Typed-based Homeomorphic Embedding for Online Termination

We introduce the type-based homeomorphic embedding relation as an extension of the standard, untyped homeomorphic embedding which allows us to obtain more precise results in the presence of infinite signatures (e.g., the integers). In particular, we show how our type-based relation can be used to improve the accuracy of online partial evaluation. For this purpose, we outline an approach to constructing suitable types for partial evaluation automatically, given an untyped program and a goal or set of goals. Our approach is based on existing analysis tools for constraint logic programs: (i) inference of a well-typing of a program and goal, and (ii) bounds analysis for numerical values. We argue that our work improves the state of the practice of online termination and it is very relevant for instance in the context of the specialization of interpreters.