  We consider history-determinism, a restricted form of non-determinism, for
Vector Addition Systems with States (VASS) when used as acceptors to recognise
languages of finite words. History-determinism requires that the
non-deterministic choices can be resolved on-the-fly; based on the past and
without jeopardising acceptance of any possible continuation of the input word.
  Our results show that the history-deterministic (HD) VASS sit strictly
between deterministic and non-deterministic VASS regardless of the number of
counters. We compare the relative expressiveness of HD systems, and
closure-properties of the induced language classes, with coverability and
reachability semantics, and with and without $\varepsilon$-labelled
transitions.
  Whereas in dimension 1, inclusion and regularity remain decidable, from
dimension two onwards, HD-VASS with suitable resolver strategies, are
essentially able to simulate 2-counter Minsky machines, leading to several
undecidability results: It is undecidable whether a VASS is
history-deterministic, or if a language equivalent history-deterministic VASS
exists. Checking language inclusion between history-deterministic 2-VASS is
also undecidable.
