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

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 discretization techniques to break time into discrete intervals, and optimal control is approximated for each interval separately. Current techniques provide an accuracy of O(epsilon^2) on each interval, which leads to an infeasibly large number of intervals. We propose a sequence of approximations that achieve accuracies of O(epsilon^3), O(epsilon^4), and O(epsilon^5), that allow us to drastically reduce the number of intervals that are considered. For CTMDPs, the resulting algorithms are comparable to the heuristic approach given by Buckholz 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 positional strategies for both players that achieve similar error bounds

Rapid Recovery for Systems with Scarce Faults

Our goal is to achieve a high degree of fault tolerance through the control of a safety critical systems. This reduces to solving a game between a malicious environment that injects failures and a controller who tries to establish a correct behavior. We suggest a new control objective for such systems that offers a better balance between complexity and precision: we seek systems that are k-resilient. In order to be k-resilient, a system needs to be able to rapidly recover from a small number, up to k, of local faults infinitely many times, provided that blocks of up to k faults are separated by short recovery periods in which no fault occurs. k-resilience is a simple but powerful abstraction from the precise distribution of local faults, but much more refined than the traditional objective to maximize the number of local faults. We argue why we believe this to be the right level of abstraction for safety critical systems when local faults are few and far between. We show that the computational complexity of constructing optimal control with respect to resilience is low and demonstrate the feasibility through an implementation and experimental results.Comment: In Proceedings GandALF 2012, arXiv:1210.202

Approximating Acceptance Probabilities of CTMC-Paths on Multi-Clock Deterministic Timed Automata

We consider the problem of approximating the probability mass of the set of timed paths under a continuous-time Markov chain (CTMC) that are accepted by a deterministic timed automaton (DTA). As opposed to several existing works on this topic, we consider DTA with multiple clocks. Our key contribution is an algorithm to approximate these probabilities using finite difference methods. An error bound is provided which indicates the approximation error. The stepping stones towards this result include rigorous proofs for the measurability of the set of accepted paths and the integral-equation system characterizing the acceptance probability, and a differential characterization for the acceptance probability

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

Efficient Approximation of Optimal Control for 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 break time into discrete intervals, and optimal control is approximated for each interval separately. Current techniques provide an accuracy of O(\epsilon^2) on each interval, which leads to an infeasibly large number of intervals. We propose a sequence of approximations that achieve accuracies of O(\epsilon^3), O(\epsilon^4), and O(\epsilon^5), 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 Buckholz 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 positional strategies for both players that achieve similar error bounds

Formal methods for motion planning and control in dynamic and partially known environments

This thesis is motivated by time and safety critical applications involving the use of autonomous vehicles to accomplish complex tasks in dynamic and partially known environments. We use temporal logic to formally express such complex tasks. Temporal logic specifications generalize the classical notions of stability and reachability widely studied within the control and hybrid systems communities. Given a model describing the motion of a robotic system in an environment and a formal task specification, the aim is to automatically synthesize a control policy that guarantees the satisfaction of the specification. This thesis presents novel control synthesis algorithms to tackle the problem of motion planning from temporal logic specifications in uncertain environments. For each one of the planning and control synthesis problems addressed in this dissertation, the proposed algorithms are implemented, evaluated, and validated thought experiments and/or simulations. The first part of this thesis focuses on a mobile robot whose success is measured by the completion of temporal logic tasks within a given period of time. In addition to such time constraints, the planning algorithm must also deal with the uncertainty that arises from the changes in the robot's workspace during task execution. In particular, we consider a robot deployed in a partitioned environment subjected to structural changes such as doors that can open and close. The motion of the robot is modeled as a continuous time Markov decision process and the robot's mission is expressed as a Continuous Stochastic Logic (CSL) formula. A complete framework to find a control strategy that satisfies a specification given as a CSL formula is introduced. The second part of this thesis addresses the synthesis of controllers that guarantee the satisfaction of a task specification expressed as a syntactically co-safe Linear Temporal Logic (scLTL) formula. In this case, uncertainty is characterized by the partial knowledge of the robot's environment. Two scenarios are considered. First, a distributed team of robots required to satisfy the specification over a set of service requests occurring at the vertices of a known graph representing the environment is examined. Second, a single agent motion planning problem from the specification over a set of properties known to be satised at the vertices of the known graph environment is studied. In both cases, we exploit the existence of o-the-shelf model checking and runtime verification tools, the efficiency of graph search algorithms, and the efficacy of exploration techniques to solve the motion planning problem constrained by the absence of complete information about the environment. The final part of this thesis extends uncertainty beyond the absence of a complete knowledge of the environment described above by considering a robot equipped with a noisy sensing system. In particular, the robot is tasked with satisfying a scLTL specification over a set of regions of interest known to be present in the environment. In such a case, although the robot is able to measure the properties characterizing such regions of interest, precisely determining the identity of these regions is not feasible. A mixed observability Markov decision process is used to represent the robot's actuation and sensing models. The control synthesis problem from scLTL formulas is then formulated as a maximum probability reachability problem on this model. The integration of dynamic programming, formal methods, and frontier-based exploration tools allow us to derive an algorithm to solve such a reachability problem

