  This paper studies the relationship between disjunctive form, a syntactic
normal form for the modal mu calculus, and the alternation hierarchy. First it
shows that all disjunctive formulas which have equivalent tableau have the same
syntactic alternation depth. However, tableau equivalence only preserves
alternation depth for the disjunctive fragment: there are disjunctive formulas
with arbitrarily high alternation depth that are tableau equivalent to
alternation-free non-disjunctive formulas. Conversely, there are
non-disjunctive formulas of arbitrarily high alternation depth that are tableau
equivalent to disjunctive formulas without alternations. This answers
negatively the so far open question of whether disjunctive form preserves
alternation depth. The classes of formulas studied here illustrate a previously
undocumented type of avoidable syntactic complexity which may contribute to our
understanding of why deciding the alternation hierarchy is still an open
problem.
