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.
Similar content being viewed by others
Notes
We do allow abbreviations, e.g. we write powers such as \(x^4\), but we understand that to be shorthand for \(x\cdot x \cdot x \cdot x\).
We will keep using both symbols, since they can easily be expressed by exchanging lhs and rhs of the inequality and flipping the sign.
This elimination result relies on the discrete setting.
As the reviewer points out, Koiran’s result is even stronger, it applies in the BSS-model.
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.
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.
References
Jungeblut, P., Kleist, L., Miltzow, T.: The complexity of the Hausdorff distance. In: 38th International Symposium on Computational Geometry. LIPIcs. Leibniz Int. Proc. Inform., vol. 224, pp. 48–17. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2022). https://doi.org/10.4230/lipics.socg.2022.48
D’Costa, J., Lefaucheux, E., Neumann, E., Ouaknine, J., Worrell, J.: On the complexity of the escape problem for linear dynamical systems over compact semialgebraic sets. In: Bonchi, F., Puglisi, S.J. (eds.) 46th International Symposium on Mathematical Foundations of Computer Science. LIPIcs. Leibniz Int. Proc. Inform., vol. 202, pp. 33–21. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2021). https://doi.org/10.4230/LIPIcs.MFCS.2021.33
Dobbins, M.G., Kleist, L., Miltzow, T., Rzążewski, P.: \(\forall \exists \mathbb{R}\)-completeness and area-universality. In: Graph-theoretic Concepts in Computer Science. Lecture Notes in Comput. Sci., vol. 11159, pp. 164–175. Springer, ??? (2018). https://doi.org/10.1007/978-3-030-00256-5_14
Dobbins, M.G., Kleist, L., Miltzow, T., Rzążewski, P.: Completeness for the complexity class \(\forall \exists \mathbb{R} \) and area-universality. Discrete Comput. Geom. (2022). https://doi.org/10.1007/s00454-022-00381-0
Blanc, M., Hansen, K.A.: Computational complexity of multi-player evolutionarily stable strategies. ArXiv e-prints. (2022) https://doi.org/10.48550/ARXIV.2203.07407
Schaefer, M., Štefankovič, D.: Fixed points, Nash equilibria, and the existential theory of the reals. Theory Comput. Syst. 60(2), 172–193 (2017). https://doi.org/10.1007/s00224-015-9662-0
Bürgisser, P., Cucker, F.: Exotic quantifiers, complexity classes, and complete problems. Found. Comput. Math. 9(2), 135–170 (2009). https://doi.org/10.1007/s10208-007-9006-9
Tarski, A.: A Decision Method for Elementary Algebra And Geometry, p. 60. The Rand Corporation, Santa Monica, Calif (1948)
Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry, 2nd edn. Algorithms and Computation in Mathematics, vol. 10, p. 662. Springer, Berlin (2006). https://doi.org/10.1007/3-540-33099-2
Abrahamsen, M., Adamaszek, A., Miltzow, T.: The art gallery problem is \(\exists \mathbb{R} \)-complete. J. ACM. 69(1), 4–70 (2022). https://doi.org/10.1145/3486220
Abrahamsen, M.: Covering polygons is even harder. In: 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science—FOCS 2021, pp. 375–386. IEEE Computer Soc., Los Alamitos, CA (2022). https://doi.org/10.1109/FOCS52979.2021.00045
Schaefer, M.: The complexity of angular resolution. J. Graph Algorithms Appl. 27(7), 565–580 (2023). https://doi.org/10.7155/jgaa.00634
Miltzow, T., Schmiermann, R.F.: On classifying continuous constraint satisfaction problems. In: 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science—FOCS 2021, pp. 781–791. IEEE Computer Soc., Los Alamitos, CA (2022). https://doi.org/10.1109/FOCS52979.2021.00081
Berthelsen, M.L.T., Hansen, K.A.: On the computational complexity of decision problems about multi-player Nash equilibria. Theory Comput. Syst. 66(3), 519–545 (2022). https://doi.org/10.1007/s00224-022-10080-1
Bertschinger, D., Hertrich, C., Jungeblut, P., Miltzow, T., Weber, S.: Training fully connected neural networks is \(\exists \mathbb{R}\)-complete. ArXiv e-prints. (2022) https://doi.org/10.48550/arXiv.2204.01368
Matoušek, J.: Intersection graphs of segments and \(\exists \mathbb{R}\). ArXiv e-prints. (2014). arXiv:1406.2636
Wikipedia: Existential theory of the reals. (Online; accessed 6-June-2022) (2012). http://en.wikipedia.org/wiki/Existential_theory_of_the_reals
Bienstock, D.: Some provably hard crossing number problems. Discrete Comput. Geom. 6(5), 443–459 (1991). https://doi.org/10.1007/BF02574701
Kratochvíl, J., Matoušek, J.: Intersection graphs of segments. J. Combin. Theory Ser. B. 62(2), 289–315 (1994). https://doi.org/10.1006/jctb.1994.1071
Mnëv, N.E.: The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In: Topology and Geometry—Rohlin Seminar. Lecture Notes in Math., vol. 1346, pp. 527–543. Springer, Berlin (1988)
