LAL Is Square: Representation and Expressiveness in Light Affine Logic

We focus on how the choice of input-output representation has a crucial impact on the expressiveness of so-called “logics of polynomial time.” Our analysis illustrates this dependence in the context of Light Affine Logic (LAL), which is both a restricted version of Linear Logic, and a primitive functional programming language with restricted sharing of arguments. By slightly relaxing representation conventions, we derive doubly-exponential expressiveness bounds for this “logic of polynomial time.” We emphasize that squaring is the unifying idea that relates upper bounds on cut elimination for LAL with lower bounds on representation. Representation issues arise in the simulation of DTIME[2 n ], where we construct a uniform family of proof-nets encoding a Turing Machine; specifically, the dependence on n only affects the number of enclosing boxes. A related technical improvement is the simulation of DTIME[n] in depth O(log k) LAL proof-nets. The resulting upper bounds on cut elimination then satisfy the properties of a first-class polynomial Turing Machine simulation, where there is a fixed polynomial slowdown in the simulation of any polynomial computation.