Lipschitz Robustness of Finite-state Transducers

We investigate the problem of checking if a finite-state transducer is robust to uncertainty in its input. Our notion of robustness is based on the analytic notion of Lipschitz continuity --- a transducer is K-(Lipschitz) robust if the perturbation in its output is at most K times the perturbation in its input. We quantify input and output perturbation using similarity functions. We show that K-robustness is undecidable even for deterministic transducers. We identify a class of functional transducers, which admits a polynomial time automata-theoretic decision procedure for K-robustness. This class includes Mealy machines and functional letter-to-letter transducers. We also study K-robustness of nondeterministic transducers. Since a nondeterministic transducer generates a set of output words for each input word, we quantify output perturbation using set-similarity functions. We show that K-robustness of nondeterministic transducers is undecidable, even for letter-to-letter transducers. We identify a class of set-similarity functions which admit decidable K-robustness of letter-to-letter transducers.Comment: In FSTTCS 201

Coordination of Multirobot Systems Under Temporal Constraints

Multirobot systems have great potential to change our lives by increasing efficiency or decreasing costs in many applications, ranging from warehouse logistics to construction. They can also replace humans in dangerous scenarios, for example in a nuclear disaster cleanup mission. However, teleoperating robots in these scenarios would severely limit their capabilities due to communication and reaction delays. Furthermore, ensuring that the overall behavior of the system is safe and correct for a large number of robots is challenging without a principled solution approach. Ideally, multirobot systems should be able to plan and execute autonomously. Moreover, these systems should be robust to certain external factors, such as failing robots and synchronization errors and be able to scale to large numbers, as the effectiveness of particular tasks might depend directly on these criteria. This thesis introduces methods to achieve safe and correct autonomous behavior for multirobot systems. Firstly, we introduce a novel logic family, called counting logics, to describe the high-level behavior of multirobot systems. Counting logics capture constraints that arise naturally in many applications where the identity of the robot is not important for the task to be completed. We further introduce a notion of robust satisfaction to analyze the effects of synchronization errors on the overall behavior and provide complexity analysis for a fragment of this logic. Secondly, we propose an optimization-based algorithm to generate a collection of robot paths to satisfy the specifications given in counting logics. We assume that the robots are perfectly synchronized and use a mixed-integer linear programming formulation to take advantage of the recent advances in this field. We show that this approach is complete under the perfect synchronization assumption. Furthermore, we propose alternative encodings that render more efficient solutions under certain conditions. We also provide numerical results that showcase the scalability of our approach, showing that it scales to hundreds of robots. Thirdly, we relax the perfect synchronization assumption and show how to generate paths that are robust to bounded synchronization errors, without requiring run-time communication. However, the complexity of such an approach is shown to depend on the error bound, which might be limiting. To overcome this issue, we propose a hierarchical method whose complexity does not depend on this bound. We show that, under mild conditions, solutions generated by the hierarchical method can be executed safely, even if such a bound is not known. Finally, we propose a distributed algorithm to execute multirobot paths while avoiding collisions and deadlocks that might occur due to synchronization errors. We recast this problem as a conflict resolution problem and characterize conditions under which existing solutions to the well-known drinking philosophers problem can be used to design control policies that prevents collisions and deadlocks. We further provide improvements to this naive approach to increase the amount of concurrency in the system. We demonstrate the effectiveness of our approach by comparing it to the naive approach and to the state-of-the-art.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/162921/1/ysahin_1.pd

Optimality and resilience in parity games

Modeling reactive systems as infinite games has yielded a multitude of results in the fields of program verification and program synthesis. The canonical parity condition, however, neither suffices to express non-functional requirements on the modeled system, nor to capture malfunctions of the deployed system. We address these issues by investigating quantitative games in which the above characteristics can be expressed. Parity games with costs are a variant of parity games in which traversing an edge incurs some nonnegative cost. The cost of a play is the limit superior of the cost incurred between answering odd colors by larger even ones. We extend that model by using integer costs, obtaining parity games with weights, and show that the problem of solving such games is in the intersection of NP and coNP and that it is PTIME-equivalent to the problem of solving energy parity games. We moreover show that Player 0 requires exponential memory to implement a winning strategy in parity games with weights. Further, we show that the problem of determining whether Player 0 can keep the cost of a play below a given bound is EXPTIME-complete for parity games with weights and PSPACE-complete for the special cases of parity games with costs and finitary parity games, i.e., it is harder than solving the game. Thus, optimality comes at a price even in finitary parity games. We further determine the complexity of computing strategies in parity games that are resilient against malfunctions. We show that such strategies can be effectively computed and that this is as hard as solving the game without disturbances. Finally, we combine all these aspects and show that Player 0 can trade memory, cost, and resilience for one another. Furthermore, we show how to compute the possible tradeoffs for a given game.Die Modellierung von reaktiven Systemen durch unendliche Spiele ermöglichte zahlreiche Fortschritte in der Programmverifikation und der Programmsynthese. Die häufig genutzte Paritätsbedingung kann jedoch weder nichtfunktionale Anforderungen ausdrücken, noch Fehlfunktionen des Systems modellieren. Wir betrachten quantitative Spiele in denen diese Merkmale ausgedrückt werden können. Paritätsspiele mit Kosten (PSK) sind eine Variante der Paritätsspiele in denen die Benutzung einer Kante nichtnegative Kosten verursacht. Die Kosten einer Partie sind der Limes Superior der Kosten zwischen ungeraden und den jeweils nächsten größeren geraden Farben. Wir erweitern dieses Modell durch ganzzahlige Kosten zu Paritätsspielen mit Gewichten (PSG). Wir zeigen, dass das Lösen dieser Spiele im Schnitt von NP und coNP liegt, dass es PTIME-äquivalent dazu ist, Energieparitätsspiele zu lösen und dass Spieler 0 exponentiellen Speicher benötigt, um zu gewinnen. Ferner zeigen wir, dass das Problem, zu entscheiden, ob Spieler 0 die Kosten eines Spiels unter einer gegebenen Schranke halten kann, EXPTIME-vollständig für PSG ist, sowie dass es PSPACE-vollständig für die Spezialfälle PSK und finitäre Paritätsspiele (FPS) ist. Optimalität ist also selbst in FPS nicht kostenlos. Außerdem bestimmen wir die Komplexität davon, Strategien in Paritätsspielen zu berechnen, die robust gegenüber Fehlfunktionen sind, zeigen, dass solche Strategien effektiv berechnet werden können und beweisen, dass dies nur linearen Mehraufwand bedeutet. Darüberhinaus kombinieren wir die oben genannten Aspekte, zeigen, dass Spieler 0 Speicher, Kosten und Robustheit gegeneinander eintauschen kann und berechnen die möglichen Kompromisse