structure HawkinsGweonGoodman2021.ObjectFeatures : Type

Object features in the simplified task (Experiment 1)

• shape : ℕ
• color : ℕ
• texture : ℕ

instance HawkinsGweonGoodman2021.instDecidableEqObjectFeatures : DecidableEq ObjectFeatures

Equations

• HawkinsGweonGoodman2021.instDecidableEqObjectFeatures = HawkinsGweonGoodman2021.instDecidableEqObjectFeatures.decEq

def HawkinsGweonGoodman2021.instDecidableEqObjectFeatures.decEq (x✝ x✝¹ : ObjectFeatures) : Decidable (x✝ = x✝¹)

Equations

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

instance HawkinsGweonGoodman2021.instReprObjectFeatures : Repr ObjectFeatures

Equations

• HawkinsGweonGoodman2021.instReprObjectFeatures = { reprPrec := HawkinsGweonGoodman2021.instReprObjectFeatures.repr }

def HawkinsGweonGoodman2021.instReprObjectFeatures.repr : ObjectFeatures → ℕ → Std.Format

Equations

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

def HawkinsGweonGoodman2021.instBEqObjectFeatures.beq : ObjectFeatures → ObjectFeatures → Bool

Equations

• HawkinsGweonGoodman2021.instBEqObjectFeatures.beq { shape := a, color := a_1, texture := a_2 } { shape := b, color := b_1, texture := b_2 } = (a == b && (a_1 == b_1 && a_2 == b_2))
• HawkinsGweonGoodman2021.instBEqObjectFeatures.beq x✝¹ x✝ = false

instance HawkinsGweonGoodman2021.instBEqObjectFeatures : BEq ObjectFeatures

Equations

• HawkinsGweonGoodman2021.instBEqObjectFeatures = { beq := HawkinsGweonGoodman2021.instBEqObjectFeatures.beq }

structure HawkinsGweonGoodman2021.Object : Type

An object in the display

• features : ObjectFeatures
• visible : Bool

def HawkinsGweonGoodman2021.instDecidableEqObject.decEq (x✝ x✝¹ : Object) : Decidable (x✝ = x✝¹)

Equations

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

instance HawkinsGweonGoodman2021.instDecidableEqObject : DecidableEq Object

Equations

• HawkinsGweonGoodman2021.instDecidableEqObject = HawkinsGweonGoodman2021.instDecidableEqObject.decEq

instance HawkinsGweonGoodman2021.instReprObject : Repr Object

Equations

• HawkinsGweonGoodman2021.instReprObject = { reprPrec := HawkinsGweonGoodman2021.instReprObject.repr }

def HawkinsGweonGoodman2021.instReprObject.repr : Object → ℕ → Std.Format

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

def HawkinsGweonGoodman2021.instBEqObject.beq : Object → Object → Bool

Equations

• HawkinsGweonGoodman2021.instBEqObject.beq { features := a, visible := a_1 } { features := b, visible := b_1 } = (a == b && a_1 == b_1)
• HawkinsGweonGoodman2021.instBEqObject.beq x✝¹ x✝ = false

instance HawkinsGweonGoodman2021.instBEqObject : BEq Object

Equations

• HawkinsGweonGoodman2021.instBEqObject = { beq := HawkinsGweonGoodman2021.instBEqObject.beq }

structure HawkinsGweonGoodman2021.Utterance : Type

Utterance: combination of features mentioned

• mentionShape : Bool
• mentionColor : Bool
• mentionTexture : Bool

def HawkinsGweonGoodman2021.instDecidableEqUtterance.decEq (x✝ x✝¹ : Utterance) : Decidable (x✝ = x✝¹)

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

instance HawkinsGweonGoodman2021.instDecidableEqUtterance : DecidableEq Utterance

