The Random Bit Complexity of Mobile Robots Scattering

We consider the problem of scattering $n$ robots in a two dimensional continuous space. As this problem is impossible to solve in a deterministic manner, all solutions must be probabilistic. We investigate the amount of randomness (that is, the number of random bits used by the robots) that is required to achieve scattering. We first prove that $n \log n$ random bits are necessary to scatter $n$ robots in any setting. Also, we give a sufficient condition for a scattering algorithm to be random bit optimal. As it turns out that previous solutions for scattering satisfy our condition, they are hence proved random bit optimal for the scattering problem. Then, we investigate the time complexity of scattering when strong multiplicity detection is not available. We prove that such algorithms cannot converge in constant time in the general case and in $o(\log \log n)$ rounds for random bits optimal scattering algorithms. However, we present a family of scattering algorithms that converge as fast as needed without using multiplicity detection. Also, we put forward a specific protocol of this family that is random bit optimal ($n \log n$ random bits are used) and time optimal ($\log \log n$ rounds are used). This improves the time complexity of previous results in the same setting by a $\log n$ factor. Aside from characterizing the random bit complexity of mobile robot scattering, our study also closes its time complexity gap with and without strong multiplicity detection (that is, $O(1)$ time complexity is only achievable when strong multiplicity detection is available, and it is possible to approach it as needed otherwise)

Deterministic meeting of sniffing agents in the plane

Two mobile agents, starting at arbitrary, possibly different times from arbitrary locations in the plane, have to meet. Agents are modeled as discs of diameter 1, and meeting occurs when these discs touch. Agents have different labels which are integers from the set of 0 to L-1. Each agent knows L and knows its own label, but not the label of the other agent. Agents are equipped with compasses and have synchronized clocks. They make a series of moves. Each move specifies the direction and the duration of moving. This includes a null move which consists in staying inert for some time, or forever. In a non-null move agents travel at the same constant speed, normalized to 1. We assume that agents have sensors enabling them to estimate the distance from the other agent (defined as the distance between centers of discs), but not the direction towards it. We consider two models of estimation. In both models an agent reads its sensor at the moment of its appearance in the plane and then at the end of each move. This reading (together with the previous ones) determines the decision concerning the next move. In both models the reading of the sensor tells the agent if the other agent is already present. Moreover, in the monotone model, each agent can find out, for any two readings in moments t1 and t2, whether the distance from the other agent at time t1 was smaller, equal or larger than at time t2. In the weaker binary model, each agent can find out, at any reading, whether it is at distance less than \r{ho} or at distance at least \r{ho} from the other agent, for some real \r{ho} > 1 unknown to them. Such distance estimation mechanism can be implemented, e.g., using chemical sensors. Each agent emits some chemical substance (scent), and the sensor of the other agent detects it, i.e., sniffs. The intensity of the scent decreases with the distance.Comment: A preliminary version of this paper appeared in the Proc. 23rd International Colloquium on Structural Information and Communication Complexity (SIROCCO 2016), LNCS 998

The locally covariant Dirac field

We describe the free Dirac field in a four dimensional spacetime as a locally covariant quantum field theory in the sense of Brunetti, Fredenhagen and Verch, using a representation independent construction. The freedom in the geometric constructions involved can be encoded in terms of the cohomology of the category of spin spacetimes. If we restrict ourselves to the observable algebra the cohomological obstructions vanish and the theory is unique. We establish some basic properties of the theory and discuss the class of Hadamard states, filling some technical gaps in the literature. Finally we show that the relative Cauchy evolution yields commutators with the stress-energy-momentum tensor, as in the scalar field case.Comment: 36 pages; v2 minor changes, typos corrected, updated references and acknowledgement

Global Overview of Flora and Plant Diversity in Togo (West Africa)

The flora constitutes an important reference tool very used in biology. Its constant update is fundamental in order to promote new directions in conservation. The current study was developed as a synthesis of the analytical flora in Togo. It aims to produce an inventory of this flora by emphasizing its distribution along ecological zones. Its purpose is to make an assessment of the greatest entities of the flora including those of introduced, threatened to disappearance plant species. This synthesis was possible thanks to the use of data retrieved from the existing analytical flora associated with some reports on this flora. Information was organized according to the junctions, of introducing and threatened to disappearances species. They were analyzed on the basis of the phytogeographical types and life forms, and then classified according to ecological zone. From this synthesis it appeared that the flora of Togo would lay out 3468 species distributed among the algae (201especes), Bryophytes (13 species), Pteridophyts (129 species), Gymnosperms (13 species), and Angiosperms (2992 species). The phanerophytes, therophytes, the lianas, and hemicryptophytes dominate the biological spectrum, whereas the guineo-congolian (GC), Sudano-Zambezian (SZ), and transition zone species (GC-SZ) represents phytogeographical types. Face to the new challenges, especially economic, and environmental which fit all in a sustainable context of resources management, an update of all the flora will be a strategic element in the framework of management and conservation of biodiversity

Want to Gather? No Need to Chatter!

