## Section: New Results

### Transformations

#### Guided Tree Automata for the XPath Containment

The XPath containment problem between two XPath expressions

We have proposed a sound and complete decision procedure for containment of XPath queries [3] . The XPath fragment considered covers most of the language features used in practice, with the only exceptions of counting and data values comparisons. Specifically, we have shown how XPath queries can be translated into the Weak Second Order Logic of Two Successors (WS2S). Using this translation, we construct an optimized logical formulation of the containment problem, which is decided using specific tree automata operations equipped with guides. When the containment relation does not hold between two XPath expressions, a counter-example XML tree is generated. We have provided detailed practical experiments that illustrate the empirical cost of the decision procedure for realistic scenarios of the XPath containment problem.

#### XML/XPath Analysis based on the Alternation Free Modal mu-calculus

We have proposed a modal logic approach for the resolution of decision problems where both XPath queries and regular tree types are translated into the mu-calculus. XML decision problems are expressed as formulas in this logic, then decided using a decision procedure for mu-calculus satisfiability. We take advantage of the expressive power of a variant of the mu-calculus, called the alternation-free modal mu-calculus with converse. We have shown how this logic can be used for reasoning on XML trees, XPath queries and XML types, then reduce several other XML decision problems to satisfiability in the mu-calculus such as coverage and overlap. We have obtained an enhanced complexity of O(*n* log(*n* )) for the decision procedure of XML related problems. We propose a system that has been fully implemented
[2] and tested on a wide range of decision problems.

#### An Efficient Tree Logic for Reasoning on XML Types and Paths

We have proposed a new logic and the corresponding satisfiability algorithm and we have shown its effectiveness in the context of XML static analysis. To this end, we have proven the decidability of a logic with converse for finite ordered trees whose time complexity is a simple exponential of the size of a formula. The logic is closed under boolean operations and has only the least fixpoint for finite recursion.

Our proof method is based on two auxiliary results. First, XML regular tree types and XPath expressions have a linear translation to cycle-free formulas. Second, the least and greatest fixpoints are equivalent for finite trees, hence the logic is closed under negation. With these proofs, we have implemented a practically effective system for solving the satisfiability of cycle-free formulas. The system has been experimented with the XML decision problems introduced above. The advantage of the approach is that the system is very effective as compared to earlier results and can be used for larger XML problems [1] .