Proofs as processes

Abstract

The main purpose of this short paper is to serve as an introduction to the following paper, “On the rr-calculus and linear logic”, by Gianluigi Bellin and Philip Scott. The circumstances from which it arises are as follows. In June 1991 I gave a lecture on “Proofs as processes” at the symposium held at Tel-Aviv University to celebrate Boris Trakhtenbrot’s 70th birthday [6]. Material from this lecture also appeared in lectures subsequently given at the International Category Theory Meeting in Montreal (June 1991) and the London Mathematical Society Symposium on Category Theory in Computer Science in Durham (July 1991). The material was also presented in my tutorial lecture on linear logic given at the International Logic Programming Symposium in San Diego in October 1991. The lecture notes for these talks, particularly the tutorial lecture, were quite widely circulated; however, I did not write up a paper for publication, for reasons shortly to be explained. My point of departure in this work was the “propositions as types” paradigm (encompassing such notions as “Curry-Howard isomorphism”, realizability, functional interpretation and BHK semantics) familiar from proof theory and typed functional programming (see e.g. [9]). In this paradigm, we have the correspondences