Recent psycholinguistic evidence suggests that human parsing of moved elements is ‘active’, and perhaps even ‘hyper-active’: it seems that a leftward-moved object is related to a verbal position rapidly, perhaps even before the transitivity information associated with the verb is available to the listener. This paper presents a formal, sound and complete parser for Minimalist Grammars whose search space contains branching points that we can identify as the locus of the decision to perform this kind of active gap-finding. This brings formal models of parsing into closer contact with recent psycholinguistic theorizing than was previously possible.