Minimizing Expected Cost Under Hard Boolean Constraints, with Applications to Quantitative Synthesis
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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.