Equations

• HawkinsGweonGoodman2021.instDecidableEqUtterance = HawkinsGweonGoodman2021.instDecidableEqUtterance.decEq

instance HawkinsGweonGoodman2021.instReprUtterance : Repr Utterance

Equations

• HawkinsGweonGoodman2021.instReprUtterance = { reprPrec := HawkinsGweonGoodman2021.instReprUtterance.repr }

def HawkinsGweonGoodman2021.instReprUtterance.repr : Utterance → ℕ → Std.Format

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

instance HawkinsGweonGoodman2021.instBEqUtterance : BEq Utterance

Equations

• HawkinsGweonGoodman2021.instBEqUtterance = { beq := HawkinsGweonGoodman2021.instBEqUtterance.beq }

def HawkinsGweonGoodman2021.instBEqUtterance.beq : Utterance → Utterance → Bool

Equations

• One or more equations did not get rendered due to their size.
• HawkinsGweonGoodman2021.instBEqUtterance.beq x✝¹ x✝ = false

def HawkinsGweonGoodman2021.allUtterances : List Utterance

All possible utterances (2³ = 8)

Equations

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

def HawkinsGweonGoodman2021.utteranceCost (u : Utterance) : ℕ

Utterance cost: number of features mentioned

Equations

• HawkinsGweonGoodman2021.utteranceCost u = ((if u.mentionShape = true then 1 else 0) + if u.mentionColor = true then 1 else 0) + if u.mentionTexture = true then 1 else 0

structure HawkinsGweonGoodman2021.Context : Type

Context: set of objects in the display

• target : Object
• distractors : List Object

instance HawkinsGweonGoodman2021.instReprContext : Repr Context

Equations

• HawkinsGweonGoodman2021.instReprContext = { reprPrec := HawkinsGweonGoodman2021.instReprContext.repr }

def HawkinsGweonGoodman2021.instReprContext.repr : Context → ℕ → Std.Format

Equations

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

def HawkinsGweonGoodman2021.utteranceApplies (u : Utterance) (targetFeatures : ObjectFeatures) (o : Object) : Bool

Does utterance literally apply to object?

Equations

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

def HawkinsGweonGoodman2021.extension (u : Utterance) (targetFeatures : ObjectFeatures) (objects : List Object) : List Object

Extension of utterance: objects it applies to

Equations

• HawkinsGweonGoodman2021.extension u targetFeatures objects = List.filter (HawkinsGweonGoodman2021.utteranceApplies u targetFeatures) objects

def HawkinsGweonGoodman2021.moreSpecific (u0 u1 : Utterance) : Bool

Utterance u₀ is more specific than u₁ if its extension is a subset

Equations

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

def HawkinsGweonGoodman2021.literalListenerProb (u : Utterance) (targetFeatures : ObjectFeatures) (o : Object) (allObjects : List Object) : ℚ

Literal listener probability: P(o | u, C) ∝ L(u, o)

Equations

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

def HawkinsGweonGoodman2021.egocentricInformativity (u : Utterance) (target : Object) (visibleObjects : List Object) : ℚ

Egocentric informativity: listener success rate in visible context only

Equations

• HawkinsGweonGoodman2021.egocentricInformativity u target visibleObjects = HawkinsGweonGoodman2021.literalListenerProb u target.features target visibleObjects

def HawkinsGweonGoodman2021.egocentricUtility (u : Utterance) (target : Object) (visibleObjects : List Object) (costWeight : ℚ := 1 / 10) : ℚ

Egocentric utility: informativity minus cost U_ego(u; o, C) = I_ego(u; o, C_visible) - λ · cost(u)

Equations

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

def HawkinsGweonGoodman2021.possibleHiddenObjects : List ObjectFeatures

Possible hidden objects: all feature combinations

Equations

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

def HawkinsGweonGoodman2021.asymmetricInformativity (u : Utterance) (target : Object) (visibleObjects : List Object) (hiddenPrior : ObjectFeatures → ℚ) : ℚ

Asymmetric informativity: marginalizes over possible hidden objects

I_asym(u; o, C) = Σ_{o_h} P(o_h) · P_L0(o | u, C ∪ {o_h})

This captures the speaker's expected listener success rate under uncertainty.

