Using Curry-Howard isomorphism, we extend the typed lambda-calculus with intersection and union types, and its corresponding proof-functional logic, previously defined by the authors, with subtyping and explicit coercions. We show the extension of the lambda-calculus to be isomorphic to the Barbanera-Dezani-de'Liguoro type assignment system and we provide a sound interpretation of the proof-functional logic with the NJ(β) logic, using Mints' realizers. We finally present a sound and complete algorithm for subtyping in presence of intersection and union types. The algorithm is conceived to work for the (sub)type theory Ξ.

We present an ongoing implementation of a dependent-type theory (∆-framework) based on the Edinburgh Logical Framework LF, extended with Proof-functional logical connectives such as intersection, union, and strong (or minimal relevant) implication. Their combination opens up new possibilities of formal reasoning on proof-theoretic semantics. We provide some examples in the extended type theory and we outline a type checker. The theory of the system is under investigation. Once validated in vitro, the proof-functional type theory could be successfully plugged into existing truth-functional proof-systems.

Session types are a well-established framework for the specification of interactions between components of a distributed systems. An important issue is how to determine the type for an open system, i.e., obtained by assembling subcomponents, some of which could be missing. To this end, we introduce partial sessions and partial (multiparty) session types. Partial sessions can be composed, and the type of the resulting system is derived from those of its components without knowing any suitable global type nor the types of missing parts. To deal with this incomplete information, partial session types represent the subjective views of the interactions from participants’ perspectives; when sessions are composed, different partial views can be merged if compatible, yielding a unified view of the session. Incompatible types, due to, e.g., miscommunications or deadlocks, are detected at the merging phase. In fact, in this theory the distinction between global and local types vanishes. We apply these types to a process calculus for which we prove subject reduction and progress, so that well-typed systems never violate the prescribed constraints. In particular, we introduce a generalization of the progress property, in order to accommodate the case when a partial session cannot progress not due to a deadlock, but because some participants are still missing. Therefore, partial session types support the development of systems by incremental assembling of components.