Usually subtyping relations are defined either syntactically by a formal system or semantically by an interpretation of types in an untyped denotational model. In this work we show how to define a subtyping relation semantically, for a language whose operational semantics is driven by types; we consider a rich type algebra, with product, arrow, recursive, intersection, union and complement types. Our approach is to "bootstrap" the subtyping relation through a notion of set-theoretic model of the type algebra.The advantages of the semantic approach are manifold. Foremost we get "for free" many properties (e.g., the transitivity of subtyping) that, with axiomatized subtyping, would require tedious and error prone proofs. Equally important is that the semantic approach allows one to derive complete algorithms for the subtyping relation or the propagation of types through patterns. As the subtyping relation has a natural (inasmuch as semantic) interpretation, the type system can give informative error messages when static type-checking fails. Last but not least the approach has an immediate impact in the definition and the implementation of languages manipulating XML documents, as this was our original motivation.
Despite the extensiveness of recent investigations on static typing for XML, parametric polymorphism has rarely been treated. This well-established typing discipline can also be useful in XML processing in particular for programs involving "parametric schemas," i.e., schemas parameterized over other schemas (e.g., SOAP). The difficulty in treating polymorphism for XML lies in how to extend the "semantic" approach used in the mainstream (monomorphic) XML type systems. A naive extension would be "semantic" quantification over all substitutions for type variables. However, this approach reduces to an NEXPTIME-complete problem for which no practical algorithm is known. In this paper, we propose a different method that smoothly extends the semantic approach yet is algorithmically easier. In this, we devise a novel and simple marking technique, where we interpret a polymorphic type as a set of values with annotations of which subparts are parameterized. We exploit this interpretation in every ingredient of our polymorphic type system such as subtyping, inference of type arguments, and so on. As a result, we achieve a sensible system that directly represents a usual expected behavior of polymorphic type systems-"values of variable types are never reconstructed"-in a reminiscence of Reynold's parametricity theory. Also, we obtain a set of practical algorithms for typechecking by local modifications to existing ones for a monomorphic system.
Abstract. This paper studies the problem of matching sequences against regular expressions in order to produce structured values.
Macro tree transducers (mtt) are an important model that both covers many useful XML transformations and allows decidable exact typechecking. This paper reports our first step toward an implementation of mtt typechecker that has a practical efficiency. Our approach is to represent an input type obtained from a backward inference as an alternating tree automaton, in a style similar to Tozawa's XSLT0 typechecking. In this approach, typechecking reduces to checking emptiness of an alternating tree automaton. We propose several optimizations (Cartesian factorization, state partitioning) on the backward inference process in order to produce much smaller alternating tree automata than the naive algorithm, and we present our efficient algorithm for checking emptiness of alternating tree automata, where we exploit the explicit representation of alternation for local optimizations. Our preliminary experiments confirm that our algorithm has a practical performance that can typecheck simple transformations with respect to the full XHTML in a reasonable time.
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