issues).

Equations

• WesterbeekKoolenMaes2015.exp1_typicality = { β := -2.36, se := 0.25, significant := true }

def WesterbeekKoolenMaes2015.exp1_correlation : Float

Exp 1: Pearson correlation between typicality score and proportion of color descriptions, r(40) = −0.86, p < .001 (Figure 3).

Equations

• WesterbeekKoolenMaes2015.exp1_correlation = -0.86

theorem WesterbeekKoolenMaes2015.exp1_typicality_negative : exp1_typicality.β < 0

theorem WesterbeekKoolenMaes2015.exp1_strong_correlation : exp1_correlation < -0.80

def WesterbeekKoolenMaes2015.exp2_atypical_rate : Float

Exp 2 descriptive rates (collapsed across shape diagnosticity): 73.5% of references to atypically colored targets contained color vs 13.1% for typically colored targets (p. 9).

Equations

• WesterbeekKoolenMaes2015.exp2_atypical_rate = 0.735

def WesterbeekKoolenMaes2015.exp2_typical_rate : Float

Equations

• WesterbeekKoolenMaes2015.exp2_typical_rate = 0.131

theorem WesterbeekKoolenMaes2015.exp2_atypical_exceeds_typical : exp2_atypical_rate > exp2_typical_rate

def WesterbeekKoolenMaes2015.exp2_atypicality : RegressionResult

Exp 2 main effect: color atypicality predicts color mention. Note sign flip vs Exp 1: Exp 2 codes the predictor as atypicality (higher = more atypical), so β is positive. Same direction: more atypical → more color mention. β = 3.55, SE = 0.96, p < .001. N = 62 participants (31 speaker–addressee pairs, 31 speakers analysed).

Equations

• WesterbeekKoolenMaes2015.exp2_atypicality = { β := 3.55, se := 0.96, significant := true }

def WesterbeekKoolenMaes2015.exp2_interaction : RegressionResult

Exp 2 interaction: color atypicality × shape diagnosticity. β = −0.56, SE = 0.32, p = .010. Negative: the atypicality effect decreases as shape diagnosticity increases (complex shapes reduce the importance of color).

Equations

• WesterbeekKoolenMaes2015.exp2_interaction = { β := -0.56, se := 0.32, significant := true }

theorem WesterbeekKoolenMaes2015.consistent_direction : exp1_typicality.β < 0 ∧ exp2_atypicality.β > 0

Both experiments find the same directional effect despite opposite predictor coding: more atypical → more color mention.

theorem WesterbeekKoolenMaes2015.prototype_predicts_color_mention : bananaPrototype.MoreTypical BananaColor.yellow BananaColor.blue ∧ bareEffectiveness (bananaPrototype.typicality BananaColor.blue) < bareEffectiveness (bananaPrototype.typicality BananaColor.yellow) ∧ S1_q φ_typical Utterance.withColor World.target < S1_q φ_atypical Utterance.withColor World.target

Full-chain monotonicity: higher Prototype typicality → higher bareEffectiveness → lower S1(withColor). Each step is derived:

1. moreTypical from Prototype (stored knowledge)
2. bareEffectiveness_strictMono (strictly increasing linear map)
3. S1_colorMention_decreasing (strictly decreasing rational function) The composition reverses the ordering: more typical → less color mention.

The Incremental Algorithm iterates through a fixed preference-ordered attribute list. We demonstrate concretely that the IA produces identical output for a typically and atypically colored target in the same scene layout, because the preference order is static.

def WesterbeekKoolenMaes2015.typicalTarget : DaleReiter1995.KBEntity

Target: a yellow banana (typical color).

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def WesterbeekKoolenMaes2015.atypicalTarget : DaleReiter1995.KBEntity

Target: a blue banana (atypical color).

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

def WesterbeekKoolenMaes2015.appleDistractor : DaleReiter1995.KBEntity

Distractor: an apple (different category).

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

def WesterbeekKoolenMaes2015.prefOrder : List DaleReiter1995.REGAttribute

Preference order: type first, then color.

Equations

• WesterbeekKoolenMaes2015.prefOrder = [DaleReiter1995.REGAttribute.headNoun, DaleReiter1995.REGAttribute.modifier Features.PropertyDomain.color]

theorem WesterbeekKoolenMaes2015.ia_same_output : DaleReiter1995.incrementalAlgorithm typicalTarget [appleDistractor] prefOrder = DaleReiter1995.incrementalAlgorithm atypicalTarget [appleDistractor] prefOrder

The IA produces the same attributes for both typical and atypical targets: {type = banana}. Color is included in neither case because "banana" alone rules out the apple distractor — the IA never reaches color in the preference order.

The IA is structurally blind to typicality: it does not know that "blue banana" would be more helpful to the addressee than "yellow banana." cs-RSA captures this through the meaning function.

theorem WesterbeekKoolenMaes2015.ia_both_succeed : DaleReiter1995.iaSuccess typicalTarget [appleDistractor] prefOrder = true ∧ DaleReiter1995.iaSuccess atypicalTarget [appleDistractor] prefOrder = true

Both IA runs succeed (the single distractor is ruled out by type).

theorem WesterbeekKoolenMaes2015.ia_vs_csrsa : DaleReiter1995.incrementalAlgorithm typicalTarget [appleDistractor] prefOrder = [(DaleReiter1995.REGAttribute.headNoun, "banana")] ∧ S1_q φ_atypical Utterance.withColor World.target > S1_q φ_typical Utterance.withColor World.target

The IA produces a single attribute (type = banana) and omits color entirely. cs-RSA, by contrast, varies the probability of color mention with typicality.

@cite{degen-etal-2020}'s cs-RSA derives the color > size asymmetry from noise discrimination parameters. This paper's contribution is orthogonal: typicality modulates the meaning function within a single feature dimension (color), varying the effectiveness of the bare noun by how well the perceived color matches stored expectations. Both mechanisms operate through continuous semantics.

theorem WesterbeekKoolenMaes2015.noise_and_typicality_complementary : RSA.Noise.colorDiscrimination > RSA.Noise.sizeDiscrimination ∧ S1_q φ_atypical Utterance.withColor World.target > S1_q φ_typical Utterance.withColor World.target

Noise and typicality both modulate continuous semantics:

• Noise: color discrimination > size discrimination (across features)
• Typicality: atypical color → more color mention (within a feature) Both are necessary: noise explains color > size; typicality explains atypical > typical within color.