How Light Is Safe Recursion? Compositional Translations Between Languages of Polynomial Time

We investigate the relationship between two polynomial time languages: Bellantoni and Cook’s functional combinator language BC, and Girard’s Light Linear Logic LAL (as amended by Asperti). It is known that both languages characterize exactly the polynomial time functions. Despite this similarity, an open question has been to provide a compositional translation between the two languages: given a ptime function f described in a BC program pf , how can we translate pf into an LAL-program? Compositionality means that the translation of a program is a function of the translation of its constituent parts. The construction of such a compiler is fundamentally an exercise in functional programming. The best known such translation, due to Murawski and Ong, compiles only the linear fragment BC, where some variables can only be referenced once. We show that this fragment can be evaluated in logspace. Next, we prove that extensions of their approach to the full BC language cannot provide a correct translation into LAL. In fact, any translation that interprets integers and primitive recursion via inductive type codings fails. Finally, we provide a correct, compositional translation between the two languages. For this translation we generalize Turing machine simulations into an SECD machine, realized in LAL.