Recursive queries have been traditionally studied in the framework of datalog, a language that restricts recursion to monotone queries over sets, which is guaranteed to converge in polynomial time in the size of the input. But modern big data systems require recursive computations beyond the Boolean space. In this article, we study the convergence of datalog when it is interpreted over an arbitrary semiring. We consider an ordered semiring, define the semantics of a datalog program as a least fixpoint in this semiring, and study the number of steps required to reach that fixpoint, if ever. We identify algebraic properties of the semiring that correspond to certain convergence properties of datalog programs. Finally, we describe a class of ordered semirings on which one can use the semi-naïve evaluation algorithm on any datalog program.

传统上，递归查询是在数据记录框架中研究的，数据记录是一种将递归限制为集合上单调查询的语言，保证在输入大小的多项式时间内收敛。但现代大数据系统需要布尔空间之外的递归计算。在本文中，我们研究了在任意半环上解释数据记录时的收敛性。我们考虑一个有序半环，将数据记录程序的语义定义为该半环中的最小固定点，并研究达到该固定点（如果有的话）所需的步骤数。我们确定了与数据记录程序的某些收敛特性相对应的半环的代数特性。最后，我们描述了一类有序半环，可以在任何数据记录程序上使用半朴素评估算法。