Ranking Functions as RSA Limits #
@cite{spohn-1988} @cite{goldszmidt-pearl-1996} @cite{frank-goodman-2012}
As the rationality parameter α → ∞, softmax-based probabilistic inference converges to ranking-based default reasoning. This file makes the connection precise.
Key results #
rankToScore: Every ranking function κ induces scores s(w) = -κ(w).softmax_concentrates_unique: softmax(s, α) → point mass on the unique rank-0 world as α → ∞.minRank_worlds_satisfy: Under ranking entailment, all minimum-rank φ-worlds satisfy σ — connecting to the softmax limit.condProb_tendsto_one: Conditional Softmax Limit Theorem — P_α(σ|φ) → 1 as α → ∞ whenever κ ⊨ φ → σ. This is the central result: RSA with infinite rationality = ranking entailment.
Mathematical background #
@cite{spohn-1988} §7 observes that rankings are ordinal probabilities: κ(w) = n corresponds to P(w) ∝ ε^n for infinitesimal ε. The finite analogue replaces ε^n with exp(-n·α) for large α:
softmax(-κ, α)(w) = exp(-α · κ(w)) / Σ_v exp(-α · κ(v))
As α → ∞, this concentrates on rank-0 (most normal) worlds — exactly the worlds that survive ranking-based default reasoning.
This makes System Z's κ^z ranking (@cite{goldszmidt-pearl-1996}) the "infinite rationality" limit of RSA pragmatic inference (@cite{frank-goodman-2012}). Qualitative default reasoning and quantitative Bayesian pragmatics are not rival frameworks but endpoints of the same rationality continuum.
Convert a ranking function to softmax scores.
s(w) = -(κ(w) : ℝ). Lower rank → higher score → more plausible.
This is the finite analogue of @cite{spohn-1988} §7's P(w) ∝ ε^{κ(w)}: here exp(α · s(w)) = exp(-α · κ(w)) plays the role of ε^{κ(w)} with ε = exp(-α).
Equations
- RSA.RankingBridge.rankToScore κ w = -↑(κ.rank w)
Instances For
rankToScore reverses the ordering: higher rank ↔ lower score.
Rank-0 worlds achieve the maximum score (0).
All scores are non-positive.
Rank-0 worlds maximize the score.
When κ has a unique rank-0 world, softmax(rankToScore κ, α) concentrates on it as α → ∞.
This is the core convergence result: the probabilistic distribution approaches a point mass on the most normal world, recovering the ranking-based notion of normality.
Proof: Apply Softmax.tendsto_softmax_infty_unique_max with
scores s = rankToScore κ. The unique rank-0 world maximizes s
(since s(w) = -κ(w) and κ(w₀) = 0 < κ(v) for v ≠ w₀).
Entropy of softmax(rankToScore κ, α) → 0 as α → ∞ (unique rank-0 case). The distribution becomes maximally concentrated.
rankEntails is equivalent to: every φ∧¬σ world has strictly higher rank than some φ∧σ world.
When φ∧σ is non-empty, this is the same as: the minimum rank among φ∧σ worlds is strictly less than the rank of every φ∧¬σ world. When φ∧¬σ is empty, rankEntails holds vacuously.
Under ranking entailment, all minimum-rank φ-worlds satisfy σ.
This is the key lemma connecting ranking entailment to softmax concentration: as α → ∞, softmax concentrates on minimum-rank worlds. If all minimum-rank φ-worlds satisfy σ, then the limiting conditional probability P(σ|φ) = 1.
Proof: If a minimum-rank φ-world w fails σ, then rankEntails gives a φ∧σ world v with κ(v) < κ(w), contradicting w having the minimum rank among φ-worlds.
The theorems above establish the formal bridge between RSA and ranking-based default reasoning:
```
α = 0 α finite α → ∞
uniform ←——— softmax(s, α) ———→ hardmax
no inference RSA pragmatics ranking entailment
```
At α = 0, the listener has no rationality assumption and
the posterior is uniform (`Softmax.tendsto_softmax_zero`).
At finite α, the listener performs soft Bayesian inference
(standard RSA). As α → ∞, the posterior concentrates on
the most normal worlds (`softmax_concentrates_unique`),
and inference becomes ranking entailment.
This justifies two practices:
1. RSA modelers can use System Z rankings as the "skeleton"
of their prior, softened by finite α for gradient predictions.
2. Default reasoning theorists can view their qualitative
inferences as the limiting case of probabilistic pragmatics.
The exponential prior induced by a ranking function.
P(w) = exp(-κ(w)). This is the natural prior for connecting ranking functions to Bayesian inference: rank-0 worlds get maximal prior probability, and the prior ordering on worlds matches the ranking ordering.
@cite{spohn-1988} §7 uses P(w) ∝ ε^{κ(w)} with ε infinitesimal; here we use the concrete parameterization exp(-κ(w)).
Equations
- RSA.RankingBridge.rankToPrior κ w = Real.exp (-↑(κ.rank w))
Instances For
The exponential prior is always positive.
The exponential prior is non-negative.
Rank-0 worlds maximize the exponential prior.
The exponential prior preserves the rank ordering: lower rank ↔ higher prior.
An RSAConfig with worldPrior = rankToPrior κ and boolean meaning
computes L1 posteriors that concentrate on the minimum-rank worlds
in the support of the utterance as α → ∞.
