Programming languages often come with type systems. Some of these are simple, others are sophisticated. As a stylistic representation of types in programming languages several versions of typed lambda calculus are studied. During the last 20 years many of these systems have appeared, so there is some need of classification. Working towards a taxonomy, Barendregt (1991) gives a fine-structure of the theory of constructions (Coquand and Huet 1988) in the form of a canonical cube of eight type systems ordered by inclusion. Berardi (1988) and Terlouw (1988) have independently generalized the method of constructing systems in the λ-cube. Moreover, Berardi (1988, 1990) showed that the generalized type systems are flexible enough to describe many logical systems. In that way the well-known propositions-as-types interpretation obtains a nice canonical form.

We present a modular proof of strong normalization for the Calculus of Constructions of Coquand and Huet (1985, 1988). This result was first proved by Coquand (1986), but our proof is more perspicious. The method consists of a little juggling with some systems in the cube of Barendregt (1989), which provides a fine structure of the calculus of constructions. It is proved that the strong normalization of the calculus of constructions is equivalent with the strong normalization of Fω.

In order to give the proof, we first establish some properties of various type systems. Therefore, we present a general framework of typed lambda calculi, including many well-known ones.

Polymorphic types are labels classifying both (a) defined components in a library and (b) contexts of free variables in partially written programs. It is proposed to help programmers make better use of software libraries by providing a system that, given (b), identifies candidates from (a) with matching types. Assuming at first that matching means unifying (i.e. having a common instance), efficient ways of implementing such a retrieval system are discussed and its likely effectiveness based on a quantitative study of currently available libraries is indicated. The applicative instance relation between types, which captures some intuitions about generalization/specialization is then introduced, and its use as the basis of a more flexible system is discussed.

This paper presents an application of functional programming: searching a domain for elements which satisfy certain constraints. We give a very general formulation of the problem and describe ‘generate and test’, ‘backtracking’ and ‘forward checking’ algorithms. We then introduce the concept of domain generating functions to capture a common optimization during the search process: using partial solutions to reduce the size of the search space. We compare the efficiency of the original algorithms and those using domain generating functions first with the ‘classical’ n-queens example, and then with a problem having larger domains to search which was inspired by an application in macromolecular structure determination. Using algorithms coded in Miranda, Haskell and Common Lisp, we show that a high order (lazy) functional language is a useful and efficient tool for prototyping search methods in large complex domains.

Programming languages which are capable of interpreting themselves have been fascinating computer scientists. Indeed, if this is possible then a ‘strange loop’ (in the sense of Hofstadter, 1979) is involved. Nevertheless, the phenomenon is a direct consequence of the existence of universal languages. Indeed, if all computable functions can be captured by a language, then so can the particular job of interpreting the code of a program of that language. Self-interpretation will be shown here to be possible in lambda calculus.

The set of λ-terms, notation Λ, is defined by the following abstract syntax

where

is the set {v, v′, v″, v′″,…} of variables. Arbitrary variables are usually denoted by x, y,z,… and λ-terms by M,N,L,…. A redex is a λ-term of the form

that is, the result of substituting N for (the free occurrences of) x in M. Stylistically, it can be said that λ-terms represent functional programs including their input. A reduction machine executes such terms by trying to reduce them to normal form; that is, redexes are continuously replaced by their contracta until hopefully no more redexes are present. If such a normal form can be reached, then this is the output of the functional program; otherwise, the program diverges.

The function remdup (also called mkset in some functional languages) removes duplicates from a given list. The following definition of remdup is standard and leads to a quadratic time algorithm

The operator (—) used in the last expression subtracts one list from another; its definition is

Defined in this way remdup x returns the list of distinct elements of x in the order in which they first appear in x. In other words, the position of remdup JC as a subsequence of x is lexically the smallest among all possible solutions.

Now let us change the problem and ask that remdup simply return the lexically least solution. Note the subtle difference between this version and the previous one: before, it was the position of the subsequence that was lexically the least, now it is the subsequence itself. To make the distinction clear, consider x = [1,4,2,4,3]. With the original definition remdup x returns [1,4,2,3]; the lexically least subsequence, however, is [1,2,4,3].

The question we are interested in is this: can we also find a quadratic time algorithm for the new problem? The answer turns out to be yes, but justifying it requires a bit of work.