Equations

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

def HawkinsGweonGoodman2021.asymmetricUtility (u : Utterance) (target : Object) (visibleObjects : List Object) (hiddenPrior : ObjectFeatures → ℚ) (costWeight : ℚ := 1 / 10) : ℚ

Asymmetric utility: informativity minus cost

U_asym(u; o, C) = I_asym(u; o, C) - λ · cost(u)

Equations

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

@[reducible, inline]

abbrev HawkinsGweonGoodman2021.PerspectiveWeight : Type

Perspective-taking weight: 0 = egocentric, 1 = full perspective-taking

Equations

• HawkinsGweonGoodman2021.PerspectiveWeight = ℚ

def HawkinsGweonGoodman2021.mixtureInformativity (u : Utterance) (target : Object) (visibleObjects : List Object) (hiddenPrior : ObjectFeatures → ℚ) (wS : PerspectiveWeight) : ℚ

Mixture informativity: interpolates between egocentric and asymmetric

I_mix(u; o, C, w_S) = w_S · I_asym(u; o, C) + (1 - w_S) · I_ego(u; o, C)

Equations

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

def HawkinsGweonGoodman2021.mixtureUtility (u : Utterance) (target : Object) (visibleObjects : List Object) (hiddenPrior : ObjectFeatures → ℚ) (wS : PerspectiveWeight) (costWeight : ℚ := 1 / 10) : ℚ

Mixture utility: interpolates between egocentric and asymmetric

U_mix(u; o, C, w_S) = w_S · U_asym(u; o, C) + (1 - w_S) · U_ego(u; o, C)

Equations

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

def HawkinsGweonGoodman2021.speakerScore (u : Utterance) (target : Object) (visibleObjects : List Object) (hiddenPrior : ObjectFeatures → ℚ) (wS : PerspectiveWeight) (alpha : ℕ) (costWeight : ℚ := 1 / 10) : ℚ

Speaker probability: P(u | o, C, w_S) ∝ exp(α · U_mix)

Equations

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

def HawkinsGweonGoodman2021.speakerDist (target : Object) (visibleObjects : List Object) (hiddenPrior : ObjectFeatures → ℚ) (wS : PerspectiveWeight) (alpha : ℕ) : List (Utterance × ℚ)

Normalize speaker scores to get probabilities

Equations

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

def HawkinsGweonGoodman2021.expectedAccuracy (target : Object) (visibleObjects : List Object) (hiddenPrior : ObjectFeatures → ℚ) (wS : PerspectiveWeight) (alpha : ℕ) : ℚ

Expected communicative accuracy at weight w_S. This is the benefit side of the cost-benefit trade-off.

Equations

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

def HawkinsGweonGoodman2021.resourceRationalUtility (target : Object) (visibleObjects : List Object) (hiddenPrior : ObjectFeatures → ℚ) (wS : PerspectiveWeight) (alpha : ℕ) (beta : ℚ) : ℚ

Resource-rational utility: accuracy - β · w

U_RR(w_S) = E[accuracy] - β · w_S

Equations

• HawkinsGweonGoodman2021.resourceRationalUtility target visibleObjects hiddenPrior wS alpha beta = HawkinsGweonGoodman2021.expectedAccuracy target visibleObjects hiddenPrior wS alpha - beta * wS

def HawkinsGweonGoodman2021.exampleTarget : Object

Example context: target is unique in shape

Equations

• HawkinsGweonGoodman2021.exampleTarget = { features := { shape := 0, color := 0, texture := 0 }, visible := true }

def HawkinsGweonGoodman2021.exampleDistractors : List Object

Equations

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

def HawkinsGweonGoodman2021.exampleVisible : List Object

Equations

• HawkinsGweonGoodman2021.exampleVisible = HawkinsGweonGoodman2021.exampleTarget :: HawkinsGweonGoodman2021.exampleDistractors

def HawkinsGweonGoodman2021.uniformHiddenPrior : ObjectFeatures → ℚ

Uniform prior over hidden objects

Equations

• HawkinsGweonGoodman2021.uniformHiddenPrior x✝ = 1 / 64

def HawkinsGweonGoodman2021.shapeOnly : Utterance

Shape-only utterance

Equations

• HawkinsGweonGoodman2021.shapeOnly = { mentionShape := true, mentionColor := false, mentionTexture := false }

def HawkinsGweonGoodman2021.shapeAndColor : Utterance

Shape + color utterance

Equations

• HawkinsGweonGoodman2021.shapeAndColor = { mentionShape := true, mentionColor := true, mentionTexture := false }

def HawkinsGweonGoodman2021.fullDescription : Utterance

Shape + color + texture utterance

Equations

• HawkinsGweonGoodman2021.fullDescription = { mentionShape := true, mentionColor := true, mentionTexture := true }

theorem HawkinsGweonGoodman2021.more_specific_higher_asymmetric_informativity : asymmetricInformativity fullDescription exampleTarget exampleVisible uniformHiddenPrior > asymmetricInformativity shapeOnly exampleTarget exampleVisible uniformHiddenPrior

Theorem 1: More specific utterance has higher INFORMATIVITY under asymmetry.

