Complexity results for checking distributed implementability

We consider the distributed implementability problem: Given a labeled transition system TS together with a distribution /spl Delta/ of its actions over a set of processes, does there exist a distributed system over /spl Delta/ such that its global transition system is 'equivalent' to TS? We work with the distributed system models of synchronous products of transition systems (Arnold, 1994) and asynchronous automata (Zielonka, 1987). 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 problems left open in Castellani et al. (1999) and Morin (1999).