Path resolution for nested recursive modules

The ML module system facilitates the modular development of large programs, through decomposition, abstraction and reuse. To increase its flexibility, much work has been devoted to extending it with recursion, which is currently prohibited. The introduction of recursion certainly adds expressivity to the module system. However it also brings out non-trivial problems that a non-recursive module system does not have. In this paper, we address one of such problems, namely resolution of path references. Paths are how one refers to nested modules in ML. Without recursion, well-typedness guarantees termination of path resolution, in other words, we can statically determine the module that a path refers to. This does not always hold with recursive module extensions, since the module system then can encode a lambda-calculus with recursion, whose termination is undecidable regardless of welltypedness. We formalize this problem of path resolution by means of a rewrite system on paths and prove that the problem is undecidable even without higher-order functors, via an encoding of the Turing machine into a calculus with just recursive modules, first-order functors, and nesting. Motivated by this result, we introduce a further restriction on first-order functors, limiting polymorphism on functor parameters by requiring signatures for functor parameters to be non-recursive, and show that this restriction is decidable and admits terminating path resolution.