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 big-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 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.

The Negligible and Yet Subtle Cost of Pattern Matching

“…Sands, Gustavsson, and Moran have then showed that ordinary abstract machines for call-by-name and call-by-need closed λ-calculi are also reasonable [34]. Similar results have also been obtained by Martini and Dal Lago (by combining the results in [22] and [21]), and then the whole question has been finely decomposed and studied by Accattoli, Barenbaum, Mazza, and Sacerdoti Coen [6,14]. It is thus fair to say that the number of β-steps is the time cost model of the closed λ-calculus.…”

“…To be formal, we should make both tasks explicit in the form of an abstract machine. The work of Accattoli and coauthors [6,9,4] has however repeatedly showed that these tasks require an overhead linear in the number of β-steps and the size of the initial term, and in some cases even logarithmic in the size of the initial term (see the companion paper [7])-in the terminology of this paper, the costs of search and α-renaming are negligible. At the technical level, for the study of cost models we mostly adopt the techniques and the terminology (linear substitution calculus, subterm invariant, harmony, etc) developed by Accattoli and his coauthors (Dal Lago, Barenbaum, Mazza, Sacerdoti Coen, Guerrieri) in [10,6,9,11,4].…”

“…The terminology multiplicative / exponential comes from the connection with linear logic, that is however kept hidden here-see [6] for more details-just bear in mind that the exponential rule does not have an exponential cost, the name is due to other reasons.…”

“…The choice of LIME for our study is justified by the similarity with the formalisms used in the studies on functional cost models [6,9,11,4] and with the one used in the Coq abstract machine designed by Barras [16]. A further justification is the fact that it is conservative with respect to CbV closed λ-calculus in a sense that we are now going to explain.…”

“…Here we adopt the presentation of CbNeed of Accattoli, Barenbaum, and Mazza [6], resting on the linear substitution calculus. With respect to LIME, the only difference is that the environment is integrated inside the term itself and the notion of program disappears-in CbNeed is not possible to disentangle the term and the environment, unless more data structures are used.…”