Implicative Formulae in the "Proofs as Computations" Analogy

Abstract

In [As87] a correspondence between the subset of Linear Logic [Gi86] involving the conjunctive tensor product only and Place/Transition Petri Nets [Rei85] is established. In this correspondence, formulae are regarded as distributed states and provable sequents are computations in the net. Developing this idea, Martì-Oliet and Meseguer [MaM89] have suggested that all the other computations of Linear Logic, which do not have an immediate correspondence with Petri Nets, should be regarded as “gedanken” or idealized processes, providing a richer language for the specification and the study of properties of distributed computations. In this paper we apply this program to the fundamental connective of linear implication. We prove that the introduction of linear implication allows us to observe the net at a lower, more decentralized level of atomicity, where the preemption of each resource needed for the firing of a transition is represented as a separate move. We give a conservative theorem relating computations at different levels of abstraction. The categorical semantics establishes a tight correspondence among Petri nets, monoidal closed categories and tensor theories, reminiscent of the well known relation among functional languages, Cartesian closed categories and intuitionistic logic [LS86]. The identification of computations in the categorical model naturally suggests the generalisation of the notion of process [DMM89] at the lower level of atomicity.