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

arXiv: https://arxiv.org/abs/1604.07064 · HTML: https://mlsanity.com/paper/1604.07064/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Apr 2016.

## Built on by (most cited first, 1 of 1)

- [Follow the STARs: Dynamic ω-Regular Shielding of Learned Policies](https://mlsanity.com/paper/2505.14689/) (2025)

## Abstract (arXiv)

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 ω-regular condition against an adversarial environment. We consider the case the ω-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 ε-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 ω-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.