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The Existential Theory of the Reals as a Complexity Class: A Compendium

Marcus Schaefer, Jean Cardinal, Tillmann Miltzow

TL;DR

This survey provides a comprehensive, dual-lensed account of the existential theory of the reals, ∃R: its foundational language through ETR, and the associated real-computation complexity via the BSS model, including universality results that drive ∃R-hardness. It compiles a broad compendium of problems that are ∃R-complete or ∃R-hard across logic, algebra, geometry, games, and machine learning, and it presents key tools (Ball Theorem, gap theorems, quantifier elimination) that enable hardness proofs and membership results. The work highlights the intricate connections between geometry, topology, and algebra, showing how universality phenomena explain why seemingly simple geometric realizability problems exhibit deep computational complexity. It also points to future directions in breadth, depth, higher levels of the real hierarchy, and potential unifying algorithmic techniques, aiming to establish ∃R as a standard framework in theoretical computer science. The treatise emphasizes the practical and theoretical significance of ∃R for understanding continuous solution spaces and their computational boundaries in diverse domains.

Abstract

We survey the complexity class $\exists \mathbb{R}$, which captures the complexity of deciding the existential theory of the reals. The class $\exists \mathbb{R}$ has roots in two different traditions, one based on the Blum-Shub-Smale model of real computation, and the other following work by Mnëv and Shor on the universality of realization spaces of oriented matroids. Over the years the number of problems for which $\exists \mathbb{R}$ rather than NP has turned out to be the proper way of measuring their complexity has grown, particularly in the fields of computational geometry, graph drawing, game theory, and some areas in logic and algebra. $\exists \mathbb{R}$ has also started appearing in the context of machine learning, Markov decision processes, and probabilistic reasoning. We have aimed at collecting a comprehensive compendium of problems complete and hard for $\exists \mathbb{R}$, as well as a long list of open problems. The compendium is presented in the third part of our survey; a tour through the compendium and the areas it touches on makes up the second part. The first part introduces the reader to the existential theory of the reals as a complexity class, discussing its history, motivation and prospects as well as some technical aspects.

The Existential Theory of the Reals as a Complexity Class: A Compendium

TL;DR

This survey provides a comprehensive, dual-lensed account of the existential theory of the reals, ∃R: its foundational language through ETR, and the associated real-computation complexity via the BSS model, including universality results that drive ∃R-hardness. It compiles a broad compendium of problems that are ∃R-complete or ∃R-hard across logic, algebra, geometry, games, and machine learning, and it presents key tools (Ball Theorem, gap theorems, quantifier elimination) that enable hardness proofs and membership results. The work highlights the intricate connections between geometry, topology, and algebra, showing how universality phenomena explain why seemingly simple geometric realizability problems exhibit deep computational complexity. It also points to future directions in breadth, depth, higher levels of the real hierarchy, and potential unifying algorithmic techniques, aiming to establish ∃R as a standard framework in theoretical computer science. The treatise emphasizes the practical and theoretical significance of ∃R for understanding continuous solution spaces and their computational boundaries in diverse domains.

Abstract

We survey the complexity class , which captures the complexity of deciding the existential theory of the reals. The class has roots in two different traditions, one based on the Blum-Shub-Smale model of real computation, and the other following work by Mnëv and Shor on the universality of realization spaces of oriented matroids. Over the years the number of problems for which rather than NP has turned out to be the proper way of measuring their complexity has grown, particularly in the fields of computational geometry, graph drawing, game theory, and some areas in logic and algebra. has also started appearing in the context of machine learning, Markov decision processes, and probabilistic reasoning. We have aimed at collecting a comprehensive compendium of problems complete and hard for , as well as a long list of open problems. The compendium is presented in the third part of our survey; a tour through the compendium and the areas it touches on makes up the second part. The first part introduces the reader to the existential theory of the reals as a complexity class, discussing its history, motivation and prospects as well as some technical aspects.
Paper Structure (74 sections, 9 theorems, 18 equations, 12 figures, 6 tables)

This paper contains 74 sections, 9 theorems, 18 equations, 12 figures, 6 tables.

Table of Contents

  1. Background and Overview
  2. A Brief Introduction to the Existential Theory of the Reals
  3. What is the Existential Theory of the Reals?
  4. Where is the Existential Theory of the Reals?
  5. Universalities
  6. Irrationality.
  7. Precision.
  8. Algebraic Universality.
  9. Proving $\exists\mathbb{R}$-Completeness
  10. Algebraic Encoding.
  11. Stretchability Encoding.
  12. What is next for $\exists\mathbb{R}$?
  13. Breadth.
  14. Depth.
  15. Parameterized Theory of $\exists\mathbb{R}$.
  16. ...and 59 more sections

Key Result

Theorem 2.1

If $f_i \in \mathbb{Z}\xspace[x_1, \ldots, x_n]$, $i \in [k]$ is a family of polynomials with coefficients of bitlength at most $L$ and total degree at most $d$, then every connected component of $\{x \in \mathbb{R}\xspace^n: f_i(x) \geq 0$ for all $i \in [k]\}$ intersects the ball of radius $2^{(L+

Figures (12)

  • Figure 1: A gallery with $16$ vertices that can be guarded with $4 < \lfloor 16/3 \rfloor$ guards.
  • Figure 2: Left: Feasibility problem in 3d; are there $(x,y) \in \mathbb{R}\xspace^2$ such that $f(x,y) = 0$? Middle: A pseudoline arrangement (top) and an isomorphic line arrangement (bottom). Right: A straight-line drawing of the complete $4$-partite graph $K_{2,2,2,2}$ with $8$ crossings, which is minimal for straight-line drawings. If the drawing does not have the be straight-line, six crossings are sufficient.
  • Figure 3: The Perles configuration.
  • Figure 4: The white guard needs to be placed anywhere on the dotted line segment in order to see all the marked vertices. We can interpret the position of the guard as a value in a real-valued interval.
  • Figure 5: Simplified von Staudt addition gadget, left, and multiplication gadget, right. The original gadgets are projective, not Euclidean.
  • ...and 7 more figures

Theorems & Definitions (12)

  • Conjecture
  • Remark : Addition with von Staudt
  • Remark : Multiplication with von Staudt
  • Theorem 2.1: Vorob ev and Grigoriev
  • Theorem 2.2: Goodman,Pollack, and Sturmfels
  • Theorem 2.3: Goodman,Pollack, and Sturmfels
  • Theorem 2.4: Schaefer
  • Theorem 2.5: Jeronimo, Perruci
  • Lemma 2.6: Schaefer, Štefankovič
  • Lemma 2.7
  • ...and 2 more