A Contraction Method to Decide Monadic Second-order Theories of Trees

We generalize the contraction method, originally proposed by Elgot
and Rabin and later extended by Carton and Thomas, from labeled
linear orderings to deterministic colored trees. We introduce a suitable
notion of indistinguishability of trees with respect to tree automata 
and we take advantage of compositional properties of trees to reduce a
number of instances of the acceptance problem for tree auotomata to
simple instances involving regular trees. We prove that such a method
works effectively for a large class of trees, which is closed under
noticeable operations and includes all deterministic trees in the
Caucal hierarchy as well as several trees outside it.