On the Relative Usefulness of Fireballs

In CSL-LICS 2014, Accattoli and Dal Lago [1] showed that there is an implementation of the ordinary (i.e. strong, pure, call-by-name) λ-calculus into models like RAM machines which is polynomial in the number of β-steps, answering a long-standing question. The key ingredient was the use of a calculus with useful sharing, a new notion whose complexity was shown to be polynomial, but whose implementation was not explored. This paper, meant to be complementary, studies useful sharing in a call-by-value scenario and from a practical point of view. We introduce the Fireball Calculus, a natural extension of call-by-value to open terms, that is an intermediary step towards the strong case, and we present three results. First, we adapt useful sharing, refining the meta-theory. Then, we introduce the GLAMOUr a simple abstract machine implementing the Fireball Calculus extended with useful sharing. Its key feature is that usefulness of a step is tested-surprisingly-in constant time. Third, we provide a further optimisation that leads to an implementation having only a linear overhead with respect to the number of β-steps.