X does action a and changes to t.
This yields an infinite state system which can be analysed by finite state methods. In a previous work with Ph. Schnoebelen, we have shown that regular tree languages and finite tree automata provide a powerful tool for this infinite state system. In this talk, we show how we can easily extend this work in two directions, getting strong decidability results for a whole bunch of temporal logics. Proofs are simple, for instance the regularity theorems of our previous work are simple byproducts of this new approach.