A team of mobile agents, starting from different nodes of an unknown network, possibly at different times, have to meet at the same node and declare that they have all met. Agents have different labels and move in synchronous rounds along links of the network. The above task is known as gathering and was traditionally considered under the assumption that when some agents are at the same node then they can talk. In this paper we ask the question of whether this ability of talking is needed for gathering. The answer turns out to be no. Our main contribution are two deterministic algorithms that always accomplish gathering in a much weaker model. We only assume that at any time an agent knows how many agents are at the node that it currently occupies but agents do not see the labels of other co-located agents and cannot exchange any information with them. They also do not see other nodes than the current one. Our first algorithm works under the assumption that agents know a priori some upper bound N on the network size, and it works in time polynomial in N and in the length l of the smallest label. Our second algorithm does not assume any a priori knowledge about the network but its complexity is exponential in the network size and in the labels of agents. Its purpose is to show feasibility of gathering under this harsher scenario. As a by-product of our techniques we obtain, in the same weak model, the solution of the fundamental problem of leader election among agents. As an application of our result we also solve, in the same model, the well-known gossiping problem: if each agent has a message at the beginning, we show how to make all messages known to all agents, even without any a priori knowledge about the network. If agents know an upper bound N on the network size then our gossiping algorithm works in time polynomial in N, in l and in the length of the largest message

Deaf, Dumb, and Chatting Robots, Enabling Distributed Computation and Fault-Tolerance Among Stigmergic Robot

We investigate ways for the exchange of information (explicit communication) among deaf and dumb mobile robots scattered in the plane. We introduce the use of movement-signals (analogously to flight signals and bees waggle) as a mean to transfer messages, enabling the use of distributed algorithms among the robots. We propose one-to-one deterministic movement protocols that implement explicit communication. We first present protocols for synchronous robots. We begin with a very simple coding protocol for two robots. Based on on this protocol, we provide one-to-one communication for any system of n \geq 2 robots equipped with observable IDs that agree on a common direction (sense of direction). We then propose two solutions enabling one-to-one communication among anonymous robots. Since the robots are devoid of observable IDs, both protocols build recognition mechanisms using the (weak) capabilities offered to the robots. The first protocol assumes that the robots agree on a common direction and a common handedness (chirality), while the second protocol assumes chirality only. Next, we show how the movements of robots can provide implicit acknowledgments in asynchronous systems. We use this result to design asynchronous one-to-one communication with two robots only. Finally, we combine this solution with the schemes developed in synchronous settings to fit the general case of asynchronous one-to-one communication among any number of robots. Our protocols enable the use of distributing algorithms based on message exchanges among swarms of Stigmergic robots. Furthermore, they provides robots equipped with means of communication to overcome faults of their communication device

Pattern formation

The Pattern Formation problem is one of the most important coordination problem for robotic systems. Initially the entities are in arbitrary positions; within finite time they must arrange themselves in the space so to form a pattern given in input. In this chapter, we will mainly deal with the problem in the OBLOT model

The Newtonian Limit for Asymptotically Flat Solutions of the Vlasov-Einstein System

It is shown that there exist families of asymptotically flat solutions of the Einstein equations coupled to the Vlasov equation describing a collisionless gas which have a Newtonian limit. These are sufficiently general to confirm that for this matter model as many families of this type exist as would be expected on the basis of physical intuition. A central role in the proof is played by energy estimates in unweighted Sobolev spaces for a wave equation satisfied by the second fundamental form of a maximal foliation.Comment: 24 pages, plain TE

Byzantine Gathering in Networks

This paper investigates an open problem introduced in [14]. Two or more mobile agents start from different nodes of a network and have to accomplish the task of gathering which consists in getting all together at the same node at the same time. An adversary chooses the initial nodes of the agents and assigns a different positive integer (called label) to each of them. Initially, each agent knows its label but does not know the labels of the other agents or their positions relative to its own. Agents move in synchronous rounds and can communicate with each other only when located at the same node. Up to f of the agents are Byzantine. A Byzantine agent can choose an arbitrary port when it moves, can convey arbitrary information to other agents and can change its label in every round, in particular by forging the label of another agent or by creating a completely new one. What is the minimum number M of good agents that guarantees deterministic gathering of all of them, with termination? We provide exact answers to this open problem by considering the case when the agents initially know the size of the network and the case when they do not. In the former case, we prove M=f+1 while in the latter, we prove M=f+2. More precisely, for networks of known size, we design a deterministic algorithm gathering all good agents in any network provided that the number of good agents is at least f+1. For networks of unknown size, we also design a deterministic algorithm ensuring the gathering of all good agents in any network but provided that the number of good agents is at least f+2. Both of our algorithms are optimal in terms of required number of good agents, as each of them perfectly matches the respective lower bound on M shown in [14], which is of f+1 when the size of the network is known and of f+2 when it is unknown

Rendezvous on a Line by Location-Aware Robots Despite the Presence of Byzantine Faults

A set of mobile robots is placed at points of an infinite line. The robots are equipped with GPS devices and they may communicate their positions on the line to a central authority. The collection contains an unknown subset of "spies", i.e., byzantine robots, which are indistinguishable from the non-faulty ones. The set of the non-faulty robots need to rendezvous in the shortest possible time in order to perform some task, while the byzantine robots may try to delay their rendezvous for as long as possible. The problem facing a central authority is to determine trajectories for all robots so as to minimize the time until the non-faulty robots have rendezvoused. The trajectories must be determined without knowledge of which robots are faulty. Our goal is to minimize the competitive ratio between the time required to achieve the first rendezvous of the non-faulty robots and the time required for such a rendezvous to occur under the assumption that the faulty robots are known at the start. We provide a bounded competitive ratio algorithm, where the central authority is informed only of the set of initial robot positions, without knowing which ones or how many of them are faulty. When an upper bound on the number of byzantine robots is known to the central authority, we provide algorithms with better competitive ratios. In some instances we are able to show these algorithms are optimal