We consider the distributed implementability problem as: Given a labeled transition system TS together with a distribution D of its actions over a set of processes, does there exist a distributed system over D such that its global transition system is `equivalent' to TS? We consider the distributed system models of synchronous products of transition systems and Zielonka's asynchronous automata. In this paper we provide complexity bounds for the above problem with three interpretations of `equivalent': as transition system isomorphism, as language equivalence, and as bisimilarity. In particular, we solve two problems left open in the literature. We also describe a logic programming implementation which complements the existing implementation for the synthesis of asynchronous automata initiated by the second author.