Generalised -reduction and Explicit Substitutions 8th International Conference on Programming

Extending the-calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently which still has many open problems. Due to this reason, the properties of a calculus combining both generalised reduction and explicit substitutions have never been studied. This paper presents such a calculus sg and shows that it is a desirable extension of the-calculus. In particular , we show that sg preserves strong normalisation, is sound and it simulates classical-reduction. Furthermore, we study the simply typed-calculus extended with both generalised reduction and explicit substitution and show that well-typed terms are strongly normalising and that other properties such as subtyping and subject reduction hold. 1 Introduction 1.1 The-calculus with generalised reduction In ((x :: y :N)P)Q, the function starting with x and the argument P result in the redex (x :: y :N)P which when contracted will turn the function starting with y and Q into a redex. This fact has been exploited by many researchers and reduction has been extended so that the future redex based on the matching y and Q is given the same priority as the other redex. Attempts at generalising reduction can be summarized by three axioms: These rules attempt to make more redexes visible. C e.g., makes sure that y and Q form a redex even before the redex based on x and P is contracted. and hence both and C put adjacently next to its matching argument.