Lexicalization: Efficient Encoding of Emerging Concepts

@cite{xu-etal-2024}

Substrate for the information-theoretic account of lexicalization in @cite{xu-etal-2024}: novel concepts enter the lexicon either by reuse (an existing word picks up a new sense — mouse → computer peripheral) or by compounding (concatenation of existing words — spreadsheet). Both strategies are shaped by the same tradeoff between speaker effort (word length) and information loss (listener confusion).

Diachronic framing

This is a model of innovation spread under variation, not a synchronic optimization of a static lexicon. @cite{xu-etal-2024} §1–§2 ground in the variation-theory tradition (@cite{weinreich-labov-herzog-1968}, @cite{milroy-milroy-1985}, @cite{labov-2011}): there is a spread interval [t₁, t₂] during which only some members of the speech community have acquired the new encoding E*. The speaker's production policy is conditioned on the expanded lexicon L' (= L ∪ E*); the listener's interpretation is conditioned on the existing lexicon L. This asymmetry is the diachronic content — it lives at the type level via Pragmatics.Communication.AsymmetricCommModel (its produce and comprehend are independent functions that may disagree).

Cline-following diachronic processes (grammaticalization, subjectification) and constructionalization are out of scope here; see sibling files in Theories/Diachronic/. The contrast is type of diachronic process (cline-following vs. punctuated lexical innovation), not diachronic vs. synchronic.

Lexicalization strategies

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).

The paper notes that borrowing (e.g., tofu) and coinage (e.g., quark) are additional lexicalization strategies excluded from its scope; this enum mirrors that scope restriction.

• reuse : Strategy

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

• compound : Strategy

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

@[implicit_reducible]

instance Diachronic.Lexicalization.instDecidableEqStrategy : DecidableEq Strategy

Equations

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

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

Equations

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

@[implicit_reducible]

instance Diachronic.Lexicalization.instReprStrategy : Repr Strategy

Equations

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

inductive Diachronic.Lexicalization.Literality : Type

Literality of the form–meaning relationship. Literal items tend to be more communicatively efficient (@cite{xu-etal-2024} §3.2).

This binary distinction is a coarsening of the continuous taxonomic- distance measures the paper also tests (Wu-Palmer 1994; Leacock-Chodorow-Miller 1998), reported in paper §3.2 final paragraph + SI §S5.E as monotonically correlated with efficiency loss. The enum captures the headline literal/non-literal contrast; a continuous version would parameterize over a distance metric.

• literal : Literality

  Form directly relates to the intended concept.

  • Reuse: intended sense is a hyponym of an existing sense.
  • Compound: endocentric (head = superordinate of intended concept).
• nonliteral : Literality

  Metaphorical or metonymic relationship.

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

@[implicit_reducible]

instance Diachronic.Lexicalization.instDecidableEqLiterality : DecidableEq Literality

Equations

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

@[implicit_reducible]

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.

structure Diachronic.Lexicalization.FormConceptPair : Type

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

The concept field is a human-readable label. In @cite{xu-etal-2024} actual use, concepts are WordNet sense IDs embedded via Sentence-BERT (paper §5.3); two distinct senses can share a surface label, so serious instantiation needs disambiguating IDs. The string here is a presentation-layer convenience for example data.

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

@[implicit_reducible]

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.FormConceptPair.formLength (p : FormConceptPair) : ℕ

Orthographic form length, used as the speaker-effort proxy in @cite{xu-etal-2024} (paper eq. 2).

Equations

• p.formLength = p.form.length

Communicative costs

def Diachronic.Lexicalization.surprisalCap : ℝ

Per-pair surprisal cap used when model.comprehend returns a non-positive value. The paper's softmax model never produces zero listener probability, so this bound is for numeric robustness only. Default is 20 nats ≈ 28.8 bits, comfortably above attested typical information loss of ~10–15 bits (@cite{xu-etal-2024} Fig. 2 axes; paper uses log₂, this file uses natural log).

Equations

• Diachronic.Lexicalization.surprisalCap = 20

noncomputable def Diachronic.Lexicalization.encodingCosts (pairs : List FormConceptPair) (needProb : String → ℝ) (model : Pragmatics.Communication.AsymmetricCommModel String String) : Pragmatics.Efficiency.CostPair

Communicative costs of an encoding under an asymmetric communication model. Cost₂ uses model.comprehend (the listener-side channel conditioned on the existing lexicon L), reflecting the diachronic asymmetry @cite{xu-etal-2024} introduces. The speaker-side produce channel is not consumed in the deterministic-policy case (see below) but lives in the same model so future non-deterministic versions can read it.

cost₁ (paper eq. 2): expected word length under needProb. cost₂ (paper eq. 3): expected surprisal under the listener distribution. The unweighted sum cost₂ + β · cost₁ recovers L_β (paper eq. 4); the per-pair, proportional rearrangement is paper eq. 5. We use natural log throughout, so cost₂ is in nats; multiply by 1 / Real.log 2 for the paper's bits convention.

Deterministic-policy assumption. The signature takes needProb : String → ℝ (a concept-only marginal) rather than a joint p(c, w | L'). This silently assumes a deterministic production policy — one form per concept in pairs. Paper §5.2 estimates p(c, w | L') = p(w | L') · p(c | w, L') separately for each language; under the assumption that each emerging form-sense pair appears with multiplicity 1 in E*, the joint reduces to needProb p.concept and these costs are exact. With non-deterministic encodings (multiple forms competing for one concept), use the joint instead and marginalize.

Equations

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

noncomputable def Diachronic.Lexicalization.unifiedObjective (pairs : List FormConceptPair) (needProb : String → ℝ) (model : Pragmatics.Communication.AsymmetricCommModel String String) (β : ℝ) : ℝ

Combined cost (paper eq. 4): L_β = info_loss + β · effort. The named hook the Studies layer invokes to ask for a particular β-scalarization of an encoding's CostPair. Parameterizes the Pareto frontier in Pragmatics.Efficiency.

Equations

• Diachronic.Lexicalization.unifiedObjective pairs needProb model β = Pragmatics.Efficiency.weightedCost (Diachronic.Lexicalization.encodingCosts pairs needProb model) β