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, both with coverability and reachability acceptance. 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, 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 an VASS is history-deterministic, or if a language equivalent history-deterministic VASS exists. Checking language inclusion between history-deterministic 2-VASS is also undecidable.

翻译：本文研究历史确定性（一种受限的非确定性）在带状态的向量加法系统（VASS）中作为接受器、识别有限词语言（覆盖可达性和可达性接受）时的性质。历史确定性要求非确定性选择能够基于输入词的过去信息在线解析，且不损害任何可能延续的接受性。结果表明，无论计数器数量如何，历史确定性VASS严格介于确定性与非确定性VASS之间。我们比较了历史确定性系统的相对表达能力、所诱导语言类的封闭性质（覆盖语义与可达语义下，是否包含ε标记转移）。在一维情况下，包含性和正则性仍可判定；但从二维起，具有合适解析策略的历史确定性VASS本质上能够模拟双计数器明斯基机，从而推导出多个不可判定结果：VASS是否为历史确定性、是否存在与某语言等价的历史确定性VASS、以及检查历史确定性二维VASS的语言包含性均不可判定。