Jumping automata are finite automata that read their input in a non-sequential manner, by allowing a reading head to ``jump'' between positions on the input, consuming a permutation of the input word. We argue that allowing the head to jump should incur some cost. To this end, we propose four quantitative semantics for jumping automata, whereby the jumps of the head in an accepting run define the cost of the run. The four semantics correspond to different interpretations of jumps: the \emph{absolute distance} semantics counts the distance the head jumps, the \emph{reversal} semantics counts the number of times the head changes direction, the \emph{Hamming distance} measures the number of letter-swaps the run makes, and the \emph{maximum jump} semantics counts the maximal distance the head jumps in a single step, We study these measures, with the main focus being the \emph{boundedness problem}: given a jumping automaton, decide whether its (quantitative) language is bounded by some given number $k$. We establish the decidability and complexity for this problem under several variants.

翻译：跳跃自动机是一种通过允许读头在输入位置上"跳跃"以消耗输入词排列的方式非顺序读取输入的有限自动机。我们认为允许读头跳跃应产生一定代价。为此，我们为跳跃自动机提出四种量化语义，其中接受运行中读头的跳跃定义了运行代价。这四种语义对应跳跃的不同解释：\emph{绝对距离}语义统计读头跳跃的距离，\emph{转向}语义统计读头改变方向的次数，\emph{汉明距离}衡量运行中字母交换的次数，而\emph{最大跳跃}语义统计单步中读头跳跃的最大距离。我们研究这些度量，主要关注\emph{有界性问题}：给定一个跳跃自动机，判定其（量化）语言是否被给定数$k$所界定。我们在若干变体下建立了该问题的可判定性与复杂度。