  For a fixed type of Petri nets $\tau$, \textsc{$\tau$-Synthesis} is the task
of finding for a given transition system $A$ a Petri net $N$ of type $\tau$
($\tau$-net, for short) whose reachability graph is isomorphic to $A$ if there
is one. The decision version of this search problem is called
\textsc{$\tau$-Solvability}. If an input $A$ allows a positive decision, then
it is called $\tau$-solvable and a sought net $N$ $\tau$-solves $A$. As a well
known fact, $A$ is $\tau$-solvable if and only if it has the so-called
$\tau$-\emph{event state separation property} ($\tau$-ESSP, for short) and the
$\tau$-\emph{state separation property} ($\tau$-SSP, for short). The question
whether $A$ has the $\tau$-ESSP or the $\tau$-SSP defines also decision
problems. In this paper, for all $b\in \mathbb{N}$, we completely characterize
the computational complexity of \textsc{$\tau$-Solvability},
\textsc{$\tau$-ESSP} and \textsc{$\tau$-SSP} for the types of pure $b$-bounded
Place/Transition-nets, the $b$-bounded Place/Transition-nets and their
corresponding $\mathbb{Z}_{b+1}$-extensions.
