Finiteness conditions for fixed point iteration

Abstract

This paper provides a link between the formulation of static program analyses using the framework of abstract interpretation (popular for functional languages) and using the more classical framework of data flow analysis (popular for imperative languages). In particular we show how the classical notions of fastness, rapidity and k-boundedness carry over to the abstract interpretation framework and how this may be used to bound the number of times a functional should be unfolded in order to yield the fixed point. This is supplemented with a number of results on how to calculate the bounds for iterative forms (as for tail recursion), for linear forms (as for one nested recursive call), and for primitive recursive forms. In some cases this improves the "worst case" results of [9], but more importantly it gives much better "average case" results.

Finiteness Conditions for Strictness Analysis

“…This result was stated but not proved in [6]. P One undesirable feature of the above lemma is that we need to take the length of a composite function.…”

“…Given a functional H as might arise from the above strictness analysis the aim now is to find sufficient conditions for H to be additive and k-bounded for some hopefully low value of k. We begin with a simple fact and a brief review of the main results from [6]; then we move on to a more general treatment of the operators and £.…”

“…Some of them are treated in greater detail in [2]. P Instead of using the measure len m , of Section 3 we shall be able to obtain better results by following [6] and defining…”

“…In [6] we developed first results along this line. Section 2 contains a brief review of the main results of [6] but with a change of emphasis that is more suited to a structural approach (for functional programs).…”

“…Section 2 contains a brief review of the main results of [6] but with a change of emphasis that is more suited to a structural approach (for functional programs). Section 3 then develops our main results for simple strictness analysis and in Section 4 we add the analysis of lists using Wadler's "inverse cons" method.…”