Completeness of Graphical Regular Computation

Prove that every regular computation of a structured kernel from a set of structured kernels can be implemented by a finite sequence of the paper’s graphical operations, including simplification, c-component decomposition, gluing, and revealing operations.

Background

Structured kernels augment ordinary kernels with graph-based sparsity and latent-variable information. The paper establishes several sound graphical transformations, but does not prove that these transformations are exhaustive. The conjecture asks whether every regular computation whose input and output are structured kernels has a corresponding finite graphical construction.

References

If a structured kernel $\mu(\graphStructural,L)$ can be regularly computed from a set $\knowledgeSet$ of structured kernels, then there is a finite sequence of graphical operations computing $\mu(\graphStructural,L)$ from $\knowledgeSet$.

— Symmetries and Causality: Causal Effect Identification Beyond IID Data  (2609.03697 - Rabel et al., 3 Sep 2026) in Conjecture 2, Section 2.3, “Graphical Operations”

The question as to whether such a revealing operation can be computed more generally is quite subtle and discussed in \ref{apdx:reveal}.

— Symmetries and Causality: Causal Effect Identification Beyond IID Data  (2609.03697 - Rabel et al., 3 Sep 2026) in Section 7.3, Appendix “Revealing Operations,” following Example “C-Subgraph Revealing”