  Families of DFAs (FDFAs) are a computational model recognizing
$\omega$-regular languages. They were introduced in the quest of finding a
Myhill-Nerode theorem for $\omega$-regular languages, and obtaining learning
algorithms. FDFAs have been shown to have good qualities in terms of the
resources required for computing Boolean operations on them (complementation,
union, and intersection) and answering decision problems (emptiness and
equivalence); all can be done in non-deterministic logspace. In this paper we
study FDFAs with a new type of acceptance condition, duo-normalization, that
generalizes the traditional normalization acceptance type. We show that
duo-normalized FDFAs are advantageous to normalized FDFAs in terms of
succinctness as they can be exponentially smaller. Fortunately this added
succinctness doesn't come at the cost of increasing the complexity of Boolean
operations and decision problems -- they can still be preformed in
non-deterministic logspace.
  An important measure of the complexity of an $\omega$-regular language, is
its position in the Wagner hierarchy. It is based on the inclusion measure of
Muller automata and for the common $\omega$-automata there exist algorithms
computing their position. We develop a similarly robust measure for
duo-normalized (and normalized) FDFAs, which we term the diameter measure. We
show that the diameter measure corresponds one-to-one to the position on the
Wagner hierarchy. We show that computing it for duo-normalized FDFAs is
PSPACE-complete, while it can be done in non-deterministic logspace for
traditional FDFAs.
