Completeness of Regular Computation

Establish whether every finite functional that uniquely computes a result from finitely many generic probability kernels can be represented using finite combinations of tensor products, left-disintegrations, right-marginalizations, and transpositions, excluding limits and fixed points.

Background

The paper defines regular functionals as finite compositions of tensor products, disintegrations, marginalizations, and transpositions. The authors conjecture that, for generic kernels without additional internal structure such as linearity, these operations exhaust all finite, uniquely defined computations. Establishing this would characterize the computational power of the kernel calculus used throughout the paper and clarify the scope of its completeness claims.

References

If a functional uniquely computes a result from finitely many generic kernels then it is regular (Def. \ref{def:regular_functionals}, Rmk. \ref{rmk:regular_limits}).

— Symmetries and Causality: Causal Effect Identification Beyond IID Data  (2609.03697 - Rabel et al., 3 Sep 2026) in Conjecture 1, Section 2.1, “Structured Kernels”

What is needed is a principled theory of polynomially-bounded stochastic kernels and sized spaces. Such a theory might use notions such as Nash, quasi-Nash, or polynomially bounded definable geometry~\citep{debrouwereNoteCompositionOperators2022,shavivTemperedDistributionsSchwartz2020}, and it could potentially have a similar structure of concrete sheaves to $DQBS$, but consider more complex compatibility conditions based on tail estimates. We leave it as another direction for future research.

— Compositional Information Geometry of Probabilistic Programs  (2610.17886 - Karwowski et al., 13 Oct 2026) in Section 8, subsection "Polynomially controlled maps"