The Machinery of Interaction

Abstract

This paper revisits the Interaction Abstract Machine (IAM), a machine based on Girard's Geometry of Interaction, introduced by Mackie and Danos & Regnier. It is an unusual machine, not relying on environments, presented on linear logic proof nets, and whose soundness proof is convoluted and passes through various other formalisms. Here we provide a new direct proof of its correctness, based on a variant of Sands's improvements, a natural notion of bisimulation. Moreover, our proof is carried out on a new presentation of the IAM, defined as a machine acting directly on λ-terms, rather than on linear logic proof nets. CCS CONCEPTS • Theory of computation → Lambda calculus; Abstract machines.

The (In)Efficiency of interaction

“…Additionally, on states such as (λx .D⟨x⟩, C, L, l•T ), the logged position l has shape (x, λx .D, L ′ ), so that transition → bt2 can always applyÐthis is a consequence of the exhaustible state invariant in Sect. 6, as shown in Accattoli et al [2020b].…”

“…In this section we provide an overview of the Interaction Abstract Machine (IAM). We adopt the λ-calculus presentation of the IAM, rather called λIAM and recently developed by Accattoli et al [2020b]Ðwe refer to their work for an in-depth study of the λIAM. The literature usually studies the (λ)IAM with respect to head evaluation of potentially open terms.…”

“…Keep in mind that the λIAM is an unusual machine, and that finding it hard to grasp is normalÐprobably, the next sections about the λJAM and the KAM provide clarifying insights. Also, in [Accattoli et al 2020b] there is an alternative explanation of the λIAM, that may be helpful, together with the relationship with proof nets, which is however not needed here.…”

“…Last, note that our example is a linear λ-term. The technical report associated to this paper [Accattoli et al 2020a] contains an example showing how the λIAM uses logged positions to deal with duplications.…”

“…If the λIAM starts on the initial state s t , the execution may either never stop or end in a state s of the shape s = (λx .u, C, L, ϵ). The fact that no other shapes are possible for s is proved in Accattoli et al [2020b]. The tape and direction invariant guarantees that the machine never stops because the log or the tape have not enough logged positions to apply a → var , → bt1 , or a → arg transition.…”