This article is a brief review of the type free λ-calculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λ-calculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λ-calculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λ-calculus is introduced and some of its advantages are outlined.

We define an equivalence relation on lambda-terms called shuffle-equivalence which attempts to capture the notion of reductional equivalence on strongly normalizing terms. The aim of reductional equivalence is to characterize the evaluation behavior of programs. The shuffle-equivalence classes are shown to divide the classes of beta-equal strongly normalising terms (programs which lead to the same final value/output) into smaller ones consisting of terms with similar evaluation behavior. We refine beta-reduction from a relation on terms to a relation on shuffle-equivalence classes, called shuffle-reduction, and show that this refinement captures existing generalisations of lambda-reduction. Shuffle-reduction allows one to make more redexes visible and to contract these newly visible redexes. Moreover, it allows more freedom in choosing the reduction path of a term, which can result in smaller terms along the reduction path if a clever reduction strategy is used. This can benefit both programming language...

In this paper, a notation influenced by de Bruijn's syntax of the λ-calculus is used to describe canonical forms of terms and an equivalence relation which divides terms into classes according to their reductional behaviour. We show that this notation helps describe canonical forms more elegantly than the classical notation. We de ne reduction modulo equivalence classes of terms up to the permutation of redexes in canonical forms and show that this reduction contains other notions of reductions in the literature including the -reduction of Regnier. We establish all the desirable properties of our reduction modulo equivalence classes. Then, we give two extensions of the Barendregt cube, one with the - reduction of Regnier and the other with our class reduction and show that the subject reduction property fails in each of these extensions. We then show that adding de nitions in the contexts of type derivations, enables each of these extensions to satisfy all the desirable properties of type systems, including subject reduction and strong normalisation.

In this paper, a notation inuenced by de Bruijn's syntax of the λ-calculus is used to describe canonical forms of terms and an equivalence relation which divides terms into classes according to their reductional behaviour. We show that this notation helps describe canonical forms more elegantly than the classical notation and we establish the desirable properties of our reduction modulo equivalence classes rather than single terms. Finally, we extend the cube consisting of eight type systems with class reduction and show that this extension satis es all the desirable properties of type systems.