Taming Infinity one Chunk at a Time: Concisely Represented Strategies in One-Counter MDPs
This work addresses a fundamental challenge in decision-making under uncertainty for infinite systems, with potential applications in verification and control, though it is incremental as it builds on existing OC-MDP theory.
The paper tackles the problem of synthesizing and verifying strategies in one-counter Markov decision processes (OC-MDPs), which are infinite MDPs with a counter, for objectives like state-reachability and selective termination. It introduces concisely represented strategies based on interval partitions and develops a generic compression method that yields decidability with all complexities within PSPACE.
Markov decision processes (MDPs) are a canonical model to reason about decision making within a stochastic environment. We study a fundamental class of infinite MDPs: one-counter MDPs (OC-MDPs). They extend finite MDPs via an associated counter taking natural values, thus inducing an infinite MDP over the set of configurations (current state and counter value). We consider two characteristic objectives: reaching a target state (state-reachability), and reaching a target state with counter value zero (selective termination). The synthesis problem for the latter is not known to be decidable and connected to major open problems in number theory. Furthermore, even seemingly simple strategies (e.g., memoryless ones) in OC-MDPs might be impossible to build in practice (due to the underlying infinite configuration space): we need finite, and preferably small, representations. To overcome these obstacles, we introduce two natural classes of concisely represented strategies based on a (possibly infinite) partition of counter values in intervals. For both classes, and both objectives, we study the verification problem (does a given strategy ensure a high enough probability for the objective?), and two synthesis problems (does there exist such a strategy?): one where the interval partition is fixed as input, and one where it is only parameterized. We develop a generic approach based on a compression of the induced infinite MDP that yields decidability in all cases, with all complexities within PSPACE.