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

Abstract

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.