# MDPs as Distribution Transformers: Affine Invariant Synthesis for Safety Objectives

arXiv: https://arxiv.org/abs/2305.16796 · HTML: https://mlsanity.com/paper/2305.16796/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 May 2023.

## Built on by (most cited first, 3 of 3)

- [Certified Policy Verification and Synthesis for MDPs under Distributional Reach-avoidance Properties](https://mlsanity.com/paper/2405.04015/) (2024)
- [Entropy Objectives in Markov Decision Processes](https://mlsanity.com/paper/2606.21726/) (2026)
- [Tensor Probabilistic Model Checking of Finite-Horizon Markov Chains (Extended Version)](https://mlsanity.com/paper/2608.00374/) (2026)

## Abstract (arXiv)

Markov decision processes can be viewed as transformers of probability distributions. While this view is useful from a practical standpoint to reason about trajectories of distributions, basic reachability and safety problems are known to be computationally intractable (i.e., Skolem-hard) to solve in such models. Further, we show that even for simple examples of MDPs, strategies for safety objectives over distributions can require infinite memory and randomization. In light of this, we present a novel overapproximation approach to synthesize strategies in an MDP, such that a safety objective over the distributions is met. More precisely, we develop a new framework for template-based synthesis of certificates as affine distributional and inductive invariants for safety objectives in MDPs. We provide two algorithms within this framework. One can only synthesize memoryless strategies, but has relative completeness guarantees, while the other can synthesize general strategies. The runtime complexity of both algorithms is in PSPACE. We implement these algorithms and show that they can solve several non-trivial examples.