Counting CSP^d is the counting constraint satisfaction problem (# CSP in short) restricted to the instances where every variable occurs a multiple of d times. This paper revisits tractable structures in # CSP and gives a complexity classification theorem for # CSP^d with algebraic complex weights. The result unifies affine functions (stabilizer states in quantum information theory) and related variants such as the local affine functions, the discovery of which leads to all the recent progress on the complexity of Holant problems. The Holant is a framework that generalizes counting CSP. In the literature on Holant problems, weighted constraints are often expressed as tensors (vectors) such that projections and linear transformations help analyze the structure. This paper gives an example showing that different classes of constraints distinguished by these algebraic operations may share the same closure property.

计数CSP ^d被计数约束满足问题（＃限于其中每个变量发生的倍数的情况下，在短CSP）d次。本文在回访听话结构＃ CSP，并给出了一个复杂的分类定理＃ CSP ^d与代数复数权重。结果统一了仿射函数（量子信息理论中的稳定器状态）和相关变体，例如局部仿射函数，其发现导致了 Holant 问题复杂性的所有最新进展。Holant 是一个概括计数 CSP 的框架。在 Holant 问题的文献中，加权约束通常表示为张量（向量），这样投影和线性变换有助于分析结构。本文给出了一个例子，说明由这些代数运算区分的不同类别的约束可能共享相同的闭包性质。