## Section: New Results

### Towards efficient abstract domains for regular language based static analysis

Participants : Thomas Genet, Valérie Murat, Yann Salmon.

We develop a specific theory and the related tools for analyzing programs whose semantics is defined using term rewriting systems. The analysis principle is based on regular approximations of infinite sets of terms reachable by rewriting. The tools we develop use, so-called, Tree Automata Completion to compute a tree automaton recognizing a superset of all reachable terms. This over-approximation is then used to prove safety properties on the program by showing that some “bad” terms, encoding dangerous or problematic configurations, are not in the superset and thus not reachable. This is a specific form of, so-called, Regular Tree Model Checking. However, when dealing with infinite-state systems, Regular Tree Model Checking approaches may have some difficulties to represent infinite sets of data. We proposed Lattice Tree Automata, an extended version of tree automata to represent complex data domains and their related operations in an efficient manner. Moreover, we introduce a new completion-based algorithm for computing the possibly infinite set of reachable states in a finite amount of time. This algorithm is independent of the lattice making it possible to seamlessly plug abstract domains into a Regular Tree Model Checking algorithm[27] . As a first instance, we implemented in Timbuk a completion with an interval abstract domain. We shown that this implementation permits to scale up regular tree model-checking of Java programs dealing with integer arithmetics. Now, we aim at applying this technique to the static analysis of programming languages whose semantics is based on terms, like functional programming languages [38] .