Linear Time Membership for a Class of XML Types with Interleaving and Counting.

Regular Expressions (REs) form the basis of most XML type lan- guages, such as DTDs, XML Schema types, and XDuce types (Thomp- son et al. 2004; Hosoya and Pierce 2003). In this context, the in- terleaving operator would be a natural addition to the language of REs, as witnessed by the presence of limited forms of interleav- ing in XSD (the all group), Relax-NG, and SGML. Unfortunately, membership checking for REs with interleaving is NP-hard in gen- eral. We present here a restricted class of REs with interleaving and counting which admits a linear membership algorithm. This restricted class is known to be expressive enough for the vast ma- jority of the content models used in real-world DTDs and XSD schemas; moreover, we have proved in (Ghelli et al. 2007) that the same class admits a polynomial algorithm for subtyping and type- equivalence, problems which are EXPSPACE-complete for the full language of REs with interleaving. We first present an algorithm for membership of a list of words into a RE with interleaving and counting, based on the translation of the RE into a set of constraints. We generalize the approach in order to check membership of XML trees into a class of EDTDs with interleaving and counting, which models the crucial aspects of DTDs and XSD schemas. Finally, we extend the approach to REs with intersection.