RSA Embedded Scalar Implicatures: Simplified Model (For Analysis)

@cite{bergen-levy-goodman-2016} @cite{geurts-2010} @cite{potts-etal-2016}

This file implements a simplified 2-lexicon model to analyze why minimal Lexical Uncertainty models fail to derive embedded implicature patterns.

Status

The ℚ-based RSA evaluation infrastructure (RSA.Eval, boolToRat, LURSA) has been removed. Type definitions and the model limitation analysis are preserved. RSA computations need to be re-implemented using the new RSAConfig framework.

This File's Purpose

Demonstrates that a minimal 2-lexicon, 3-world model gives inverted predictions, motivating the richer structure in the full model.

inductive RSA.EmbeddedScalars.EmbeddedWorld : Type

World states for embedded scalar scenarios.

• none: Nobody solved any problems
• someNotAll: Someone solved some-but-not-all problems
• someAll: Someone solved all problems

• none : EmbeddedWorld
• someNotAll : EmbeddedWorld
• someAll : EmbeddedWorld

instance RSA.EmbeddedScalars.instDecidableEqEmbeddedWorld : DecidableEq EmbeddedWorld

Equations

• RSA.EmbeddedScalars.instDecidableEqEmbeddedWorld x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def RSA.EmbeddedScalars.instReprEmbeddedWorld.repr : EmbeddedWorld → ℕ → Std.Format

Equations

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

instance RSA.EmbeddedScalars.instReprEmbeddedWorld : Repr EmbeddedWorld

Equations

• RSA.EmbeddedScalars.instReprEmbeddedWorld = { reprPrec := RSA.EmbeddedScalars.instReprEmbeddedWorld.repr }

instance RSA.EmbeddedScalars.instBEqEmbeddedWorld : BEq EmbeddedWorld

Equations

• RSA.EmbeddedScalars.instBEqEmbeddedWorld = { beq := RSA.EmbeddedScalars.instBEqEmbeddedWorld.beq }

def RSA.EmbeddedScalars.instBEqEmbeddedWorld.beq : EmbeddedWorld → EmbeddedWorld → Bool

Equations

• RSA.EmbeddedScalars.instBEqEmbeddedWorld.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance RSA.EmbeddedScalars.instInhabitedEmbeddedWorld : Inhabited EmbeddedWorld

Equations

• RSA.EmbeddedScalars.instInhabitedEmbeddedWorld = { default := RSA.EmbeddedScalars.instInhabitedEmbeddedWorld.default }

def RSA.EmbeddedScalars.instInhabitedEmbeddedWorld.default : EmbeddedWorld

Equations

• RSA.EmbeddedScalars.instInhabitedEmbeddedWorld.default = RSA.EmbeddedScalars.EmbeddedWorld.none

inductive RSA.EmbeddedScalars.DEUtterance : Type

Utterances for DE context: "No one solved {some/all} problems"

We need scalar alternatives for RSA to reason about informativity.

• noSome : DEUtterance
• noAll : DEUtterance
• null : DEUtterance

instance RSA.EmbeddedScalars.instDecidableEqDEUtterance : DecidableEq DEUtterance

Equations

• RSA.EmbeddedScalars.instDecidableEqDEUtterance x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance RSA.EmbeddedScalars.instReprDEUtterance : Repr DEUtterance

Equations

• RSA.EmbeddedScalars.instReprDEUtterance = { reprPrec := RSA.EmbeddedScalars.instReprDEUtterance.repr }

def RSA.EmbeddedScalars.instReprDEUtterance.repr : DEUtterance → ℕ → Std.Format

Equations

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

instance RSA.EmbeddedScalars.instBEqDEUtterance : BEq DEUtterance

Equations

• RSA.EmbeddedScalars.instBEqDEUtterance = { beq := RSA.EmbeddedScalars.instBEqDEUtterance.beq }

def RSA.EmbeddedScalars.instBEqDEUtterance.beq : DEUtterance → DEUtterance → Bool

Equations

• RSA.EmbeddedScalars.instBEqDEUtterance.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

def RSA.EmbeddedScalars.instInhabitedDEUtterance.default : DEUtterance

Equations

• RSA.EmbeddedScalars.instInhabitedDEUtterance.default = RSA.EmbeddedScalars.DEUtterance.noSome

instance RSA.EmbeddedScalars.instInhabitedDEUtterance : Inhabited DEUtterance

Equations

• RSA.EmbeddedScalars.instInhabitedDEUtterance = { default := RSA.EmbeddedScalars.instInhabitedDEUtterance.default }

The Lexical Uncertainty Model

Each lexicon L assigns meanings to "some":

