We describe a framework for finding counter-examples in a transition system. This framework constructs an on-the-fly heuristic for choosing states that are good candidates for exploration. and refines this heuristic when it realizes it made mistakes. The refinements are always relative to the mistakes made, and therefore the amount of overhead needed for constructing the heuristic is minimized. This heuristic is used for partial exploration of the state space, directed toward reaching a user specified set of states. Empirical results show this method to be a strong competitor to the standard BFS based methods.