  We answer an open complexity question by Hofman, Lasota, Mayr, Totzke (LMCS
2016) for simulation preorder on the class of succinct one-counter nets (i.e.,
one-counter automata with no zero tests where counter increments and decrements
are integers written in binary); the problem was known to be PSPACE-hard and in
EXPSPACE. We show that all relations between bisimulation equivalence and
simulation preorder are EXPSPACE-hard for these nets; simulation preorder is
thus EXPSPACE-complete. The result is proven by a reduction from reachability
games whose EXPSPACE-completeness in the case of succinct one-counter nets was
shown by Hunter (RP 2015), by using other results. We also provide a direct
self-contained EXPSPACE-completeness proof for a special case of such
reachability games, namely for a modification of countdown games that were
shown EXPTIME-complete by Jurdzinski, Sproston, Laroussinie (LMCS 2008); in our
modification the initial counter value is not given but is freely chosen by the
first player. We also present an alternative proof for the upper bound by
Hofman et al. In particular, we give a new simplified proof of the belt theorem
that yields a simple graphic presentation of simulation preorder on
(non-succinct) one-counter nets and leads to a polynomial-space algorithm
(which is trivially extended to an exponential-space algorithm for succinct
one-counter nets).