• L_base: "some" = at-least-one (literal)
• L_refined: "some" = some-but-not-all (Neo-Gricean strengthened)

The listener reasons over which lexicon the speaker is using.

def RSA.EmbeddedScalars.lexBaseMeaning : DEUtterance → EmbeddedWorld → Bool

Base lexicon meaning: "some" = at-least-one

"No one solved some problems" under L_base:

• True only when nobody solved any problems

Equations

• RSA.EmbeddedScalars.lexBaseMeaning RSA.EmbeddedScalars.DEUtterance.noSome RSA.EmbeddedScalars.EmbeddedWorld.none = true
• RSA.EmbeddedScalars.lexBaseMeaning RSA.EmbeddedScalars.DEUtterance.noSome RSA.EmbeddedScalars.EmbeddedWorld.someNotAll = false
• RSA.EmbeddedScalars.lexBaseMeaning RSA.EmbeddedScalars.DEUtterance.noSome RSA.EmbeddedScalars.EmbeddedWorld.someAll = false
• RSA.EmbeddedScalars.lexBaseMeaning RSA.EmbeddedScalars.DEUtterance.noAll RSA.EmbeddedScalars.EmbeddedWorld.none = true
• RSA.EmbeddedScalars.lexBaseMeaning RSA.EmbeddedScalars.DEUtterance.noAll RSA.EmbeddedScalars.EmbeddedWorld.someNotAll = true
• RSA.EmbeddedScalars.lexBaseMeaning RSA.EmbeddedScalars.DEUtterance.noAll RSA.EmbeddedScalars.EmbeddedWorld.someAll = false
• RSA.EmbeddedScalars.lexBaseMeaning RSA.EmbeddedScalars.DEUtterance.null x✝ = true

def RSA.EmbeddedScalars.lexRefinedMeaning : DEUtterance → EmbeddedWorld → Bool

Refined lexicon meaning: "some" = some-but-not-all

"No one solved some problems" under L_refined:

• True when nobody solved "some-but-not-all"
• This is TRUE when someone solved ALL (they didn't solve "some-but-not-all")!

Equations

• RSA.EmbeddedScalars.lexRefinedMeaning RSA.EmbeddedScalars.DEUtterance.noSome RSA.EmbeddedScalars.EmbeddedWorld.none = true
• RSA.EmbeddedScalars.lexRefinedMeaning RSA.EmbeddedScalars.DEUtterance.noSome RSA.EmbeddedScalars.EmbeddedWorld.someNotAll = false
• RSA.EmbeddedScalars.lexRefinedMeaning RSA.EmbeddedScalars.DEUtterance.noSome RSA.EmbeddedScalars.EmbeddedWorld.someAll = true
• RSA.EmbeddedScalars.lexRefinedMeaning RSA.EmbeddedScalars.DEUtterance.noAll RSA.EmbeddedScalars.EmbeddedWorld.none = true
• RSA.EmbeddedScalars.lexRefinedMeaning RSA.EmbeddedScalars.DEUtterance.noAll RSA.EmbeddedScalars.EmbeddedWorld.someNotAll = true
• RSA.EmbeddedScalars.lexRefinedMeaning RSA.EmbeddedScalars.DEUtterance.noAll RSA.EmbeddedScalars.EmbeddedWorld.someAll = false
• RSA.EmbeddedScalars.lexRefinedMeaning RSA.EmbeddedScalars.DEUtterance.null x✝ = true

inductive RSA.EmbeddedScalars.UEUtterance : Type

Utterances for UE context

• someSome : UEUtterance
• someAll : UEUtterance
• null : UEUtterance

instance RSA.EmbeddedScalars.instDecidableEqUEUtterance : DecidableEq UEUtterance

Equations

• RSA.EmbeddedScalars.instDecidableEqUEUtterance x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance RSA.EmbeddedScalars.instReprUEUtterance : Repr UEUtterance

Equations

• RSA.EmbeddedScalars.instReprUEUtterance = { reprPrec := RSA.EmbeddedScalars.instReprUEUtterance.repr }

def RSA.EmbeddedScalars.instReprUEUtterance.repr : UEUtterance → ℕ → Std.Format

Equations

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

instance RSA.EmbeddedScalars.instBEqUEUtterance : BEq UEUtterance

Equations

• RSA.EmbeddedScalars.instBEqUEUtterance = { beq := RSA.EmbeddedScalars.instBEqUEUtterance.beq }

def RSA.EmbeddedScalars.instBEqUEUtterance.beq : UEUtterance → UEUtterance → Bool

Equations

• RSA.EmbeddedScalars.instBEqUEUtterance.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance RSA.EmbeddedScalars.instInhabitedUEUtterance : Inhabited UEUtterance

