A Backward Algorithm for the Multiprocessor Online Feasibility of Sporadic Tasks

The online feasibility problem (for a set of sporadic tasks) asks whether there is a scheduler that always prevents deadline misses (if any), whatever the sequence of job releases, which is a priori} unknown to the scheduler. In the multiprocessor setting, this problem is notoriously difficult. The only exact test for this problem has been proposed by Bonifaci and Marchetti-Spaccamela: it consists in modelling all the possible behaviours of the scheduler and of the tasks as a graph; and to interpret this graph as a game between the tasks and the scheduler, which are seen as antagonistic players. Then, computing a correct scheduler is equivalent to finding a winning strategy for the `scheduler player', whose objective in the game is to avoid deadline misses. In practice, however this approach is limited by the intractable size of the graph. In this work, we consider the classical attractor algorithm to solve such games, and introduce antichain techniques to optimise its performance in practice and overcome the huge size of the game graph. These techniques are inspired from results from the formal methods community, and exploit the specific structure of the feasibility problem. We demonstrate empirically that our approach allows to dramatically improve the performance of the game solving algorithm.Comment: Long version of a conference paper accepted to ACSD 201

Dynamics and Coalitions in Sequential Games

We consider N-player non-zero sum games played on finite trees (i.e., sequential games), in which the players have the right to repeatedly update their respective strategies (for instance, to improve the outcome wrt to the current strategy profile). This generates a dynamics in the game which may eventually stabilise to a Nash Equilibrium (as with Kukushkin's lazy improvement), and we argue that it is interesting to study the conditions that guarantee such a dynamics to terminate. We build on the works of Le Roux and Pauly who have studied extensively one such dynamics, namely the Lazy Improvement Dynamics. We extend these works by first defining a turn-based dynamics, proving that it terminates on subgame perfect equilibria, and showing that several variants do not terminate. Second, we define a variant of Kukushkin's lazy improvement where the players may now form coalitions to change strategies. We show how properties of the players' preferences on the outcomes affect the termination of this dynamics, and we thereby characterise classes of games where it always terminates (in particular two-player games).Comment: In Proceedings GandALF 2017, arXiv:1709.0176

Safe and Optimal Scheduling for Hard and Soft Tasks

We consider a stochastic scheduling problem with both hard and soft tasks on a single machine. Each task is described by a discrete probability distribution over possible execution times, and possible inter-arrival times of the job, and a fixed deadline. Soft tasks also carry a penalty cost to be paid when they miss a deadline. We ask to compute an online and non-clairvoyant scheduler (i.e. one that must take decisions without knowing the future evolution of the system) that is safe and efficient. Safety imposes that deadline of hard tasks are never violated while efficient means that we want to minimise the mean cost of missing deadlines by soft tasks. First, we show that the dynamics of such a system can be modelled as a finite Markov Decision Process (MDP). Second, we show that our scheduling problem is PP-hard and in EXPTime. Third, we report on a prototype tool that solves our scheduling problem by relying on the Storm tool to analyse the corresponding MDP. We show how antichain techniques can be used as a potential heuristic

Models and Algorithms for Chronology

The last decades have seen the rise of many fundamental chronological debates in Old World archaeology, with far-reaching historical implications. Yet, outside of radiocarbon dating - where Bayesian formal tools and models are applied - these chronological debates are still relying on non-formal models, and dates are mostly derived by hand, without the use of mathematical or computational tools, albeit the large number of complex constraints to be taken into account. This article presents formal models and algorithms for encoding archaeologically-relevant chronological constraints, computing optimal chronologies in an automated way, and automatically checking for chronological properties of a given model

