It is shown that the general inference problem for embedded implicational dependencies (EIDs) is undecidable. For the more important case of finite inference (i.e., inference for finite data bases), the problem is not even recursively enumerable (r.e.); rather, it is complete in co-r.e. These results hold even for typed EIDs without equality, as well as for (untyped) template dependencies. The case for typed template dependencies remains open. The complexity of the inference problem for full dependencies has also been characterized - it is complete in exponential time for full implicational dependencies, and even for full typed template dependencies.