Abstract
Bertrand et al. introduced a model of parameterised systems, where each agent is represented by a finite state system, and studied the following control problem: for any number of agents, does there exist a controller able to bring all agents to a target state? They showed that the problem is decidable and EXPTIME-complete in the adversarial setting, and posed as an open problem the stochastic setting, where the agent is represented by a Markov decision process. In this paper, we show that the stochastic control problem is decidable. Our solution makes significant uses of well quasi orders, of the max-flow min-cut theorem, and of the theory of regular cost functions. We introduce an intermediate problem of independence interest called the sequential flow problem and study its complexity.
Highlights
The control problem for populations of identical agents
We study the stochastic setting, where each agent evolves independently according to a probabilistic distribution, i.e. the finite state system modelling an agent is a Markov decision process
We showed the decidability of the stochastic control problem
Summary
The control problem for populations of identical agents. The model we study was introduced in [BDGG17] (see the journal version [BDG+19]): a population of agents are controlled uniformly, meaning that the controller applies the same action to every agent. We study the stochastic setting, where each agent evolves independently according to a probabilistic distribution, i.e. the finite state system modelling an agent is a Markov decision process. In the paragraphs we discuss four motivations for studying this problem: control of biological systems, parameterised verification and control, distributed computing, and automata theory. As discussed in the conclusions of [BDGG17, BDG+19], the stochastic semantics is more satisfactory than the adversarial one for representing the behaviours of chemical reactions, so our decidability result is a step towards a better understanding of the modelling of biological systems as populations of arbitrarily many agents represented by finite state systems. The first and most widely studied is population protocols, introduced in [AAD+06]: the agents are modelled by finite state systems and interact by pairs drawn at random.
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