Maximal Cost-Bounded Reachability Probability on Continuous-Time Markov Decision Processes

In this paper, we consider multi-dimensional maximal cost-bounded reachability probability over continuous-time Markov decision processes (CTMDPs). Our major contributions are as follows. Firstly, we derive an integral characterization which states that the maximal cost-bounded reachability probability function is the least fixed point of a system of integral equations. Secondly, we prove that the maximal cost-bounded reachability probability can be attained by a measurable deterministic cost-positional scheduler. Thirdly, we provide a numerical approximation algorithm for maximal cost-bounded reachability probability. We present these results under the setting of both early and late schedulers

Bisimulations and Logical Characterizations on Continuous-time Markov Decision Processes

In this paper we study strong and weak bisimulation equivalences for continuous-time Markov decision processes (CTMDPs) and the logical characterizations of these relations with respect to the continuous-time stochastic logic (CSL). For strong bisimulation, it is well known that it is strictly finer than CSL equivalence. In this paper we propose strong and weak bisimulations for CTMDPs and show that for a subclass of CTMDPs, strong and weak bisimulations are both sound and complete with respect to the equivalences induced by CSL and the sub-logic of CSL without next operator respectively. We then consider a standard extension of CSL, and show that it and its sub-logic without X can be fully characterized by strong and weak bisimulations respectively over arbitrary CTMDPs.Comment: The conference version of this paper was published at VMCAI 201

A Hierarchy of Scheduler Classes for Stochastic Automata

Stochastic automata are a formal compositional model for concurrent stochastic timed systems, with general distributions and non-deterministic choices. Measures of interest are defined over schedulers that resolve the nondeterminism. In this paper we investigate the power of various theoretically and practically motivated classes of schedulers, considering the classic complete-information view and a restriction to non-prophetic schedulers. We prove a hierarchy of scheduler classes w.r.t. unbounded probabilistic reachability. We find that, unlike Markovian formalisms, stochastic automata distinguish most classes even in this basic setting. Verification and strategy synthesis methods thus face a tradeoff between powerful and efficient classes. Using lightweight scheduler sampling, we explore this tradeoff and demonstrate the concept of a useful approximative verification technique for stochastic automata

Efficient approximation of optimal control for continuous-time Markov games

We study the time-bounded reachability problem for continuous-time Markov decision processes (CTMDPs) and games (CTMGs). Existing techniques for this problem use discretisation techniques to partition time into discrete intervals of size ε, and optimal control is approximated for each interval separately. Current techniques provide an accuracy of on each interval, which leads to an infeasibly large number of intervals. We propose a sequence of approximations that achieve accuracies of , , and , that allow us to drastically reduce the number of intervals that are considered. For CTMDPs, the performance of the resulting algorithms is comparable to the heuristic approach given by Buchholz and Schulz, while also being theoretically justified. All of our results generalise to CTMGs, where our results yield the first practically implementable algorithms for this problem. We also provide memoryless strategies for both players that achieve similar error bounds

A uniform framework for modelling nondeterministic, probabilistic, stochastic, or mixed processes and their behavioral equivalences

Labeled transition systems are typically used as behavioral models of concurrent processes, and the labeled transitions define the a one-step state-to-state reachability relation. This model can be made generalized by modifying the transition relation to associate a state reachability distribution, rather than a single target state, with any pair of source state and transition label. The state reachability distribution becomes a function mapping each possible target state to a value that expresses the degree of one-step reachability of that state. Values are taken from a preordered set equipped with a minimum that denotes unreachability. By selecting suitable preordered sets, the resulting model, called ULTraS from Uniform Labeled Transition System, can be specialized to capture well-known models of fully nondeterministic processes (LTS), fully probabilistic processes (ADTMC), fully stochastic processes (ACTMC), and of nondeterministic and probabilistic (MDP) or nondeterministic and stochastic (CTMDP) processes. This uniform treatment of different behavioral models extends to behavioral equivalences. These can be defined on ULTraS by relying on appropriate measure functions that expresses the degree of reachability of a set of states when performing single-step or multi-step computations. It is shown that the specializations of bisimulation, trace, and testing equivalences for the different classes of ULTraS coincide with the behavioral equivalences defined in the literature over traditional models

On Decidability of Time-Bounded Reachability in CTMDPs

We consider the time-bounded reachability problem for continuous-time Markov decision processes. We show that the problem is decidable subject to Schanuel's conjecture. Our decision procedure relies on the structure of optimal policies and the conditional decidability (under Schanuel's conjecture) of the theory of reals extended with exponential and trigonometric functions over bounded domains. We further show that any unconditional decidability result would imply unconditional decidability of the bounded continuous Skolem problem, or equivalently, the problem of checking if an exponential polynomial has a non-tangential zero in a bounded interval. We note that the latter problems are also decidable subject to Schanuel's conjecture but finding unconditional decision procedures remain longstanding open problems

Policy learning in Continuous-Time Markov Decision Processes using Gaussian Processes

Continuous-time Markov decision processes provide a very powerful mathematical framework to solve policy-making problems in a wide range of applications, ranging from the control of populations to cyber\u2013physical systems. The key problem to solve for these models is to efficiently compute an optimal policy to control the system in order to maximise the probability of satisfying a set of temporal logic specifications. Here we introduce a novel method based on statistical model checking and an unbiased estimation of a functional gradient in the space of possible policies. Our approach presents several advantages over the classical methods based on discretisation techniques, as it does not assume the a-priori knowledge of a model that can be replaced by a black-box, and does not suffer from state-space explosion. The use of a stochastic moment-based gradient ascent algorithm to guide our search considerably improves the efficiency of learning policies and accelerates the convergence using the momentum term. We demonstrate the strong performance of our approach on two examples of non-linear population models: an epidemiology model with no permanent recovery and a queuing system with non-deterministic choice

Behavioural Preorders on Stochastic Systems - Logical, Topological, and Computational Aspects

Computer systems can be found everywhere: in space, in our homes, in our cars, in our pockets, and sometimes even in our own bodies. For concerns of safety, economy, and convenience, it is important that such systems work correctly. However, it is a notoriously difficult task to ensure that the software running on computers behaves correctly. One approach to ease this task is that of model checking, where a model of the system is made using some mathematical formalism. Requirements expressed in a formal language can then be verified against the model in order to give guarantees that the model satisfies the requirements. For many computer systems, time is an important factor. As such, we need our formalisms and requirement languages to be able to incorporate real time. We therefore develop formalisms and algorithms that allow us to compare and express properties about real-time systems. We first introduce a logical formalism for reasoning about upper and lower bounds on time, and study the properties of this formalism, including axiomatisation and algorithms for checking when a formula is satisfied. We then consider the question of when a system is faster than another system. We show that this is a difficult question which can not be answered in general, but we identify special cases where this question can be answered. We also show that under this notion of faster-than, a local increase in speed may lead to a global decrease in speed, and we take step towards avoiding this. Finally, we consider how to compare the real-time behaviour of systems not just qualitatively, but also quantitatively. Thus, we are interested in knowing how much one system is faster or slower than another system. This is done by introducing a distance between systems. We show how to compute this distance and that it behaves well with respect to certain properties.Comment: PhD dissertation from Aalborg Universit