# Fast Left Kan Extensions Using The Chase

arXiv: https://arxiv.org/abs/2205.02425 · HTML: https://mlsanity.com/paper/2205.02425/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 May 2022.

## Built on by (most cited first, 2 of 2)

- [Learning Is a Kan Extension](https://mlsanity.com/paper/2502.13810/) (2025)
- [Learning Transfers: Kan Extensions of Topological Invariants of Learned Representations](https://mlsanity.com/paper/2606.07627/) (2026)

## Abstract (arXiv)

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.