An Ode to Arrows

We study a number of embedded DSLs for autonomous ordinary differential equations (autonomous ODEs) in Haskell. A naive implementation based on the lazy tower of derivatives is straightforward but has serious time and space leaks due to the loss of sharing when handling cyclic and infinite data structures. In seeking a solution to fix this problem, we explore a number of DSLs ranging from shallow to deep embeddings, and middle-grounds in between. We advocate a solution based on arrows, an abstract notion of computation that offers both a succinct representation and an effective implementation. Arrows are ubiquitous in their combinator style that happens to capture both sharing and recursion elegantly. We further relate our arrow-based DSL to a more constrained form of arrows called causal commutative arrows, the normalization of which leads to a staged compilation technique improving ODE performance by orders of magnitude.