Equations

• RSA.EmbeddedScalars.instInhabitedUEUtterance = { default := RSA.EmbeddedScalars.instInhabitedUEUtterance.default }

def RSA.EmbeddedScalars.instInhabitedUEUtterance.default : UEUtterance

Equations

• RSA.EmbeddedScalars.instInhabitedUEUtterance.default = RSA.EmbeddedScalars.UEUtterance.someSome

def RSA.EmbeddedScalars.lexBaseUEMeaning : UEUtterance → EmbeddedWorld → Bool

Base lexicon meaning for UE: "some" = at-least-one

Equations

• RSA.EmbeddedScalars.lexBaseUEMeaning RSA.EmbeddedScalars.UEUtterance.someSome RSA.EmbeddedScalars.EmbeddedWorld.none = false
• RSA.EmbeddedScalars.lexBaseUEMeaning RSA.EmbeddedScalars.UEUtterance.someSome RSA.EmbeddedScalars.EmbeddedWorld.someNotAll = true
• RSA.EmbeddedScalars.lexBaseUEMeaning RSA.EmbeddedScalars.UEUtterance.someSome RSA.EmbeddedScalars.EmbeddedWorld.someAll = true
• RSA.EmbeddedScalars.lexBaseUEMeaning RSA.EmbeddedScalars.UEUtterance.someAll RSA.EmbeddedScalars.EmbeddedWorld.none = false
• RSA.EmbeddedScalars.lexBaseUEMeaning RSA.EmbeddedScalars.UEUtterance.someAll RSA.EmbeddedScalars.EmbeddedWorld.someNotAll = false
• RSA.EmbeddedScalars.lexBaseUEMeaning RSA.EmbeddedScalars.UEUtterance.someAll RSA.EmbeddedScalars.EmbeddedWorld.someAll = true
• RSA.EmbeddedScalars.lexBaseUEMeaning RSA.EmbeddedScalars.UEUtterance.null x✝ = true

def RSA.EmbeddedScalars.lexRefinedUEMeaning : UEUtterance → EmbeddedWorld → Bool

Refined lexicon meaning for UE: "some" = some-but-not-all

Equations

• RSA.EmbeddedScalars.lexRefinedUEMeaning RSA.EmbeddedScalars.UEUtterance.someSome RSA.EmbeddedScalars.EmbeddedWorld.none = false
• RSA.EmbeddedScalars.lexRefinedUEMeaning RSA.EmbeddedScalars.UEUtterance.someSome RSA.EmbeddedScalars.EmbeddedWorld.someNotAll = true
• RSA.EmbeddedScalars.lexRefinedUEMeaning RSA.EmbeddedScalars.UEUtterance.someSome RSA.EmbeddedScalars.EmbeddedWorld.someAll = false
• RSA.EmbeddedScalars.lexRefinedUEMeaning RSA.EmbeddedScalars.UEUtterance.someAll RSA.EmbeddedScalars.EmbeddedWorld.none = false
• RSA.EmbeddedScalars.lexRefinedUEMeaning RSA.EmbeddedScalars.UEUtterance.someAll RSA.EmbeddedScalars.EmbeddedWorld.someNotAll = false
• RSA.EmbeddedScalars.lexRefinedUEMeaning RSA.EmbeddedScalars.UEUtterance.someAll RSA.EmbeddedScalars.EmbeddedWorld.someAll = true
• RSA.EmbeddedScalars.lexRefinedUEMeaning RSA.EmbeddedScalars.UEUtterance.null x✝ = true

Analysis of Results

With α = 1 and uniform priors, the simplified 2-lexicon model gives INVERTED predictions compared to the empirical pattern. This motivates the need for richer model structure.

DE Context ("No one solved some"):

• The simple model predicts L_refined (local) wins -- WRONG

UE Context ("Someone solved some"):

• The simple model predicts L_base (global) wins -- WRONG

Why This Happens

The key asymmetry is world coverage:

In DE:

• L_base: noSome true in {none} -- 1 world
• L_refined: noSome true in {none, someAll} -- 2 worlds

L_refined makes the utterance true in MORE worlds, so even though L_base is more informative, L_refined gets extra probability mass.

What Potts et al. Actually Does

The paper succeeds because of richer model structure:

1. Multiple refinable items: Not just "some", but also proper names, predicates like "scored" vs "aced" (equation 14)

2. Richer world space: 3 players × 3 outcomes = 10 equivalence classes

3. Message alternatives: Full cross-product of quantifiers and predicates

4. Low λ = 0.1: Speaker nearly uniform, so implicatures emerge from lexicon structure, not informativity