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.
翻译：我们证明,在更高级别上,真实理论的完整性是一个强有力的概念(在改变签名和约束量化标准领域的情况下 ) 。 这种修补认识到了等级体系中的差距,并导致各种计算问题更加完整的结果。 我们展示了几个完整的问题家族,这些问题可以在未来实现完整性时在真正的等级体系中产生结果。 作为B\"{u}rgisser和Cucker的应用程序,我们用B\\"{u}rgiser和Cucker来放大一些关于半地理数据集性质复杂性的结果,包括Jungeblut、Kleist和Miltzow也研究过的Hausdorff距离问题。
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