We propose regular expression types as a foundation for statically typed XML processing languages. Regular expression types, like most schema languages for XML, introduce regular expression notations such as repetition (*), alternation (|), etc., to describe XML documents. The novelty of our type system is a semantic presentation of subtyping, as inclusion between the sets of documents denoted by two types. We give several examples illustrating the usefulness of this form of subtyping in XML processing.The decision problem for the subtype relation reduces to the inclusion problem between tree automata, which is known to be exptime-complete. To avoid this high complexity in typical cases, we develop a practical algorithm that, unlike classical algorithms based on determinization of tree automata, checks the inclusion relation by a top-down traversal of the original type expressions. The main advantage of this algorithm is that it can exploit the property that type expressions being compared often share portions of their representations. Our algorithm is a variant of Aiken and Murphy's set-inclusion constraint solver, to which are added several new implementation techniques, correctness proofs, and preliminary performance measurements on some small programs in the domain of typed XML processing.

We propose a type system based on regular tree grammars, where algebraic datatypes are interpreted in a structural way. Thus, the same constructors can be reused for different types and a flexible subtyping relation can be defined between types, corresponding to the inclusion of their semantics. For instance, one can define a type for lists and a subtype of this type corresponding to lists of even length. Patterns are simply types annotated with binders. This provides a generalization of algebraic patterns with the ability of matching arbitrarily deep in a value. Our main contribution, compared to languages such as XDuce and CDuce, is that we are able to deal with both polymorphism and function types.