From Witness-Space Sharpness To Family-Pointwise Exactness For The Solvability Complexity Index
Christopher Sorg

TL;DR
This paper explores how to formulate and upgrade the Solvability Complexity Index (SCI) for families of problems, establishing conditions for exactness and analyzing transport preorders with applications to integration and spectral decision problems.
Contribution
It formalizes the distinctions in exactness notions for problem families, proves upgrade theorems, and analyzes transport preorders with practical examples.
Findings
Witness-space sharpness coincides with worst-case exactness.
Upgrade theorems guarantee conditions for exactness enhancement.
Transport degrees may not form a lattice in general.
Abstract
We study how exact Solvability Complexity Index (SCI) statements should be formulated for families of computational problems rather than for single problems. While the equality \(\mathrm{SCI}_G (\mathcal P)=k\) is unambiguous for an individual computational problem \(\mathcal P\), the family setting requires one to distinguish family-pointwise exactness, witness-space sharpness, and worst-case exactness. We formalize this trichotomy, prove that witness-space sharpness coincides with worst-case exactness but is, in general, strictly weaker than family-pointwise exactness, and give a canonical source-family example witnessing the strictness. We then establish two positive upgrade theorems: an abstract pullback principle and a concrete finite-query criterion guaranteeing that witness-space sharpness upgrades to family-pointwise exactness. Next, we introduce a decoder-regular finite-query…
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