Although type reconstruction for dependently typed languages is common in practical systems, it is still ill-understood. Detailed descriptions of the issues around it are hard to find and formal descriptions together with correctness proofs are non-existing. In this paper, we discuss a one-pass type reconstruction for objects in the logical framework LF, describe formally the type reconstruction process using the framework of contextual modal types, and prove correctness of type reconstruction. Since type reconstruction will find most general types and may leave free variables, we in addition describe abstraction which will return a closed object where all free variables are bound at the outside. We also implemented our algorithms as part of the Beluga language, and the performance of our type reconstruction algorithm is comparable to type reconstruction in existing systems such as the logical framework Twelf.

This paper presents a formal definition and machine-checked soundness proof for a very expressive type-and-capability system, that is, a low-level type system that keeps precise track of ownership and side effects. The programming language has first-class functions and references. The type system's features include the following: universal, existential, and recursive types; subtyping; a distinction between affine and unrestricted data; support for strong updates; support for naming values and heap fragments via singleton and group regions; a distinction between ordinary values (which exist at runtime) and capabilities (which do not); support for dynamic reorganizations of the ownership hierarchy by disassembling and reassembling capabilities; and support for temporarily or permanently hiding a capability via frame and anti-frame rules. One contribution of the paper is the definition of the type-and-capability system itself. We present the system as modularly as possible. In particular, at the core of the system, the treatment of affinity, in the style of dual intuitionistic linear logic, is formulated in terms of an arbitrary monotonic separation algebra, a novel axiomatization of resources, ownership, and the manner in which they evolve with time. Only the peripheral layers of the system are aware that we are dealing with a specific monotonic separation algebra, whose resources are references and regions. This semi-abstract organization should facilitate further extensions of the system with new forms of resources. The other main contribution is a machine-checked proof of type soundness. The proof is carried out in the Wright and Felleisen's syntactic style. This offers an evidence that this relatively simple-minded proof technique can scale up to systems of this complexity, and constitutes a viable alternative to more sophisticated semantic proof techniques. We do not claim that the syntactic technique is superior: We simply illustrate how it is used and highlight its strengths and shortcomings.