  We study a variant of the classical membership problem in automata theory,
which consists of deciding whether a given input word is accepted by a given
automaton. We do so under a different perspective, that is, we consider a
dynamic version of the problem, called monitoring problem, where the automaton
is fixed and the input is revealed as in a stream, one symbol at a time
following the natural order on positions. The goal here is to design a dynamic
data structure that can be queried about whether the word consisting of symbols
revealed so far is accepted by the automaton, and that can be efficiently
updated when the next symbol is revealed. We provide complexity bounds for this
monitoring problem, by considering timed automata that process symbols
interleaved with timestamps. The main contribution is that monitoring of a
one-clock timed automaton, with all its components but the clock constants
fixed, can be done in amortised constant time per input symbol.
