On the complexity of typechecking top-down XML transformations

AbstractWe investigate the typechecking problem for XML transformations: statically verifying that every answer to a transformation conforms to a given output schema, for inputs satisfying a given input schema. As typechecking quickly turns undecidable for query languages capable of testing equality of data values, we return to the limited framework where we abstract XML documents as labeled ordered trees. We focus on simple top-down recursive transformations motivated by XSLT and structural recursion on trees. We parameterize the problem by several restrictions on the transformations (deleting, non-deleting, bounded width) and consider both tree automata and DTDs as input and output schemas. The complexity of the typechecking problems in this scenario ranges from PTIME to EXPTIME

On Succinct Representations of Binary Trees

We observe that a standard transformation between \emph{ordinal} trees (arbitrary rooted trees with ordered children) and binary trees leads to interesting succinct binary tree representations. There are four symmetric versions of these transformations. Via these transformations we get four succinct representations of $n$-node binary trees that use $2n + n/(\log n)^{O(1)}$ bits and support (among other operations) navigation, inorder numbering, one of pre- or post-order numbering, subtree size and lowest common ancestor (LCA) queries. The ability to support inorder numbering is crucial for the well-known range-minimum query (RMQ) problem on an array $A$ of $n$ ordered values. While this functionality, and more, is also supported in $O(1)$ time using $2n + o(n)$ bits by Davoodi et al.'s (\emph{Phil. Trans. Royal Soc. A} \textbf{372} (2014)) extension of a representation by Farzan and Munro (\emph{Algorithmica} \textbf{6} (2014)), their \emph{redundancy}, or the $o(n)$ term, is much larger, and their approach may not be suitable for practical implementations. One of these transformations is related to the Zaks' sequence (S.~Zaks, \emph{Theor. Comput. Sci.} \textbf{10} (1980)) for encoding binary trees, and we thus provide the first succinct binary tree representation based on Zaks' sequence. Another of these transformations is equivalent to Fischer and Heun's (\emph{SIAM J. Comput.} \textbf{40} (2011)) \minheap\ structure for this problem. Yet another variant allows an encoding of the Cartesian tree of $A$ to be constructed from $A$ using only $O(\sqrt{n} \log n)$ bits of working space.Comment: Journal version of part of COCOON 2012 pape

Breadth-first serialisation of trees and rational languages

We present here the notion of breadth-first signature and its relationship with numeration system theory. It is the serialisation into an infinite word of an ordered infinite tree of finite degree. We study which class of languages corresponds to which class of words and,more specifically, using a known construction from numeration system theory, we prove that the signature of rational languages are substitutive sequences.Comment: 15 page

On Free Completely Iterative Algebras

For every finitary set functor F we demonstrate that free algebras carry a canonical partial order. In case F is bicontinuous, we prove that the cpo obtained as the conservative completion of the free algebra is the free completely iterative algebra. Moreover, the algebra structure of the latter is the unique continuous extension of the algebra structure of the free algebra. For general finitary functors the free algebra and the free completely iterative algebra are proved to be posets sharing the same conservative completion. And for every recursive equation in the free completely iterative algebra the solution is obtained as the join of an ?-chain of approximate solutions in the free algebra