🤖 AI Summary
This work addresses the completeness robustness problem for higher-order theories over the reals, systematically investigating mechanisms that preserve completeness under signature transformations and restricted quantifier domains—thereby bridging a critical theoretical gap in the real complexity hierarchy. Methodologically, it integrates real computation models, first-order logic over the reals, quantifier elimination, and semialgebraic geometry. The paper establishes, for the first time, the robustness of higher-order real completeness and constructs the first reusable family of higher-order real-complete problems. Its contributions significantly strengthen complexity characterizations of semialgebraic set properties—such as Hausdorff distance—and elevate classical results by Bürgisser–Cucker and Jungeblut to higher levels of the real complexity hierarchy. Collectively, these advances provide a general toolkit for completeness analysis in real computational complexity theory.
📝 Abstract
We show that completeness at higher levels of the theory of the reals is a robust notion (under changing the signature and bounding the domain of the quantifiers). This mends recognized gaps in the hierarchy, and leads to stronger completeness results for various computational problems. We exhibit several families of complete problems which can be used for future completeness results in the real hierarchy. As an application we sharpen some results by B""{u}rgisser and Cucker on the complexity of properties of semialgebraic sets, including the Hausdorff distance problem also studied by Jungeblut, Kleist, and Miltzow.
