Satisfiability of ECTL* with Tree Constraints

Recently, we have shown that satisfiability for ECTL∗ with constraints over Z is decidable using a new technique. This approach reduces the satisfiability problem of ECTL∗ with constraints over some structure A (or class of structures) to the problem whether A has a certain model theoretic property that we called EHD (for “existence of homomorphisms is decidable”). Here we apply this approach to concrete domains that are tree-like and obtain several results. We show that satisfiability of ECTL∗ with constraints is decidable over (i) semi-linear orders (i.e., tree-like structures where branches form arbitrary linear orders), (ii) ordinal trees (semi-linear orders where the branches form ordinals), and (iii) infinitely branching trees of height h for each fixed h ∈ N. We prove that all these classes of structures have the property EHD. In contrast, we introduce Ehrenfeucht-Fräıssé-games for WMSO+B (weak MSO with the bounding quantifier) and use them to show that the infinite (order) tree does not have property EHD. As a consequence, a different approach has to be taken in order to settle the question whether satisfiability of ECTL ∗ (or even LTL) with constraints over the infinite (order) tree is decidable.