  Counter reachability games are played by two players on a graph with labelled
edges. Each move consists in picking an edge from the current location and
adding its label to a counter vector. The objective is to reach a given counter
value in a given location. We distinguish three semantics for counter
reachability games, according to what happens when a counter value would become
negative: the edge is either disabled, or enabled but the counter value becomes
zero, or enabled. We consider the problem of deciding the winner in counter
reachability games and show that, in most cases, it has the same complexity
under all semantics. Surprisingly, under one semantics, the complexity in
dimension one depends on whether the objective value is zero or any other
integer.
