Theory of Computing Systems

, Volume 61, Issue 2, pp 689–720 | Cite as

Satisfiability of ECTL with Local Tree Constraints

  • Claudia Carapelle
  • Shiguang Feng
  • Alexander Kartzow
  • Markus Lohrey
Article

Abstract

Recently, we have shown that satisfiability for the temporal logic ECTL with local constraints over (ℤ, <, =) is decidable using a new technique (Carapelle et al., 2013). This approach reduces the satisfiability problem of ECTL 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 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 \(h\in \mathbb {N}\). 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 the EHD-property. As a consequence, our technique cannot be used to establish whether satisfiability of ECTL 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 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institut für Theoretische InformatikTU DresdenDresdenGermany
  2. 2.Institut für InformatikUniversität LeipzigLeipzigGermany
  3. 3.Department für Elektrotechnik und InformatikUniversität SiegenSiegenGermany

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