Semantics of lambda-I and of other substructure lambda calculi

Abstract

The ordinary untyped λ-calculus (the main object of study in [3]) will be denoted here by λK. Church originally introduced the λI-calculus, which can be understood as the λK-calculus without weakening: one cannot throw away variables. Similarly there is a affine calculus λA without contraction: there, one cannot duplicate variables. There is also a linear calculus λL in which one has neither weakening nor contraction. In λL variables occur precisely once.

We give a systematic description of the semantics of these four calculi. It starts with two sorts of domain theoretic models: graph models and filter models (of intersection types) are constructed for each of these calculi. Later on, we describe an appropriate categorical way to capture such structures in terms of monoidal categories (with diagonals or projections).

The research reported here was done during the academic year '91–'92 at the Pure Maths Dep., Univ. of Cambridge, UK. An early version is [5].