Characterizing EF and EX Tree Logics

We describe the expressive power of temporal branching time logics that use the modalities EX and EF. We give a forbidden pattern characterization of the tree languages definable in three logics: EX, EF and EX+EF. The properties in these characterizations can be verified in polynomial time when given a minimal deterministic bottom-up tree automaton. We consider the definability problem for logics over binary trees: given a regular tree language decide if it can be expressed by a formula of the logic in question. The main motivation for considering this problem is to understand the expressive power of tree logics. Although a very old question, definability has gained new relevance with the XML community’s burgeoning interest in tree models [8]. Indeed, numerous new formalisms for describing tree properties have been recently proposed. For words the definability question is well studied and understood. Starting from the celebrated Schutzenberger theorem [12], characterizing star-free word languages by aperiodicity, numerous other language classes have been classified. In particular, we now have a good understanding of the expressive power of LTL and its fragments [14, 18]. This is in sharp contrast with the case of trees where much less is known. We feel that the major goal in the study of the definability problem for trees is to characterize the expressive power of first-order logic, or equivalently CTL∗[1] (we consider finite binary trees here). It seems however that this is a difficult problem whose solution demands new tools and expertise. This is why we have decided to consider fragments of CTL∗ where the problem is easier. The fragments in question use the modalities EX (there is a successor) and EF (there is a descendant). Apart from being a step towards solving the first-order definability problem, these fragments are interesting on their own. The model-checking problem for them is easier than for CTL∗, 1 ha l-0 03 35 73 6, v er si on 1 30 O ct 2 00 8 Author manuscript, published in "Journal of Theoretical Computer Science (TCS) 358, 2-3 (2006) 255-272"