Supermartingale Certificates for Parametric MDPs

In this paper, we consider the parameter synthesis and optimization problem for parametric Markov decision processes (pMDPs), the extension of classical MDPs where exact probability values are replaced by parametric expressions. Computing the rational function $f_{\lsf}$ that maps parameter valuations to the satisfaction value of a PRCTL property $\lsf$ is a computationally expensive task, particularly for pMDPs where the optimal policy may vary across the parameter space. We adopt the \emph{scenario approach} to efficiently synthesize a probably approximately correct (PAC) approximation $\ApproxFunOfProperty{f}$ of $f_{\lsf}$: by sampling parameter configurations and solving a linear program, we obtain a polynomial approximation whose error margin $\margin$ is guaranteed, with prescribed confidence, for all but an $\errorRate$-fraction of the parameter domain under the sampling distribution. We further show how this PAC framework can be combined with statistical model checking (SMC), enabling the analysis of black-box parametric models. Building on the PAC approximation, we integrate the DIRECT (DIviding RECTangles) algorithm for derivative-free global optimization over the parameter space. We establish conditional optimality-gap guarantees: under explicit Lipschitz and PAC-good-set assumptions, the difference between the true optimum $f_{\lsf}(\parameters^{*})$ and the value found by DIRECT is bounded by a partition-diameter term and, in the PAC case, an additional approximation-error term. An empirical evaluation on 2997 benchmarks focuses on the new DIRECT-based optimization component. The results show that DIRECT variants solve fewer instances than the scenario optimizer, but on their common successful instances they often return slightly better objective values and usually run faster, while remaining close to the scenario values within the PAC margin.

Learning-based approaches to verifying unknown Markov decision processes (MDPs) often employ uncertain MDPs. These models use, for example, confidence intervals to capture transition uncertainty and allow synthesis of policies that are robust to this uncertainty. However, this approach typically quantifies uncertainty independently for individual transition probabilities, ignoring dependencies due to shared latent quantities. We propose to learn such models using parametric MDPs (pMDPs), where transition probabilities are expressions over a set of parameters. We project statistical uncertainty from empirical transition frequencies onto the pMDP's parameter space, yielding a probably approximately correct (PAC) uncertainty model for the underlying MDP that respects the algebraic dependencies between transitions. The resulting models are algorithmically challenging to solve, so we propose a hierarchy of sound polytopic outer approximations of the induced confidence set. We implement and evaluate our approach, demonstrating substantially tighter uncertainty estimates than classical interval-based uncertain MDP learning techniques.

Certification methods for stochastic systems provide sufficient proof rules, based on real-valued supermartingale certificates, to determine the almost-sure satisfaction of $ω$-regular properties (and therefore of linear temporal logic) over general state spaces, encompassing both countably infinite and continuous state spaces. Conversely, reinforcement learning (RL) methods for $ω$-regular tasks have received considerable attention, but they typically lack formal guarantees that the learned policy satisfies the specification, except possibly for finite state and action spaces. We bridge these two lines of research by establishing a novel theoretical connection: under an appropriate reward, the value function associated to a policy that almost surely satisfies an $ω$-regular property encodes a Streett supermartingale certificate for that specification. Our results, validated experimentally on finite Markov decision processes, hold for finite, countably infinite, and continuous state spaces, suggesting a principled route to certificate synthesis via RL.