Source
Beckers, Eberhardt and Halpern (2020; arXiv:1906.11583, 2019), LIT-1310, read in full in NOTE-1045: Definitions 2.4–2.5, 3.1–3.2, Propositions 3.3, 5.1 and 5.2, Example 5.3. τ-abstraction is Beckers and Halpern’s (LIT-1316, NOTE-1032).
What was actually shown
Let (M_L, I_L) and (M_H, I_H) be finite deterministic causal models, τ-consistent: τ surjective, I_H = ω_τ(I_L), |R_L(U_L)| ≥ |R_H(U_H)|. With a metric d_H on high-level states,
d_τ(M_L, M_H) = min over surjective τ_U of max over i ∈ I_L and u ∈ R_L(U_L) of d_H(τ(M_L(u, i)), M_H(τ_U(u), ω_τ(i))).
Zero is exact abstraction (Proposition 3.3). The minimum is over a finite set, so d_τ = 0 iff some surjective τ_U makes every term zero, iff τ(M_L(u, i)) = M_H(τ_U(u), ω_τ(i)) for all u and i, which with τ-consistency is Definition 2.5’s τ-abstraction.
Abstraction, then approximation (Proposition 5.1). Let M_H be a τ-abstraction of M_L via τ_U, and M_H′ be similar to M_H with max_{i,u_H} d_H(M_H′(u_H, i), M_H(u_H, i)) ≤ α. Using the same τ_U, each term d_H(τ(M_L(u, i)), M_H′(τ_U(u), ω_τ(i))) equals d_H(M_H(τ_U(u), ω_τ(i)), M_H′(τ_U(u), ω_τ(i))) ≤ α. The minimum is no larger.
Approximation, then abstraction (Proposition 5.2). If instead M_L is a d_max-α approximation of M_L′ and M_H a τ-abstraction of M_L, then d_H(τ(M_L′(u, i)), M_H(τ_U(u), ω_τ(i))) = d_H(τ(M_L′(u, i)), τ(M_L(u, i))) ≤ k · d_L(M_L′(u, i), M_L(u, i)) ≤ kα, with k the maximum of the ratio of the two distances. The bound is what τ does to distances, which can be anything: “d_VL(v_L, v_L′) and d_VH(τ(v_L), τ(v_L′)) may be unrelated.”
No factoring (Example 5.3), checked. M_L has A = B = C = U with U ≡ 0, and allowed interventions ∅, A ← 0, A ← 1, B ← 0, B ← 1; τ(a, b, c) = (a ⊕ b, b ⊕ c). Each single-variable intervention leaves every high-level pair reachable, so ω_τ sends all of them to the empty intervention, and I_H = {∅}; M_H has D = E = U ≡ 0. Under B ← 1 the low state is (0, 1, 0), lifted to (1, 1), at Euclidean distance √2 from (0, 0); under A ← 1 it is (1, 0, 0), lifted to (1, 0), at distance 1. So d_τ = √2. For M_H to be an exact abstraction of a model M_L′ with the same signature, every allowed intervention must yield a state lifted to (0, 0), so a = b = c. Then B ← 0 and B ← 1 must give A = 0 and A = 1, so A depends on B; and A ← 0 and A ← 1 must give B = 0 and B = 1, so B depends on A. No recursive M_L′ does both. (The paper ends “Thus, M_H cannot be recursive”; the argument is about M_L′.)
What could have failed: errors could have failed to compose at all once the two levels have unrelated metrics; they compose in one order with no loss and in the other only through k.
What this does not say
- Not a measure of whether the coarse state keeps a system’s response. The minimum over τ_U lets high-level noise hold what τ drops, so for CASE-125 with interventions on H only, d_τ = 0 while the governance deficiency is ρ(h = 1) bits (NOTE-1045). The deficiency of TERM-067 would need τ_U fixed to the coarse state, an average over interventions rather than a maximum, and an information measure rather than a metric.
- Not a scale independent of the metric. “the choice of distance function makes a crucial difference.”
- Not the probabilistic variants, which replace the maximum over contexts by an expectation (Definition 4.4) and add an intervention distribution over implementations (Definition 4.8); they are recorded in NOTE-1045 and not re-derived here.
- Not a bound when k is undefined, as it is wherever M_L′ and M_L agree.
Connections
- THEORY-849. Its zero set.
- TERM-043. D_causal is a different graded relation: an average over interventions of a distance between push-forward distributions.