The work described in this paper belongs to the general context of distributed program debugging. For the goal of verification, the expected behaviour or the suspected errors of the system under test , is described by a global property : basically a global predicate on variables, or the possible occurences orders of observable events. The problem is to verify wether this property is satisfied or not during the execution. In the context of asynchronous parallelism on distributed architectures that we are considering, the correct evaluation of global properties requires a careful analysis of the causal structure of the execution. The basis of the method is a reachability analysis of the state space associated to a distributed execution. We study on-line algorithms dedicated to trace checking which are efficient both in time and memory. We show that the state space can be linearly built due to its algebraic structure of lattice.