  Cousot and Cousot introduced and studied a general past/future-time
specification language, called mu*-calculus, featuring a natural time-symmetric
trace-based semantics. The standard state-based semantics of the mu*-calculus
is an abstract interpretation of its trace-based semantics, which turns out to
be incomplete (i.e., trace-incomplete), even for finite systems. As a
consequence, standard state-based model checking of the mu*-calculus is
incomplete w.r.t. trace-based model checking. This paper shows that any
refinement or abstraction of the domain of sets of states induces a
corresponding semantics which is still trace-incomplete for any propositional
fragment of the mu*-calculus. This derives from a number of results, one for
each incomplete logical/temporal connective of the mu*-calculus, that
characterize the structure of models, i.e. transition systems, whose
corresponding state-based semantics of the mu*-calculus is trace-complete.
