  A landmark result in the study of logics for formal verification is Janin &
Walukiewicz's theorem, stating that the modal $\mu$-calculus ($\mu\mathrm{ML}$)
is equivalent modulo bisimilarity to standard monadic second-order logic (here
abbreviated as $\mathrm{smso}$), over the class of labelled transition systems
(LTSs for short). Our work proves two results of the same kind, one for the
alternation-free fragment of $\mu\mathrm{ML}$ ($\mu_D\mathrm{ML}$) and one for
weak $\mathrm{mso}$ ($\mathrm{wmso}$). Whereas it was known that
$\mu_D\mathrm{ML}$ and $\mathrm{wmso}$ are equivalent modulo bisimilarity on
binary trees, our analysis shows that the picture radically changes once we
reason over arbitrary LTSs. The first theorem that we prove is that, over LTSs,
$\mu_D\mathrm{ML}$ is equivalent modulo bisimilarity to noetherian
$\mathrm{mso}$ ($\mathrm{nmso}$), a newly introduced variant of $\mathrm{smso}$
where second-order quantification ranges over "well-founded" subsets only. Our
second theorem starts from $\mathrm{wmso}$, and proves it equivalent modulo
bisimilarity to a fragment of $\mu_D\mathrm{ML}$ defined by a notion of
continuity. Analogously to Janin & Walukiewicz's result, our proofs are
automata-theoretic in nature: as another contribution, we introduce classes of
parity automata characterising the expressiveness of $\mathrm{wmso}$ and
$\mathrm{nmso}$ (on tree models) and of $\mu_C\mathrm{ML}$ and
$\mu_D\mathrm{ML}$ (for all transition systems).