Simple Priced Timed Games Are Not That Simple

Priced timed games are two-player zero-sum games played on priced timed automata (whose locations and transitions are labeled by weights modeling the costs of spending time in a state and executing an action, respectively). The goals of the players are to minimise and maximise the cost to reach a target location, respectively. We consider priced timed games with one clock and arbitrary (positive and negative) weights and show that, for an important subclass of theirs (the so-called simple priced timed games), one can compute, in exponential time, the optimal values that the players can achieve, with their associated optimal strategies. As side results, we also show that one-clock priced timed games are determined and that we can use our result on simple priced timed games to solve the more general class of so-called reset-acyclic priced timed games (with arbitrary weights and one-clock)

Efficient Energy Distribution in a Smart Grid using Multi-Player Games

Algorithms and models based on game theory have nowadays become prominent techniques for the design of digital controllers for critical systems. Indeed, such techniques enable automatic synthesis: given a model of the environment and a property that the controller must enforce, those techniques automatically produce a correct controller, when it exists. In the present paper, we consider a class of concurrent, weighted, multi-player games that are well-suited to model and study the interactions of several agents who are competing for some measurable resources like energy. We prove that a subclass of those games always admit a Nash equilibrium, i.e. a situation in which all players play in such a way that they have no incentive to deviate. Moreover, the strategies yielding those Nash equilibria have a special structure: when one of the agents deviate from the equilibrium, all the others form a coalition that will enforce a retaliation mechanism that punishes the deviant agent. We apply those results to a real-life case study in which several smart houses that produce their own energy with solar panels, and can share this energy among them in micro-grid, must distribute the use of this energy along the day in order to avoid consuming electricity that must be bought from the global grid. We demonstrate that our theory allows one to synthesise an efficient controller for these houses: using penalties to be paid in the utility bill as an incentive, we force the houses to follow a pre-computed schedule that maximises the proportion of the locally produced energy that is consumed.Comment: In Proceedings Cassting'16/SynCoP'16, arXiv:1608.0017

Quantitative Games under Failures

We study a generalisation of sabotage games, a model of dynamic network games introduced by van Benthem. The original definition of the game is inherently finite and therefore does not allow one to model infinite processes. We propose an extension of the sabotage games in which the first player (Runner) traverses an arena with dynamic weights determined by the second player (Saboteur). In our model of quantitative sabotage games, Saboteur is now given a budget that he can distribute amongst the edges of the graph, whilst Runner attempts to minimise the quantity of budget witnessed while completing his task. We show that, on the one hand, for most of the classical cost functions considered in the literature, the problem of determining if Runner has a strategy to ensure a cost below some threshold is EXPTIME-complete. On the other hand, if the budget of Saboteur is fixed a priori, then the problem is in PTIME for most cost functions. Finally, we show that restricting the dynamics of the game also leads to better complexity

To Reach or not to Reach? Efficient Algorithms for Total-Payoff Games

International audienceQuantitative games are two-player zero-sum games played on directed weighted graphs. Total-payoff games – that can be seen as a refinement of the well-studied mean-payoff games – are the variant where the payoff of a play is computed as the sum of the weights. Our aim is to describe the first pseudo-polynomial time algorithm for total-payoff games in the presence of arbitrary weights. It consists of a non-trivial application of the value iteration paradigm. Indeed, it requires to study, as a milestone, a refinement of these games, called min-cost reachability games, where we add a reachability objective to one of the players. For these games, we give an efficient value iteration algorithm to compute the values and optimal strategies (when they exist), that runs in pseudo-polynomial time. We also propose heuristics to speed up the computations

Timed-Automata-Based Verification of MITL over Signals

It has been argued that the most suitable semantic model for real-time formalisms is the non-negative real line (signals), i.e. the continuous semantics, which naturally captures the continuous evolution of system states. Existing tools like UPPAAL are, however, based on omega-sequences with timestamps (timed words), i.e. the pointwise semantics. Furthermore, the support for logic formalisms is very limited in these tools. In this article, we amend these issues by a compositional translation from Metric Temporal Interval Logic (MITL) to signal automata. Combined with an emptiness-preserving encoding of signal automata into timed automata, we obtain a practical automata-based approach to MITL model-checking over signals. We implement the translation in our tool MightyL and report on case studies using LTSmin as the back-end