# Satisfiability of ECTL^{∗} with Local Tree Constraints

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## 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-Fraïssé-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).

## Keywords

Temporal logics ECTL^{∗}Concrete domains Local constraints Semi-linear orders Ordinal trees WMSO+B EF-games

## Notes

### Acknowledgements

We thank Manfred Droste for fruitful discussions on universal structures and semi-linear orders.

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