This short note aims at proving that the isolation problem is undecidable for probabilistic automata with only one probabilistic transition. This problem is known to be undecidable for general probabilistic automata, without restriction on the number
of probabilistic transitions. In this note, we develop a simulation technique that allows to simulate any probabilistic automaton with one having only one probabilistic transition.
Games on graphs provide a natural and powerful model for reactive systems. In this paper, we consider generalized reachability objectives, defined as conjunctions of reachability objectives. We first prove that deciding the winner in such games is $P
SPACE$-complete, although it is fixed-parameter tractable with the number of reachability objectives as parameter. Moreover, we consider the memory requirements for both players and give matching upper and lower bounds on the size of winning strategies. In order to allow more efficient algorithms, we consider subclasses of generalized reachability games. We show that bounding the size of the reachability sets gives two natural subclasses where deciding the winner can be done efficiently.
