In case an ACTL formula phi fails over a transition graph M, it is most use
ful to provide a counterexample, i.e., a computation tree of M, witnessing
the failure. If there exists a single path in M which by itself witnesses t
he failure of phi, then phi has a linear counterexample. We show that, give
n M and phi, where M does not satisfy phi, it is NP-hard to determine wheth
er there exists a linear counterexample. Moreover, it is PSPACE-hard to dec
ide whether an ACTL formula phi always admits a linear counterexample if it
fails. This means that there exists no simple characterization of the ACTL
formulas that guarantee linear counterexamples. Consequently, we study tem
plates of ACTL formulas, i.e., skeletons of modal formulas whose atoms are
disregarded. We identify the (unique) maximal set LIN of templates whose in
stances (obtained by replacing atoms with arbitrary pure state formulas) al
ways guarantee linear counterexamples. We show that For each ACTL formula g
which is an instance of a template gamma (star)* epsilon LIN, and for each
Kripke structure M such that M does not satisfy phi, a single path of M wi
tnessing the failure by itself can be computed in polynomial time. (C) 2001
Academic Press.