On the Parity Problem in One-Dimensional Cellular Automata

We consider the parity problem in one-dimensional, binary, circular cellular automata: if the initial configuration contains an odd number of 1s, the lattice should converge to all 1s; otherwise, it should converge to all 0s. It is easy to see that the problem is ill-defined for even-sized lattices (which, by definition, would never be able to converge to 1). We then consider only odd lattices. We are interested in determining the minimal neighbourhood that allows the problem to be solvable for any initial configuration. On the one hand, we show that radius 2 is not sufficient, proving that there exists no radius 2 rule that can possibly solve the parity problem from arbitrary initial configurations. On the other hand, we design a radius 4 rule that converges correctly for any initial configuration and we formally prove its correctness. Whether or not there exists a radius 3 rule that solves the parity problem remains an open problem.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249

Population protocols with faulty interactions: The impact of a leader

We consider the problem of simulating traditional popula-tion protocols under weaker models of communication, which include one-way interactions (as opposed to two-way interactions) and omission faults (i.e., failure by an agent to read its partner’s state during an inter-action), which in turn may be detectable or undetectable. We focus on the impact of a leader, and we give a complete characterization of the models in which the presence of a unique leader in the system allows the construction of simulators: when simulations are possible, we give explicit protocols; when they are not, we give proofs of impossibility. Specifically, if each agent has only a finite amount of memory, the simulation is pos-sible only if there are no omission faults. If agents have an unbounded amount of memory, the simulation is possible as long as omissions are detectable. If an upper bound on the number of omissions involving the leader is known, the simulation is always possible, except in the one-way model in which one side is unable to detect the interaction

Oblivious permutations on the plane

We consider a distributed system of n identical mobile robots operating in the two dimensional Euclidian plane. As in the previous studies, we consider the robots to be anonymous, oblivious, dis-oriented, and without any communication capabilities, operating based on the Look-Compute-Move model where the next location of a robot depends only on its view of the current configuration. Even in this seemingly weak model, most formation problems which require constructing specific configurations, can be solved quite easily when the robots are fully synchronized with each other. In this paper we introduce and study a new class of problems which, unlike the studied formation problems, cannot always be solved even in the fully synchronous model with atomic and rigid moves. This class of problems requires the robots to permute their locations in the plane. In particular, we are interested in implementing two special types of permutations - permutations without any fixed points and permutations of order n. The former (called Move-All) requires each robot to visit at least two of the initial locations, while the latter (called Visit-All) requires every robot to visit each of the initial locations in a periodic manner. We provide a characterization of the solvability of these problems, showing the main challenges in solving this class of problems for mobile robots. We also provide algorithms for the feasible cases, in particular distinguishing between one-step algorithms (where each configuration must be a permutation of the original configuration) an

Ping Pong in dangerous graphs: Optimal black hole search with pure tokens

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Effective Decentralized Energy Restoration by a Mobile Robot

As most existing sensors are powered by batteries, the coverage provided by a sensor network degrades over time and eventually disappears if energy is not restored. A popular approach to energy restoration is to use a robot acting as a mobile battery charger/changer. The robot decides where to move next according to a predefined on-line energy restoration strategy. Since the goal of an energy restoration strategy is to maintain as much as possible of the network operational at any time, its effectiveness depends on the number of nodes it is able to maintain operational at any given time, as well as on for how long a node battery remains depleted. The ideal optimal on-line strategy (called OPTIMAL) occurs when the robot knows at any time the current status of all sensors, and it computes the best request to satisfy next, based on this information. Although optimal in terms of effectiveness, this centralized strategy constantly requires up-to-date global information; hence its high computational and communication costs make it not feasible. We consider a drastically different on-line strategy (called LIC), which is simple and fully decentralized, uses only local communication, requires no computations, and is highly scalable. In our strategy, the robot visits the sensors in a predefined circular order, moving in a 'clockwise' direction and only when aware of a pending request. A sensor whose battery is about to become depleted originates a recharging request and waits for the robot; the request is forwarded in a 'counter-clockwise' direction until it reaches either the robot or another sensor waiting for the robot. We show the perhaps unexpected result that, once the system becomes stable, in most networks the effectiveness of LIC is equivalent to that of OPTIMAL. In other words, in most cases, in spite of its simplicity and its extremely small (communication and computation) costs, the proposed decentralized strategy is as effective as the optimal centralized one. We augment our theoretical results with experimental analysis, showing among other things that the system stabilizes very quickly

Online Energy Restoration by a Mobile Robot in a Ring of Sensors

As most existing sensors are powered by batteries, the coverage provided by a sensor network degrades over time and eventually disappears if energy is not restored. A common approach to energy restoration is to use a robot acting as a mobile battery charger/changer. The goal is to maintain the best level of coverage; that is, to constantly minimize the number of coverage holes and their duration. Depending on the strategy employed by the robot, clearly different results can be obtained. In this paper, we focus on sensors arranged in a ring topology and we consider the intuitive online ON-DEMAND strategy where: the robot visits the sensors in clockwise direction when aware of a pending request; a sensor whose battery is about to become depleted originates a recharging request and waits for the robot; the request is forwarded along the ring in counter-clockwise direction until it reaches either the robot or another sensor waiting for the robot. We also consider the simplest possible online strategy, BLIND, where no requests are sent and the robot automatically moves along the ring looking for needing sensors. We experimentally study the efficiency of the two strategies, and we make the counter-intuitive discovery that BLIND, which does not require communication nor computation, is as efficient as On Demand

On the Parity Problem in One-Dimensional Cellular Automata

We consider the parity problem in one-dimensional, binary, circular cellular automata: if the initial configuration contains an odd number of 1s, the lattice should converge to all 1s; otherwise, it should converge to all 0s. It is easy to see that the problem is ill-defined for even-sized lattices (which, by definition, would never be able to converge to 1). We then consider only odd lattices. We are interested in determining the minimal neighbourhood that allows the problem to be solvable for any initial configuration. On the one hand, we show that radius 2 is not sufficient, proving that there exists no radius 2 rule that can possibly solve the parity problem from arbitrary initial configurations. On the other hand, we design a radius 4 rule that converges correctly for any initial configuration and we formally prove its correctness. Whether or not there exists a radius 3 rule that solves the parity problem remains an open problem

On memory, communication, and synchronous Schedulers when moving and computing

We investigate the computational power of distributed systems whose autonomous computational entities, called robots, move and operate in the 2-dimensional Euclidean plane in synchronous Look-Compute-Move (LCM) cycles. Specifically, we focus on the power of persistent memory and that of explicit communication, and on their computational relationship. In the most common model, OBLOT , the robots are oblivious (no persistent memory) and silent (no explicit means of communication). In contrast, in the LUMI model, each robot is equipped with a constant-sized persistent memory (called light), visible to all the robots; hence, these luminous robots are capable in each cycle of both remembering and communicating. Since luminous robots are computationally more powerful than the standard oblivious one, immediate important questions are about the individual computational power of persistent memory and of explicit communication. In particular, which of the two capabilities, memory or communication, is more important? in other words, is it better to remember or to communicate ? In this paper we address these questions, focusing on two sub-models of LUMI: FST A, where the robots have a constant-size persistent memory but are silent; and FCOM, where the robots can communicate a constant number of bits but are oblivious. We analyze the relationship among all these models and provide a complete exhaustive map of their computational relationship. Among other things, we prove that communication is more powerful than persistent memory under the fully synchronous scheduler Fsynch, while they are incomparable under the semi-synchronous scheduler Ssynch