Abstract We present an O(n 3 ) time type inference algorithm for a type system with a largest type Τ , a smallest type ⊥, and the usual ordering between function types. The algorithm infers type annotations of least shape, and it works equally well for recursive types. For the problem of typability, our algorithm is simpler than the one of Kozen, Palsberg, and Schwartzbach for type inference without ⊥. This may be surprising, especially because the system with ⊥ is strictly more powerful.

Flow-based safety analysis of higher-order languages has been studied by Shivers, and Palsberg and Schwartzbach. Open until now is the problem of finding a type system that accepts exactly the same programs as safety analysis.

We study a type system with a notion of subtyping that involves a largest type ⊤, a smallest type ⊥, atomic coercions between base types, and the usual ordering of function types. We prove that any λ-term typable in this system is strongly normalizing, which solves an open problem of Thatte. We also prove that the fragment without ⊥ types has strictly fewer terms. This demonstrates that ⊥ adds power to a type system.

Flow-based safety analysis of higher-order languages has been studied by Shivers, and Palsberg and Schwartzbach. Open until now is the problem of finding a type system that accepts exactly the same programs as safety analysis. In this article we prove that Amadio and Cardelli's type system with subtyping and recursive types accepts the same programs as a certain safety analysis. The proof involves mappings from types to flow information and back. As a result, we obtain an inference algorithm for the type system, thereby solving an open problem.