We investigate type inference for programming languages with subtypes. As described in previous work, there are several type inference problems for any given expression language, depending on the form of the subtype partial order and the ability to define new subtypes in programs. Our first main result is that for any specific subtype partial order, the problem of determining whether a lambda term is typable is algorithmically (polynomial-time) equivalent to a form of satisfiability problem over the same partial order. This gives the first exact characterization of the problem that is independent of the syntax of expressions. In addition, since this form of satisfiability problem is PSPACE-hard over certain partial orders, this equivalence strengthens the previous lower bound of NP-hard to PSPACE-hard. Our second main result is a lower bound on the length of most general types when the subtype hierarchy may change as a result of additional type declarations within the program. More specifically, given any input expression, a type inference algorithm tries to find a most general (or principal) typing. The property of a most general typing is that it has all other possible typings as instances. However, there are several sound notions of instance in the presence of subtyping. Our lower bound is that no sound definition of instance would allow the set of additional subtyping hypotheses about a term to grow less than linearly in the size of the term.

Standard ML of New Jersey (SML-NJ) uses weak-type variables to restrict the polymorphic use of functions that may allocate reference cells, manipulate continuations, or use exceptions. However, the type system used in the SML-NJ compiler has not been presented in a form other than source code and has not been proved correct. A type system, in the form of typing rules and an equivalent algorithm, that appears to subsume the implemented algorithm is presented. Both use type variables of only a slightly more general nature than the compiler. One insight in the analysis is that the indexed type of a free variable is used in two ways, once in describing the applicative behavior of the variable itself and once in describing the larger term containing the variable. Taking this into account, an application rule that is more general than SML-NJ is formulated for applications of polymorphic functions to imperative arguments. The soundness of the type system is proved for imperative code using operational semantics.>