  In Boolean synthesis, we are given an LTL specification, and the goal is to
construct a transducer that realizes it against an adversarial environment.
Often, a specification contains both Boolean requirements that should be
satisfied against an adversarial environment, and multi-valued components that
refer to the quality of the satisfaction and whose expected cost we would like
to minimize with respect to a probabilistic environment.
  In this work we study, for the first time, mean-payoff games in which the
system aims at minimizing the expected cost against a probabilistic
environment, while surely satisfying an $\omega$-regular condition against an
adversarial environment. We consider the case the $\omega$-regular condition is
given as a parity objective or by an LTL formula. We show that in general,
optimal strategies need not exist, and moreover, the limit value cannot be
approximated by finite-memory strategies. We thus focus on computing the
limit-value, and give tight complexity bounds for synthesizing
$\epsilon$-optimal strategies for both finite-memory and infinite-memory
strategies.
  We show that our game naturally arises in various contexts of synthesis with
Boolean and multi-valued objectives. Beyond direct applications, in synthesis
with costs and rewards to certain behaviors, it allows us to compute the
minimal sensing cost of $\omega$-regular specifications -- a measure of quality
in which we look for a transducer that minimizes the expected number of signals
that are read from the input.
