Automatic amortised analysis of dynamic memory allocation for lazy functional programs

Abstract

This paper describes the first successful attempt, of which we are aware, to define an automatic, type-based static analysis of resource bounds for lazy functional programs. Our analysis uses the automatic amortisation approach developed by Hofmann and Jost, which was previously restricted to eager evaluation. In this paper, we extend this work to a lazy setting by capturing the costs of unevaluated expressions in type annotations and by amortising the payment of these costs using a notion of lazy potential. We present our analysis as a proof system for predicting heap allocations of a minimal functional language (including higher-order functions and recursive data types) and define a formal cost model based on Launchbury's natural semantics for lazy evaluation. We prove the soundness of our analysis with respect to the cost model. Our approach is illustrated by a number of representative and non-trivial examples that have been analysed using a prototype implementation of our analysis.

Type-Based Allocation Analysis for Co-recursion in Lazy Functional Languages

“…A significant difference from our previous work [3] lies in the typing of letexpressions. Typing let x = e 1 in e 2 allows lower costs for the bound variable x within the recursively-defined expression e 1 .…”

“…For concreteness, we choose to bound the number of heap allocations performed by a standard operational semantics for lazy evaluation. The work presented here complements our previous analysis for lazy functional programs [3]. We have previously shown that amortisation allows cost bounds to be determined for recursive definitions over finite data, but also that it does not contribute to the analysis of co-recursion over infinite data.…”

“…A publicly accessible web version with several editable examples (including the ones presented here) is available at http://kashmir.dcc.fc.up.pt/cgi/lazy.cgi. The implementation combines the analysis presented in this paper with our earlier amortised analysis [3]. These techniques complement each other: amortisation deals with recursive definitions over finite data, while our new system deals with co-recursive definitions on infinite data.…”

“…For simplicity, we consider only single-variable let-bindings: multiple let-bindings can be encoded, if needed, using pairs and projections. Unlike [3], we do not require a distinguished let-construct for introducing constructors here. Figure 1 defines an instrumented version of Launchbury's semantics, using a simple cost counting mechanism, against which we prove the soundness of our cost analysis.…”

“…In fact, these two examples do not admit annotated types using just the amortised analysis described in [3]. Table 1 presents a short assessment of the quality of the upper-bounds obtained from our analysis by comparison with the exact costs obtained from an implementation of the operational semantics of Section 2.…”