Manipulating Trees with Hidden Labels (Fossacs Updated Version).fm

XML and semistructured data [1] are inspiring a new generation of programming and query languages based on more flexible type systems [26, 5, 6, 15]. Traditional type systems are grounded on mathematical constructions such as cartesian products, disjoint unions, function spaces, and recursive types. The type systems for semistructured data, in contrast, resemble grammars or logics, with untagged unions, associative products, and Kleene star operators. The theory of formal languages, for strings and trees, provides a wealth of ready results, but it does not account, in particular, for functions. Some integration of the two approaches to type systems is necessary [26, 5]. While investigating semistructured data models and associated languages, we became aware of the need for manipulating private data elements, such as XML identifiers, unique node identifiers in graph models [7], and even heap locations. Such private resources can be modeled using names and name hiding notions arising from the π-calculus [27]: during data manipulation, the identity of a private name is not important as long as the distinctions between it and other (public or private) names are preserved. Recent progress has been made in handling private resources in programming. FreshML [21] pioneers the transposition [30], or swapping, of names, within a type systems that prevents the disclosure of private names. Other recent techniques can be useful for our purposes. The spatial logics of concurrency devised to cope with π-calculus restriction and scope extrusion [27], and the separation logics used to describe data structures [28,29], provide novel logical operators that can be used also in type systems. Moreover, the notion of dependent types, when the dependence is restricted to names, is tractable [25]. In this paper we bring together a few current threads of development: the effort to devise new languages, type systems, and logics for data structures, the logical operators that come from spatial and nominal logics for private resources, the techniques of transpositions, and the necessity to handle name-dependent types when manipulating private resources. We study these issues in the context of a simplified data model: simple labeled trees with hidden labels, and programs that manipulate such trees. The edges of such trees are labeled with names. Our basic techniques can be applied to related data models, such as graphs with hidden node and edge labels, which will be the subject of further work.