Abstract.We study the problems of existence and construction of infinite schedules for finite weighted automata and one-clock weighted timed automata, subject to boundary constraints on the accumulated weight. More specifically, we consider automata equipped with positive and negative weights on transitions and locations, corresponding to the production and consumption of some resource (e.g. energy). We ask the question whether there exists an infinite path for which the accumulated weight for any finite prefix satisfies certain constraints (e.g. remains between 0 and some given upper-bound). We also consider a game version of the above, where certain transitions may be uncontrollable.

Abstract. Energy games have recently attracted a lot of attention. These are games played on finite weighted automata and concern the existence of infinite runs subject to boundary constraints on the accumulated weight, allowing e.g. only for behaviours where a resource is always available (nonnegative accumulated weight), yet does not exceed a given maximum capacity. We extend energy games to a multiweighted and parameterized setting, allowing us to model systems with multiple quantitative aspects. We present reductions between Petri nets and multiweighted automata and among different types of multiweighted automata and identify new complexity and (un)decidability results for both one-and two-player games. We also investigate the tractability of an extension of multiweighted energy games in the setting of timed automata.

We present a distance-agnostic approach to quantitative verification. Taking as input an unspecified distance on system traces, or executions, we develop a game-based framework which allows us to define a spectrum of different interesting system distances corresponding to the given trace distance. Thus we extend the classic linear-time-branching-time spectrum to a quantitative setting, parametrized by trace distance. We also provide fixed-point characterizations of all system distances, and we prove a general transfer principle which allows us to transfer counterexamples from the qualitative to the quantitative setting, showing that all system distances are mutually topologically inequivalent. ACM Subject Classification IntroductionFor rigorous design and verification of embedded systems, both qualitative and quantitative information and constraints have to be taken into account [16,18,20]. This applies to the models considered, to the properties one wishes to be satisfied, and to the verification itself. Hence the question asked in quantitative verification is not "Does the system satisfy the requirements?", but rather "To which extent does the system satisfy the requirements?" Standard qualitative verification techniques are inherently fragile: either the requirements are satisfied, or they are not, regardless of how close the actual system might come to the specification. To overcome this lack of robustness, notions of distance between systems are essential.As pointed out in [16], qualitative and quantitative aspects of verification should be treated orthogonally in any theory of quantitative verification (of course they can hardly be separated in practice, but that is not of our concern here). The formalism we propose in this paper addresses this orthogonality by modeling qualitative aspects using standard labeled transition systems and expressing the quantitative aspects using trace distances, or distances on system executions. Based on these ingredients, we develop a comprehensive theory of system distances which generalizes the standard linear-time-branching-time spectrum [12,13,24] to a quantitative setting, see Figure 1.Similarly to [3], our theory relies on Ehrenfeucht-Fraïssé games and allows for a more refined analysis of systems. More precisely, our parametrized framework forms a hierarchy of games, for each trace distance used in its instantiation. In the quantitative setting, using games with real-valued outcomes, as opposed to discrete games, effectively allows us obtain © U. Fahrenberg, A. Legay, and C. Thrane; licensed under Creative Commons License NC-ND 31

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