Source
Beckers and Halpern (2019), LIT-1316, read in full in NOTE-1032: Definitions 3.1, 3.5, 3.13 and 3.15, Proposition 3.6, Theorem 3.8 and the examples named below. Exact transformation is Rubenstein et al.’s (LIT-1334, NOTE-1037).
What was actually shown
Fix deterministic models M_L, M_H, a map τ from low- to high-level endogenous states, and a surjective, order-preserving ω from allowed low-level interventions I_L to high-level ones. (M_H, Pr_H) is an exact transformation of (M_L, Pr_L) when Pr_H^{ω(i)} = τ(Pr_L^i) for every i ∈ I_L. M_H is a uniform transformation of M_L when for every Pr_L some Pr_H makes the pair exact.
(b) ⇒ (a). Suppose τ_U satisfies τ(M_L(u, i)) = M_H(τ_U(u), ω(i)) for all contexts u and all i ∈ I_L. Given Pr_L, put Pr_H = τ_U(Pr_L). For a high-level state v_H, Pr_H^{ω(i)}(v_H) = Pr_H{u_H : M_H(u_H, ω(i)) = v_H} = Pr_L{u : M_H(τ_U(u), ω(i)) = v_H} = Pr_L{u : τ(M_L(u, i)) = v_H} = τ(Pr_L
(a) ⇒ (b), with I_L countable. Fix a context u and let Pr_L be the point mass at u. Then τ(Pr_L^i) is the point mass at τ(M_L(u, i)), so the Pr_H that uniformity supplies gives probability one to A_i = {u_H : M_H(u_H, ω(i)) = τ(M_L(u, i))} for each i. A countable intersection of probability-one sets has probability one, so ⋂ᵢ A_i is nonempty; let τ_U(u) be any of its points. Then τ_U commutes with every allowed intervention at u, and u was arbitrary. The paper’s Theorem 3.8 is stated without the countability that this step, its Proposition 3.6, uses.
Strictness of the steps.
- Exact but not uniform (Example 3.4). Two unrelated models, each with only the empty intervention and a point-mass distribution, with τ sending every state of each to the other’s one realized state, are exact transformations of each other. With another distribution they are not.
- Uniform but not a τ-abstraction (Examples 3.10–3.11). Choosing ω freely lets M₁ and M₂ be uniform transformations of each other; and a high-level variable X* that changes every equation only when U* = 0, a context never given positive probability, is uniform with τ not surjective. τ-abstraction requires τ surjective, a surjective τ_U, and ω = ω_τ, the map τ itself induces.
- A τ-abstraction but not strong (Example 3.16). With energy computed from velocity, height and mass, M ← m leaves every energy pair reachable, so ω_τ sends it to the empty intervention while it changes the outcome; strong abstraction, which allows every intervention ω_τ is defined on, fails, and removing M ← m from I_L gives a τ-abstraction.
What could have failed: uniformity could have required more than a single context map, for example one map per intervention. Countability lets one point serve every intervention at once.
What this does not say
- Not that the high-level context is the coarse state. τ_U is any map that works. In the hidden bit (CASE-125), dropping K is a τ-abstraction for interventions on H alone, with τ_U(u) = (u_t, u_H, (u_K, u_{t+1})): the high-level noise carries K. So the ladder up to τ-abstraction accepts a coarse-graining whose coarse state keeps nothing of the response to h. Governance-response sufficiency (TERM-067) is the compatibility condition with τ_U required to be the coarse state itself, compared on the outcome only (NOTE-1032, Bearing on the record, item 1).
- Not anything approximate; that is LIT-1310 (THEORY-843).
- Not for uncountable intervention sets without a further argument.
- Not that every strong abstraction is constructive; the paper conjectures it.
Connections
- THEORY-848. Exact transformation’s acceptance of marginalising a never-intervened variable, here seen to survive uniformity.
- CLAIM-350. Its fibre-wise factorization of a judgment is the shape of the compatibility condition with the context map fixed to the coarse map.
- THEORY-243. Interchange interventions test a commuting condition of this kind on chosen inputs.