When there may be hidden objects, being more specific guards against more possible distractors. This tests informativity (without cost).

def HawkinsGweonGoodman2021.specificityGain_asym : ℚ

Theorem 2: Asymmetric INFORMATIVITY favors specificity more than egocentric.

This is the key qualitative prediction: the GAIN from being more specific is larger under asymmetric reasoning than egocentric reasoning.

Equations

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

def HawkinsGweonGoodman2021.specificityGain_ego : ℚ

Equations

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

theorem HawkinsGweonGoodman2021.asymmetry_increases_specificity_gain : specificityGain_asym > specificityGain_ego

def HawkinsGweonGoodman2021.preferFullDescription_wS1 : Bool

Theorem 3: Full description has highest informativity at high perspective-taking weight.

When w_S = 1 (full perspective-taking), more informative utterances maximize expected listener success rate.

Equations

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

theorem HawkinsGweonGoodman2021.full_description_preferred_at_wS1 : preferFullDescription_wS1 = true

def HawkinsGweonGoodman2021.shapeOnlySufficient_wS0 : Bool

Theorem 4: Shape-only has same informativity as full description at egocentric weight.

When w_S = 0 (pure egocentric), shape alone is equally informative in visible context (target is unique in shape among visible objects).

Equations

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

theorem HawkinsGweonGoodman2021.shape_only_sufficient_at_wS0 : shapeOnlySufficient_wS0 = true

Compositional Grounding

The utterance semantics derive from predicate modification (H&K Ch. 4):

⟦α β⟧ = λx. ⟦α⟧(x) ∧ ⟦β⟧(x)

Each feature mention (shape, color, texture) is an intersective adjective that denotes a characteristic function of type e → t:

• ⟦square⟧ = λx. shape(x) = square
• ⟦blue⟧ = λx. color(x) = blue
• ⟦checked⟧ = λx. texture(x) = checked

Composing via predicate modification: ⟦blue checked square⟧ = λx. blue(x) ∧ checked(x) ∧ square(x)

This is exactly Semantics.Montague.Modification.intersectiveMod applied iteratively.

def HawkinsGweonGoodman2021.MontaguGrounding.shapePred (targetShape : ℕ) : Object → Bool

Feature predicates are Montague-style intersective adjectives (e → t).

Each feature denotes a characteristic function from entities (Objects) to truth values. These are the basic building blocks for compositional utterance semantics.

Equations

• HawkinsGweonGoodman2021.MontaguGrounding.shapePred targetShape o = (o.features.shape == targetShape)

def HawkinsGweonGoodman2021.MontaguGrounding.colorPred (targetColor : ℕ) : Object → Bool

Equations

• HawkinsGweonGoodman2021.MontaguGrounding.colorPred targetColor o = (o.features.color == targetColor)

def HawkinsGweonGoodman2021.MontaguGrounding.texturePred (targetTexture : ℕ) : Object → Bool

Equations

• HawkinsGweonGoodman2021.MontaguGrounding.texturePred targetTexture o = (o.features.texture == targetTexture)

def HawkinsGweonGoodman2021.MontaguGrounding.compositionalDenotation (u : Utterance) (targetFeatures : ObjectFeatures) : Object → Bool

Compositionally derived utterance denotation.

An utterance mentions some subset of {shape, color, texture}. The denotation is the conjunction of all mentioned feature predicates, using predMod from Semantics.Montague.Modification:

⟦blue checked square⟧ = predMod (predMod ⟦blue⟧ ⟦checked⟧) ⟦square⟧ = λx. blue(x) ∧ checked(x) ∧ square(x)

Equations

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

def HawkinsGweonGoodman2021.MontaguGrounding.directDenotation (u : Utterance) (targetFeatures : ObjectFeatures) : Object → Bool

Direct (ad-hoc) utterance denotation from Part 2

Equations

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

theorem HawkinsGweonGoodman2021.MontaguGrounding.grounding_compositional_equals_direct (u : Utterance) (tf : ObjectFeatures) (o : Object) : compositionalDenotation u tf o = directDenotation u tf o

Grounding theorem: Direct denotation equals compositional derivation.

