Complete Fusion for Stateful Streams: Equational Theory of Stateful Streams and Fusion as Normalization-by-Evaluation
Processing large amounts of data fast, in constant and small space is the point of stream processing and the reason for its increasing use. Alas, the most performant, imperative processing code tends to be almost impossible to read, let alone modify, reuse -- or write correctly. We present both a stream compilation theory and its implementation as a portable stream processing library Strymonas that lets us assemble complex stream pipelines just by plugging in simple combinators, and yet attain the performance of hand-written imperative loops and state machines. The library supports finite and infinite streams and offers a rich set of combinators. They may be freely composed, and yet the resulting convoluted imperative code has no traces of combinator abstractions: no closures or intermediate objects. The high-performance is portable and statically guaranteed, without relying on compiler or black-box optimizations. We greatly exceed in performance the available stream processing libraries in OCaml. The library generates C and OCaml code. The declaratively built Strymonas pipelines are all stateful. The stream state introduced in the library is not directly observable. Therefore, the Strymonas API looks like the familiar interface of `pure functional' combinators. Programmers may introduce their own stream state and share it across the pipeline. Strymonas has been developed in tandem with the equational theory of stateful streams. Our theoretical model represents all desired pipelines and guarantees the existence of unique normal forms, which are mappable to (fused) state machines. We describe the normalization algorithm, as a form of normalization-by-evaluation. The equational theory lets us state and prove the correctness of the complete fusion optimization.