We investigate the decidability status of some problems for parallel commun
icating finite automata systems. We show that the universe and the emptines
s problems are undecidable for cpcfa and pcfa with at least five components
and rpcfa and cpcfa with at least four components, respectively. Consequen
tly, the equivalence and inclusion problems are also undecidable for these
automata systems. Some open problems are finally formulated. (C) 2001 Elsev
ier Science B.V. All rights reserved.