@cite{cohn-gordon-goodman-potts-2019} — Incremental Iterated Response Model #
Cohn-Gordon, R., Goodman, N. D., & Potts, C. (2019). An Incremental Iterated Response Model of Pragmatics. Proceedings of the Society for Computation in Linguistics (SCiL) 2, 81–90.
The Model #
The incremental RSA model extends the standard RSA framework to word-by-word production. The speaker produces referring expressions incrementally, choosing each word to maximize the listener's posterior probability for the target:
S1^WORD(wₖ | [w₁,...,wₖ₋₁], r) ∝ L0(r | w₁,...,wₖ)^α
The full utterance probability factors via the chain rule:
S1^UTT-IP(w₁,...,wₙ | r) = ∏ₖ S1^WORD(wₖ | [w₁,...,wₖ₋₁], r)
L0 uses extension-based incremental semantics (§2.2): given prefix c,
⟦c⟧(w) = |{u ∈ U : c ⊑ u ∧ ⟦u⟧(w) = 1}| / |{u ∈ U : c ⊑ u ∧ ∃w'. ⟦u⟧(w') = 1}|
where U is the set of complete utterances and ⊑ is the prefix relation.
Formalization via the IncrementalSemantics bundle #
Each scene in this file is a single value of RSA.IncrementalSemantics U W
(in Theories/Pragmatics/RSA/Incremental.lean), specifying just the lexicon
(wordApplies), the closed set of complete utterances, and the world set.
The bundle's toRSAConfig builder produces the full RSAConfig with chain-
rule speaker, α = 1, no cost, uniform priors, and extension-based L0 — so
the three scenes (Figure 1, the @cite{sedivy-2007} reference game, the
@cite{rubio-fernandez-2016} display) share machinery rather than duplicating it.
The bundle exposes a single deep theorem, l0Utt_ge_inv_card, proving
the §2.4 weakly-informative bound generically: any complete utterance true
of r ∈ worlds yields a literal posterior at least 1 / worlds.length.
The Figure 1 application (greedyUnroll_weakly_informative) below
discharges only the r ∈ worlds and uttSem utt r = true premises;
the bound follows.
Findings #
| # | Finding | Status |
|---|---|---|
| 1 | Adjective-first preferred for target R1 (Figure 1c) | rsa_predict |
| 2 | Noun preferred after adjective for R1 (Figure 1c) | rsa_predict |
| 3 | R2 must start with "dress" (Figure 1c) | rsa_predict |
| 4 | R3 must start with "red" (Figure 1c) | rsa_predict |
| 5 | Uniform fallback after "red" for R2 (§2.2) | cases w <;> rsa_predict |
| 6 | L1 anticipatory implicature: "red" → R3 (Figure 1d) | rsa_predict |
| 7 | Incremental model prefers bare noun over modified NP (Figure 1e) | rsa_predict |
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- CohnGordonEtAl2019.instDecidableEqWord x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- CohnGordonEtAl2019.instReprWord = { reprPrec := CohnGordonEtAl2019.instReprWord.repr }
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- CohnGordonEtAl2019.instDecidableEqReferent x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- CohnGordonEtAl2019.instReprReferent = { reprPrec := CohnGordonEtAl2019.instReprReferent.repr }
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The Figure 1 reference scene as an IncrementalSemantics bundle:
three words ("red", "dress", "object"), three complete utterances
("dress", "red dress", "red object"), three referents (R1, R2, R3).
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Qualitative findings from the incremental RSA model.
- adj_first_for_target : Finding
The incremental speaker prefers the adjective "red" first when referring to the target R1 (red dress).
- noun_after_adj : Finding
After producing "red", the speaker prefers the type noun "dress" over the generic "object".
- noun_only_for_r2 : Finding
For R2 (blue dress), the speaker must start with "dress" — "red" has zero incremental semantics for R2.
- adj_only_for_r3 : Finding
For R3 (red hat), the speaker must start with "red" — "dress" has zero incremental semantics for R3.
