Section: New Results

Tree automata theory

Participants : Luc Segoufin, Serge Abiteboul, M Praveen.

Tree automata

We studied the expressive power of a subclass of regular tree languages. We gave a decidable characterization of those languages that are “piecewise testable”, i.e. definable using boolean combination of existential first-order formulas [12] .

Automata with counters.

We studied extending techniques used in standard Petri nets to other models. We extended the Rackoff technique to decide coverability and boundedness problems for Strongly Increasing Affine nets, a subclass of Affine nets [20] .

Languages on trees.

We studied in [18] highly expressive query languages for unordered data trees, using as formal vehicles Active XML and extensions of languages in the while family. All languages may be seen as adding some form of control on top of a set of basic pattern queries. The results highlight the impact and interplay of different factors: the expressive power of basic queries, the embedding of computation into data (as in Active XML), and the use of deterministic vs. nondeterministic control. All languages are Turing complete, but not necessarily query complete in the sense of Chandra and Harel. Indeed, we show that some combinations of features yield serious limitations, analogous to FOk definability in the relational context. On the other hand, the limitations come with benefits such as the existence of powerful normal forms. Other languages are “almost” complete, but fall short because of subtle limitations reminiscent of the copy elimination problem in object databases.

Probabilistic XML.

In [15] , we study the problem of, given a corpus of XML documents and its schema, finding an optimal (generative) probabilistic model, where optimality here means maximizing the like- lihood of the particular corpus to be generated. Focusing first on the structure of documents, we present an efficient algorithm for finding the best generative probabilistic model, in the absence of constraints. We further study the problem in the presence of integrity constraints, namely key, inclusion, and domain constraints. We study in this case two different kinds of generators. First, we consider a continuation-test generator that performs, while generating documents, tests of schema satisfiability ; these tests prevent from generating a document violating the constraints but, as we will see, they are computationally expensive. We also study a restart generator that may generate an invalid document and, when this is the case, restarts and tries again. Finally, we consider the injection of data values into the structure, to obtain a full XML document. We study different approaches for generating these values.

Infinite alphabet.

We studied the complexity of satisfiability of linear temporal logics extended to reason about repetitions of values from an infinite data domain. We refined an existing result that reduced this problem to Petri net reachability, and showed that it can be reduced to the coverability problem. Using this refinement, we gave the precise complexity of the satisfiability problem. We also characterized the complexity of satisfiability for many fragments and extensions of the logic.