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

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 transport preorder on SCI computational problems, prove that it is a preorder, derive a transport-saturation sufficient criterion extending the principal-source package, and show that the associated transport degrees need not form a lattice in full generality. We analyze the natural decoder classes \(\mathscr R_{\mathrm{cont}}\) and \(\mathscr R_{\mathrm{Bor}}\): on the full class the corresponding quotients are not upper semilattices, while on the nondegenerate subclass the preorder is upward and downward directed. Finally, we exhibit two natural positive families realizing the principal transport mechanism: exact integration on compact intervals and a fixed-window spectral decision family obtained by block-diagonal stabilization.