The weak call-by-value λ-calculus is reasonable for both time and space

Abstract

We study the weak call-by-value λ-calculus as a model for computational complexity theory and establish the natural measures for time and space -the number of beta-reductions and the size of the largest term in a computation -as reasonable measures with respect to the invariance thesis of Slot and van Emde Boas [STOC 84]. More precisely, we show that, using those measures, Turing machines and the weak call-by-value λ-calculus can simulate each other within a polynomial overhead in time and a constant factor overhead in space for all computations that terminate in (encodings) of "true" or "false". We consider this result as a solution to the long-standing open problem, explicitly posed by Accattoli [ENTCS 18], of whether the natural measures for time and space of the λ-calculus are reasonable, at least in case of weak call-by-value evaluation.Our proof relies on a hybrid of two simulation strategies of reductions in the weak call-by-value λ-calculus by Turing machines, both of which are insufficient if taken alone. The first strategy is the most naive one in the sense that a reduction sequence is simulated precisely as given by the reduction rules; in particular, all substitutions are executed immediately. This simulation runs within a constant overhead in space, but the overhead in time might be exponential. The second strategy is heap-based and relies on structure sharing, similar to existing compilers of eager functional languages. This strategy only has a polynomial overhead in time, but the space consumption might require an additional factor of log n, which is essentially due to the size of the pointers required for this strategy. Our main contribution is the construction and verification of a space-aware interleaving of the two strategies, which is shown to yield both a constant overhead in space and a polynomial overhead in time. ACM Subject Classification Theory of computation → Computational complexity and cryptography; Mathematics of computing → Lambda calculusKeywords and phrases invariance thesis, lambda calculus, weak call-by-value reduction, time and space complexity, abstract machines Supplement Material A Coq formalisation of the results in Section 2 and Section 3 is available at https://ps.uni-saarland.de/extras/wcbv-reasonable.

The Space of Interaction (long version)

“…First of all, studying interesting space complexity classes such as L, requires being able to measure sub-linear space. There is a recent result in the literature about reasonable space for the λ-calculus, by Forster, Kunze, and Roth [24], but their cost model-namely, the size of the termcan only measure linear and super-linear space, and thus it is not a solution for the general problem. Now, showing that a cost model is reasonable requires studying the relationship with another known-to-be reasonable model, typically Turing machines.…”

“…The specific operator we type is the one used in the encoding of Turing machines in the λ-calculus used by Accattoli and Dal Lago in their study of reasonable time [3], [15], as well as by Forster, Kunze, and Roth in [24]. Seeing it as the natural way of encoding tail recursion, it follows that the IAM space performance is poor with tail recursion, and, in turn, with the natural way of encoding Turing machines.…”

Reasonable Space for the $λ$-Calculus, Logarithmically

“…The literature about the -calculus does offer results about space complexity, but they are all partial, as they either concern variants of the -calculus (Dal Lago and Schöpp [16,35], Mazza [30] and Ghica [22]), or they are not valid when the bounds in spaces are sub-linear (Forster et al [19]).…”

The Machinery of Interaction

“…For space, the situation is different. Only recently the problem has been tackled [31] and some preliminary and limited results have appeared. Then, environment machines store information for every β-step, therefore using space linear in time, which is the worst possible use of space 1 .…”