We study the computational complexity of counting constraint satisfaction problems (#CSPs) whose constraints assign complex numbers to Boolean inputs when the corresponding constraint hypergraphs are acyclic. These problems are called acyclic #CSPs or succinctly, #ACSPs. We wish to determine the computational complexity of all such #ACSPs when arbitrary unary constraints are freely available. Depending on whether we further allow or disallow the free use of the specific constraint XOR (binary disequality), we present two complexity classifications of the #ACSPs according to the types of constraints used for the problems. When XOR is freely available, we first obtain a complete dichotomy classification. On the contrary, when XOR is not available for free, we then obtain a trichotomy classification. To deal with an acyclic nature of constraints in those classifications, we develop a new technical tool called acyclic-T-constructibility or AT-constructibility, and we exploit it to analyze a complexity upper bound of each #ACSPs.

翻译：我们研究计数约束满足问题（#CSPs）的计算复杂度，其中约束对布尔输入分配复数值，且对应约束超图是无环的。此类问题称为无环#CSPs，简记为#ACSPs。我们旨在确定当任意一元约束可自由使用时，所有此类#ACSPs的计算复杂度。根据是否进一步允许或禁止自由使用特定约束XOR（二元不等关系），我们依据问题所用约束类型，给出了#ACSPs的两类复杂度分类。当XOR可自由使用时，我们首先获得完整的二分分类。反之，当XOR不可自由使用时，我们进而得到三分分类。为处理这些分类中约束的无环特性，我们开发了一种称为“无环T-可构造性”（AT-可构造性）的新技术工具，并利用该工具分析每个#ACSPs的复杂度上界。