We revisit the parameterized model checking problem for token-passing systems
and specifications in indexed CTL∗\X.
Emerson and Namjoshi (1995, 2003) have shown that parameterized model checking
of indexed CTL∗\X in uni-directional token
rings can be reduced to checking rings up to some \emph{cutoff} size. Clarke et
al. (2004) have shown a similar result for general topologies and indexed
LTL\X, provided processes cannot choose the
directions for sending or receiving the token.
We unify and substantially extend these results by systematically exploring
fragments of indexed CTL∗\X with respect to
general topologies. For each fragment we establish whether a cutoff exists, and
for some concrete topologies, such as rings, cliques and stars, we infer small
cutoffs. Finally, we show that the problem becomes undecidable, and thus no
cutoffs exist, if processes are allowed to choose the directions in which they
send or from which they receive the token.Comment: We had to remove an appendix until the proofs and notations there is
cleare