The ability to compute reward-optimal policies for given and known finite Markov decision processes (MDPs) underpins a variety of applications across planning, controller synthesis, and verification. However, we often want policies (1) to be robust, i.e., they perform well on perturbations of the MDP and (2) to satisfy additional structural constraints regarding, e.g., their representation or implementation cost. Computing such robust and constrained policies is indeed computationally more challenging. This paper contributes the first approach to effectively compute robust policies subject to arbitrary structural constraints using a flexible and efficient framework. We achieve flexibility by allowing to express our constraints in a first-order theory over a set of MDPs, while the root for our efficiency lies in the tight integration of satisfiability solvers to handle the combinatorial nature of the problem and probabilistic model checking algorithms to handle the analysis of MDPs. Experiments on a few hundred benchmarks demonstrate the feasibility for constrained and robust policy synthesis and the competitiveness with state-of-the-art methods for various fragments of the problem.

Abstract Computing optimal conditional reachability probabilities in Markov decision processes (MDPs) is tractable by a reduction to reachability probabilities. Yet, this reduction yields cyclic, challenging MDPs that are often notoriously hard to solve. We present an alternative, practically efficient method to compute optimal conditional reachabilities. This new method is numerically stable, can decide the threshold problem in linear time on acyclic MDPs, and yields performance comparable to standard reachability queries. We also integrate the method in an abstraction-refinement framework to analyse millions of Markov chains at once. We demonstrate the efficacy of the new methods on benchmarks from Bayesian network analysis, probabilistic programs, and runtime monitoring and show speed-ups up to multiple orders of magnitude.

Abstract Shielding is a prominent model-based technique to ensure safety of autonomous agents. Classical shielding aims to ensure that nothing bad ever happens and comes with strong guarantees about safety and maximal permissiveness. However, shielding systems for probabilistic safety, where something bad is allowed to happen with an acceptable probability, has proven to be more intricate. This paper presents a formal framework that conservatively extends classical shields to probabilistic safety. In this framework, we (i) demonstrate the impossibility of preserving the strong guarantees on safety and permissiveness, (ii) provide natural shields with weaker guarantees, and (iii) introduce offline and online shield constructions ensuring strong safety guarantees. The empirical evaluation highlights the practical advantages of the new shields, as well as their computational feasibility.

Abstract This paper marries two state-of-the-art controller synthesis methods for partially observable Markov decision processes (POMDPs), a prominent model in sequential decision making under uncertainty. A central issue is to find a POMDP controller—that solely decides based on the observations seen so far—to achieve a total expected reward objective. As finding optimal controllers is undecidable, we concentrate on synthesising good finite-state controllers (FSCs). We do so by tightly integrating two modern, orthogonal methods for POMDP controller synthesis: a belief-based and an inductive approach. The former method obtains an FSC from a finite fragment of the so-called belief MDP, an MDP that keeps track of the probabilities of equally observable POMDP states. The latter is an inductive search technique over a set of FSCs, e.g., controllers with a fixed memory size. The key result of this paper is a symbiotic anytime algorithm that tightly integrates both approaches such that each profits from the controllers constructed by the other. Experimental results indicate a substantial improvement in the value of the controllers while significantly reducing the synthesis time and memory footprint.

Abstract Markov decision processes (MDPs) describe decision making subject to probabilistic uncertainty. A classical problem on MDPs is to compute a policy, selecting actions in every state, that maximizes the probability of reaching a dedicated set of target states. Computing such policies in tabular form is efficiently possible via standard algorithms. However, for further processing by either humans or machines, policies should be represented concisely, e.g., as a decision tree. This paper considers finding (almost) optimal decision trees of minimal depth and contributes a deductive synthesis approach. Technically, we combine pruning the space of concise policies with an abstraction-refinement loop with an SMT-encoding that maps candidate policies into decision trees. Our experiments show that this approach beats the state-of-the-art solver using an MILP encoding by orders of magnitude. The approach also pairs well with heuristic approaches that map a fixed policy into a decision tree: for an MDP with 1.5M states, our approach reduces the size of the given tree by 90%, while sacrificing only 1% of the optimal performance.

We study decision making policies in Markov decision processes (MDPs). Two key performance indicators of such policies are their value and their interpretability. On the one hand, policies that optimize value can be efficiently computed via a plethora of standard methods. However, the representation of these policies may prevent their interpretability. On the other hand, policies with good interpretability, such as policies represented by a small decision tree, are computationally hard to obtain. This paper contributes a local search approach to find policies with good value, represented by small decision trees. Our local search symbiotically combines learning decision trees from value-optimal policies with symbolic approaches that optimize the size of the decision tree within a constrained neighborhood. Our empirical evaluation shows that this combination provides drastically smaller decision trees for MDPs that are significantly larger than what can be handled by optimal decision tree learners.

Dealing with aleatoric uncertainty is key in many domains involving sequential decision making, e.g., planning in AI, network protocols, and symbolic program synthesis. This paper presents a general-purpose model-based framework to obtain policies operating in uncertain environments in a fully automated manner. The new concept of coloured Markov Decision Processes (MDPs) enables a succinct representation of a wide range of synthesis problems. A coloured MDP describes a collection of possible policy configurations with their structural dependencies. The framework covers the synthesis of (a) programmatic policies from probabilistic program sketches and (b) finite-state controllers representing policies for partially observable MDPs (POMDPs), including decentralised POMDPs as well as constrained POMDPs. We show that all these synthesis problems can be cast as exploring memoryless policies in the corresponding coloured MDP. This exploration uses a symbiosis of two orthogonal techniques: abstraction refinement—using a novel refinement method—and counter-example generalisation. Our approach outperforms dedicated synthesis techniques on some problems and significantly improves an earlier version of this framework.

Abstract This paper marries two state-of-the-art controller synthesis methods for partially observable Markov decision processes (POMDPs), a prominent model in sequential decision making under uncertainty. A central issue is to find a POMDP controller—that solely decides based on the observations seen so far—to achieve a total expected reward objective. As finding optimal controllers is undecidable, we concentrate on synthesising good finite-state controllers (FSCs). We do so by tightly integrating two modern, orthogonal methods for POMDP controller synthesis: a belief-based and an inductive approach. The former method obtains an FSC from a finite fragment of the so-called belief MDP, an MDP that keeps track of the probabilities of equally observable POMDP states. The latter is an inductive search technique over a set of FSCs, e.g., controllers with a fixed memory size. The key result of this paper is a symbiotic anytime algorithm that tightly integrates both approaches such that each profits from the controllers constructed by the other. Experimental results indicate a substantial improvement in the value of the controllers while significantly reducing the synthesis time and memory footprint.