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

Synchronization and Control in Intrinsic and Designed Computation: An Information-Theoretic Analysis of Competing Models of Stochastic Computation

We adapt tools from information theory to analyze how an observer comes to synchronize with the hidden states of a finitary, stationary stochastic process. We show that synchronization is determined by both the process's internal organization and by an observer's model of it. We analyze these components using the convergence of state-block and block-state entropies, comparing them to the previously known convergence properties of the Shannon block entropy. Along the way, we introduce a hierarchy of information quantifiers as derivatives and integrals of these entropies, which parallels a similar hierarchy introduced for block entropy. We also draw out the duality between synchronization properties and a process's controllability. The tools lead to a new classification of a process's alternative representations in terms of minimality, synchronizability, and unifilarity.Comment: 25 pages, 13 figures, 1 tabl

High-level Counterexamples for Probabilistic Automata

Providing compact and understandable counterexamples for violated system properties is an essential task in model checking. Existing works on counterexamples for probabilistic systems so far computed either a large set of system runs or a subset of the system's states, both of which are of limited use in manual debugging. Many probabilistic systems are described in a guarded command language like the one used by the popular model checker PRISM. In this paper we describe how a smallest possible subset of the commands can be identified which together make the system erroneous. We additionally show how the selected commands can be further simplified to obtain a well-understandable counterexample

A Compositional Semantics for Stochastic Reo Connectors

In this paper we present a compositional semantics for the channel-based coordination language Reo which enables the analysis of quality of service (QoS) properties of service compositions. For this purpose, we annotate Reo channels with stochastic delay rates and explicitly model data-arrival rates at the boundary of a connector, to capture its interaction with the services that comprise its environment. We propose Stochastic Reo automata as an extension of Reo automata, in order to compositionally derive a QoS-aware semantics for Reo. We further present a translation of Stochastic Reo automata to Continuous-Time Markov Chains (CTMCs). This translation enables us to use third-party CTMC verification tools to do an end-to-end performance analysis of service compositions.Comment: In Proceedings FOCLASA 2010, arXiv:1007.499

Markov two-components processes

We propose Markov two-components processes (M2CP) as a probabilistic model of asynchronous systems based on the trace semantics for concurrency. Considering an asynchronous system distributed over two sites, we introduce concepts and tools to manipulate random trajectories in an asynchronous framework: stopping times, an Asynchronous Strong Markov property, recurrent and transient states and irreducible components of asynchronous probabilistic processes. The asynchrony assumption implies that there is no global totally ordered clock ruling the system. Instead, time appears as partially ordered and random. We construct and characterize M2CP through a finite family of transition matrices. M2CP have a local independence property that guarantees that local components are independent in the probabilistic sense, conditionally to their synchronization constraints. A synchronization product of two Markov chains is introduced, as a natural example of M2CP.Comment: 34 page

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.Comment: In Proceedings iWIGP 2011, arXiv:1102.374