Many applications require infinite plans ---i.e. an infinite sequence of actions--- in order to carry out some given process indefinitely. In addition, it is desirable to guarantee optimality. In this paper, we address this problem in the setting of doubly-priced timed automata, where we show how to efficiently compute ratio-optimal cycles for optimal infinite plans. For efficient computation, we present symbolic λ-deduction (S-λD), an any-time algorithm that uses a symbolic representation (priced zones) to search the state-space with a compact representation of the time constraints. Our approach guarantees termination while arriving at an optimal solution. Our experimental evaluation shows that S-λD outperforms the alternative of searching in the concrete state space; is very robust with respect to fine-grained temporal constraints; and has a very good anytime behaviour.

Modern computer networks based on the software-defined networking (SDN) paradigm are becoming increasingly complex and often require frequent configuration changes in order to react to traffic fluctuations. It is essential that forwarding policies are preserved not only before and after the configuration update but also at any moment during the inherently distributed execution of such an update. We present Kaki, a Petri game based tool for automatic synthesis of switch batches which can be updated in parallel without violating a given (regular) forwarding policy like waypointing or service chaining. Kaki guarantees to find the minimum number of concurrent batches and supports both splittable and nonsplittable flow forwarding. In order to achieve optimal performance, we introduce two novel optimisation techniques based on static analysis: decomposition into independent subproblems and identification of switches that can be collectively updated in the same batch. These techniques considerably improve the performance of our tool Kaki, relying on TAPAAL’s verification engine for Petri games as its backend. Experiments on a large benchmark of real networks from the Internet Topology Zoo database demonstrate that Kaki outperforms the state-of-the-art tools Netstack and FLIP. Kaki computes concurrent update synthesis significantly faster than Netstack and compared to FLIP, it provides shorter (and provably optimal) concurrent update sequences at similar runtimes.