Calculi of Generalized beta-Reduction and Explicit Substitutions: The Type-Free and Simply Typed Versions

Extending the λ-calculus with either explicit substitution or generalized reduction has been the subject of extensive research recently, and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalized reduction and explicit substitutions. We present a calculus, λgs, that combines a calculus of explicit substitution, λs, and a calculus with generalized reduction, λg. We believe that λgs is a useful extension of the λcalculus, because it allows postponement of work in two different but complementary ways. Moreover, λgs (and also λs) satisfies properties desirable for calculi of explicit substitutions and generalized reductions. In particular, we show that λgs preserves strong normalization, is a conservative extension of λg, and simulates β-reduction of λg and the classical λ-calculus. Furthermore, we study the simply typed versions of λs and λgs, and show that well-typed terms are strongly normalizing and that other properties, such as typing of subterms and subject reduction, hold. Our proof of the preservation of strong normalization (PSN) is based on the minimal derivation method. It is, however, much simpler, because we prove the commutation of arbitrary internal and external reductions. Moreover, we use one proof to show both the preservation of λ-strong normalization in λs and the preservation of λg-strong normalization in λgs. We remark that the technique of these proofs is not suitable for calculi without explicit substitutions (e.g., the preservation of λ-strong normalization in λg requires a different technique). 1 The Journal of Functional and Logic Programming 1998-2 Kamareddine, et al. Calculi of Generalized β-Reduction §1.1