We examine some flexible notions of constraint satisfaction, observing some relationships between model theoretic notions of universal Horn class membership and robust satisfiability. We show the \texttt{NP}-completeness of $2$-robust monotone 1-in-3 3SAT in order to give very small examples of finite algebras with \texttt{NP}-hard variety membership problem. In particular we give a $3$-element algebra with this property, and solve a widely stated problem by showing that the $6$-element Brandt monoid has \texttt{NP}-hard variety membership problem. These are the smallest possible sizes for a general algebra and a semigroup to exhibit \texttt{NP}-hardness for the membership problem of finite algebras in finitely generated varieties.

翻译：我们研究了一些柔性约束可满足性概念，观察到通用 Horn 类成员关系的模型论概念与鲁棒可满足性之间的某些联系。通过证明 $2$-鲁棒单调 1-in-3 3SAT 的 \texttt{NP} 完全性，我们构造了具有 \texttt{NP} 困难簇成员关系问题的有限代数的极小实例。特别地，我们给出了一个具有该性质的 $3$ 元代数，并通过证明 $6$ 元 Brandt 幺半群具有 \texttt{NP} 困难的簇成员关系问题，解决了一个被广泛提及的公开问题。对于一般代数与半群而言，这是在有限生成簇中有限代数的成员关系问题呈现 \texttt{NP} 困难性的最小可能规模。