Environments and the complexity of abstract machines

Abstract

machines for functional languages rely on the notion of environment, a data structure storing the previously encountered and delayed beta-redexes. This paper provides a close analysis of the different approaches to define and implement environments. There are two main styles. The most common one is to have many local environments, one for every piece of code in the data structures of the machine. A minority of works instead uses a single global environment. Up to now, the two approaches have been considered equivalent, in particular at the level of the complexity of the overhead: they have both been used to obtain bilinear bounds, that is, linear in the number of beta steps and in the size of the initial term.We start by having a close look on global environments and how to implement them. Then we show that local environments admit implementations that are asymptotically faster than global environments, lowering the dependency from the size of the initial term from linear to logarithmic, thus improving the bounds in the literature. We then focus on a third style, split environments, that are in between local and global ones, and have the benefits of both. Finally, we provide a call-by-need machine with split environments for which we prove the new improved bounds on the overhead.Here we introduce the Milner Abstract Machine (MAM) [4,38], a machine with a single global environment for call-by-name evaluation.Machine components. The MAM is defined in Fig. 1. A machine state s is a triple (t, π , E) given by:• Code t: a term not considered up to α-equivalence, which is why it is over-lined;

The Negligible and Yet Subtle Cost of Pattern Matching

“…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].…”

Types by Need

“…Finally, we put Φ, Ψ and Ξ together in the following derivation Θ for t = (s(II))(II), where s := λx.λy.xx and n [n] (2,3) II : [n, n [n] ] app (4,5) s(II) : 0 n many (0,0) II : 0 app (5,5)…”

“…The three evaluations are presented as strategies of Accattoli and Kesner's Linear Substitution Calculus (shortened to LSC) [1,6], a calculus with a simple but expressive form of explicit sharing. The LSC is strongly related to linear logic [2], and provides a neat and manageable presentation of CbNeed, introduced by Accattoli, Barenbaum, and Mazza in [3], and further developed by various authors in [4,5,10,14,15,40,42]. Our type systems count evaluation steps by annotating typing rules in the exact same way, and the proofs of correctness and completeness all follow the exact same structure.…”

Sharing Equality is Linear

“…Our algorithm requires λ-terms to be represented as graphs. is choice is fair, since abstract machines with global environments such as those described by Acca oli and Barras in [2] do manipulate similar representations, and thus produce λ-graphs to be compared for sharing equality-moreover, it is essentially the same representation induced by the translation of λ-calculus into linear logic proof nets [1,29] up to explicit sharing nodes-namely ?-links-and explicit boxes. In this paper we do not consider explicit sharing nodes (that in λ-calculus syntax correspond to have sharing only on variables), but one can translate in linear time between that representation to ours (and viceversa) by collapsing these variables-as-sharing on their child if they are the child of some other node.…”