The ad-hoc semantics in utteranceApplies are exactly what we get from applying predicate modification (from Semantics.Montague.Modification) to individual feature predicates.

theorem HawkinsGweonGoodman2021.MontaguGrounding.utteranceApplies_grounded (u : Utterance) (tf : ObjectFeatures) (o : Object) : utteranceApplies u tf o = compositionalDenotation u tf o

Grounding theorem: utteranceApplies = compositional denotation

theorem HawkinsGweonGoodman2021.MontaguGrounding.rsa_meaning_compositional (u : Utterance) (tf : ObjectFeatures) (o : Object) : utteranceApplies u tf o = true ↔ compositionalDenotation u tf o = true

The RSA meaning function φ is grounded in compositional semantics

def HawkinsGweonGoodman2021.utteranceDenotation (u : Utterance) (targetFeatures : ObjectFeatures) : Object → Bool

Equations

• HawkinsGweonGoodman2021.utteranceDenotation = HawkinsGweonGoodman2021.MontaguGrounding.directDenotation

theorem HawkinsGweonGoodman2021.semantics_grounded (u : Utterance) (target : ObjectFeatures) (o : Object) : utteranceApplies u target o = utteranceDenotation u target o

Grounding: utteranceApplies equals compositional denotation

Asymmetric Case via Unified API

The full perspective-taking case (w_S = 1) maps to RSAConfig with latent variables:

• Latent = Speaker's visual access (which objects they)
• World = Full context (visible objects + hidden object features)
• speakerCredence = P(world | speaker's visual access)

The mixture model (w_S ∈ (0,1)) and resource-rational optimization (finding w*) are implementation-specific extensions that sit outside the unified API.

structure HawkinsGweonGoodman2021.WorldState : Type

World state: visible objects + one hidden object behind occlusion

• visible : List Object
• hidden : ObjectFeatures
• target : Object

instance HawkinsGweonGoodman2021.instDecidableEqWorldState : DecidableEq WorldState

Equations

• HawkinsGweonGoodman2021.instDecidableEqWorldState = HawkinsGweonGoodman2021.instDecidableEqWorldState.decEq

def HawkinsGweonGoodman2021.instDecidableEqWorldState.decEq (x✝ x✝¹ : WorldState) : Decidable (x✝ = x✝¹)

Equations

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

def HawkinsGweonGoodman2021.instBEqWorldState.beq : WorldState → WorldState → Bool

Equations

• HawkinsGweonGoodman2021.instBEqWorldState.beq { visible := a, hidden := a_1, target := a_2 } { visible := b, hidden := b_1, target := b_2 } = (a == b && (a_1 == b_1 && a_2 == b_2))
• HawkinsGweonGoodman2021.instBEqWorldState.beq x✝¹ x✝ = false

instance HawkinsGweonGoodman2021.instBEqWorldState : BEq WorldState

Equations

• HawkinsGweonGoodman2021.instBEqWorldState = { beq := HawkinsGweonGoodman2021.instBEqWorldState.beq }

instance HawkinsGweonGoodman2021.instReprWorldState : Repr WorldState

Equations

• HawkinsGweonGoodman2021.instReprWorldState = { reprPrec := HawkinsGweonGoodman2021.instReprWorldState.repr }

def HawkinsGweonGoodman2021.instReprWorldState.repr : WorldState → ℕ → Std.Format

Equations

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

structure HawkinsGweonGoodman2021.VisualAccess : Type

Speaker's visual access: what objects they can see

• visibleObjects : List Object
• targetObject : Object

def HawkinsGweonGoodman2021.instDecidableEqVisualAccess.decEq (x✝ x✝¹ : VisualAccess) : Decidable (x✝ = x✝¹)

Equations

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

instance HawkinsGweonGoodman2021.instDecidableEqVisualAccess : DecidableEq VisualAccess

