Jens Palsberg
Abstract
Flow analyses of untyped higher-order functional programs have in the past decade been presented by Ayers, Bondorf, Consel, Jones, Heintze, Sestoft, Shivers, Steckler, Wand, and others. The analyses are usually defined as abstract interpretations and are used for rather different tasks such as type recovery, globalization, and binding-time analysis. The analyses all contain a global closure analysis that computes information about higher-order control-flow. Sestoft proved in 1989 and 1991 that closure analysis is correct with respect to call-by-name and call-by-value semantics, but it remained open if correctness holds for arbitrary beta-reduction.
This article answers the question; both closure analysis and others are correct with respect to arbitrary beta-reduction. We also prove a subject-reduction result: closure information is still valid after beta-reduction. The core of our proof technique is to define closure analysis using a constraint system. The constraint system is equivalent to the closure analysis of Bondorf, which in turn is based on Sestoft's.
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Index Terms
- Closure analysis in constraint form
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Reviews
Pierre Jouvelot
Closure analysis extends the classical algorithm that computes function call graphs to the realm of higher-order first-class functions. This analysis is important when optimization of such languages is called for, for instance when using partial evaluation or code inlining. Even when limited to untyped functional languages, the literature offers many different ways to tackle this problem. This paper shows that Sestoft's closure analysis, which has already been proven correct with respect to both call-by-name and call-by-value semantics, is valid for arbitrary reduction orders. This proof relies on a constraint-based formulation, related to Bondorf's, of such a closure analysis. Although this paper is easy to read, I would only recommend it to researchers already involved in closure analysis or, at least, the static semantics of functional languages. Indeed, although the result is new, it appears to be a mostly theoretical extension of existing research and, as such, seems of limited practical import to practitioners.
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