Foundational Analysis Of The Solvability Complexity Index: The Weihrauch-SCI Intermediate Hierarchy And A Koopman Operator Example
Christopher Sorg
Abstract
The Solvability Complexity Index (SCI) provides an abstract notion of computing a target map $Ξ$ from finitely many oracle evaluations $Λ\subseteq \mathbb{C}$ via finite-height towers of pointwise limits. We first give a foundational analysis of what this extensional framework does and does not determine. We show that the SCI separation consistency is equivalent to a factorization of $Ξ$ through the full evaluation table, and we isolate the minimal logical role of $Λ$ as an information interface. To connect the SCI to Type-2 computability and Weihrauch reducibility, we give an effective enrichment for countable $Λ$ by viewing the evaluation table image $I_Λ\subseteq\mathbb{C}^{\mathbb{N}}$ as a represented space and factoring $Ξ$ as $\widehatΞ$. We then define the Weihrauch-SCI rank of a problem as the least number of iterated limit-oracles needed to compute it in the Weihrauch sense, i.e.\ the least $k$ such that $\widehatΞ\le_{W}\lim^{(k)}$, and prove well-posedness and representation invariance of this rank. A central negative result is that the unrestricted type-$G$ SCI model (arbitrary post-processing of finite oracle transcripts) is generally not comparable to Weihrauch/Type-2 complexity: finite-query factorizations collapse type-$G$ height, and analytic (non-Borel) decision problems yield examples with $\mathrm{SCI}_{G}=0$ but infinite Weihrauch-SCI rank. To recover a robust bridge, we introduce an intermediate SCI hierarchy by restricting the admissible base-level post-processing to regularity classes (continuous/Borel/Baire) and, optionally, to fixed-query versus adaptive-query policies. We prove that these restrictions form genuine hierarchies, and we establish comparison theorems showing what each restriction logically enforces (e.g.\ Borel towers compute only Borel targets; continuous-base towers yield finite Baire class).