Launchbury and Peyton Jones came up with an ingenious idea for embedding regions of imperative programming in a pure functional language like Haskell. The key idea was based on a simple modification of Hindley-Milner's type system. Our first contribution is to propose a more natural encapsulation construct exploiting higher-order kinds, which achieves the same encapsulation effect, but avoids the ad hoc type parameter of the original proposal. The second contribution is a type safety result for encapsulation of strict state using both the original encapsulation construct and the newly introduced one. We establish this result in a more expressive context than the original proposal, namely in the context of the higher-order lambda-calculus. The third contribution is a type safety result for encapsulation of lazy state in the higher-order lambda-calculus. This result resolves an outstanding open problem on which previous proof attempts failed. In all cases, we formalize the intended implementations as simple big-step operational semantics on untyped terms, which capture interesting implementation details not captured by the reduction semantics proposed previously. 1

In this article we consider the polymorphic type checking of an imperative language. Our language contains variables, first-class references (pointers), and first-class functions. Variables, as in traditional imperative languages, are implicitly dereferenced, and their addresses (L-values) are not first-class values. Variables are easier to type check than references and, in many cases, lead to more general polymorphic types. We present a polymorphic type system for our language and prove that it is sound. Programs that use variables sometimes require weak types, as in Tofte’s type system for Standard ML, but such weak types arise far less frequently with variables than with references.