Devil's Games and $\text{Q}\mathbb{R}$: Continuous Games complete for the First-Order Theory of the Reals

TL;DR

This work defines the complexity class Quantified Reals (Qℝ) as problems polynomial-time reducible to the true sentences of the first-order theory of the reals. It introduces a robust reduction framework for devil's games—two players alternating moves with continuum choices—showing several natural games (FOTRINV, Packing, Planar Extension, Order Type) are Qℝ-complete. The core technical engine is a compactification technique that replaces unbounded first-order formulas with fully closed, bounded equivalents, enabling polynomial-time reductions via real RAM/real Turing machine reasoning. The results place these geometric-continuation games on par with existential real problems in hardness, while also providing a machine-model characterization and extensive discussion of implications for the real polynomial hierarchy and related decision problems. Altogether, the paper establishes a unified pathway to prove Qℝ-hardness across diverse continuous-game settings and highlights the profound computational implications of real-analytic, infinite-move structures.

Abstract

We introduce the complexity class Quantified Reals ($\text{Q}\mathbb{R}$). Let FOTR be the set of true sentences in the first-order theory of the reals. A language $L$ is in $\text{Q}\mathbb{R}$, if there is a polynomial time reduction from $L$ to FOTR. This seems the first time this complexity class is studied. We show that $\text{Q}\mathbb{R}$ can also be defined using real Turing machines. It is known that deciding FOTR requires at least exponential time unconditionally [Berman, 1980]. We focus on devil's games with two defining properties: (1) Players (human and devil) alternate turns and (2) each turn has a continuum of options. First, we show that FOTRINV is $\text{Q}\mathbb{R}$-complete. FOTRINV has only inversion and addition constraints and all variables are in a compact interval. FOTRINV is a stepping stone for further reductions. Second, we show that the Packing Game is $\text{Q}\mathbb{R}$-complete. In the Packing Game we are given a container and two sets of pieces. One set of pieces for the human and one set for the devil. The human and the devil alternate by placing a piece into the container. Both rotations and translations are allowed. The first player that cannot place a piece loses. Third, we show that the Planar Extension Game is $\text{Q}\mathbb{R}$-complete. We are given a partially drawn plane graph and the human and the devil alternate by placing vertices and the corresponding edges in a straight-line manner. The vertices and edges to be placed are prescribed before hand. The first player that cannot place a vertex loses. Finally, we show that the Order Type Game is $\text{Q}\mathbb{R}$-complete. We are given an order-type together with a linear order. The human and the devil alternate in placing a point in the Euclidean plane following the linear order. The first player that cannot place a point correctly loses.

Figures (37)

• Figure 1: The different levels of the real polynomial hierarchy with inclusions indicated by arrows.
• Figure 2: An illustration of a real RAM machine with a program/algorithm.
• Figure 3: Left: The human and the devil place alternatingly vertices in the plane and the incident straight edges. The black vertices and thick edges were part of the input. The vertices $2,3,4$ still need to be placed. Right: We see a square container and a set of pieces both for the human and the devil. Both players have already placed some pieces and the first player who cannot place a piece will loses.
• Figure 4: The orientation of a triple of points is either positive, negative, or zero.
• Figure 5: Left: The tightest known way to pack $11$ unit squares into a square container Gensane2005. Right: How would the devil and the human need to play if the table had the shape of a pentagon?

Theorems & Definitions (92)

• Definition 1: $\textsc{FOTRINV}$
• Theorem 2
• Definition 3: Real Polynomial Hierarchy
• Definition 4: $\forall_{k}\textsc{INV}$
• Theorem 5
• Theorem 6
• Definition 7: $\textsc{Packing Game}$
• Theorem 8
• Definition 9: $\textsc{Planar }\textsc{Extension }\textsc{Game}$
• Theorem 10