CONTROLLING A POPULATION

Abstract : We introduce a new setting where a population of agents, each modelled by a finite-state system, are controlled uniformly: the controller applies the same action to every agent. The framework is largely inspired by the control of a biological system, namely a population of yeasts, where the controller may only change the environment common to all cells. We study a synchronisation problem for such populations: no matter how individual agents react to the actions of the controller, the controller aims at driving all agents synchronously to a target state. The agents are naturally represented by a non-deterministic finite state automaton (NFA), the same for every agent, and the whole system is encoded as a 2-player game. The first player (Controller) chooses actions, and the second player (Agents) resolves non-determinism for each agent. The game with m agents is called the m-population game. This gives rise to a parameterized control problem (where control refers to 2 player games), namely the population control problem: can Controller control the m-population game for all m ∈ N whatever Agents does? In this paper, we prove that the population control problem is decidable, and it is a EXPTIME-complete problem. As far as we know, this is one of the first results on the control of parameterized systems. Our algorithm, which is not based on cutoff techniques, produces winning strategies which are symbolic, that is, they do not need to count precisely how the population is spread between states. The winning strategies produced by our algorithm are optimal with respect to the synchronisation time: the maximal number of steps before synchronisation of all agents in the target state is at most polynomial in the number of agents m, and exponential in the size of the NFA. We also show that if there is no winning strategy, then there is a population size M such that Controller wins the m-population game if and only if m ≤ M. Surprisingly, M can be doubly exponential in the number of states of the NFA, with tight upper and lower bounds.
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Nathalie Bertrand, Miheer Dewaskar, Blaise Genest, Hugo Gimbert, Adwait Godbole. CONTROLLING A POPULATION. Logical Methods in Computer Science, Logical Methods in Computer Science Association, 2019, 15, ⟨10.23638/LMCS-15(3:6)2019⟩. ⟨hal-02350251⟩

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