Combining linear logic and size types for implicit complexity

Abstract

Several type systems have been proposed to statically control the time complexity of lambda-calculus programs and characterize complexity classes such as FPTIME or FEXPTIME. A first line of research stems from linear logic and defines type systems based on restricted versions of the "!" modality controlling duplication. An instance of this is light linear logic for polynomial time computation [Girard98]. A second perspective relies on the idea of tracking the size increase between input and output, and together with a restricted use of recursion, to deduce from that time complexity bounds. This second approach is illustrated for instance by non-size-increasing types [Hofmann99]. However both approaches suffer from limitations. The first one, that of linear logic, has a limited intensional expressivity, that is to say some natural polynomial time programs are not typable. As to the second approach it is essentially linear, more precisely it does not allow for a non-linear use of functional arguments. In the present work we adress the problem of incorporating both approaches into a common type system. The source language we consider is a lambda-calculus with data-types and iteration, that is to say a variant of Godel's system T. Our goal is to design a system for this language allowing both to handle non-linear functional arguments and to keep a good intensional expressivity. We illustrate our methodology by choosing the system of elementary linear logic (ELL) and combining it with a system of linear size types. We discuss the expressivity of this new type system and prove that it gives a characterization of the complexity classes FPTIME and 2k-FEXPTIME, for k ě 0.