- uniform_after_red_for_r2 : Finding
After "red" for R2, no extension is true — the speaker is indifferent between "dress" and "object" (uniform fallback).
- listener_anticipation : Finding
After hearing "red", L1 infers the target is more likely R3 (red hat) than R1 (red dress) — an anticipatory implicature.
- incremental_prefers_bare_noun : Finding
At the utterance level, the incremental model assigns higher probability to "dress" than to "red dress" for R1 — diverging from the global model which prefers "red dress".
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- CohnGordonEtAl2019.instDecidableEqFinding x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- CohnGordonEtAl2019.instReprFinding = { reprPrec := CohnGordonEtAl2019.instReprFinding.repr }
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Finding 1 (Figure 1c): The incremental speaker prefers "red" first when referring to R1 (red dress).
S1(red | [], R1) = 4/7 ≈ 0.57 > S1(dress | [], R1) = 3/7 ≈ 0.43
Mechanism: "red" narrows the extension set to {red dress, red object}, both true of R1 (trueExtCount = 2, viableExtCount = 2 → meaning = 1). "dress" narrows to {dress}, true of R1 (meaning = 1) but the L0 posterior for R1 is lower because "dress" also applies to R2.
Finding 2 (Figure 1c): After producing "red", the speaker prefers "dress" over "object" for R1.
S1(dress | [red], R1) = 2/3 ≈ 0.67 > S1(object | [red], R1) = 1/3 ≈ 0.33
"red dress" uniquely identifies R1 (only R1 is a red dress), while "red object" is ambiguous between R1 and R3.
Finding 3 (Figure 1c): For R2 (blue dress), the speaker must start with "dress" — "red" never applies to R2 (it's a blue dress), so all extensions of "red" have zero semantics for R2.
S1(dress | [], R2) > S1(red | [], R2)
Finding 4 (Figure 1c): For R3 (red hat), the speaker must start with "red" — "dress" never applies to R3 (it's a hat), so the only extension of "dress" (= "dress" itself) has zero semantics for R3.
S1(red | [], R3) > S1(dress | [], R3)
Finding 5 (§2.2, uniform fallback): After "red" for R2, no complete utterance extension of "red" is true of R2 (blue dress). The paper states: "probability is evenly distributed over all choices of word."
S1(dress | [red], R2) = S1(object | [red], R2)
Both equal 1/2 because the meaning function returns 0 for all R2 extensions of "red", yielding uniform L0 → uniform S1.
Finding 6 (Figure 1d): After hearing "red", the pragmatic listener L1 infers that the target is more likely R3 (red hat) than R1 (red dress).
L1(R3 | red) = 7/11 ≈ 0.64 > L1(R1 | red) = 4/11 ≈ 0.36
This is an anticipatory implicature: "red" is the ONLY word available for R3 (S1(red|[],R3) = 1), so hearing "red" raises R3's probability. For R1, the speaker could have said "dress" instead, so "red" is less diagnostic. We pick this up below as a structural foreshadowing of @cite{sedivy-etal-1999}'s contrastive-inference findings; CG themselves cite @cite{sedivy-2007} for the same effect.
Finding 7 (Figure 1e): The incremental model prefers "dress" over "red dress" for R1 — the key divergence from the global RSA model.
S1^UTT-IP(dress | R1) = 3/7 ≈ 0.43 > S1^UTT-IP(red dress | R1) = 8/21 ≈ 0.38
The global model prefers "red dress" (more informative). The incremental model prefers "dress" because it is produced in one step with probability 3/7, while "red dress" requires two steps: S1(red|[],R1) · S1(dress|[red],R1) = 4/7 · 2/3 = 8/21 < 9/21 = 3/7.
Map each finding to the model prediction that accounts for it.
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All 7 findings verified.