Specifically, for an utterance u with boolean denotation ⟦u⟧:
- L1(w|u) ∝ rankToPrior(κ, w) · S1(u|w)
- As α → ∞, S1 concentrates on the most informative true utterance
- The prior rankToPrior(κ, ·) breaks ties among equally informative
utterances in favor of lower-rank worlds
- The net effect: L1 concentrates on minimum-rank ⟦u⟧-worlds
This is exactly what `rankEntails κ ⟦u⟧ σ` characterizes: the
most normal worlds satisfying the utterance also satisfy σ.
The conditional limit theorem `condProb_tendsto_one` (§7)
formalizes this precisely: P_α(σ|φ) → 1 as α → ∞ whenever
κ ⊨ φ → σ.
The softmax distribution with ranking scores is exactly the exponential prior (up to normalization).
softmax(rankToScore κ, 1)(w) = exp(-κ(w)) / Σ_v exp(-κ(v)) = rankToPrior(κ, w) / Σ_v rankToPrior(κ, v)
At α = 1, softmax with ranking scores IS the normalized exponential prior. At other α, it's a tempered version: softmax(rankToScore κ, α)(w) = exp(-α·κ(w)) / Σ_v exp(-α·κ(v)).
@cite{spohn-1988} §7 observes that ranking functions are the ordinal analogue of probability measures. This is not an analogy — it is an exact algebraic identity via the tropical semiring.
The **tropical semiring** `Tropical ℕ` has:
- Addition = min (the rank of a disjunction)
- Multiplication = + (the rank of a conjunction under independence)
The map ε^{·} (for 0 < ε < 1) is a homomorphism from the tropical
semiring to (ℝ₊, max, ×):
- ε^{a * b} = ε^{a + b} = ε^a · ε^b (tropical × → real ×)
- ε^{a + b} = ε^{min(a,b)} = max(ε^a, ε^b) (tropical + → real max)
The only "approximation" is that real addition (Σ) approximates max
in the ε → 0 limit. This is the dequantization: as ε → 0 (α → ∞),
the sum Σ_w ε^{κ(w)} is dominated by its largest term max_w ε^{κ(w)},
and probabilistic inference degenerates to tropical (ranking) algebra.
The exponential map sends tropical multiplication (= underlying +) to real multiplication: ε^{a ×_trop b} = ε^{a + b} = ε^a · ε^b.
This is exact, not approximate. It formalizes @cite{spohn-1988} §7's observation that ranking independence (κ(A∩B) = κ(A) + κ(B)) corresponds to probabilistic independence (P(A∩B) = P(A)·P(B)).
The exponential map sends tropical addition (= min) to real max: ε^{a +_trop b} = ε^{min(a,b)} = max(ε^a, ε^b) for 0 < ε ≤ 1.
This formalizes @cite{spohn-1988} §7's observation that ranking disjunction (κ(A∪B) = min(κ(A), κ(B))) corresponds to the dominant term in P(A∪B) = P(A) + P(B) − P(A∩B).
Ranking independence is tropical multiplicativity.
κ(φ ∧ ψ) = κ(φ) + κ(ψ) ↔ trop(κ(φ∧ψ)) = trop(κ(φ)) ×_trop trop(κ(ψ))
This is not an analogy: "P(A∩B) = P(A)·P(B)" and "κ(A∩B) = κ(A)+κ(B)" are literally the same equation in two semirings.
The softmax distribution is the tropical-to-probabilistic functor applied to ranking values, with ε = exp(-α).
softmax(-κ, α)(w) = exp(-α·κ(w)) / Σ_v exp(-α·κ(v)) = ε^{κ(w)} / Σ_v ε^{κ(v)} where ε = exp(-α)
As ε → 0 (α → ∞), the normalizing sum Σ_v ε^{κ(v)} is dominated by its largest term ε^{min_v κ(v)} = ε^0 = 1 (for rank-0 worlds). In this limit, the probabilistic sum degenerates to the tropical sum, and Bayesian inference becomes ranking-based reasoning.
The conditional limit theorem: as α → ∞, the conditional probability P_α(σ|φ) → 1 whenever ranking entailment κ ⊨ φ → σ holds.
This completes the RSA–ranking bridge:
- §2 shows the *prior* concentrates on rank-0 worlds
- §7 shows the *posterior* (conditioned on φ) concentrates on σ-worlds
The proof uses exponential decay of non-minimal-rank worlds:
under ranking entailment, every φ∧¬σ world has rank strictly above
the minimum-rank φ∧σ world, so its softmax weight decays
exponentially faster, driving P(σ|φ) → 1.
rankEntails implies the existence of a φ∧σ witness.
Conditional probability of σ given φ under scores s at rationality α. P_α(σ|φ) = Σ_{w: φ∧σ} exp(α·s(w)) / Σ_{w: φ} exp(α·s(w))
Equations
- RSA.RankingBridge.condProb s α φ σ = (∑ w : W with φ w ∧ σ w, Real.exp (α * s w)) / ∑ w : W with φ w, Real.exp (α * s w)
Instances For
Conditional Softmax Limit Theorem. Under ranking entailment κ ⊨ φ → σ, the conditional probability P_α(σ|φ) converges to 1 as α → ∞.
This is the central theorem connecting RSA to default reasoning: an RSA listener with ranking-derived scores and infinite rationality draws exactly the conclusions that ranking entailment sanctions.