Causal commutative arrows

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 noninterference property of concurrent computations. We also add an init operator 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 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 compiled with the Glasgow Haskell Compiler (GHC), the overall methodology can result in significant speedups.

Settable and non-interfering signal functions for FRP

“…Although stateful signal functions can be achieved in a variety of ways, we follow Liu et al [14] in the use of a delay 1 operator along with loop. In this model, we use the loop as a feedback mechanism, allowing an auxiliary output containing the state to be fed back as an input, and we use the delay to prevent an infinite feedback loop.…”

“…In this model, we use the loop as a feedback mechanism, allowing an auxiliary output containing the state to be fed back as an input, and we use the delay to prevent an infinite feedback loop. Indeed, Liu et al [14] even demonstrate that integral can be defined using this method:…”

“…Therefore, regardless of what data the signals contain, the arrow's overall behavior is fixed. For example, CCA [14] relies on this restriction to optimize an FRP program and often improve its performance in GHC by an order of magnitude.…”

“…These signals may also be streams of events, and indeed, by allowing signal functions themselves to be the values carried by these events (in essence, signals of signal functions), one can conveniently make discrete changes in program behavior by "switching" into and out of these signal functions. This higherorder notion of switching is common among many FRP systems, in particular those based on arrows, such as Yampa.Although convenient, the power of switching is often an overkill and can pose problems for certain types of program optimization (such as causal commutative arrows [14]), as it causes the structure of the program to change dynamically at run-time. Without a notion of just-in-time compilation or related idea, which itself is beset with problems, such optimizations are not possible at compile time.…”

“…Although convenient, the power of switching is often an overkill and can pose problems for certain types of program optimization (such as causal commutative arrows [14]), as it causes the structure of the program to change dynamically at run-time. Without a notion of just-in-time compilation or related idea, which itself is beset with problems, such optimizations are not possible at compile time.…”