Documentation

Linglib.Studies.ZaslavskyKempRegierTishby2018

Zaslavsky, Kemp, Regier & Tishby (2018): efficient compression in color naming #

[ZKRT18] [BK69] [DH13b]

[ZKRT18] argue that color-naming systems efficiently compress meanings into words by optimizing the Information Bottleneck (IB) trade-off between lexicon complexity (the information rate I(M;W)) and accuracy (I(W;U)). Cross-language variation is captured by a single trade-off parameter β, and the Berlin & Kay evolutionary sequence ([BK69]: dark/light, then red, then green/yellow, then blue, …) falls out as motion up the complexity axis — successive systems carve color space more finely, paying complexity for accuracy.

This file formalizes the per-language WALS color profiles (the eight WALS-sourced ColorProfiles below, derived via ColorProfile.fromWALS) observable through the efficient-communication framework in Pragmatics.Efficiency (CostPair, weightedCost = cost₂ + β·cost₁, efficiencyLossAt).

What is derived vs. cited #

Derived (from the WALS-sourced color profiles): the Berlin-Kay complexity coordinate of each sampled language — the number of basic color categories (WALS Ch 133). The IB complexity I(M;W) of a deterministic K-word system is bounded by log K, so the category count is a monotone handle on the complexity axis (not I(M;W) itself).
Bridge (via Pragmatics.Efficiency): the β-scalarized IB objective weightedCost is monotone in this complexity coordinate, so the Berlin-Kay category ordering is exactly an ordering on the IB complexity axis.
Cited stimulus (from the paper, not formalized): real languages are near-optimal — they lie close to the IB curve with small efficiency loss ε_l, at a fitted β_l ≳ 1 (e.g. English β_l ≈ 1.085, Fig. 4). The formal anchor here is only the idealized optimum (zero loss at coincidence); the empirical near-optimality is the paper's measured result.

Main results #

bk_complexity_strictMono: the category-count handle is strictly monotone in the WALS Ch 133 ordering (the Berlin-Kay sequence).
weightedCost_mono_in_complexity: more categories ⇒ higher β-scalarized IB complexity cost, for every β ≥ 0 — the structural bridge.
sample_all_warm_and_cool_split: every sampled language distinguishes both red/yellow and green/blue (all are high-complexity, late-sequence systems).

WALS Ch 132: Number of non-derived basic color categories #

inductive ZaslavskyKempRegierTishby2018.NonDerivedColorCount :

Number of non-derived basic color categories (WALS Ch 132, [KM13c]). Ranges from 3 to 6 along the Berlin & Kay sequence; transitional half-values represent languages with one composite category undergoing splitting.

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@[implicit_reducible]

instance ZaslavskyKempRegierTishby2018.instDecidableEqNonDerivedColorCount :

DecidableEq NonDerivedColorCount

Equations

ZaslavskyKempRegierTishby2018.instDecidableEqNonDerivedColorCount x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance ZaslavskyKempRegierTishby2018.instReprNonDerivedColorCount :

Repr NonDerivedColorCount

Equations

ZaslavskyKempRegierTishby2018.instReprNonDerivedColorCount = { reprPrec := ZaslavskyKempRegierTishby2018.instReprNonDerivedColorCount.repr }

def ZaslavskyKempRegierTishby2018.instReprNonDerivedColorCount.repr :

NonDerivedColorCount → ℕ → Std.Format

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

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WALS Ch 133: Total number of basic color categories #

inductive ZaslavskyKempRegierTishby2018.BasicColorCount :

Total number of basic color categories including derived ones (WALS Ch 133, [KM13b]). Ranges from 3–4 (minimal systems) to the top bucket, WALS's "more than 10" — canonically 11 basic terms, the Berlin & Kay Stage-VII maximum (e.g., English, Russian).

