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Register a new userRuntime Monitoring for Markov Decision Processes
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Added by
Sebastian Junges
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We investigate the problem of monitoring partially observable systems with nondeterministic and probabilistic dynamics. In such systems, every state may be associated with a risk, e.g., the probability of an imminent crash. During runtime, we obtain partial information about the system state in form of observations. The monitor uses this information to estimate the risk of the (unobservable) current system state. Our results are threefold. First, we show that extensions of state estimation approaches do not scale due the combination of nondeterminism and probabilities. While convex hull algorithms improve the practical runtime, they do not prevent an exponential memory blowup. Second, we present a tractable algorithm based on model checking conditional reachability probabilities. Third, we provide prototypical implementations and manifest the applicability of our algorithms to a range of benchmarks. The results highlight the possibilities and boundaries of our novel algorithms.
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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.
Model checking probabilistic CTL properties of Markov decision processes with convex uncertainties has been recently investigated by Puggelli et al. Such model checking algorithms typically suffer from the state space explosion. In this paper, we address probabilistic bisimulation to reduce the size of such an MDP while preserving the probabilistic CTL properties it satisfies. In particular, we discuss the key ingredients to build up the operations of parallel composition for composing interval MDP components at run-time. More precisely, we investigate how the parallel composition operator for interval MDPs can be defined so as to arrive at a congruence closure. As a result, we show that probabilistic bisimulation for interval MDPs is congruence with respect to two facets of parallelism, namely synchronous product and interleaving.
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.
This article presents the complexity of reachability decision problems for parametric Markov decision processes (pMDPs), an extension to Markov decision processes (MDPs) where transitions probabilities are described by polynomials over a finite set of parameters. In particular, we study the complexity of finding values for these parameters such that the induced MDP satisfies some maximal or minimal reachability probability constraints. We discuss different variants depending on the comparison operator in the constraints and the domain of the parameter values. We improve all known lower bounds for this problem, and notably provide ETR-completeness results for distinct variants of this problem.
This paper studies parametric Markov decision processes (pMDPs), an extension to Markov decision processes (MDPs) where transitions probabilities are described by polynomials over a finite set of parameters. Fixing values for all parameters yields MDPs. In particular, this paper studies the complexity of finding values for these parameters such that the induced MDP satisfies some reachability constraints. We discuss different variants depending on the comparison operator in the constraints and the domain of the parameter values. We improve all known lower bounds for this problem, and notably provide ETR-completeness results for distinct variants of this problem. Furthermore, we provide insights in the functions describing the induced reachability probabilities, and how pMDPs generalise concurrent stochastic reachability games.
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