@cite{cohn-gordon-goodman-potts-2019} — Incremental Iterated Response Model #
@cite{dale-reiter-1995}
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 #
This is the first formalization to use RSAConfig's sequential infrastructure:
Ctx = List Word— the prefix produced so fartransition ctx w = ctx ++ [w]— append the new wordinitial = []— start with empty prefixmeaning ctx _ w r= extension-based incremental semantics ofctx ++ [w]
The domain is Figure 1 from the paper: 3 referents (red dress, blue dress, red hat), 3 words (red, dress, object), 3 complete utterances (dress, red dress, red object). Costs are 0 for all words.
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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Whether a word is veridically true of a referent.
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- Phenomena.Reference.Studies.CohnGordonEtAl2019.wordApplies Phenomena.Reference.Studies.CohnGordonEtAl2019.Word.red Phenomena.Reference.Studies.CohnGordonEtAl2019.Referent.redDress = true
- Phenomena.Reference.Studies.CohnGordonEtAl2019.wordApplies Phenomena.Reference.Studies.CohnGordonEtAl2019.Word.red Phenomena.Reference.Studies.CohnGordonEtAl2019.Referent.redHat = true
- Phenomena.Reference.Studies.CohnGordonEtAl2019.wordApplies Phenomena.Reference.Studies.CohnGordonEtAl2019.Word.dress Phenomena.Reference.Studies.CohnGordonEtAl2019.Referent.redDress = true
- Phenomena.Reference.Studies.CohnGordonEtAl2019.wordApplies Phenomena.Reference.Studies.CohnGordonEtAl2019.Word.dress Phenomena.Reference.Studies.CohnGordonEtAl2019.Referent.blueDress = true
- Phenomena.Reference.Studies.CohnGordonEtAl2019.wordApplies Phenomena.Reference.Studies.CohnGordonEtAl2019.Word.object x✝ = true
- Phenomena.Reference.Studies.CohnGordonEtAl2019.wordApplies x✝¹ x✝ = false
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The three complete utterances in the scene (Figure 1a): "dress", "red dress", "red object".
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Count of viable extensions: complete utterances extending pfx that are
true of at least one referent.
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Extension-based incremental semantics (§2.2):
⟦pfx⟧(r) = trueExtCount(pfx, r) / viableExtCount(pfx)
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Incremental RSA for the Figure 1 reference game.
This is the first RSAConfig to use the sequential infrastructure
(Ctx, transition, initial). The model produces referring expressions
word-by-word, with each step choosing the next word to maximize L0's
posterior for the target referent.
Architecture:
- L0_at(ctx): literal listener given prefix ctx + next word
- S1_at(ctx): speaker choosing next word given prefix ctx
- trajectoryProb: chain-rule product of S1_at probabilities
Parameters: α = 1, cost = 0 for all words, uniform priors.
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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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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. This foreshadows @cite{sedivy-etal-1999}'s finding that listeners draw contrastive inferences from prenominal adjectives.
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.
The incremental RSA model and @cite{dale-reiter-1995}'s Incremental Algorithm (IA) solve the same problem — generating referring expressions for a target among distractors — via structurally parallel mechanisms:
| Property | D&R IA | Incremental RSA |
|---|---|---|
| Processing | Sequential (attribute-by-attr) | Sequential (word-by-word) |
| Selection | Deterministic (fixed order) | Probabilistic (soft-max) |
| Q2/Cost | None (No Brevity) | None (s1Score = L0) |
| State | Remaining distractors | Ctx = word prefix |
| Termination | All distractors ruled out | Chain rule product over words |
Both operate in the No-Brevity regime: D&R's IA includes any
discriminating attribute without brevity optimization; the incremental
RSA's s1Score l0 _ _ w u := l0 u w has no cost term. D&R's fixed
PreferredAttributes order is generalized by RSA's probabilistic
ranking, which emerges from the L0 semantics at each step.
The key difference: D&R is deterministic and may produce non-minimal
descriptions (as shown in DaleReiter1995.cups_non_minimal), while
the incremental RSA is probabilistic and assigns lower total probability
to longer utterances via the chain rule product (Finding 3:
incremental_prefers_bare_noun).
Both the incremental RSA and @cite{dale-reiter-1995}'s Incremental Algorithm operate in the No-Brevity regime (strength = 0) — the weakest Q2 interpretation. Both enforce Q1 (each word/attribute must contribute to identifying the referent) without Q2 (brevity) pressure. D&R's deterministic fixed-order selection is generalized by the incremental RSA's probabilistic word-by-word production.