Satisfiability of ECTL* with tree constraints
Abstract
Recently, we have shown that satisfiability for $\mathsf{ECTL}^*$ with constraints over $\mathbb{Z}$ is decidable using a new technique. This approach reduces the satisfiability problem of $\mathsf{ECTL}^*$ 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 treelike and obtain several results. We show that satisfiability of $\mathsf{ECTL}^*$ with constraints is decidable over (i) semilinear orders (i.e., treelike structures where branches form arbitrary linear orders), (ii) ordinal trees (semilinear 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 EhrenfeuchtFraissegames for $\mathsf{WMSO}+\mathsf{B}$ (weak $\mathsf{MSO}$ 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 $\mathsf{ECTL}^*$ (or even $\mathsf{LTL}$) with constraints over the infinite (order) tree is decidable.
 Publication:

arXiv eprints
 Pub Date:
 December 2014
 arXiv:
 arXiv:1412.2905
 Bibcode:
 2014arXiv1412.2905C
 Keywords:

 Computer Science  Logic in Computer Science;
 Computer Science  Formal Languages and Automata Theory;
 Mathematics  Logic