@cite{cohn-gordon-goodman-potts-2019} §2.4 establishes a weakly
informative lower bound on greedy unrolling: even though the
incremental speaker has no global view of the utterance space, the
greedy choice at each step yields a complete utterance under which the
literal listener's posterior for the target is at least 1 / |W|. The
bound itself — l0Utt_ge_inv_card — is proved generically over the
IncrementalSemantics bundle in Theories/Pragmatics/RSA/Incremental.lean.
What's left for this study is to (i) define the greedy unroller for
Figure 1's three referents and (ii) verify that each output is a complete
utterance true of the target; the §2.4 bound then follows.
Greedy unrolling for Figure 1's scene: at each step pick the word maximizing L0(r | ctx ++ [w]); stop when ctx is a complete utterance. Tabulated by case for the three Figure 1 referents.
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- CohnGordonEtAl2019.greedyUnroll CohnGordonEtAl2019.Referent.redDress = [CohnGordonEtAl2019.Word.red, CohnGordonEtAl2019.Word.dress]
- CohnGordonEtAl2019.greedyUnroll CohnGordonEtAl2019.Referent.blueDress = [CohnGordonEtAl2019.Word.dress]
- CohnGordonEtAl2019.greedyUnroll CohnGordonEtAl2019.Referent.redHat = [CohnGordonEtAl2019.Word.red, CohnGordonEtAl2019.Word.object]
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§2.4 weakly informative bound, instantiated for Figure 1.
Each of the three greedy outputs is a complete utterance true of its
target referent, so the generic l0Utt_ge_inv_card theorem from
Incremental.lean immediately gives the 1/|worlds| = 1/3 bound. The
actual values for this scene are 1, 1/2, 1/2 — the bound is loose here
by design: it certifies architectural sanity, not optimality.
The global RSA model treats each complete utterance as an atomic
option, normalizing over the whole utterance space rather than chaining
word-by-word. The divergence between global and incremental predictions
for R1 is a central result of @cite{cohn-gordon-goodman-potts-2019} §2.4:
the global model prefers the more-informative "red dress" over the bare
"dress" (standard RSA Q-implicature), but the incremental model prefers
"dress" because chain-rule products penalize longer trajectories
(Finding 7, incremental_prefers_bare_noun).
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- CohnGordonEtAl2019.instDecidableEqUtterance x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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Boolean truth of a complete utterance for a referent.
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- CohnGordonEtAl2019.uttApplies CohnGordonEtAl2019.Utterance.dress CohnGordonEtAl2019.Referent.redDress = true
- CohnGordonEtAl2019.uttApplies CohnGordonEtAl2019.Utterance.dress CohnGordonEtAl2019.Referent.blueDress = true
- CohnGordonEtAl2019.uttApplies CohnGordonEtAl2019.Utterance.redDress CohnGordonEtAl2019.Referent.redDress = true
- CohnGordonEtAl2019.uttApplies CohnGordonEtAl2019.Utterance.redObject CohnGordonEtAl2019.Referent.redDress = true
- CohnGordonEtAl2019.uttApplies CohnGordonEtAl2019.Utterance.redObject CohnGordonEtAl2019.Referent.redHat = true
- CohnGordonEtAl2019.uttApplies x✝¹ x✝ = false
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Global RSA model for the Figure 1 reference game: U = full utterances, one-shot normalization, α = 1 with no cost term. The same model class @cite{frank-goodman-2012} would write for a non-incremental speaker.
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Divergence from incremental (§2.4): the global RSA prefers the
fully-modified "red dress" over the bare "dress" for R1, because
"red dress" uniquely identifies R1 while "dress" leaves R1/R2
ambiguous. Compare Finding 7 (incremental_prefers_bare_noun),
where the incremental trajectory probability has the opposite
preference: chain-rule products discount longer trajectories enough
to flip the ordering. This is the central empirical wedge between
the two architectures the paper articulates.
