A weighted version of McNaughton’s theorem

We investigate weighted finite state automata with the Büchi-acceptance condition. The costs of computations depend on the state of the computation and on the time when it is executed. We characterize those systems whose behaviour is continuous. Then we introduce weighted Muller-automata and show that their behaviour is even Lipschitz-continuous. Under some not to hard restrictions we show that if the behaviour of a finite weighted Büchi-automaton is Lipschitzcontinuous, then it is also Muller-recognizable. This is a weighted version of McNaughton’s classical theorem. Using this result we show that under the same restrictions it is decidable whether the behaviour of two weighted Muller-automata is equal.