Concurrent Construction of Proof-Nets

Abstract

The functional paradigm of computation has been widely investigated and given a solid mathematical foundation, initiated with the Curry-Howard isomorphism, then elaborated and extended in multiple ways. However, this paradigm is inadequate to capture many useful programming intuitions, arising in particular in the development of applications integrating distributed, autonomous components. Indeed, in this context, non-determinism and true concurrency are the rule, whereas functional programming stresses determinism, and, although it allows some degree of concurrency, it is more as a “nice feature to have” rather than a primary assumption.

This paper is part of a program the ambition of which is to provide a logical foundation to a set of programming intuitions which, until now, have not been adequately accounted for. In particular, we are interested in the intuitions which lie behind the concept of transaction, a powerful and essential concept in distributed component-based application development. This concept is independent of the application domain and usually captured in an abstract form in middleware architectural layers.

We claim here that proof-construction, and more precisely proof-net construction in Linear Logic, offers the adequate basis for our purpose. We outline the relation, which is of the same nature as the Curry-Howard isomorphism, between transactional concepts and mechanisms on one hand, and proof-net construction on the other. Finally, we describe an algorithm which performs concurrent proof-net construction, where each expansion step is viewed as a transaction. Conflicts between such transactions are minimised using general topological properties of proof-nets, based on a variant of the notion of “domination tree”, introduced and proved here.