一句话：将FO2上WFOMC的多项式时间结果扩展到唯一存在量词扩展与U1片段，并给出前缀类复杂度完全分类。
本文研究对称加权一阶模型计数（WFOMC）的复杂度边界。已知FO2句子的WFOMC可在多项式时间内求解，而某些FO3句子是#P1完全的。作者将FO2的多项式时间结果沿两个独立方向推广：一是形如φ∧∀x∃=1 y ψ(x,y)的句子，其中φ、ψ均为FO2，这推广了带函数关系符号的FO2；二是均匀一维片段U1，它是二变量逻辑的扩展，可处理任意元数关系符号。论文证明这些扩展的WFOMC仍具多项式时间算法，并进一步给出一阶前缀类关于WFOMC是多项式时间还是#P1完全的一个完整分类。
It was recently shown by van den Broeck at al. that the symmetric weighted first-order model counting problem (WFOMC) for sentences of two-variable logic FO2 is in polynomial time, while it is #P1-complete for some FO3-sentences. We extend the result for FO2 in two independent directions: to sentences of the form φ∧∀x∃=1 y ψ (x, y) with φ and ψ formulated in FO2 and to sentences of the uniform one-dimensional fragment U1 of FO, a recently introduced extension of two-variable logic with the capacity to deal with relation symbols of all arities. We note that the former generalizes the extension of FO2 with a functional relation symbol. We also identify a complete classification of first-order prefix classes according to whether WFOMC is in polynomial time or #P1-complete.