Instances For

@[implicit_reducible]

instance ZaslavskyKempRegierTishby2018.instDecidableEqBasicColorCount :

DecidableEq BasicColorCount

Equations

ZaslavskyKempRegierTishby2018.instDecidableEqBasicColorCount x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def ZaslavskyKempRegierTishby2018.instReprBasicColorCount.repr :

BasicColorCount → ℕ → Std.Format

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

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@[implicit_reducible]

instance ZaslavskyKempRegierTishby2018.instReprBasicColorCount :

Repr BasicColorCount

Equations

ZaslavskyKempRegierTishby2018.instReprBasicColorCount = { reprPrec := ZaslavskyKempRegierTishby2018.instReprBasicColorCount.repr }

WALS Ch 134: Green and blue #

inductive ZaslavskyKempRegierTishby2018.GreenBlueRelation :

How a language treats the green-blue region of color space (WALS Ch 134, [KM13a]). The classic grue / green-blue composite distinction, with several other composite patterns (with black, with yellow).

distinct : GreenBlueRelation
Separate terms for green and blue.
merged : GreenBlueRelation
A single grue term covering both green and blue.
blackGreenBlue : GreenBlueRelation
A single term covering black, green, and blue.
blackBlueVsGreen : GreenBlueRelation
Black/blue merged, green separate.
yellowGreenBlue : GreenBlueRelation
Yellow, green, blue all merged.
yellowGreenVsBlue : GreenBlueRelation
Yellow/green merged, blue separate.
noTerm : GreenBlueRelation
No green or blue term at all.

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@[implicit_reducible]

instance ZaslavskyKempRegierTishby2018.instDecidableEqGreenBlueRelation :

DecidableEq GreenBlueRelation

Equations

ZaslavskyKempRegierTishby2018.instDecidableEqGreenBlueRelation x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance ZaslavskyKempRegierTishby2018.instReprGreenBlueRelation :

Repr GreenBlueRelation

Equations

ZaslavskyKempRegierTishby2018.instReprGreenBlueRelation = { reprPrec := ZaslavskyKempRegierTishby2018.instReprGreenBlueRelation.repr }

def ZaslavskyKempRegierTishby2018.instReprGreenBlueRelation.repr :

GreenBlueRelation → ℕ → Std.Format

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

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WALS Ch 135: Red and yellow #

inductive ZaslavskyKempRegierTishby2018.RedYellowRelation :

How a language treats the red-yellow region of color space (WALS Ch 135, [KM13d]).

distinct : RedYellowRelation
Separate terms for red and yellow.
merged : RedYellowRelation
A single term covering both red and yellow.
yellowGreenBlueVsRed : RedYellowRelation
Yellow/green/blue merged, vs red.
yellowGreenVsRed : RedYellowRelation
Yellow/green merged, vs red.
noTerm : RedYellowRelation
No red or yellow term at all.

Instances For

@[implicit_reducible]

instance ZaslavskyKempRegierTishby2018.instDecidableEqRedYellowRelation :

DecidableEq RedYellowRelation

Equations

ZaslavskyKempRegierTishby2018.instDecidableEqRedYellowRelation x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def ZaslavskyKempRegierTishby2018.instReprRedYellowRelation.repr :

RedYellowRelation → ℕ → Std.Format

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

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@[implicit_reducible]

instance ZaslavskyKempRegierTishby2018.instReprRedYellowRelation :

Repr RedYellowRelation

Equations

ZaslavskyKempRegierTishby2018.instReprRedYellowRelation = { reprPrec := ZaslavskyKempRegierTishby2018.instReprRedYellowRelation.repr }

Per-language profile #

structure ZaslavskyKempRegierTishby2018.ColorProfile :

A language's color-naming profile across [DH13b] Chs 132–135. Coverage is sparse (~120 languages); fields are optional.

language : String
iso : String
family : String
nonDerived : Option NonDerivedColorCount
Ch 132: non-derived basic color categories.
basic : Option BasicColorCount
Ch 133: total basic color categories.
greenBlue : Option GreenBlueRelation
Ch 134: green-blue relation.
redYellow : Option RedYellowRelation
Ch 135: red-yellow relation.

