Word Reuse and Combination Support Efficient Communication

[XKFX24]

Xu, A., Kemp, C., Frermann, L., & Xu, Y. (2024). Word reuse and combination support efficient communication of emerging concepts. PNAS 121(46), e2406971121.

Empirical contributions

Using WordNet data from English, French, and Finnish (1900–2000):

1. Both reuse items and compounds sit near the Pareto frontier of communicative efficiency (Fig. 2).
2. Attested encodings are more efficient than random and near-synonym baselines (Fig. 3).
3. Literal items (hyponymic reuse, endocentric compounds) tend to be more efficient than nonliteral counterparts (paper §3.2; significant for French and Finnish reuse, with English reuse supplemented by compound head words because WordNet does not directly classify English-reuse literality).
4. Reuse items tend shorter than compounds across all three languages; compounds tend more informative than reuse items in English and French only (paper §3.3 — Finnish does not show the informativeness asymmetry).

Connection to polysemy

Word reuse is a polysemy-generating process: when mouse acquires the sense "computer peripheral", the word becomes polysemous. This study provides an information-theoretic account of why productive polysemy exists — it is communicatively efficient under a tradeoff between length and listener confusion. Bridges synchronic copredication judgments ([Ash11], [Got17]) to a diachronic functional account.

§0. Lexicalization substrate

The information-theoretic model of lexicalization: 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).

This is a model of innovation spread under variation, not a synchronic optimization of a static lexicon. [XKFX24] §1–§2 ground in the variation-theory tradition ([weinreich-labov-herzog-1968], [milroy-milroy-1985], [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).

inductive XuEtAl2024.Polysemy.Strategy : Type

Strategy by which a novel concept enters the lexicon ([XKFX24] 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 XuEtAl2024.Polysemy.instDecidableEqStrategy : DecidableEq Strategy

Equations

• XuEtAl2024.Polysemy.instDecidableEqStrategy x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance XuEtAl2024.Polysemy.instReprStrategy : Repr Strategy

Equations

• XuEtAl2024.Polysemy.instReprStrategy = { reprPrec := XuEtAl2024.Polysemy.instReprStrategy.repr }

def XuEtAl2024.Polysemy.instReprStrategy.repr : Strategy → ℕ → Std.Format

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inductive XuEtAl2024.Polysemy.Literality : Type

Literality of the form–meaning relationship. Literal items tend to be more communicatively efficient ([XKFX24] §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 XuEtAl2024.Polysemy.instDecidableEqLiterality : DecidableEq Literality

Equations

• XuEtAl2024.Polysemy.instDecidableEqLiterality x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def XuEtAl2024.Polysemy.instReprLiterality.repr : Literality → ℕ → Std.Format

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@[implicit_reducible]

instance XuEtAl2024.Polysemy.instReprLiterality : Repr Literality

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• XuEtAl2024.Polysemy.instReprLiterality = { reprPrec := XuEtAl2024.Polysemy.instReprLiterality.repr }

structure XuEtAl2024.Polysemy.FormConceptPair : Type

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

The concept field is a human-readable label. In [XKFX24] 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 XuEtAl2024.Polysemy.instReprFormConceptPair : Repr FormConceptPair

Equations

• XuEtAl2024.Polysemy.instReprFormConceptPair = { reprPrec := XuEtAl2024.Polysemy.instReprFormConceptPair.repr }

def XuEtAl2024.Polysemy.instReprFormConceptPair.repr : FormConceptPair → ℕ → Std.Format

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def XuEtAl2024.Polysemy.FormConceptPair.formLength (p : FormConceptPair) : ℕ

Orthographic form length, used as the speaker-effort proxy in [XKFX24] (paper eq. 2).

Equations

• p.formLength = p.form.length

def XuEtAl2024.Polysemy.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 ([XKFX24] Fig. 2 axes; paper uses log₂, this file uses natural log).

Equations

• XuEtAl2024.Polysemy.surprisalCap = 20

noncomputable def XuEtAl2024.Polysemy.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 [XKFX24] 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.

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noncomputable def XuEtAl2024.Polysemy.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 β-scalarization of an encoding's CostPair; parameterizes the Pareto frontier in Pragmatics.Efficiency.

Equations

• XuEtAl2024.Polysemy.unifiedObjective pairs needProb model β = Pragmatics.Efficiency.weightedCost (XuEtAl2024.Polysemy.encodingCosts pairs needProb model) β

§1. Example data (Table 1)

def XuEtAl2024.Polysemy.englishReuse : List FormConceptPair

English reuse items from paper Table 1.

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def XuEtAl2024.Polysemy.englishCompounds : List FormConceptPair

English compounds from paper Table 1.

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def XuEtAl2024.Polysemy.frenchReuse : List FormConceptPair

French reuse items.

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def XuEtAl2024.Polysemy.frenchCompounds : List FormConceptPair

French compounds.

