Local Type Reconstruction by Means of Symbolic Fixed Point Iteration

We convert, via a version that uses constraints, a type inference system for strictness analysis into an algorithm which given an expression finds the set of possible typings. Although this set in general does not possess a minimal element, it can be represented compactly by means of symbolic expressions in normal form – such expressions have the property that once values for the constraint variables with negative polarity have been supplied it is straight-forward to compute the minimal values for the constraint variables with positive polarity. The normalization process works on the fly, i.e. by a leaf-to-root traversal of the inference tree. 1 Background and Motivation Recently much interest has been devoted to the formulation of program analysis in terms of inference systems, as opposed to e.g. abstract interpretation (for the relationship between those methods see e.g. [Jen91]). This approach is appealing since it separates the question “what is done?” from the question “how is it done?”. Of course, the latter issue (that is, implementation of the inference system) has to be dealt with, and a very popular method (often inspired by [Hen91]) is to (re)formulate the inference system in terms of constraints and then come up with an algorithm for solving these. In this paper we shall continue this trend, the type system in question being one for strictness analysis (the system has also been presented in [Amt93a]). The characteristic features of our approach are: – for constraints we define a notion of normal form which distinguishes between constraint variables according to their polarity (i.e. whether they occur in co/contravariant position in the type). – the constraints are normalized on the fly, that is during a leaf-to-root traversal of the inference tree (as opposed to first collecting them all and then solve them). – during the normalization process, some approximations are made – without losing precision, however, since the approximation only concerns suboptimal solutions. – the normalization process involves symbolic fixed point iteration.