Distilling abstract machines

Abstract

It is well-known that many environment-based abstract machines can be seen as strategies in lambda calculi with explicit substitutions (ES). Recently, graphical syntaxes and linear logic led to the linear substitution calculus (LSC), a new approach to ES that is halfway between small-step calculi and traditional calculi with ES. This paper studies the relationship between the LSC and environment-based abstract machines. While traditional calculi with ES simulate abstract machines, the LSC rather distills them: some transitions are simulated while others vanish, as they map to a notion of structural congruence. The distillation process unveils that abstract machines in fact implement weak linear head reduction, a notion of evaluation having a central role in the theory of linear logic. We show that such a pattern applies uniformly in call-by-name, call-by-value, and call-by-need, catching many machines in the literature. We start by distilling the KAM, the CEK, and a sketch of the ZINC, and then provide simplified versions of the SECD, the lazy KAM, and Sestoft's machine. Along the way we also introduce some new machines with global environments. Moreover, we show that distillation preserves the time complexity of the executions, i.e. the LSC is a complexity-preserving abstraction of abstract machines.

Efficient Implementation of Evaluation Strategies via Token-Guided Graph Rewriting

“…The proof is by developing the one-to-one correspondence between an evaluation by the sub-machine semantics and a "derivation" in the linear substitution calculus. This goes in the same way Accattoli et al analyse various abstract machines[2], especially the proof of the second equation[2, Thm. 11.3 & Thm.…”

“…We analyse how time-efficiently the token-guided graph-rewriting machine implements evaluation strategies, following the methodology developed by Accattoli et al [2,6,1]. The methodology tracks the number of beta-reduction steps in an evaluation in three steps: (I) bound the number of transitions required in implementing evaluation strategies, (II) estimate time cost of each transition, and (III) bound overall time cost of implementing evaluation strategies, by multiplying the number of transitions with time cost for each transition.…”

“…The challenge here is how to produce a unified framework which is flexible enough to analyse and guide the choices required by this trade-off. Recent studies by Accattoli et al [4,2,1] clearly establish classes of efficiency for a given evaluation strategy. They characterise efficiency by means of the number of beta-reduction applications required by the strategy, and introduce two efficiency classes, namely "efficient" and "reasonable."…”

Types by Need

“…Multi Types. Our typing framework is given by multi types, which is an alternative name for non-idempotent intersection types 3 . Multi types characterise termination properties exactly as intersection types, having moreover the advantages that they are closely related to (the relational semantics of) linear logic, their type derivations provide quantitative information about evaluation lengths, and the proof techniques are simpler-no need for the reducibility method.…”

Proof nets and the call-by-value λ-calculus

“…This paper is a longer version of the workshop paper [3]. Apart from updating the notation of [3] to that of other recent papers using the same formalism [5,9,6], it extends it with the detailed relationship between the value substitution calculus and the kernel calculus in Sect. 8.…”