@cite{cohn-gordon-goodman-potts-2019} §3.2 reanalyses @cite{sedivy-2007}'s review of contrastive-inference findings within the incremental RSA framework. The scene contains a target tall cup, a contrasting short cup (same category, opposite scale pole), a tall pitcher (cross-category competitor at the same scale pole), and an unrelated key. After the listener hears just "tall", the pragmatic listener L1 prefers the tall cup over the tall pitcher — even though extension semantics treats both as equally compatible — because a speaker referring to the pitcher would have said "pitcher" alone (no need for "tall" to disambiguate from the only other pitcher: there isn't one). The "tall" is therefore diagnostic of the cup with a same-category contrast.
The original empirical effect is from @cite{sedivy-etal-1999}; CG cite the @cite{sedivy-2007} review article that summarizes it.
This file formalises both contrast cells. The contrast scene is the
five-word, four-referent sedivyBundle from CG's text. The no-contrast
scene is a four-word, three-referent companion bundle
(SedivyScene_NoContrast.bundle) — .short is omitted because the
shortCup referent it would describe is absent from the display, leaving
the speaker with no scene-anchored use for the word. The two scenes
share a Referent type so a single Cell-typed
LookProportion SedivyEtAl1999.Cell ℝ projection can read off both.
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- CohnGordonEtAl2019.SedivyScene.instDecidableEqWord x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- CohnGordonEtAl2019.SedivyScene.instDecidableEqReferent x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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The Sedivy scene as an IncrementalSemantics bundle: 5 words,
6 complete utterances (3 adj+noun phrases + 3 bare nouns; the
bare-noun option is essential — without it "tall" is no longer
diagnostic of the cup), 4 referents.
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No-contrast variant of the Sedivy scene, sharing
SedivyScene.Referent but with a smaller word inventory. Empirically
this is the no-contrast cell of @cite{sedivy-etal-1999}'s 2 × 2 × 2
design: the same-category contrast object (the short cup) is removed
from the visual display.
The companion bundle drops .short from Word and the [short, cup]
utterance from completeUtterances. Justification: with no shortCup
in the display, a cooperative speaker has no scene-anchored use for
.short, and CG's IncrementalSemantics is a scene-specific
production model rather than a lexicon-wide one. (The listener's
standing mental lexicon still contains short; the bundle here is a
model of speaker production for this scene, not of mental
inventories.)
Why the fresh Word type rather than a {sedivyBundle with worlds := …}
update? Keeping .short in the lexicon while removing all of its
referents leaves incrementalSem [.short] _ = 0/0, which mathlib treats
as 0 but which the rsa_predict reflection evaluator cannot reduce.
The fresh type sidesteps the divide-by-zero pattern at the cost of mild
bundle duplication.
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- CohnGordonEtAl2019.SedivyScene_NoContrast.instDecidableEqWord x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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Cohn-Gordon §3.2 prediction: after hearing "tall", L1 favours the tall cup over the tall pitcher (3/5 vs 2/5). The mechanism is the contrastive inference: a speaker referring to the pitcher would use "pitcher" alone (S1(pitcher | tallPitcher) = 2/3); the only referent for which "tall" is the speaker's preferred first word is the tall cup, where "cup" alone leaves shortCup ambiguous.
This formalises @cite{sedivy-2007}'s anticipatory contrast effect
within the incremental RSA framework (and indirectly captures the
@cite{sedivy-etal-1999} empirical pattern Sedivy 2007 reviews).
The paradigm-level statement (Sedivy Pattern 2,
Paradigms.VisualWorld.ContrastReducesCompetitorLooks) requires a
contrast vs no-contrast comparison; this theorem captures the
contrast-condition direction only.
Cell-typed look projection for the Sedivy paradigm under the incremental-RSA model.
Linking hypothesis (load-bearing, editorial) #
The incremental RSA model produces a posterior over referents,
L1 : Word → Referent → ℝ. Visual-world data are fixation
proportions. Mapping the former to the latter requires a linking
hypothesis. This file makes the simplest one explicit:
Bayesian posterior linking hypothesis — the proportion of looks to an object equals the listener's posterior probability of that object being the referent at the same point in the unfolding utterance.
