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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. 

Modern data analytics applications, such as knowledge graph reasoning and machine learning, typically involve recursion through aggregation. Such computations pose great challenges to both system builders and theoreticians: first, to derive simple yet powerful abstractions for these computations; second, to define and study the semantics for the abstractions; third, to devise optimization techniques for these computations. In recent work we presented a generalization of Datalog called Datalog, which addresses these challenges. Datalog is a simple abstraction, which allows aggregates to be interleaved with recursion, and retains much of the simplicity and elegance of Datalog. We define its formal semantics based on an algebraic structure called Partially Ordered Pre-Semirings, and illustrate through several examples how Datalog can be used for a variety of applications. Finally, we describe a new optimization rule for Datalog, called the FGH-rule, then illustrate the FGH-rule on several examples, including a simple magic-set rewriting, generalized semi-naïve evaluation, and a bill-of-material example, and briefly discuss the implementation of the FGH-rule and present some experimental validation of its effectiveness.