Characterizing Constructive Causal Abstraction via Intervention Composition

Causal abstraction relates causal models across levels of description. Constructive abstractions are widely used and provide useful structural properties. Beckers and Halpern conjectured that strong $\tau$-abstraction would imply constructivity under minor technical conditions. Existing positive results concern linear structural causal models under additional assumptions. We address the general case. First, a fixed edgeless low-level model admits strong abstractions with arbitrary high-level DAGs, even with analytic mechanisms, independent full-support noise, and a full-rank abstraction mapping. Second, for strong $\tau$-abstractions, preservation of intervention composition on the full canonical domain is sufficient for constructivity. This condition is stronger than constructivity of the mapping: natural aggregations admit canonical interventions that violate composition preservation. Such interventions also cause failures in published abstraction results. We restrict the low-level domain while preserving every realizable high-level intervention. For a surjective mapping with full high-level intervention coverage, composition preservation on either restricted domain is then equivalent to constructivity. Finally, a controlled nonlinear learning experiment uses composition constraints to improve prediction over unregularized learning when input correlations change.