Tail Recursion Modulo Context: An Equational Approach

Abstract

The tail-recursion modulo cons transformation can rewrite functions that are not quite tail-recursive into a tail-recursive form that can be executed efficiently. In this article we generalize tail recursion modulo cons (TRMc) to modulo contexts (TRMC), and calculate a general TRMC algorithm from its specification. We can instantiate our general algorithm by providing an implementation of application and composition on abstract contexts, and showing that our context laws hold. We provide some known instantiations of TRMC, namely modulo evaluation contexts (CPS), and associative operations, and further instantiantions not so commonly associated with TRMC, such as defunctionalized evaluation contexts, monoids, semirings, exponents, and cons products. We study the modulo cons instantiation in particular and prove that an instantiation using Minamide's hole calculus is sound. We also calculate a second instantiation in terms of the Perceus heap semantics to precisely reason about the soundness of in-place update. While all previous approaches to TRMc fail in the presence of non-linear control (for example induced by call/cc, shift/reset or algebraic effect handlers), we can elegantly extend the heap semantics to a hybrid approach which dynamically adapts to non-linear control flow. We have a full implementation of hybrid TRMc in the Koka language and our benchmark shows the TRMc transformed functions are always as fast or faster than using manual alternatives.