Verifying monadic second order graph properties with tree automata

We address the concrete problem of verifying graph properties expressed in Monadic Second Order (MSO) logic. It is well-known that the model-checking problem for MSO logic on graphs is fixed-parameter tractable (FPT) [Cou09, Chap 6] with respect to tree-width and cliquewidth. The proof uses tree-decompositions (for tree-width as parameter) and clique-decompositions (for clique-width as parameter), and the construction of a finite tree automaton from an MSO sentence, expressing the property to check. However, this construction may fail because either the intermediate automata are too big even though the final automaton has a reasonable size or the final automaton itself is too big to be constructed:the sizes of automata depend, exponentially in most cases, on the tree-width or the clique-width of the graphs to be verified. We present ideas to overcome these two causes of failure. The first idea is to give a direct construction of the automaton in order to avoid explosion in the intermediate steps of the general algorithm. When the final automaton is still too big, the second idea is to represent the transition function by a function instead of computing explicitly the set of transitions; this entirely solves the space problem. All these ideas have been implemented in Common Lisp.