Instances For

@[implicit_reducible]

instance ZaslavskyKempRegierTishby2018.instReprColorProfile :

Repr ColorProfile

Equations

ZaslavskyKempRegierTishby2018.instReprColorProfile = { reprPrec := ZaslavskyKempRegierTishby2018.instReprColorProfile.repr }

def ZaslavskyKempRegierTishby2018.instReprColorProfile.repr :

ColorProfile → ℕ → Std.Format

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

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WALS converters #

def ZaslavskyKempRegierTishby2018.fromWALS132A :

Data.WALS.F132A.NumberOfNonDerivedBasicColourCategories → NonDerivedColorCount

Convert WALS 132A non-derived-color-count values into the substrate enum.

Equations

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def ZaslavskyKempRegierTishby2018.fromWALS133A :

Data.WALS.F133A.NumberOfBasicColourCategories → BasicColorCount

Convert WALS 133A basic-color-count values into the substrate enum.

Equations

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def ZaslavskyKempRegierTishby2018.fromWALS134A :

Data.WALS.F134A.GreenAndBlue → GreenBlueRelation

Convert WALS 134A green-blue values into the substrate enum.

Equations

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def ZaslavskyKempRegierTishby2018.fromWALS135A :

Data.WALS.F135A.RedAndYellow → RedYellowRelation

Convert WALS 135A red-yellow values into the substrate enum.

Equations

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def ZaslavskyKempRegierTishby2018.ColorProfile.fromWALS (language iso family : String) :

Build a ColorProfile from the WALS Chs 132–135 rows for an ISO 639-3 code, mapping each chapter's datapoint through its converter; a field for which WALS has no row is none. Makes the per-language Fragment profiles true-by-construction from the auto-generated WALS tables rather than hand-transcribed literals.

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The eight WALS-sourced sample profiles #

def ZaslavskyKempRegierTishby2018.english :

Equations

ZaslavskyKempRegierTishby2018.english = ZaslavskyKempRegierTishby2018.ColorProfile.fromWALS "English" "eng" "Indo-European"

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def ZaslavskyKempRegierTishby2018.french :

Equations

ZaslavskyKempRegierTishby2018.french = ZaslavskyKempRegierTishby2018.ColorProfile.fromWALS "French" "fra" "Indo-European"

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def ZaslavskyKempRegierTishby2018.german :

Equations

ZaslavskyKempRegierTishby2018.german = ZaslavskyKempRegierTishby2018.ColorProfile.fromWALS "German" "deu" "Indo-European"

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def ZaslavskyKempRegierTishby2018.japanese :

Equations

ZaslavskyKempRegierTishby2018.japanese = ZaslavskyKempRegierTishby2018.ColorProfile.fromWALS "Japanese" "jpn" "Japonic"

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def ZaslavskyKempRegierTishby2018.korean :

Equations

ZaslavskyKempRegierTishby2018.korean = ZaslavskyKempRegierTishby2018.ColorProfile.fromWALS "Korean" "kor" "Koreanic"

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def ZaslavskyKempRegierTishby2018.mandarin :

Equations

ZaslavskyKempRegierTishby2018.mandarin = ZaslavskyKempRegierTishby2018.ColorProfile.fromWALS "Mandarin Chinese" "cmn" "Sino-Tibetan"

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def ZaslavskyKempRegierTishby2018.russian :

Equations

ZaslavskyKempRegierTishby2018.russian = ZaslavskyKempRegierTishby2018.ColorProfile.fromWALS "Russian" "rus" "Indo-European"

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def ZaslavskyKempRegierTishby2018.spanish :

Equations

ZaslavskyKempRegierTishby2018.spanish = ZaslavskyKempRegierTishby2018.ColorProfile.fromWALS "Spanish" "spa" "Indo-European"

Instances For

The Berlin-Kay complexity coordinate #

def ZaslavskyKempRegierTishby2018.basicCount :

BasicColorCount → ℕ

A representative basic-category count for each WALS Ch 133 bucket (its lower bound). Only the ordering matters: this is a monotone handle on the IB complexity axis I(M;W) ≤ log K, where K is the number of color words.

