The BG-simulation for Byzantine Mobile Robots
📝 Abstract
This paper investigates the task solvability of mobile robot systems subject to Byzantine faults. We first consider the gathering problem, which requires all robots to meet in finite time at a non-predefined location. It is known that the solvability of Byzantine gathering strongly depends on a number of system attributes, such as synchrony, the number of Byzantine robots, scheduling strategy, obliviousness, orientation of local coordinate systems and so on. However, the complete characterization of the attributes making Byzantine gathering solvable still remains open. In this paper, we show strong impossibility results of Byzantine gathering. Namely, we prove that Byzantine gathering is impossible even if we assume one Byzantine fault, an atomic execution system, the n-bounded centralized scheduler, non-oblivious robots, instantaneous movements and a common orientation of local coordinate systems (where n denote the number of correct robots). Those hypotheses are much weaker than used in previous work, inducing a much stronger impossibility result. At the core of our impossibility result is a reduction from the distributed consensus problem in asynchronous shared-memory systems. In more details, we newly construct a generic reduction scheme based on the distributed BG-simulation. Interestingly, because of its versatility, we can easily extend our impossibility result for general pattern formation problems.
💡 Analysis
This paper investigates the task solvability of mobile robot systems subject to Byzantine faults. We first consider the gathering problem, which requires all robots to meet in finite time at a non-predefined location. It is known that the solvability of Byzantine gathering strongly depends on a number of system attributes, such as synchrony, the number of Byzantine robots, scheduling strategy, obliviousness, orientation of local coordinate systems and so on. However, the complete characterization of the attributes making Byzantine gathering solvable still remains open. In this paper, we show strong impossibility results of Byzantine gathering. Namely, we prove that Byzantine gathering is impossible even if we assume one Byzantine fault, an atomic execution system, the n-bounded centralized scheduler, non-oblivious robots, instantaneous movements and a common orientation of local coordinate systems (where n denote the number of correct robots). Those hypotheses are much weaker than used in previous work, inducing a much stronger impossibility result. At the core of our impossibility result is a reduction from the distributed consensus problem in asynchronous shared-memory systems. In more details, we newly construct a generic reduction scheme based on the distributed BG-simulation. Interestingly, because of its versatility, we can easily extend our impossibility result for general pattern formation problems.
📄 Content
arXiv:1106.0113v1 [cs.DC] 1 Jun 2011 The BG-simulation for Byzantine Mobile Robots1 Taisuke Izumi24 Zohir Bouzid3 S´ebastien Tixeuil3 Koichi Wada2 Abstract This paper investigates the task solvability of mobile robot systems subject to Byzantine faults. We first consider the gathering problem, which requires all robots to meet in finite time at a non-predefined location. It is known that the solvability of Byzantine gathering strongly depends on a number of system attributes, such as synchrony, the number of Byzantine robots, scheduling strategy, obliviousness, orientation of local coordinate systems and so on. However, the complete characterization of the attributes making Byzantine gathering solvable still remains open. In this paper, we show strong impossibility results of Byzantine gathering. Namely, we prove that Byzantine gathering is impossible even if we assume one Byzantine fault, an atomic exe- cution system, the n-bounded centralized scheduler, non-oblivious robots, instantaneous move- ments and a common orientation of local coordinate systems (where n denote the number of correct robots). Those hypotheses are much weaker than used in previous work, inducing a much stronger impossibility result. At the core of our impossibility result is a reduction from the distributed consensus problem in asynchronous shared-memory systems. In more details, we newly construct a generic reduc- tion scheme based on the distributed BG-simulation. Interestingly, because of its versatility, we can easily extend our impossibility result for general pattern formation problems. 1This work was supported in part by ANR projects, KAKENHI no.21500013 and no.22700010. 2Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, Aichi, 466-8555, Japan. E-mail: {t- izumi,wada}@nitech.ac.jp. 3Universit´e Pierre et Marie Curie - Paris 6, LIP6 CNRS 7606, France. E-mail: zohir.bouzid@gmail.com, Se- bastien.Tixeuil@lip6.fr. 4Corresponding Author. E-mail: t-izumi@nitech.ac.jp. Tel:+81-52-735-5567, Fax:+81-52-735-5408 1 Introduction Motivation Robot networks have recently become a challenging research area for distributed computing researchers. At the core of scientific studies lies the characterization of the minimum robots capabilities that are necessary to achieve non-trivial tasks, such as the formation of geo- metric patterns, scattering, gathering, etc. The considered robots are often very weak: They are anonymous (i.e. that do not have any means to perform distinct tasks based on a distinguishable identifier), oblivious (i.e. they cannot remember past observations, computations, or movements), disoriented (i.e. they share neither a common coordinate system nor a common length unit), and most importantly dumb (i.e. they don’t have any explicit mean of communication). The last property means that robots cannot communicate explicitly by sending messages to one another. Instead, their communication is indirect (or spatial): a robot ’writes’ a value to the network by moving toward a certain position, and a robot ’reads’ the state of the network by observing the positions of other robots in terms of its local coordinate system. The problem we consider in this paper is the gathering of fault-prone robots [20]. Given a set of oblivious robots with arbitrary initial locations and no agreement on a global coordinate system, the gathering problem requires that all correct robots reach and stabilize the same, but unknown beforehand, location. A number of solvability issues about the gathering problem are studied in previous works because of its fundamental importance in both theory and practice. One can easily find an analogy of the gathering problem to the consensus problem, and thus may think that its solvability issue are straightforwardly deduced from the known results about the consensus solvability (e.g., FLP impossibility). However, many differences lies between those two problems and the solvability of the gathering problem is still non-trivial. We can enumerate at least three factors that strongly affect the solvability of the gathering problem: (i) the absence of a common coordinate system, (ii) the fact that there is no explicit termination, and (iii) the lack of a validity requirement. In fault-free environments, the non-triviality of the existence of a solution mainly results from (i) that hardens symmetry breaking. Actually, gathering is known to be impossible to solve with n = 2 robots in atomic-execution (ATOM) models1. One direction of the study of gathering is to explore the weaker assumptions breaking this hardness. For example, endowing robots with a small amount of memory [5, 20], or weak agreement of local coordinate systems [15, 16, 19]. On the other hand, in fault-prone environments, the remaining two factors arise as the primary difference to the consensus in classical computation models. An important witness encouraging the difference is that the gathering problem can be solved in a certain kind of crash-prone asynchr
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