Terminating Distributed Construction of Shapes and Patterns in a Fair Solution of Automata

In this work, we consider a solution of automata similar to Population Protocols and Network Constructors. The au-tomata, also called nodes, move passively in a well-mixed solution and can cooperate by interacting in pairs. Dur-ing every such interaction, the nodes, apart from updating their states, may also choose to connect to each other in order to start forming some required structure. The model introduced here is a more applied version of Network Con-structors, imposing geometrical constraints on the permissi-ble connections. Each node can connect to other nodes only via a very limited number of local ports, which implies that at any given time it has only a bounded number of neigh-bors. Connections are always made at unit distance and are perpendicular to connections of neighboring ports. Thoug

Terminating Distributed Construction of Shapes and Patterns in a Fair Solution of Automata

In this work, we consider a solution of automata (or nodes) that move passively in a well-mixed solution without being capable of controlling their movement. Nodes can cooperate by interacting in pairs and every such interaction may result in an update of their local states. Additionally, the nodes may also choose to connect to each other in order to start forming some required structure. Such nodes can be thought of as small programmable pieces of matter, like tiny nanorobots or programmable molecules. The model that we introduce here is a more applied version of network constructors, imposing physical (or geometric) constraints on the connections that the nodes are allowed to form. Each node can connect to other nodes only via a very limited number of local ports. Connections are always made at unit distance and are perpendicular to connections of neighboring ports, which makes the model capable of forming 2D or 3D shapes. We provide direct constructors for some basic shape construction problems, like spanning line, spanning square, and self-replication. We then develop new techniques for determining the computational and constructive capabilities of our model. One of the main novelties of our approach is that of exploiting the assumptions that the system is well-mixed and has a unique leader, in order to give terminating protocols that are correct with high probability. This allows us to develop terminating subroutines that can be sequentially composed to form larger modular protocols. One of our main results is a terminating protocol counting the size n of the system with high probability. We then use this protocol as a subroutine in order to develop our universal constructors, establishing that it is possible for the nodes to become self-organized with high probability into arbitrarily complex shapes while still detecting termination of the construction

Connectivity Preserving Network Transformers

The Population Protocol model is a distributed model that concerns systems of very weak computational entities that cannot control the way they interact. The model of Network Constructors is a variant of Population Protocols capable of (algorithmically) constructing abstract networks. Both models are characterized by a fundamental inability to terminate. In this work, we investigate the minimal strengthenings of the latter that could overcome this inability. Our main conclusion is that initial connectivity of the communication topology combined with the ability of the protocol to transform the communication topology plus a few other local and realistic assumptions are sufficient to guarantee not only termination but also the maximum computational power that one can hope for in this family of models. The technique is to transform any initial connected topology to a less symmetric and detectable topology without ever breaking its connectivity during the transformation. The target topology of all of our transformers is the spanning line and we call Terminating Line Transformation the corresponding problem. We first study the case in which there is a pre-elected unique leader and give a time-optimal protocol for Terminating Line Transformation. We then prove that dropping the leader without additional assumptions leads to a strong impossibility result. In an attempt to overcome this, we equip the nodes with the ability to tell, during their pairwise interactions, whether they have at least one neighbor in common. Interestingly, it turns out that this local and realistic mechanism is sufficient to make the problem solvable. In particular, we give a very efficient protocol that solves Terminating Line Transformation when all nodes are initially identical. The latter implies that the model computes with termination any symmetric predicate computable by a Turing Machine of space $\Theta(n^2)$

Connectivity preserving network transformers

The Population Protocol model is a distributed model that concerns systems of very weak computational entities that cannot control the way they interact. The model of Network Constructors is a variant of Population Protocols capable of (algorithmically) constructing abstract networks. Both models are characterized by a fundamental inability to terminate. In this work, we investigate the minimal strengthenings of the latter that could overcome this inability. Our main conclusion is that initial connectivity of the communication topology combined with the ability of the protocol to transform the communication topology plus a few other local and realistic assumptions are sufficient to guarantee not only termination but also the maximum computational power that one can hope for in this family of models. The technique is to transform any initial connected topology to a less symmetric and detectable topology without ever breaking its connectivity during the transformation. The target topology of all of our transformers is the spanning line and we call Terminating Line Transformation the corresponding problem. We first study the case in which there is a pre-elected unique leader and give a time-optimal protocol for Terminating Line Transformation. We then prove that dropping the leader without additional assumptions leads to a strong impossibility result. In an attempt to overcome this, we equip the nodes with the ability to tell, during their pairwise interactions, whether they have at least one neighbor in common. Interestingly, it turns out that this local and realistic mechanism is sufficient to make the problem solvable. In particular, we give a very efficient protocol that solves Terminating Line Transformation when all nodes are initially identical. The latter implies that the model computes with termination any symmetric predicate computable by a Turing Machine of space $\Theta(n^2)$

Simple and efficient local codes for distributed stable network construction

In this work, we study protocols so that populations of distributed processes can construct networks. In order to highlight the basic principles of distributed network construction, we keep the model minimal in all respects. In particular, we assume finite-state processes that all begin from the same initial state and all execute the same protocol. Moreover, we assume pairwise interactions between the processes that are scheduled by a fair adversary. In order to allow processes to construct networks, we let them activate and deactivate their pairwise connections. When two processes interact, the protocol takes as input the states of the processes and the state of their connection and updates all of them. Initially all connections are inactive and the goal is for the processes, after interacting and activating/deactivating connections for a while, to end up with a desired stable network. We give protocols (optimal in some cases) and lower bounds for several basic network construction problems such as spanning line, spanning ring, spanning star, and regular network. The expected time to convergence of our protocols is analyzed under a uniform random scheduler. Finally, we prove several universality results by presenting generic protocols that are capable of simulating a Turing Machine (TM) and exploiting it in order to construct a large class of networks. We additionally show how to partition the population into k supernodes, each being a line of logk nodes, for the largest such k. This amount of local memory is sufficient for the supernodes to obtain unique names and exploit their names and their memory to realize nontrivial constructions