Shor, P.W.: Stretchability of pseudolines is NP-hard. In: Applied Geometry and Discrete Mathematics. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4, pp. 531–554. Amer. Math. Soc., Providence, RI (1991). https://doi.org/10.1090/dimacs/004/41
Shpilka, A., Volkovich, I.: Read-once polynomial identity testing. Comput. Complexity. 24(3), 477–532 (2015). https://doi.org/10.1007/s00037-015-0105-8
Sontag, E.D.: Real addition and the polynomial hierarchy. Inform. Process. Lett. 20(3), 115–120 (1985). https://doi.org/10.1016/0020-0190(85)90076-6
Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation, p. 453. Springer, New York (1998). https://doi.org/10.1007/978-1-4612-0701-6
Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull. Amer. Math. Soc. (N.S.). 21(1), 1–46 (1989) https://doi.org/10.1090/S0273-0979-1989-15750-9
Bürgisser, P., Cucker, F.: Counting complexity classes for numeric computations. II. Algebraic and semialgebraic sets. J. Complexity. 22(2), 147–191 (2006) https://doi.org/10.1145/1007352.1007425
Basu, S., Zell, T.: Polynomial hierarchy, Betti numbers, and a real analogue of Toda’s theorem. Found. Comput. Math. 10(4), 429–454 (2010). https://doi.org/10.1007/s10208-010-9062-4
Solernó, P.: Effective Łojasiewicz inequalities in semialgebraic geometry. Appl. Algebra Engrg. Comm. Comput. 2(1), 2–14 (1991). https://doi.org/10.1007/BF01810850
Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals. I. Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the existential theory of the reals. J. Symbolic Comput. 13(3), 255–299 (1992) https://doi.org/10.1016/S0747-7171(10)80003-3
Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals. II. The general decision problem. Preliminaries for quantifier elimination. J. Symbolic Comput. 13(3), 301–327 (1992) https://doi.org/10.1016/S0747-7171(10)80004-5
Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals. III. Quantifier elimination. J. Symbolic Comput. 13(3), 329–352 (1992). https://doi.org/10.1016/S0747-7171(10)80005-7
Jeronimo, G., Perrucci, D.: On the minimum of a positive polynomial over the standard simplex. J. Symbolic Comput. 45(4), 434–442 (2010). https://doi.org/10.1016/j.jsc.2010.01.001
Ouaknine, J., Worrell, J.: Ultimate positivity is decidable for simple linear recurrence sequences. ArXiv e-prints. (2017) https://doi.org/10.48550/arXiv.1309.1914, arXiv:1309.1914
Schaefer, M.: Realizability of graphs and linkages. In: Thirty Essays on Geometric Graph Theory, pp. 461–482. Springer, New York (2013). https://doi.org/10.1007/978-1-4614-0110-0_24
Cucker, F., Rosselló, F.: On the complexity of some problems for the Blum, Shub & Smale model. In: LATIN ’92 (São Paulo, 1992). Lecture Notes in Comput. Sci., vol. 583, pp. 117–129. Springer, ??? (1992). https://doi.org/10.1007/BFb0023823
Boyd, S., Vandenberghe, L.: Convex Optimization, p. 716. Cambridge University Press, Cambridge (2004). https://doi.org/10.1017/CBO9780511804441
Koiran, P.: The real dimension problem is \({\rm NP}_{\mathbb{R} }\)-complete. J. Complexity. 15(2), 227–238 (1999). https://doi.org/10.1006/jcom.1999.0502
Szeider, S.: Generalizations of matched CNF formulas. Ann. Math. Artif. Intell. 43(1–4), 223–238 (2005). https://doi.org/10.1007/s10472-005-0432-6
Le, H.P., Safey El Din, M., Wolff, T.: Computing the real isolated points of an algebraic hypersurface. In: ISSAC’20—Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation, pp. 297–304. ACM, New York (2020). https://doi.org/10.1145/3373207.3404049
Lovász, L.: Semidefinite programs and combinatorial optimization. In: Recent Advances in Algorithms and Combinatorics. CMS Books Math./Ouvrages Math. SMC, vol. 11, pp. 137–194. Springer,??? (2003). https://doi.org/10.1007/0-387-22444-0_6
Ramana, M.V.: An exact duality theory for semidefinite programming and its complexity implications. Math. Programming. 77(2, Ser. B), 129–162 (1997) https://doi.org/10.1016/S0025-5610(96)00082-2
Goemans, M.X.: Semidefinite programming and combinatorial optimization. In: Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), vol. Extra Vol. III, pp. 657–666 (1998). https://doi.org/10.4171/DMS/1-3/63
Allender, E., Bürgisser, P., Kjeldgaard-Pedersen, J., Miltersen, P.B.: On the complexity of numerical analysis. SIAM J. Comput. 38(5), 1987–2006 (2009). https://doi.org/10.1137/070697926
Erickson, J., Hoog, I., Miltzow, T.: Smoothing the gap between NP and ER. In: 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, pp. 1022–1033. IEEE Computer Soc., Los Alamitos, CA (2020). https://doi.org/10.1109/FOCS46700.2020.00099
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00224-023-10151-x
Keywords
- Existential theory of the real numbers
- Theory of the reals
- Real hierarchy
- Computational complexity
- Semialgebraic sets
- Hausdorff distance