Equations

• HawkinsGweonGoodman2021.instDecidableEqVisualAccess = HawkinsGweonGoodman2021.instDecidableEqVisualAccess.decEq

instance HawkinsGweonGoodman2021.instBEqVisualAccess : BEq VisualAccess

Equations

• HawkinsGweonGoodman2021.instBEqVisualAccess = { beq := HawkinsGweonGoodman2021.instBEqVisualAccess.beq }

def HawkinsGweonGoodman2021.instBEqVisualAccess.beq : VisualAccess → VisualAccess → Bool

Equations

• HawkinsGweonGoodman2021.instBEqVisualAccess.beq { visibleObjects := a, targetObject := a_1 } { visibleObjects := b, targetObject := b_1 } = (a == b && a_1 == b_1)
• HawkinsGweonGoodman2021.instBEqVisualAccess.beq x✝¹ x✝ = false

def HawkinsGweonGoodman2021.instReprVisualAccess.repr : VisualAccess → ℕ → Std.Format

Equations

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

instance HawkinsGweonGoodman2021.instReprVisualAccess : Repr VisualAccess

Equations

• HawkinsGweonGoodman2021.instReprVisualAccess = { reprPrec := HawkinsGweonGoodman2021.instReprVisualAccess.repr }

def HawkinsGweonGoodman2021.allWorldStates (visible : List Object) (target : Object) : List WorldState

All world states: each possible hidden object configuration

Equations

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

def HawkinsGweonGoodman2021.visualAccessCredence (access : VisualAccess) (world : WorldState) : ℚ

Speaker credence: uniform over hidden objects given visual access.

P(world | access) = 1/64 if world.visible matches access, else 0. This encodes that speaker knows what's visible but is uncertain about hidden.

Equations

• HawkinsGweonGoodman2021.visualAccessCredence access world = if (world.visible == access.visibleObjects && world.target == access.targetObject) = true then 1 / 64 else 0

def HawkinsGweonGoodman2021.worldMeaning (u : Utterance) (world : WorldState) : ℚ

Literal meaning: utterance applies to target in this world context

Equations

• HawkinsGweonGoodman2021.worldMeaning u world = HawkinsGweonGoodman2021.literalListenerProb u world.target.features world.target ({ features := world.hidden, visible := false } :: world.visible)

theorem HawkinsGweonGoodman2021.unified_worldMeaning_grounded (u : Utterance) (world : WorldState) : worldMeaning u world = literalListenerProb u world.target.features world.target ({ features := world.hidden, visible := false } :: world.visible)

Grounding: The unified API's worldMeaning computes the same listener probability as our manual literalListenerProb for each world configuration.

theorem HawkinsGweonGoodman2021.unified_credence_matches_prior : visualAccessCredence { visibleObjects := exampleVisible, targetObject := exampleTarget } { visible := exampleVisible, hidden := { shape := 0, color := 0, texture := 0 }, target := exampleTarget } = 1 / 64

Grounding: Speaker credence in unified API marginalizes uniformly over hidden objects, matching the manual uniformHiddenPrior.

Mixture Model (Implementation-Specific)

The mixture model w_S · U_asym + (1-w_S) · U_ego and resource-rational optimization for finding optimal w* are handled in Parts 6-8 above.

These are implementation-specific extensions that:

1. Blend two reasoning modes (asymmetric vs egocentric)
2. Find optimal effort allocation via cost-benefit analysis

The unified API handles the asymmetric case directly; the mixture and meta-cognitive choice of w* sit outside the core RSA loop.

structure HawkinsGweonGoodman2021.ListenerBeliefs : Type

Listener's belief about speaker's weight after observing utterances

• wS_expectation : ℚ
• observations : ℕ

def HawkinsGweonGoodman2021.instReprListenerBeliefs.repr : ListenerBeliefs → ℕ → Std.Format

Equations

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

instance HawkinsGweonGoodman2021.instReprListenerBeliefs : Repr ListenerBeliefs

Equations

• HawkinsGweonGoodman2021.instReprListenerBeliefs = { reprPrec := HawkinsGweonGoodman2021.instReprListenerBeliefs.repr }

def HawkinsGweonGoodman2021.initialBeliefs : ListenerBeliefs

Initial uniform belief about speaker's weight

Equations

• HawkinsGweonGoodman2021.initialBeliefs = { wS_expectation := 1 / 2, observations := 0 }

def HawkinsGweonGoodman2021.updateBeliefs (beliefs : ListenerBeliefs) (shortUtterance : Bool) : ListenerBeliefs

Update beliefs after observing speaker use short utterances. If speaker consistently uses minimal descriptions, listener infers low w_S.

