We investigate expressiveness of a fragment of the ambient calculus, a formalism for describing distributed and mobile computations. More precisely, we study expressiveness of the pure and public ambient calculus from which the capability open has been removed, in terms of the reachability problem of the reduction relation. Surprisingly, we show that even for this very restricted fragment, the reachability problem is not decidable. At a second step, for a slightly weaker reduction relation, we prove that reachability can be decided by reducing this problem to markings reachability for Petri nets. Finally, we show that the name-convergence problem as well as the model-checking problem turn out to be undecidable for both the original and the weaker reduction relation. The authors are grateful to S. Tison and Y. Roos for fruitful discussions and thank the anony mous ferees for valuable comments. This work is supported by an ATIP grant from CNRS.