Devil's Games and $\text{Q}\mathbb{R}$: Continuous Games complete for the First-Order Theory of the Reals
Authors
Lucas Meijer
Arnaud de Mesmay
Tillmann Miltzow
Marcus Schaefer
Jack Stade
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.
