Strong Call-by-Value is Reasonable, Implosively

Abstract

Whether the number of β-steps in the λ-calculus can be taken as a reasonable cost model (that is, polynomially related to the one of Turing machines) is a delicate problem, which depends on the notion of evaluation strategy. Since the nineties, it is known that weak (that is, out of abstractions) call-by-value evaluation is a reasonable strategy while Lévy's optimal parallel strategy, which is strong (that is, it reduces everywhere), is not. The strong case turned out to be subtler than the weak one. In 2014 Accattoli and Dal Lago have shown that strong call-by-name is reasonable, by introducing a new form of useful sharing and, later, an abstract machine with an overhead quadratic in the number of β-steps.Here we show that strong call-by-value evaluation is also reasonable, via a new abstract machine realizing useful sharing and having a linear overhead. Moreover, our machine uses a new mix of sharing techniques, adding on top of useful sharing a form of implosive sharing, which on some terms brings an exponential speed-up. We give an example of family that the machine executes in time logarithmic in the number of β-steps.

A strong call-by-need calculus

“…There are also some potential improvements to consider. For example, as made apparent by the rules of the abstract machine, our calculus could avoid substituting variables that are not applied (rule lsv-base), following [Yos93,ACSC21], but it opens the question of how to characterize the normal forms then. Another venue for investigation is how this work interacts with fully lazy sharing, which avoids more duplications but whose properties are tightly related to weak reduction [Bal12].…”