Finite horizon analysis of Markov automata

Markov automata constitute an expressive continuous-time compositional modelling formalism, featuring stochastic timing and nondeterministic as well as probabilistic branching, all supported in one model. They span as special cases, the models of discrete and continuous-time Markov chains, as well as interactive Markov chains and probabilistic automata. Moreover, they might be equipped with reward and resource structures in order to be used for analysing quantitative aspects of systems, like performance metrics, energy consumption, repair and maintenance costs. Due to their expressive nature, they serve as semantic backbones of engineering frameworks, control applications and safety critical systems. The Architecture Analysis and Design Language (AADL), Dynamic Fault Trees (DFT) and Generalised Stochastic Petri Nets (GSPN) are just some examples. Their expressiveness thus far prevents them from efficient analysis by stochastic solvers and probabilistic model checkers. A major problem context of this thesis lies in their analysis under some budget constraints, i.e. when only a finite budget of resources can be spent by the model. We study mathematical foundations of Markov automata since these are essential for the analysis addressed in this thesis. This includes, in particular, understanding their measurability and establishing their probability measure. Furthermore, we address the analysis of Markov automata in the presence of both reward acquisition and resource consumption within a finite budget of resources. More specifically, we put the problem of computing the optimal expected resource-bounded reward in our focus. In our general setting, we support transient, instantaneous and final reward collection as well as transient resource consumption. Our general formulation of the problem encompasses in particular the optimal time-bound reward and reachability as well as resource-bounded reachability. We develop a sound theory together with a stable approximation scheme with a strict error bound to solve the problem in an efficient way. We report on an implementation of our approach in a supporting tool and also demonstrate its effectiveness and usability over an extensive collection of industrial and academic case studies.Markov-Automaten bilden einen mächtigen Formalismus zur kompositionellen Modellierung mit kontinuierlicher stochastischer Zeit und nichtdeterministischer sowie probabilistischer Verzweigung, welche alle in einem Modell unterstützt werden. Sie enthalten als Spezialfälle die Modelle diskreter und kontinuierlicher Markov-Ketten sowie interaktive Markov-Ketten und probabilistischer Automaten. Darüber hinaus können sie mit Belohnungs- und Ressourcenstrukturen ausgestattet werden, um quantitative Aspekte von Systemen wie Leistungsfähigkeit, Energieverbrauch, Reparatur- und Wartungskosten zu analysieren. Sie dienen aufgrund ihrer Ausdruckskraft als semantisches Rückgrat von Engineering Frameworks, Steuerungsanwendungen und sicherheitskritischen Systemen. Die Architekturanalyse und Designsprache (AADL), Dynamic Fault Trees (DFT) und Generalized Stochastic Petri Nets (GSPN) sind nur einige Beispiele dafür. Ihre Aussagekraft verhindert jedoch bisher eine effiziente Analyse durch stochastische Löser und probabilistische Modellprüfer. Ein wichtiger Problemzusammenhang dieser Arbeit liegt in ihrer Analyse unter Budgetbeschränkungen, das heisst wenn nur ein begrenztes Budget an Ressourcen vom Modell aufgewendet werden kann. Wir studieren mathematische Grundlagen von Markov-Automaten, da diese für die in dieser Arbeit angesprochene Analyse von wesentlicher Bedeutung sind. Dazu gehört insbesondere das Verständnis ihrer Messbarkeit und die Festlegung ihrer Wahrscheinlichkeitsmaßes. Darüber hinaus befassen wir uns mit der Analyse von Markov-Automaten in Bezug auf Belohnungserwerb sowie Ressourcenverbrauch innerhalb eines begrenzten Ressourcenbudgets. Genauer gesagt stellen wir das Problem der Berechnung der optimalen erwarteten Ressourcen-begrenzte Belohnung in unserem Fokus. Dieser Fokus umfasst transiente, sofortige und endgültige Belohnungssammlung sowie transienten Ressourcenverbrauch. Unsere allgemeine Formulierung des Problems beinhalet insbesondere die optimale zeitgebundene Belohnung und Erreichbarkeit sowie ressourcenbeschränkte Erreichbarkeit. Wir entwickeln die grundlegende Theorie dazu. Zur effizienten Lösung des Problems entwerfen wir ein stabilen Approximationsschema mit einer strikten Fehlerschranke. Wir berichten über eine Umsetzung unseres Ansatzes in einem Software-Werkzeug und zeigen seine Wirksamkeit und Verwendbarkeit anhand einer umfangreichen Sammlung von industriellen und akademischen Fallstudien