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§2. Strategy properties (verified on example data)

The full Pareto-efficiency claims (Figs. 2–3) depend on a fitted sentence-encoder embedding for the listener's prototype distribution (paper §5.3) and 100,000 random/near-synonym baseline encodings per language–interval cell (paper §5.5); they are not reduced to decide-checkable form here. The claims that ARE decide-checkable on the Table-1 examples are about word length — the speaker-effort axis.

theorem XuEtAl2024.Polysemy.english_reuse_shorter : (List.map (fun (x : FormConceptPair) => x.formLength) englishReuse).sum / englishReuse.length < (List.map (fun (x : FormConceptPair) => x.formLength) englishCompounds).sum / englishCompounds.length

Reuse items are shorter on average than compounds (paper §3.3: holds across all three languages and all time intervals).

theorem XuEtAl2024.Polysemy.french_reuse_shorter : (List.map (fun (x : FormConceptPair) => x.formLength) frenchReuse).sum / frenchReuse.length < (List.map (fun (x : FormConceptPair) => x.formLength) frenchCompounds).sum / frenchCompounds.length

French reuse items are also shorter on average than French compounds.

theorem XuEtAl2024.Polysemy.both_literalities : (englishReuse.any fun (x : FormConceptPair) => x.literality == Literality.literal) = true ∧ (englishReuse.any fun (x : FormConceptPair) => x.literality == Literality.nonliteral) = true ∧ (englishCompounds.any fun (x : FormConceptPair) => x.literality == Literality.literal) = true ∧ (englishCompounds.any fun (x : FormConceptPair) => x.literality == Literality.nonliteral) = true

Both strategies include literal and nonliteral items in the paper's Table 1 sample.

§3. Substrate witnesses

Concrete instantiations of encodingCosts and unifiedObjective demonstrate the Theory-layer substrate is operationally consumed. The toy needProb and model below are not the paper's actual fitted distributions; they are stipulated only to anchor the type-checking. Real instantiation requires the WordNet+Sentence-BERT pipeline of paper §5.3 + §5.4.

noncomputable def XuEtAl2024.Polysemy.uniformNeed (n : ℕ) : String → ℝ

A toy uniform-need distribution: 1/n for each concept in a list-derived encoding, 0 elsewhere. Constant function for the witness; serious use would derive from corpus frequencies.

Equations

• XuEtAl2024.Polysemy.uniformNeed n x✝ = 1 / ↑n

noncomputable def XuEtAl2024.Polysemy.stipulatedModel : Pragmatics.Communication.AsymmetricCommModel String String

A toy symmetric communication model with constant listener score 1/2. Makes encodingCosts.cost₂ a determinate value for the witness theorems below.

Equations

• XuEtAl2024.Polysemy.stipulatedModel = Pragmatics.Communication.AsymmetricCommModel.symmetric fun (x x_1 : String) => 1 / 2

noncomputable def XuEtAl2024.Polysemy.englishReuseCosts : Pragmatics.Efficiency.CostPair

The English-reuse encoding's costs under the toy model.

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noncomputable def XuEtAl2024.Polysemy.englishCompoundsCosts : Pragmatics.Efficiency.CostPair

The English-compound encoding's costs under the toy model.

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theorem XuEtAl2024.Polysemy.unifiedObjective_eq_weightedCost (pairs : List FormConceptPair) (np : String → ℝ) (m : Pragmatics.Communication.AsymmetricCommModel String String) (β : ℝ) : unifiedObjective pairs np m β = Pragmatics.Efficiency.weightedCost (encodingCosts pairs np m) β

unifiedObjective decomposes into weightedCost (encodingCosts ...) β. This rfl theorem witnesses that the named unifiedObjective hook is the β-scalarization of the cost pair the substrate computes — no extra arithmetic, no glue.

§4. Reuse as polysemy generation

def XuEtAl2024.Polysemy.reuseIsPolysemyGeneration : List FormConceptPair → List String

Word reuse creates polysemy: the reused word acquires a new sense alongside its existing one. Connects the diachronic process of lexicalization to the synchronic phenomenon of polysemy.

Copredication ([Ash11], [Got17]) is the synchronic consequence of reuse (multiple aspects coexist); this paper's account explains the diachronic cause (efficiency pressure).

Caveat on the copredication bridge. Xu's reuse polysemy and logical polysemy are not the same phenomenon. Logical polysemy involves sortally-compatible aspects with a shared individuation ground (book = phys × info, both individuating one volume); Xu's mouse → peripheral generates two unrelated sortal categories with no shared ground. The honest bridge: Xu's literal reuse (hyponymic, e.g. car narrowed from wheeled cart) is compatible with logical polysemy (shared ground); Xu's non-literal reuse (metaphorical, e.g. mouse) is not. The Literality enum in the Theory file is the partition this distinction lives on.

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theorem XuEtAl2024.Polysemy.all_english_reuse_creates_polysemy : (reuseIsPolysemyGeneration englishReuse).length = englishReuse.length

All reuse items in the English data generate polysemy.