We consider one of the weakest variants of cost register automata over a tropical semiring, namely copyless cost register automata over [Formula: see text] with updates using [Formula: see text] and increments. We show that this model can simulate, in some sense, the runs of counter machines with zero-tests. We deduce that a number of problems pertaining to that model are undecidable, namely equivalence, upperboundedness, and semilinearity. In particular, the undecidability of equivalence disproves a conjecture of Alur et al. from 2012. To emphasize how weak these machines are, we also show that they can be expressed as a restricted form of linearly-ambiguous weighted automata.

中文翻译：

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弱成本寄存器自动机仍然强大

我们考虑热带半环上成本寄存器自动机的最弱变体之一，即[公式：见文本]上的无副本成本寄存器自动机，使用[公式：见文本]和增量进行更新。我们展示了这个模型在某种意义上可以模拟零测试计数器的运行。我们推断与该模型有关的许多问题是不可判定的，即等价性、上限和半线性。特别是，等价的不可判定性反驳了 Alur 等人的猜想。从 2012 年开始。为了强调这些机器的弱点，我们还表明它们可以表示为线性模糊加权自动机的受限形式。