  Weighted timed automata (WTA) model quantitative aspects of real-time systems
like continuous consumption of memory, power or financial resources. They
accept quantitative timed languages where every timed word is mapped to a
value, e.g., a real number. In this paper, we prove a Nivat theorem for WTA
which states that recognizable quantitative timed languages are exactly those
which can be obtained from recognizable boolean timed languages with the help
of several simple operations. We also introduce a weighted extension of
relative distance logic developed by Wilke, and we show that our weighted
relative distance logic and WTA are equally expressive. The proof of this
result can be derived from our Nivat theorem and Wilke's theorem for relative
distance logic. Since the proof of our Nivat theorem is constructive, the
translation process from logic to automata and vice versa is also constructive.
This leads to decidability results for weighted relative distance logic.