Equations

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def ZaslavskyKempRegierTishby2018.ibComplexity (p : ColorProfile) :

ℕ

IB complexity handle for a color profile: its basic-category count (0 when WALS records no Ch 133 datum).

Equations

ZaslavskyKempRegierTishby2018.ibComplexity p = (Option.map ZaslavskyKempRegierTishby2018.basicCount p.basic).getD 0

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The category-count handle is strictly monotone along the WALS Ch 133 ordering — i.e. along the Berlin & Kay evolutionary sequence ([BK69]).

Bridge to the Information-Bottleneck objective #

def ZaslavskyKempRegierTishby2018.ibCost (p : ColorProfile) (acc : ℝ) :

Pragmatics.Efficiency.CostPair

A color system as a Pragmatics.Efficiency.CostPair: cost₁ is the IB complexity handle (category count), cost₂ is the system's accuracy/ distortion component, left abstract (WALS does not record it per language).

Equations

ZaslavskyKempRegierTishby2018.ibCost p acc = { cost₁ := ↑(ZaslavskyKempRegierTishby2018.ibComplexity p), cost₂ := acc }

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theorem ZaslavskyKempRegierTishby2018.weightedCost_mono_in_complexity {p q : ColorProfile} (acc β : ℝ) (hβ : 0 ≤ β) (h : ibComplexity p ≤ ibComplexity q) :

Pragmatics.Efficiency.weightedCost (ibCost p acc) β ≤ Pragmatics.Efficiency.weightedCost (ibCost q acc) β

Structural bridge. Under the β-scalarized IB objective weightedCost, a system with more basic categories has at least as high a complexity cost, for every β ≥ 0 and any fixed accuracy. The Berlin-Kay category ordering is therefore an ordering on the IB complexity axis the paper plots.

theorem ZaslavskyKempRegierTishby2018.optimal_system_zero_loss (c : Pragmatics.Efficiency.CostPair) (β : ℝ) :

Pragmatics.Efficiency.efficiencyLossAt c c β = 0

The idealized anchor of the paper's near-optimality finding: a color system that coincides with the IB-optimal system at its fitted β has zero efficiency loss. Real languages are near-optimal (small nonzero ε_l), which is the paper's measured result rather than a theorem here.

The Fragment sample #

def ZaslavskyKempRegierTishby2018.sample :

List ColorProfile

The eight WALS-sourced color profiles formalized as Fragments. All are industrialized-language systems near the top of the Berlin & Kay sequence.

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

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theorem ZaslavskyKempRegierTishby2018.sample_all_warm_and_cool_split (p : ColorProfile) :

p ∈ sample → p.redYellow = some RedYellowRelation.distinct ∧ p.greenBlue = some GreenBlueRelation.distinct

Every sampled language draws both the warm (red/yellow) and cool (green/blue) boundaries — all are high-complexity, late-sequence systems, consistent with their high category counts and fitted β_l > 1.

theorem ZaslavskyKempRegierTishby2018.english_more_complex_than_mandarin :

ibComplexity mandarin < ibComplexity english

Concrete complexity contrast: English (11 basic terms) sits higher on the IB complexity axis than Mandarin (8–8.5), as the Berlin-Kay sequence predicts.

theorem ZaslavskyKempRegierTishby2018.english_weightedCost_ge_mandarin (acc β : ℝ) (hβ : 0 ≤ β) :

Pragmatics.Efficiency.weightedCost (ibCost mandarin acc) β ≤ Pragmatics.Efficiency.weightedCost (ibCost english acc) β

The contrast lifts to the IB objective: for every β ≥ 0 (and any fixed accuracy), English's β-scalarized complexity cost is at least Mandarin's.