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Beyond the Existential Theory of the Reals

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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 Bürgisser and Cucker on the complexity of properties of semialgebraic sets, including the Hausdorff distance problem also studied by Jungeblut, Kleist, and Miltzow.

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Notes

  1. We do allow abbreviations, e.g. we write powers such as \(x^4\), but we understand that to be shorthand for \(x\cdot x \cdot x \cdot x\).

  2. We will keep using both symbols, since they can easily be expressed by exchanging lhs and rhs of the inequality and flipping the sign.

  3. This elimination result relies on the discrete setting.

  4. As the reviewer points out, Koiran’s result is even stronger, it applies in the BSS-model.

  5. At the second level we do not need Koiran’s method for quantifier elimination [37], we can argue directly: \(\forall ^*\) can be rewritten as \(\forall \exists \), and the additional existential quantifiers merged with the existing ones.

  6. We’d like to write \(\exists \mathbb {R}^{\textbf{coNP}^{\exists \mathbb {R}}}\) for this class, but this notation suggests an oracle model for \(\exists \mathbb {R}\), and the details of that would still need to be worked out.

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Acknowledgements

We would like to thank an anonymous reader who suggested the paper by Bürgisser and Cucker [7], which led to the addition of the results in Section 3.

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  1. School of Computing, DePaul University, Chicago, 60604, IL, USA

    Marcus Schaefer

  2. Computer Science Department, University of Rochester, Rochester, 14627-0226, NY, USA

    Daniel Štefankovič

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Schaefer, M., Štefankovič, D. Beyond the Existential Theory of the Reals. Theory Comput Syst 68, 195–226 (2024). https://doi.org/10.1007/s00224-023-10151-x

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