Equations

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

def HawkinsGweonGoodman2021.beliefsAfterShortUtterances : ListenerBeliefs

After seeing short utterances, listener expects lower w_S

Equations

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

theorem HawkinsGweonGoodman2021.listener_infers_low_wS_from_short_utterances : beliefsAfterShortUtterances.wS_expectation < initialBeliefs.wS_expectation

def HawkinsGweonGoodman2021.optimalListenerWeight (speakerWS beta : ℚ) : ℚ

Resource-rational listener response: increase own perspective-taking when speaker is under-informative.

Equations

• HawkinsGweonGoodman2021.optimalListenerWeight speakerWS beta = 1 ⊓ (0 ⊔ (1 - speakerWS + beta))

theorem HawkinsGweonGoodman2021.listener_compensates_for_low_speaker_effort : optimalListenerWeight (3 / 10) (2 / 10) > optimalListenerWeight (7 / 10) (2 / 10)

Listener increases effort when speaker decreases theirs

Key Predictions from Paper (Section 2.4.1)

The paper identifies four key qualitative predictions, which we verify as theorems:

1. speakersHedgeUnknowns: Speakers increase informativity with occlusions
2. divisionDependsOnPartner: Optimal effort depends on expected partner effort
3. listenersAdaptOverTime: Listeners update beliefs about speaker from observations
4. intermediateWeightsOptimal: Partial perspective-taking when cost > 0

theorem HawkinsGweonGoodman2021.paper_prediction_1_speakers_hedge : asymmetricInformativity fullDescription exampleTarget exampleVisible uniformHiddenPrior > asymmetricInformativity shapeOnly exampleTarget exampleVisible uniformHiddenPrior

Paper Prediction 1: Speakers hedge against known unknowns.

From the paper: "speakers will anticipate possible confusion from the listener's perspective, and produce additional information beyond what would be necessary from their own viewpoint."

Verified by: asymmetric informativity favors more specific utterances.

def HawkinsGweonGoodman2021.utilityAt_wL0 : ℚ

Paper Prediction 2: Division of labor depends on partner's expected effort.

From the paper: "The effort one participant ought to exert depends on how much effort they expect others to exert."

Verified by: at different listener weights, speaker utility differs. This shows speaker decisions depend on beliefs about listener.

Equations

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

def HawkinsGweonGoodman2021.utilityAt_wL1 : ℚ

Equations

• HawkinsGweonGoodman2021.utilityAt_wL1 = HawkinsGweonGoodman2021.egocentricUtility HawkinsGweonGoodman2021.shapeOnly HawkinsGweonGoodman2021.exampleTarget HawkinsGweonGoodman2021.exampleVisible

theorem HawkinsGweonGoodman2021.paper_prediction_2_division_depends_on_partner : utilityAt_wL0 ≠ utilityAt_wL1

theorem HawkinsGweonGoodman2021.paper_prediction_3_listeners_adapt : beliefsAfterShortUtterances.wS_expectation < initialBeliefs.wS_expectation

Paper Prediction 3: Listeners adapt over time.

From the paper: "listeners used violations to adaptively make fewer errors over time" (z = 2.6, p < 0.01)

Verified by: beliefs about speaker weight decrease when observing short utterances.

def HawkinsGweonGoodman2021.rrUtility (wS beta : ℚ) : ℚ

Resource-rational utility at a given perspective weight and cost coefficient

Equations

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

def HawkinsGweonGoodman2021.rrUtility_at_0 : ℚ

Paper Prediction 4: Intermediate weights are optimal when β > 0.

From the paper (Figure 2): "Above a certain β (i.e., if perspective-taking is sufficiently effortful), an intermediate weighting of perspective-taking is boundedly optimal."

