Satisfiability of ECTL∗ with Local Tree Constraints

Abstract

Recently, we have shown that satisfiability for the temporal logic E C T L ∗ with local constraints over (ℤ, <, =) is decidable using a new technique (Carapelle et al., 2013). This approach reduces the satisfiability problem of E C T L ∗ with constraints over some structure $\mathcal {A}$ (or class of structures) to the problem whether $\mathcal {A}$ has a certain model theoretic property that we called EHD (for “existence of homomorphisms is definable”). Here we apply this approach to structures that are tree-like and obtain several results. We show that satisfiability of E C T L ∗ 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 $h\in \mathbb {N}$ . We prove that all these classes of structures have the property EHD. In contrast, we introduce Ehrenfeucht-Fraisse-games for W M S O + B (weak M S O with the bounding quantifier) and use them to show that the infinite (order) tree does not have the EHD-property. As a consequence, our technique cannot be used to establish whether satisfiability of E C T L ∗ with constraints over the infinite (order) tree is decidable. A preliminary version of this paper has appeared as (Carapelle et al., 2015).