  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 three
quantitative semantics for jumping automata, whereby the jumps of the head in
an accepting run define the cost of the run. The three 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, and the \emph{Hamming distance}
measures the number of letter-swaps the run makes.
  We study these measures, with the main focus being the \emph{boundedness
problem}: given a jumping automaton, decide whether its (quantitative)
languages is bounded by some given number $k$. We establish the decidability
and complexity for this problem under several variants.