At β = 0.2: w*_S = 0.36, w*_L = 0.51

Note: At β = 0, egocentric may have higher raw informativity (since it doesn't average over hidden distractors). But at β > 0, the cost term creates a trade-off where intermediate weights become optimal. The key insight is that speakers should choose MORE INFORMATIVE utterances (like fullDescription) rather than shapeOnly when doing perspective-taking - that's where the benefit comes.

Equations

• HawkinsGweonGoodman2021.rrUtility_at_0 = HawkinsGweonGoodman2021.rrUtility 0 (1 / 2)

def HawkinsGweonGoodman2021.rrUtility_at_half : ℚ

Equations

• HawkinsGweonGoodman2021.rrUtility_at_half = HawkinsGweonGoodman2021.rrUtility (1 / 2) (1 / 2)

def HawkinsGweonGoodman2021.rrUtility_at_1 : ℚ

Equations

• HawkinsGweonGoodman2021.rrUtility_at_1 = HawkinsGweonGoodman2021.rrUtility 1 (1 / 2)

theorem HawkinsGweonGoodman2021.high_cost_penalizes_full_perspective_taking : rrUtility_at_1 < rrUtility_at_0

theorem HawkinsGweonGoodman2021.paper_prediction_4_intermediate_weights_optimal : rrUtility_at_1 < rrUtility_at_0 ∧ rrUtility (1 / 4) (1 / 2) > rrUtility_at_1

Paper Prediction 4 (continued): Intermediate weights optimal.

When cost is moderate, the optimal weight is strictly between 0 and 1. This matches Figure 2 of the paper where w*_S ≈ 0.36 at β = 0.2.

Empirical Findings from Paper

Experiment 1 (Speaker Production, N=83 dyads)

• Occlusion effect: +1.3 words, t(120.3) = 8.8, p < .001
• Distractor effect: +0.6 words, t(206) = 5.7, p < .001

Experiment 2 (Listener Comprehension, N=116 dyads)

• Scripted: 51% critical errors
• Unscripted: 20% critical errors
• χ²(1) = 43, p < .001

def HawkinsGweonGoodman2021.empirical_scripted_error_rate : ℚ

Equations

• HawkinsGweonGoodman2021.empirical_scripted_error_rate = 51 / 100

def HawkinsGweonGoodman2021.empirical_unscripted_error_rate : ℚ

Equations

• HawkinsGweonGoodman2021.empirical_unscripted_error_rate = 20 / 100

theorem HawkinsGweonGoodman2021.model_predicts_informativity_reduces_errors : asymmetricInformativity fullDescription exampleTarget exampleVisible uniformHiddenPrior > asymmetricInformativity shapeOnly exampleTarget exampleVisible uniformHiddenPrior

Model correctly predicts that more informative speakers lead to fewer errors

def HawkinsGweonGoodman2021.informativity_error_correlation : ℚ

Informativity-error correlation from paper: ρ = -0.81

Equations

• HawkinsGweonGoodman2021.informativity_error_correlation = -81 / 100

theorem HawkinsGweonGoodman2021.model_direction_matches_correlation : asymmetricInformativity fullDescription exampleTarget exampleVisible uniformHiddenPrior > asymmetricInformativity shapeAndColor exampleTarget exampleVisible uniformHiddenPrior ∧ asymmetricInformativity shapeAndColor exampleTarget exampleVisible uniformHiddenPrior > asymmetricInformativity shapeOnly exampleTarget exampleVisible uniformHiddenPrior

Model Summary

Key model predictions verified as theorems:

1. more_specific_higher_asymmetric_informativity: More specific utterances have higher informativity when considering hidden objects

2. asymmetry_increases_specificity_gain: The asymmetric model predicts LARGER informativity gain from specificity than egocentric

3. full_description_preferred_at_wS1: At full perspective-taking, more specific utterances maximize listener success

4. shape_only_sufficient_at_wS0: At pure egocentric, minimal description is equally informative (target unique in shape)

5. listener_infers_low_wS_from_short_utterances: Listeners infer speaker's low effort from under-informative utterances

6. listener_compensates_for_low_speaker_effort: Optimal listener effort increases when speaker effort is low

7. semantics_grounded: Utterance semantics grounded in compositional (Montague) denotations