@cite{cohn-gordon-goodman-potts-2019} do not state this assumption;
they discuss the contrastive-inference effect at the level of L1
posteriors and treat empirical contact with @cite{sedivy-2007}'s
look data informally. The Bayesian linking hypothesis used here is
the strongest natural choice given a single normalised posterior.
A weaker alternative would be a Luce-choice rule over a
monotone-in-posterior activation; that weakening preserves the
qualitative inequality patterns this file proves. If a second linking hypothesis enters the codebase, the
paradigm contract should grow a typed LinkingHypothesis API and
the bridge theorem statement should mention which hypothesis is
in force.
Construction #
cgSedivyLooks role c selects the appropriate scene config based on
c.contrast (incRSA_sedivy for the contrast cell;
SedivyScene_NoContrast.incRSA_sedivy_noContrast for the no-contrast
cell) and reads off L1 .tall · at the referent corresponding to
role. Other factors of the cell (typicality, task) are ignored —
the incremental RSA model has no internal representation of
typicality or task, so the projection is constant in those factors.
Cells in the contrast condition cover four roles; cells in the
no-contrast condition omit .contrastingObject (no shortCup is on
display) and so collapse to 0 for that role.
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- CohnGordonEtAl2019.cgSedivyLooks role✝ c = 0
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Paradigm Pattern 2 verified for Cohn-Gordon's incremental RSA:
swapping the contrast factor from contrast to noContrast
strictly increases looks to the cross-category competitor (the tall
pitcher), under the Bayesian posterior linking hypothesis stated on
cgSedivyLooks.
Mechanism: in the contrast scene, L1(.tall, tallPitcher) = 2/5
because a speaker referring to the pitcher would prefer "pitcher"
alone (the shortCup distractor pulls "tall" toward the cup). In
the no-contrast scene there is no shortCup, "tall" is uninformative
between the two extant scale-pole objects, and L1(.tall, tallPitcher) = 1/2.
Discharges
SedivyEtAl1999.SatisfiesSedivyPattern.contrast_reduces_competitor_looks
for this model. The proof reduces — via the HasContrastCondition
lens applied to a destructured cell — to the per-cell L1 inequality,
dispatched by rsa_predict.
@cite{cohn-gordon-goodman-potts-2019} §3.1 reanalyses
@cite{rubio-fernandez-2016}'s finding that English speakers
over-modify (saying "the red dress" when "the dress" suffices in a
display with one dress). The mechanism: an explicit STOP token
marks the end of the utterance, so trajectories of different lengths
([dress, STOP] vs [red, dress, STOP]) become directly comparable
under the chain rule. Without STOP, the chain rule penalizes longer
trajectories monotonically (Finding 7); with STOP and a per-word
cost the over-modification preference can emerge in the right cost
regime.
This formalisation establishes the model's lexicon and complete-
utterance set with STOP, and proves the structural invariants
(every complete utterance ends in STOP; STOP does not apply
mid-utterance). The cost-dependent comparison theorem
S1^UTT-IP(red dress, STOP | R1) > S1^UTT-IP(dress, STOP | R1) is
left as future work — formalising it requires Real.exp over a cost
schedule and a quantitative argument that does not reduce via
rsa_predict.
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- CohnGordonEtAl2019.RubioFernandezScene.instDecidableEqWord x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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Display referents: a red dress and a blue hat, the canonical minimal pair from @cite{rubio-fernandez-2016}'s display.
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- CohnGordonEtAl2019.RubioFernandezScene.instDecidableEqReferent x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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The Rubio-Fernández scene as an IncrementalSemantics bundle:
five words including stop, four complete utterances all ending
in stop, two referents. stop does not apply to any referent —
it is a structural marker, not a content word.
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Every complete utterance terminates with stop. This is the
structural invariant the STOP machinery enforces.
stop never applies to any referent — it is a structural marker,
not a content word. This is what makes a STOP-augmented utterance
u ++ [.stop] veridically equivalent to the underlying content
sequence u.