Causal commutative arrows and their optimization

Abstract

Arrows 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 called Yampa, 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 of commutative arrows that capture a kind of non-interference property of concurrent computations. We also add an init operator, and identify a crucial law that captures the causal nature of arrow effects. We call the resulting computational model causal commutative arrows.To study this class of computations in more detail, we define an extension to the simply typed lambda calculus called causal commutative arrows (CCA), and study its properties. Our key contribution is the identification of a normal form for CCA called causal 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 combined with stream fusion, the overall methodology can result in speed-ups of greater than two orders of magnitude.

A semantic model for graphical user interfaces

“…The restriction to arrows is roughly equivalent to first-order functional programming, though Sculthorpe and Nilsson [23] introduced additional combinators to recover higher-order and dynamic behavior while using dependent types to retain causality. In contrast, Liu, Cheng and Hudak [18] showed how to exploit this first-order structure to develop a very efficient compilation scheme (reminiscent of the Bohm-Jacopini theorem) from arrow programs to single-loop code.…”

“…There have been several attempts to resolve this difficulty [18,20], all of which build on the basic idea of using data abstraction to restrict the definable functions to the causal ones. In recent work [17], we have described a new solution to this problem which still blocks non-causal definitions, but does not surrender the flexibility of the original FRP model -in particular, its support for making free use of higher-order functions for building abstractions.…”

Asynchronous functional reactive programming for GUIs

“…Arrowized FRP (AFRP) [20,21,24,25,29] Signal functions can be embedded in Elm (see Section 4.3), which brings the benefits of discrete AFRP to Elm programmers, allowing programs to dynamically create graphical components, without requiring signals of signals. Elm's discrete semantics make it clear how to allow asynchrony and concurrency, which appears to be semantically problematic in a continuous AFRP.…”

Efficient and Compositional Higher-Order Streams

“…However, fitting the switching combinators into the arrow framework used by Yampa is not without problems, e.g. they have no place in the causal commutative normal form [9] of a signal function. Reactive [5] is a recent reformulation of Fran that fixes the starting time of all varying quantities to zero, therefore its general higher-order capabilities (i.e.…”