Causal commutative arrows

Abstract

AbstractArrows are a popular form of abstract computation. Being more general than monads, they are more broadly applicable, and, in particular, are a good abstraction for signal processing and dataflow computations. Most notably, arrows form the basis for a domain-specific language calledYampa, which has been used in a variety of concrete applications, including animation, robotics, sound synthesis, control systems, and graphical user interfaces. Our primary interest is in better understanding the class of abstract computations captured by Yampa. Unfortunately, arrows are not concrete enough to do this with precision. To remedy this situation, we introduce the concept ofcommutative arrowsthat capture a noninterference property of concurrent computations. We also add aninitoperator that captures the causal nature of arrow effects, and identify its associated law. To study this class of computations in more detail, we define an extension to arrows calledcausal commutative arrows(CCA), and study its properties. Our key contribution is the identification of a normal form for CCA calledcausal commutative normal form(CCNF). By defining a normalization procedure, we have developed an optimization strategy that yields dramatic improvements in performance over conventional implementations of arrows. We have implemented this technique in Haskell, and conducted benchmarks that validate the effectiveness of our approach. When compiled with the Glasgow Haskell Compiler (GHC), the overall methodology can result in significant speedups.