Optimal Constrained sc-LTL Planning in MDPs via Switching Policies

This work studies model-free reinforcement learning for co-safe linear temporal logic (sc-LTL) objectives in finite Markov decision processes, which can be reduced to maximal reachability objectives via the standard product construction. For this problem, direct sample-based bootstrap methods (e.g., TD or Q-learning) may fail to converge to optimal policies due to the noncontractive nature and nonuniqueness of solutions to the Bellman equation. We develop a new two-step model-free reinforcement learning method that first uses a discounted surrogate to identify a clamp set that resolves this nonuniqueness, and then applies undiscounted policy evaluation and greedy policy improvement with guarantees of finding an optimal solution. We prove almost-sure convergence of the policy evaluation step and finite termination of the policy iteration algorithm at an optimal policy. These theoretical results are validated through numerical experiments on a stochastic grid world.

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.

We present a novel end-to-end model-based Reinforcement Learning (RL) algorithm for efficient policy synthesis under given Linear Temporal Logic (LTL) specifications (e.g., safety or reachability) in unknown environments. To do so, a Limit-Deterministic B{ü}chi Automaton (LDBA) representation of the LTL task is synchronised with a Bayes-Adaptive Markov Decision Process (BAMDP) representation of the environment, which allows us to leverage an enhanced exploration-exploitation trade-off that is achieved via Bayesian RL, as opposed to traditional non-Bayesian approaches. We further propose a novel Bayes-Adaptive Monte-Carlo Planning (BAMCP) algorithm to allow for approximate Bayes-optimal strategy synthesis in the synchronised BAMDP construct. A range of finite- and infinite-horizon task experiments demonstrate the effectiveness of our approach in terms of both property satisfaction and sample efficiency, when compared to traditional model-free approaches. Additional ablation studies also successfully highlight the value of the novel BAMCP algorithm in comparison to classical BAMCP for LTL task satisfaction. Finally, we also showcase a successful application of our approach for \textit{cautious} RL, namely to reduce the number of task violations incurred during policy training.