We study the problem of certifying programs combining imperative and functional features within the general framework of type theory. Type theory is a powerful specification language which is naturally suited for the proof of purely functional programs. To deal with imperative programs, we propose a logical interpretation of an annotated program as a partial proof of its specification. The construction of the corresponding partial proof term is based on a static analysis of the effects of the program which excludes aliases. The missing subterms in the partial proof term are seen as proof obligations, whose actual proofs are left to the user. We show that the validity of those proof obligations implies the total correctness of the program. This work has been implemented in the Coq proof assistant. It appears as a tactic taking an annotated program as argument and generating a set of proof obligations. Several nontrivial algorithms have been certified using this tactic.

We study a typing scheme derived from a semantic situation where a single category possesses several closed structures, corresponding to different varieties of function type. In this scheme typing contexts are trees built from two (or more) binary combining operations, or in short, bunches. Bunched typing and its logical counterpart, bunched implications, have arisen in joint work of the author and David Pym. The present paper gives a basic account of the type system, and then focusses on concrete models that illustrate how it may be understood in terms of resource access and sharing. The most basic system has two context-combining operations, and the structural rules of Weakening and Contraction are allowed for one but not the other. This system includes a multiplicative, or substructural, function type −∗ alongside the usual (additive) function type $\rightarrow$; it is dubbed the $\alpha\lambda$-calculus after its binders, $\alpha$ for the $\alpha$dditive binder and $\lambda$ for the multiplicative, or $\lambda$inear, binder. We show that the features of this system are, in a sense, complementary to calculi based on linear logic; it is incompatible with an interpretation where a multiplicative function uses its argument once, but perfectly compatible with a reading based on sharing of resources. This sharing interpretation is derived from syntactic control of interference, a type-theoretic method of controlling sharing of storage, and we show how bunch-based management of Contraction can be used to provide a more flexible type system for interference control.

Fusion is the process of removing intermediate data structures from modularly constructed functional programs. Short cut fusion is a particular fusion technique which uses a single, local transformation rule to fuse compositions of list-processing functions. Short cut fusion has traditionally been treated purely syntactically, and justifications for it have appealed either to intuition or to “free theorems” – even though the latter have not been known to hold in languages supporting higher-order polymorphic functions and fixpoint recursion. In this paper we use Pitts' recent demonstration that contextual equivalence in such languages is parametric to provide the first formal proof of the correctness of short cut fusion for them. In particular, we show that programs which have undergone short cut fusion are contextually equivalent to their unfused counterparts.

A combinator expression is flat if it can be written without parentheses, that is, if all applications nest to the left, never to the right. This note explores a simple method for flattening combinator expressions involving arbitrary combinators.