Satisfiability of ECTL* with Tree Constraints

Abstract

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-Fraïssé-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.

Playing Games in the Baire Space

“…Thus, N-memory automata can recognize some interesting ω-languages over N. The ability to compare successive (and also "distant") letters and to define properties of unboundedness seems to be a feature that is missing in known models of automata over the alphabet N. Let us note that in the context of temporal logic, a related idea appears in [5]; however there equality and incremental change of values from N is restriced to occurrences within a bounded (time-)interval -so a language such as N * 1N * 2N * 3N * . .…”

Model Checking Constraint LTL over Trees

“…Contrary, temporal logics with constraints over trees have not yet been investigated much, although questions concerning decidability of the satisfiability problem for LTL or CTL * with such constraints have been asked for instance in [10,6]. A first (negative) result by Carapelle et al [5] shows that a technique developed in [6,4] for satisfiability results of branching-time logics (like CTL * or ECTL * ) with integer constraints cannot be used to resolve the satisfiability status of temporal logics with constraints over trees.…”