Lexicalization: Efficient Encoding of Emerging Concepts

@cite{xu-etal-2024}

Inaugural module of Theories/Diachronic/: formal theories of language change.

Xu et al. (2024) unify word reuse and combination (compounding) under a single information-theoretic account. Both strategies for encoding novel concepts are shaped by the same tradeoff between minimizing speaker effort (word length) and minimizing information loss (listener confusion). Attested encodings in English, French, and Finnish sit near the Pareto frontier of this tradeoff.

Architecture

The model extends RSA's speaker-listener framework to lexicon evolution:

• The listener uses prototype-based categorization (Eq. 1) — an L0 whose meaning function is exp(-γ · d(c, q_w)).
• The speaker trades off informativity against word-length cost — an S1 with beliefAction scoring where cost(w) = β · l(w).
• The encoding E* is the set of (concept, form) pairs for emerging concepts. Its efficiency is measured against the Pareto frontier of possible encodings (Core.Efficiency).

Connection to LCEC

The LCEC (Morphology.WP.LCEC) states that I-complexity (conditional entropy across paradigm cells) is uniformly low despite high E-complexity. The analog here: lexicons maintain low information loss despite high polysemy, because reuse and compounding are informationally efficient. Both are instances of a general principle: natural languages achieve low integrative complexity despite high enumerative complexity, via structured redundancy.

inductive Diachronic.Lexicalization.Strategy : Type

Strategy by which a novel concept enters the lexicon. @cite{xu-etal-2024} Table 1: reuse items (R) vs. compounds (C).

• reuse : Strategy

  Reuse an existing word for a new meaning. E.g., "mouse" (rodent → peripheral), "dish" (plate → antenna).

• combination : Strategy

  Combine existing words into a compound. E.g., "birthday card", "spreadsheet", "urban renewal".

instance Diachronic.Lexicalization.instDecidableEqStrategy : DecidableEq Strategy

Equations

• Diachronic.Lexicalization.instDecidableEqStrategy x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance Diachronic.Lexicalization.instReprStrategy : Repr Strategy

Equations

• Diachronic.Lexicalization.instReprStrategy = { reprPrec := Diachronic.Lexicalization.instReprStrategy.repr }

def Diachronic.Lexicalization.instReprStrategy.repr : Strategy → ℕ → Std.Format

Equations

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

instance Diachronic.Lexicalization.instBEqStrategy : BEq Strategy

Equations

• Diachronic.Lexicalization.instBEqStrategy = { beq := Diachronic.Lexicalization.instBEqStrategy.beq }

def Diachronic.Lexicalization.instBEqStrategy.beq : Strategy → Strategy → Bool

Equations

• Diachronic.Lexicalization.instBEqStrategy.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

inductive Diachronic.Lexicalization.Literality : Type

Literality of the form-meaning relationship. Literal items are semantically transparent and tend to be more communicatively efficient (@cite{xu-etal-2024} §Item-Level Variation).

• literal : Literality

  Literal: form directly relates to the intended concept.

  • Reuse: intended meaning is a hyponym of the existing sense.
  • Combination: endocentric compound (head = superordinate).
• nonliteral : Literality

  Nonliteral: metaphorical or metonymic relationship.

  • Reuse: e.g., "mouse" for computer peripheral.
  • Combination: exocentric, e.g., "boîte noire" = flight recorder.

instance Diachronic.Lexicalization.instDecidableEqLiterality : DecidableEq Literality

Equations

• Diachronic.Lexicalization.instDecidableEqLiterality x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance Diachronic.Lexicalization.instReprLiterality : Repr Literality

Equations

• Diachronic.Lexicalization.instReprLiterality = { reprPrec := Diachronic.Lexicalization.instReprLiterality.repr }

def Diachronic.Lexicalization.instReprLiterality.repr : Literality → ℕ → Std.Format

Equations

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

instance Diachronic.Lexicalization.instBEqLiterality : BEq Literality

Equations

• Diachronic.Lexicalization.instBEqLiterality = { beq := Diachronic.Lexicalization.instBEqLiterality.beq }

def Diachronic.Lexicalization.instBEqLiterality.beq : Literality → Literality → Bool

Equations

• Diachronic.Lexicalization.instBEqLiterality.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

structure Diachronic.Lexicalization.FormConceptPair : Type

A form-concept pair in an emerging encoding (one entry in E*).

