  "Quantitative languages are extension of boolean languages that assign to
each word a real number. Mean-payoff automata are finite automata with
numerical weights on transitions that assign to each infinite path the long-run
average of the transition weights. The class of \emph{mean-payoff automaton
expressions}, introduced in [1], is a class of quantitative languages, which is
robust: it is closed under the four pointwise operations of max, min, sum and
numerical complement."[1] In this paper we improve the computational complexity
for solving the classical decision problems for mean-payoff automaton
expressions: while the previously best known upper bound was 4EXPTIME, and no
lower bound was known, we give an optimal PSPACE complete bound. As a
consequence we also obtain a conceptually simple algorithm to solve the
classical decision problems for mean-payoff automaton expressions.
