A continuous constraint satisfaction problem (CCSP) is a constraint satisfaction problem (CSP) with an interval domain $U \subset \mathbb{R}$. We engage in a systematic study to classify CCSPs that are complete of the Existential Theory of the Reals, i.e., ER-complete. To define this class, we first consider the problem ETR, which also stands for Existential Theory of the Reals. In an instance of this problem we are given some sentence of the form $\exists x_1, \ldots, x_n \in \mathbb{R} : \Phi(x_1, \ldots, x_n)$, where $\Phi$ is a well-formed quantifier-free formula consisting of the symbols $\{0, 1, +, \cdot, \geq, >, \wedge, \vee, \neg\}$, the goal is to check whether this sentence is true. Now the class ER is the family of all problems that admit a polynomial-time many-one reduction to ETR. It is known that NP $\subseteq$ ER $\subseteq$ PSPACE. We restrict our attention on CCSPs with addition constraints ($x + y = z$) and some other mild technical condition. Previously, it was shown that multiplication constraints ($x \cdot y = z$), squaring constraints ($x^2 = y$), or inversion constraints ($x\cdot y = 1$) are sufficient to establish ER-completeness. We extend this in the strongest possible sense for equality constraints as follows. We show that CCSPs (with addition constraints and some other mild technical condition) that have any one well-behaved curved equality constraint ($f(x,y) = 0$) are ER-complete. We further extend our results to inequality constraints. We show that any well-behaved convexly curved and any well-behaved concavely curved inequality constraint ($f(x,y) \geq 0$ and $g(x,y) \geq 0$) imply ER-completeness on the class of such CCSPs.

翻译：连续约束满足问题(CCSP)是定义在区间域(domain) $U \subset \mathbb{R}$ 上的约束满足问题(CSP)。我们开展系统性研究，以分类那些属于实数存在性理论(Existential Theory of the Reals)完备类(即ER完备类)的CCSP。为定义该类别，我们首先考虑ETR问题(即实数存在性理论)。在该问题的实例中，给定形如 $\exists x_1, \ldots, x_n \in \mathbb{R} : \Phi(x_1, \ldots, x_n)$ 的语句，其中$\Phi$是由符号$\{0, 1, +, \cdot, \geq, >, \wedge, \vee, \neg\}$构成的无量词合式公式，目标是判定该语句是否为真。ER类定义为所有可多项式时间多一归约至ETR的问题族。已知NP $\subseteq$ ER $\subseteq$ PSPACE。我们将注意力限制在具有加法约束($x + y = z$)及其他温和技术性条件的CCSP上。此前研究表明，乘法约束($x \cdot y = z$)、平方约束($x^2 = y$)或倒数约束($x\cdot y = 1$)足以建立ER完备性。我们针对等式约束在最强的意义上对此进行推广：证明包含任意良态弯曲等式约束($f(x,y) = 0$)的CCSP(含加法约束及其他温和技术性条件)是ER完备的。进一步将结果推广至不等式约束：证明任意良态凸弯曲不等式约束和任意良态凹弯曲不等式约束($f(x,y) \geq 0$ 与 $g(x,y) \geq 0$)均可导出该类CCSP的ER完备性。