We show how computation of left Kan extensions can be reduced to computation of free models of cartesian (finite-limit) theories. We discuss how the standard and parallel chase compute weakly free models of regular theories and free models of cartesian theories and compare the concept of “free model” with a similar concept from database theory known as “universal model”. We prove that, as algorithms for computing finite-free models of cartesian theories, the standard and parallel chase are complete under fairness assumptions. Finally, we describe an optimized implementation of the parallel chase specialized to left Kan extensions that achieves an order of magnitude improvement in our performance benchmarks compared to the next fastest left Kan extension algorithm we are aware of.

我们展示了如何将左 Kan 扩展的计算简化为笛卡尔（有限极限）理论的自由模型的计算。我们讨论了标准和并行追逐如何计算常规理论的弱自由模型和笛卡尔理论的自由模型，并将“自由模型”的概念与数据库理论中称为“通用模型”的类似概念进行比较。我们证明，作为计算笛卡尔理论的有限自由模型的算法，标准和并行追逐在公平假设下是完备的。最后，我们描述了一种专门针对左 Kan 扩展的并行追踪的优化实现，与我们所知道的下一个最快的左 Kan 扩展算法相比，它在我们的性能基准上实现了一个数量级的改进。