Many polyvariant program analyses have been studied in the 1990s, including k-CFA, polymorphic splitting, and the cartesian product algorithm. The idea of polyvariance is to analyze functions more than once and thereby obtain better precision for each call site. In this paper we present an equivalence theorem which relates a co-inductively-defined family of polyvariant flow analyses and a standard type system. The proof embodies a way of understanding polyvariant flow information in terms of union and intersection types, and, conversely, a way of understanding union and intersection types in terms of polyvariant flow information. We use the theorem as basis for a new flow-type system in the spirit of the λCIL-calculus of Wells, Dimock, Muller and Turbak, in which types are annotated with flow information. A flow-type system is useful as an interface between a flow-analysis algorithm and a program optimizer. Derived systematically via our equivalence theorem, our flow-type system should be a good interface to the family of polyvariant analyses that we study.

Destructive array update optimization is critical for writing scientific codes in functional languages. We present set constraints for an interprocedural update optimization that runs in polynomial time. This is a multi-pass optimization, involving interprocedural flow analyses for aliasing and liveness. We characterize and prove the soundness of these analyses using small-step operational semantics. We also prove that any sound liveness analysis induces a correct program transformation.

A fair amount has been written on the subject of reasoning about pointer algorithms. There was a peak about 1980 when everyone seemed to be tackling the formal verification of the Schorr–Waite marking algorithm, including Gries (1979, Morris (1982) and Topor (1979). Bornat (2000) writes: “The Schorr–Waite algorithm is the first mountain that any formalism for pointer aliasing should climb”. Then it went more or less quiet for a while, but in the last few years there has been a resurgence of interest, driven by new ideas in relational algebras (Möeller, 1993), in data refinement Butler (1999), in type theory (Hofmann, 2000; Walker and Morrisett, 2000), in novel kinds of assertion (Reynolds, 2000), and by the demands of mechanised reasoning (Bornat, 2000). Most approaches end up being based in the Floyd–Dijkstra–Hoare tradition with loops and invariant assertions. To be sure, when dealing with any recursively-defined linked structure some declarative notation has to be brought in to specify the problem, but no one to my knowledge has advocated a purely functional approach throughout. Mason (1988) comes close, but his Lisp expressions can be very impure. Möller (1999) also exploits an algebraic approach, and the structure of his paper has much in common with what follows.

This pearl explores the possibility of a simple functional approach to pointer manipulation algorithms.