Robust Synchronization in Markov Decision Processes

We consider synchronizing properties of Markov decision processes (MDP), viewed as generators of sequences of probability distributions over states. A probability distribution is p-synchronizing if the probability mass is at least p in some state, and a sequence of probability distributions is weakly p-synchronizing, or strongly p-synchronizing if respectively infinitely many, or all but finitely many distributions in the sequence are p-synchronizing. For each synchronizing mode, an MDP can be (i) sure winning if there is a strategy that produces a 1-synchronizing sequence; (ii) almost-sure winning if there is a strategy that produces a sequence that is, for all {epsilon} > 0, a (1-{epsilon})-synchronizing sequence; (iii) limit-sure winning if for all {epsilon} > 0, there is a strategy that produces a (1-{epsilon})-synchronizing sequence. For each synchronizing and winning mode, we consider the problem of deciding whether an MDP is winning, and we establish matching upper and lower complexity bounds of the problems, as well as the optimal memory requirement for winning strategies: (a) for all winning modes, we show that the problems are PSPACE-complete for weakly synchronizing, and PTIME-complete for strongly synchronizing; (b) we show that for weakly synchronizing, exponential memory is sufficient and may be necessary for sure winning, and infinite memory is necessary for almost-sure winning; for strongly synchronizing, linear-size memory is sufficient and may be necessary in all modes; (c) we show a robustness result that the almost-sure and limit-sure winning modes coincide for both weakly and strongly synchronizing.

Limit Synchronization in Markov Decision Processes

Markov decision processes (MDP) are finite-state systems with both strategic and probabilistic choices. After fixing a strategy, an MDP produces a sequence of probability distributions over states. The sequence is eventually synchronizing if the probability mass accumulates in a single state, possibly in the limit. Precisely, for 0 <= p <= 1 the sequence is p-synchronizing if a probability distribution in the sequence assigns probability at least p to some state, and we distinguish three synchronization modes: (i) sure winning if there exists a strategy that produces a 1-synchronizing sequence; (ii) almost-sure winning if there exists a strategy that produces a sequence that is, for all epsilon > 0, a (1-epsilon)-synchronizing sequence; (iii) limit-sure winning if for all epsilon > 0, there exists a strategy that produces a (1-epsilon)-synchronizing sequence. We consider the problem of deciding whether an MDP is sure, almost-sure, limit-sure winning, and we establish the decidability and optimal complexity for all modes, as well as the memory requirements for winning strategies. Our main contributions are as follows: (a) for each winning modes we present characterizations that give a PSPACE complexity for the decision problems, and we establish matching PSPACE lower bounds; (b) we show that for sure winning strategies, exponential memory is sufficient and may be necessary, and that in general infinite memory is necessary for almost-sure winning, and unbounded memory is necessary for limit-sure winning; (c) along with our results, we establish new complexity results for alternating finite automata over a one-letter alphabet.

Infinite Synchronizing Words for Probabilistic Automata (Erratum)

In [1], we introduced the weakly synchronizing languages for probabilistic automata. In this report, we show that the emptiness problem of weakly synchronizing languages for probabilistic automata is undecidable. This implies that the decidability result of [1-3] for the emptiness problem of weakly synchronizing language is incorrect.

Synchronizing Objectives for Markov Decision Processes

We introduce synchronizing objectives for Markov decision processes (MDP). Intuitively, a synchronizing objective requires that eventually, at every step there is a state which concentrates almost all the probability mass. In particular, it implies that the probabilistic system behaves in the long run like a deterministic system: eventually, the current state of the MDP can be identified with almost certainty. We study the problem of deciding the existence of a strategy to enforce a synchronizing objective in MDPs. We show that the problem is decidable for general strategies, as well as for blind strategies where the player cannot observe the current state of the MDP. We also show that pure strategies are sufficient, but memory may be necessary.