Monadic Intersection Types, Relationally, and Ordered

We extend intersection types to a computational \(\lambda\) -calculus with algebraic operations à la Plotkin and Power. We achieve this by considering monadic intersections—whereby computational effects appear not only in the operational semantics but also in the type system . Since in the effectful setting, termination is not anymore the only property of interest, we want to analyze the interactive behavior of typed programs with the environment. Indeed, our type system can characterize the natural notion of observation, both in the finitary and in the infinitary setting. In a second phase, we extend our system with subtyping to incorporate a richer class of effects via monads on preorders instead of sets allowing us to model in particular non-determinism. The main technical tool is a novel combination of syntactic techniques with abstract relational reasoning, which allows us to lift all the required notions, for example, of typability and logical relation, to the monadic setting.