Asymmetric Communication Models

@cite{xu-etal-2024} @cite{milroy-milroy-1985} @cite{labov-2011}

A communicative interaction in which speaker and listener operate on different lexicons / different channels. The canonical case is variation/innovation diffusion: a speaker has acquired a novel encoding E*, the listener has not yet, so the speaker's production policy is conditioned on L' while the listener's interpretation is conditioned on L (@cite{xu-etal-2024} §2.1).

This generalizes the standard symmetric Shannon setup (@cite{kemp-regier-2012}, @cite{zaslavsky-kemp-regier-tishby-2018}, where speaker and listener share one channel) and serves as substrate for any linguistic-modeling case in which the two ends of a communication act have different generative models. Concrete uses:

• Lexicalization spread (@cite{xu-etal-2024})
• Iterated learning (parent / child generation mismatch; @cite{kirby-tamariz-cornish-smith-2015})
• L1 / L2 communicative accommodation
• Native / non-native listener mismatch in adaptation studies

Two layers

ChannelAccess is a bare-bones non-finite, non-normalized score function C → W → ℝ — the most general view of "what a participant returns when asked about (concept, form)". CommChannel (in InformationTheory/Channel.lean) is the Shannon-strength version (finite alphabets + row-stochastic). CommChannel.toAccess projects the latter onto the former. AsymmetricCommModel holds two ChannelAccesses so it can be instantiated either by lifting two CommChannels (ofCommChannels) or by stipulating two arbitrary score functions over a non-finite type (e.g., String-valued lexicons).

@[reducible, inline]

abbrev Pragmatics.Communication.ChannelAccess (C : Type u_1) (W : Type u_2) : Type (max u_1 u_2)

Bare-bones channel access: a score function C → W → ℝ with no finiteness or normalization constraints. The minimal interface needed to compute per-pair surprisals.

Equations

• Pragmatics.Communication.ChannelAccess C W = (C → W → ℝ)

def Pragmatics.InformationTheory.CommChannel.toAccess {C W : Type} [Fintype C] [Fintype W] (ch : CommChannel C W) : Communication.ChannelAccess C W

Project a Shannon CommChannel onto its bare access function. Lives in Pragmatics.InformationTheory so dot notation ch.toAccess resolves.

Equations

• ch.toAccess = ch.encode

structure Pragmatics.Communication.AsymmetricCommModel (C : Type u_1) (W : Type u_2) : Type (max u_1 u_2)

A two-channel communication model: speaker uses produce, listener uses comprehend. The asymmetry is structural — produce and comprehend are independent fields and may disagree.

@cite{xu-etal-2024}'s headline conceptual move over @cite{kemp-regier-2012} and @cite{zaslavsky-kemp-regier-tishby-2018}: the speaker's production policy is conditioned on the expanded lexicon L' while the listener's interpretation is conditioned on the existing lexicon L. This structural distinction lives in the variation-theory tradition of @cite{labov-2011} and @cite{milroy-milroy-1985}.

• produce : ChannelAccess C W

  Speaker channel p(w | c, L').

• comprehend : ChannelAccess C W

  Listener channel p(w | c, L) (or, equivalently, the listener's score function for inferring c from w under their lexicon).

def Pragmatics.Communication.AsymmetricCommModel.symmetric {C : Type u_1} {W : Type u_2} (ch : ChannelAccess C W) : AsymmetricCommModel C W

Symmetric special case: speaker and listener share one channel. Recovers the Kemp-Regier / Zaslavsky setup as AsymmetricCommModel.symmetric ch.

Equations

• Pragmatics.Communication.AsymmetricCommModel.symmetric ch = { produce := ch, comprehend := ch }

def Pragmatics.Communication.AsymmetricCommModel.ofCommChannels {C W : Type} [Fintype C] [Fintype W] (speakerCh listenerCh : InformationTheory.CommChannel C W) : AsymmetricCommModel C W

Construct an asymmetric model from two finite-alphabet CommChannels — one for the speaker on L', one for the listener on L.

Equations

• Pragmatics.Communication.AsymmetricCommModel.ofCommChannels speakerCh listenerCh = { produce := speakerCh.toAccess, comprehend := listenerCh.toAccess }

def Pragmatics.Communication.AsymmetricCommModel.IsSymmetric {C : Type u_1} {W : Type u_2} (m : AsymmetricCommModel C W) : Prop

An asymmetric model is symmetric iff its two channels coincide as functions.

Equations

• m.IsSymmetric = (m.produce = m.comprehend)

@[simp]

theorem Pragmatics.Communication.AsymmetricCommModel.symmetric_isSymmetric {C : Type u_1} {W : Type u_2} (ch : ChannelAccess C W) : (symmetric ch).IsSymmetric

@[simp]

theorem Pragmatics.Communication.AsymmetricCommModel.symmetric_produce {C : Type u_1} {W : Type u_2} (ch : ChannelAccess C W) : (symmetric ch).produce = ch

@[simp]

theorem Pragmatics.Communication.AsymmetricCommModel.symmetric_comprehend {C : Type u_1} {W : Type u_2} (ch : ChannelAccess C W) : (symmetric ch).comprehend = ch