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

TL;DR

This paper introduces a novel approach to Boolean synthesis by studying mean-payoff games that aim to minimize expected costs under probabilistic environments while satisfying strict Boolean constraints, with implications for synthesis with costs and rewards.

Contribution

It is the first to analyze mean-payoff games in the context of Boolean synthesis with probabilistic environments and provides complexity bounds for synthesizing near-optimal strategies.

Findings

Optimal strategies may not always exist.

Finite-memory strategies cannot approximate the limit value.

Complexity bounds for synthesizing ε-optimal strategies.

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.