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

2025-12-02

AI Summary

This paper investigates the computational complexity of the first-order theory of the reals (FOTR) and introduces the quantumized real complexity class $ ext{Q}mathbb{R}$, systematically formalizing the difficulty of alternating games over continuous domains. To analyze the “Devil’s Game”—a two-player adversarial model with continuous action spaces—the authors employ polynomial-time reductions, real Turing machines, and geometric constraint analysis. Their main contributions are: (1) establishing that the restricted FOTR subclass FOTR$_ ext{INV}$—featuring only reciprocal and addition constraints with variables confined to compact intervals—is $ ext{Q}mathbb{R}$-complete; and (2) proving $ ext{Q}mathbb{R}$-completeness for three classical geometric games: bin-packing games, planar expansion games, and order-type games. This work establishes $ ext{Q}mathbb{R}$ as a natural and complete class for characterizing the complexity of continuous-strategy games, and for the first time constructs rigorous completeness bridges between FOTR and classical geometric games, providing a foundational framework at the intersection of real-number logic and computational game theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Game Theory and Economic Paradigms: Mechanism Design
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Security and Privacy: Large-scale security measurements

Abstract

We introduce the complexity class Quantified Reals ($ ext{Q}mathbb{R}$). Let FOTR be the set of true sentences in the first-order theory of the reals. A language $L$ is in $ ext{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 $ ext{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 $ ext{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 $ ext{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 $ ext{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 $ ext{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.

Problem

• Introduces and studies the complexity class Qℝ related to real number theory.
• Shows several continuous games are Qℝ-complete, linking complexity to game theory.
• Focuses on devil's games with alternating turns and continuum move options.

Innovation

• Introduces complexity class Qℝ for first-order theory of reals
• Proves Qℝ-completeness for Packing Game with continuous moves
• Shows Qℝ-completeness for Planar Extension Game with straight-line placement