• form : String
• concept : String
• strategy : Strategy
• literality : Literality
• formLength : ℕ

instance Diachronic.Lexicalization.instReprFormConceptPair : Repr FormConceptPair

Equations

• Diachronic.Lexicalization.instReprFormConceptPair = { reprPrec := Diachronic.Lexicalization.instReprFormConceptPair.repr }

def Diachronic.Lexicalization.instReprFormConceptPair.repr : FormConceptPair → ℕ → Std.Format

Equations

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

def Diachronic.Lexicalization.encodingCosts (pairs : List FormConceptPair) (needProb : String → ℚ) (listenerScore : String → String → ℚ) : Core.Efficiency.CostPair

Communicative costs of an encoding, parameterized by a listener model.

listenerScore concept form is the probability that the listener recovers concept from form. In @cite{xu-etal-2024}, this is the prototype-based categorization model (Eq. 1): m̂_{w,L}(c) ∝ exp{-γ · d(c, q_w)}.

Effort (Eq. 2) = expected word length. Information loss (Eq. 3) = expected surprisal under listener distribution. The aggregate is weighted by concept need probability.

Equations

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

def Diachronic.Lexicalization.unifiedObjective (pairs : List FormConceptPair) (needProb : String → ℚ) (listenerScore : String → String → ℚ) (β : ℚ) : ℚ

Unified objective (Eq. 5): L_β = info_loss + β · effort. Parameterizes the Pareto frontier.

Equations

• Diachronic.Lexicalization.unifiedObjective pairs needProb listenerScore β = Core.Efficiency.weightedCost (Diachronic.Lexicalization.encodingCosts pairs needProb listenerScore) β

def Diachronic.Lexicalization.asS1ScoreSpec (β : ℚ) (length : String → ℚ) : RSA.S1ScoreSpec String String Unit

The prototype-based listener IS an RSA L0, and the unified objective IS an S1 with beliefAction scoring.

To instantiate: set RSAConfigData with

• U := Form, W := Concept
• meaning _ c w := exp(-γ · d(c, prototype(w)))
• s1Spec := .beliefAction (fun w => β * length(w))

This function constructs the corresponding S1 scoring rule.

Equations

• Diachronic.Lexicalization.asS1ScoreSpec β length = RSA.S1ScoreSpec.beliefAction fun (w : String) => β * length w

def Diachronic.Lexicalization.moreEfficientThan (attested baseline : Core.Efficiency.CostPair) (optimalAt : ℚ → Core.Efficiency.CostPair) (βs : List ℚ) : Prop

Efficiency Claim (Figs. 2–3): attested encodings are closer to the Pareto frontier than baseline encodings (random or near-synonym).

Equations

• Diachronic.Lexicalization.moreEfficientThan attested baseline optimalAt βs = (Core.Efficiency.efficiencyLoss attested optimalAt βs < Core.Efficiency.efficiencyLoss baseline optimalAt βs)

def Diachronic.Lexicalization.strategyTradeoff (reuseCosts compoundCosts : Core.Efficiency.CostPair) : Prop

Strategy Tradeoff (§Strategy Comparison): reuse items tend shorter; compounds tend more informative. The two strategies occupy complementary regions of the effort-informativity space.

Equations

• Diachronic.Lexicalization.strategyTradeoff reuseCosts compoundCosts = (reuseCosts.cost₁ ≤ compoundCosts.cost₁ ∧ compoundCosts.cost₂ ≤ reuseCosts.cost₂)

def Diachronic.Lexicalization.literalAdvantage (literalCosts nonliteralCosts : Core.Efficiency.CostPair) (optimalAt : ℚ → Core.Efficiency.CostPair) (βs : List ℚ) : Prop

Literal Advantage (§Item-Level Variation): literal items (hyponymic reuse, endocentric compounds) are more efficient than nonliteral ones, because semantic transparency reduces information loss.

Equations

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