# Temporal Logic Control for Stochastic Linear Systems using Abstraction Refinement of Probabilistic Games

arXiv: https://arxiv.org/abs/1410.5387 · HTML: https://mlsanity.com/paper/1410.5387/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Oct 2014.

## Built on by (most cited first, 2 of 2)

- [Learning Optimal Strategies for Temporal Tasks in Stochastic Games](https://mlsanity.com/paper/2102.04307/) (2021)
- [Provably Correct Controller Synthesis of Switched Stochastic Systems with Metric Temporal Logic Specifications: A Case Study on Power Systems](https://mlsanity.com/paper/2103.14264/) (2021)

## Abstract (arXiv)

We consider the problem of computing the set of initial states of a dynamical system such that there exists a control strategy to ensure that the trajectories satisfy a temporal logic specification with probability 1 (almost-surely). We focus on discrete-time, stochastic linear dynamics and specifications given as formulas of the Generalized Reactivity(1) fragment of Linear Temporal Logic over linear predicates in the states of the system. We propose a solution based on iterative abstraction-refinement, and turn-based 2-player probabilistic games. While the theoretical guarantee of our algorithm after any finite number of iterations is only a partial solution, we show that if our algorithm terminates, then the result is the set of satisfying initial states. Moreover, for any (partial) solution our algorithm synthesizes witness control strategies to ensure almost-sure satisfaction of the temporal logic specification. We demonstrate our approach on an illustrative case study.