Type inference in intuitionistic linear logic

We study the type checking and type inference problems for intuitionistic linear logic: given a System F typed λ-term, (i) for an alleged linear logic type, determine whether there exists a corresponding typing derivation in linear logic (type checking) ii) provide a concise description of all possible corresponding linear logic typings (type inference). We solve these problems using a novel algorithmic type system for linear logic whose typing rules carry arithmetic side conditions describing essentially the nesting depth of (proof-net) boxes. By understanding these side conditions as unknowns we then reduce type inference to solving a system of arithmetic constraints. We show that these constraint systems fall into a tractable class hence leading to an efficient (polynomial-time) solution. There are two important restrictions: first, our source language is typed System F rather than untyped lambda calculus; this is necessary because type inference for System F is known to be undecidable. Second, we assume that sharing is made explicit in the input, thus we do not try to automatically infer opportunities for sharing identical subterms. Relieving the latter restriction is left as a challenge for future work.