Whether the number of β -steps in the λ-calculus can be taken as a reasonable time 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 also strong call-by-value evaluation is reasonable for time, 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 examples of families that the machine executes in time logarithmic in the number of β-steps.

Extending the λ-calculus with a construct for sharing, such as let expressions, enables a special representation of terms: iterated applications are decomposed by introducing sharing points in between any two of them, reducing to the case where applications have only values as immediate subterms.

The λ-calculus is a handy formalism to specify the evaluation of higher-order programs. It is not very handy, however, when one interprets the specification as an execution mechanism, because terms can grow exponentially with the number of β-steps. This is why implementations of functional languages and proof assistants always rely on some form of sharing of subterms.