Authors: Alessandro Abate, Daniel Contro, Mirco Giacobbe, Agustín Martínez-Suñé, Diptarko Roy
Organizations: University of Oxford, UK · University of Birmingham, UK
Abstract
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.
Safe offline reinforcement learning assumes a cost function on every transition. We ask what remains possible when safety can be judged only by comparing short clips and occasionally asking whether an episode exceeded its budget. Certified safety curation answers with a filter-then-clone pipeline: a state-only value trained from segment comparisons scores whole trajectories, Learn-then-Test calibration certifies a selection threshold under a distribution-free (α,δ) bound on the unsafe fraction of the selection, and behavior cloning follows. We are not aware of prior work certifying the composition of a training set for offline RL or imitation. Oracle controls justify the design: reweighting individual transitions fails even with an exact value, so the value selects whole trajectories. The policies satisfy the cost budget on eleven of fifteen DSRL tasks, one short of cloning the ground-truth safe subset, which needs a label on every trajectory; the uncertified variant reaches twelve. Retrained on the certified selection, the strongest full-label method becomes safe where no setting of its own cost target rescues it. Refusal is predictable: the certificate's probability has a closed form in the purity the pool attains, which the calibration sample estimates and the scorer enters only through.
We consider the problems of formal verification and synthesis in parametric Markov decision processes (MDPs) with general measurable state and action spaces. The heart of our approach is a parameter flattening transformation, which allows us to transform parametric MDPs into semantically equivalent non-parametric MDPs. Building on this transformation, we introduce the novel notion of parametric supermartingale certificates, which generalize the traditional supermartingale certificates---used for non-parametric MDPs---to the parametric setting. We use our parametric supermartingale certificates to design algorithms for verification and approximate synthesis in polynomial arithmetic parametric MDPs. This leads to the first verification and synthesis algorithms for parametric MDPs with general state and action spaces. We implement our algorithms and experimentally evaluate them on several continuous parametric random walk benchmarks.
While Bellman equations for basic reach, avoid, and reach-avoid problems are well studied, the relationship between value optimality and policy optimality becomes subtle in the undiscounted infinite-horizon setting, particularly for more complicated tasks. Greedily maximizing the Q-function can produce policies that indefinitely defer task completion for reach-avoid problems, or equivalently, Until specifications, even when the value function is optimal. Building upon recent results decomposing the value function for temporal logic (TL) into a graph of constituent value functions, we construct non-Markovian policies based on state history that avoid this pathology and prove their optimality with respect to the quantitative robustness score for nested Until, Globally, and Globally-Until specifications. We further show how the Q function can serve as a safety filter for complex TL specifications, extending prior results beyond simple avoid or reach-avoid tasks.