One of the goals of this paper is to demonstrate that denotational semantics is useful for operational issues like implementation of functional languages by abstract machines. This is exemplified in a tutorial way by studying the case of extensional untyped call-by-name λ-calculus with Felleisen's control operator [Cscr ]. We derive the transition rules for an abstract machine from a continuation semantics which appears as a generalization of the ¬¬-translation known from logic. The resulting abstract machine appears as an extension of Krivine's machine implementing head reduction. Though the result, namely Krivine's machine, is well known our method of deriving it from continuation semantics is new and applicable to other languages (as e.g. call-by-value variants). Further new results are that Scott's D∞-models are all instances of continuation models. Moreover, we extend our continuation semantics to Parigot's λμ-calculus from which we derive an extension of Krivine's machine for λμ-calculus. The relation between continuation semantics and the abstract machines is made precise by proving computational adequacy results employing an elegant method introduced by Pitts.

We present an extension of the Hindley–Milner type system that supports a generous class of type constructors called functors, and provide a parametrically polymorphic algorithm for their mapping, i.e. for applying a function to each datum appearing in a value of constructed type. The algorithm comes from shape theory, which provides a uniform method for locating data within a shape. The resulting system is Church–Rosser and strongly normalizing, and supports type inference. Several different semantics are possible, which affects the choice of constants in the language, and are used to illustrate the relationship to polytypic programming.

A string-formatting function such as printf in C seemingly requires dependent types, because its control string determines the rest of its arguments. Examples:

formula here

We show how changing the representation of the control string makes it possible to program printf in ML (which does not allow dependent types). The result is well typed and perceptibly more efficient than the corresponding library functions in Standard ML of New Jersey and in Caml.

In this paper we describe the discrete interval encoding tree for storing subsets of types having a total order and a predecessor and a successor function. In the following, we consider for simplicity only the case for integer sets; the generalization is not difficult.

The discrete interval encoding tree is based on the observation that the set of integers {i[mid ]a[les ]i[les ]b} can be perfectly represented by the closed interval [a, b]. The general idea is to represent a set by a binary search tree of integers in which maximal adjacent subsets are each represented by an interval. For example, inserting the sequence of numbers 6, 9, 2, 13, 8, 14, 10, 7, 5 into a binary search tree, respectively, into a discrete interval encoding tree results in the tree structures shown in figure 1.

The efficiency of the interval representation, both in terms of space and time, improves with the density of the set, i.e. with the number of adjacencies between set elements. So what we propose is a ‘diet’ (discrete interval encoding tree) for ‘fat’ sets in the sense of ‘the same amount of information with less nodes’.