Learning Algorithms for Verification of Markov Decision Processes (2403.09184v3)

Abstract: We present a general framework for applying learning algorithms and heuristical guidance to the verification of Markov decision processes (MDPs). The primary goal of our techniques is to improve performance by avoiding an exhaustive exploration of the state space, instead focussing on particularly relevant areas of the system, guided by heuristics. Our work builds on the previous results of Br{\'{a}}zdil et al., significantly extending it as well as refining several details and fixing errors. The presented framework focuses on probabilistic reachability, which is a core problem in verification, and is instantiated in two distinct scenarios. The first assumes that full knowledge of the MDP is available, in particular precise transition probabilities. It performs a heuristic-driven partial exploration of the model, yielding precise lower and upper bounds on the required probability. The second tackles the case where we may only sample the MDP without knowing the exact transition dynamics. Here, we obtain probabilistic guarantees, again in terms of both the lower and upper bounds, which provides efficient stopping criteria for the approximation. In particular, the latter is an extension of statistical model-checking (SMC) for unbounded properties in MDPs. In contrast to other related approaches, we do not restrict our attention to time-bounded (finite-horizon) or discounted properties, nor assume any particular structural properties of the MDP.

• The paper develops novel learning algorithms that iteratively refine upper and lower bounds for reachability probabilities in MDP verification.
• It contrasts white-box models that leverage known internal structures with black-box models relying on probabilistic sampling and observed executions.
• Empirical evaluations demonstrate robust performance and significant computational efficiency improvements over traditional verification methods.

Learning Algorithms for Verification of Markov Decision Processes

Introduction

The paper presents a comprehensive paper on the development and evaluation of learning algorithms tailored for the verification of Markov Decision Processes (MDPs). With a focus on both white-box and black-box models, the research explores algorithms that efficiently estimate upper and lower bounds for reachability probabilities. The significance of this work lies in its potential to enhance the verification of MDPs, which are pivotal in modeling decision-making scenarios that involve uncertainty.

Markov Decision Processes and Verification

MDPs represent systems characterized by probabilistic and non-deterministic behaviors. Verification of MDPs involves determining whether a system meets certain specifications, commonly expressed in terms of reachability probabilities. Traditional verification methods are often computationally intensive, highlighting the need for innovative approaches that leverage learning algorithms to approximate verification results efficiently.

Learning Algorithms in Verification

The core contribution of this paper revolves around the development of learning algorithms optimized for verifying reachability probabilities in MDPs. These algorithms operate by iteratively refining estimates of reachability probabilities until they converge within a specified tolerance level. The researchers differentiate their approaches based on the type of model access provided:

• White-box models, where the internal structure of the MDP is known and can be directly manipulated.
• Black-box models, where the MDP's structure is hidden, and information can only be inferred through observation of its execution.

White-Box versus Black-Box Models

The research meticulously compares the efficiency and accuracy of learning algorithms in both white-box and black-box settings. For white-box models, the algorithms can exploit the known structure to efficiently identify and collapse strongly connected components (SCCs) and end components (ECs), significantly improving the convergence rate of the probability estimates. Conversely, in black-box models, the algorithms rely on sampled executions of the MDP, employing statistical methods to gradually refine their probability bounds.

Evaluation and Results

Empirical evaluations underscore the practical efficacy of the proposed algorithms, demonstrating their ability to produce accurate bounds on reachability probabilities with significantly reduced computational overhead compared to traditional verification methods. The algorithms exhibit robust performance across a range of scenarios, adapting dynamically to the complexity of the MDP and the opacity of its model.

Theoretical Implications

Beyond the immediate practical benefits, this research enriches the theoretical groundwork for learning-based verification. It offers a deeper understanding of the interplay between learning dynamics and verification accuracy, paving the way for further innovations in algorithmic design. Notably, the paper provides rigorous bounds on the convergence properties of the algorithms, contributing valuable insights into their reliability and efficiency.

Future Directions

The paper concludes with a discussion on prospective research avenues, emphasizing potential enhancements to the algorithms' efficiency and scalability. Future work might explore adaptive sampling techniques to optimize exploration and exploitation, the integration of reinforcement learning principles to better navigate the state space, and the development of parallelization strategies to leverage contemporary high-performance computing architectures.

Conclusion

The paper on learning algorithms for the verification of MDPs marks a significant step forward in the quest to balance computational feasibility with verification accuracy. By tailoring algorithms to the constraints of white-box and black-box models, this work delivers vital tools for advancing the reliability of systems modeled by MDPs. The practical outcomes evidenced by this research, combined with its theoretical contributions, set a new benchmark for future investigations in the field of probabilistic model checking.