Abstract
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 model 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 and the ability of a node to detect when its degree is equal to a small constant, plus either a unique leader or the ability of detecting common neighbors, are sufficient to guarantee not only termination but also the maximum computational power that one can hope for in this family of models. In particular, the model, under these minimal assumptions, computes with termination any symmetric predicate computable by a Turing Machine of space \(\varTheta (n^2)\).
Supported in part by (i) the project “Foundations of Dynamic Distributed Computing Systems” (FOCUS) which is implemented under the “ARISTEIA” Action of the Operational Programme “Education and Lifelong Learning” and is co-funded by the European Union (European Social Fund) and Greek National Resources and (ii) the FET EU IP project MULTIPLEX under contract no 317532. The full paper on which this chapter is based, can be found at http://arxiv.org/abs/1512.02832.
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- 1.
A convenient way to think of this setting is to imagine a clique graph with its edges labeled from \(\{0,1\}\). Then, in this case, the clique is the interaction network while its subgraph induced by the edges labeled 1 is the constructed network.
- 2.
On the other hand, it is possible, and plausible w.r.t. several application scenarios, that the set of available interactions at a given step actually depends on the constructed network. Such a case was considered in [14], where the constructed network is always a subnetwork of the grid network and two processes can only interact if a connection between them would preserve this requirement. So, in that case, the set of available interactions is, in every step, constrained by the network that has been constructed by the protocol so far.
- 3.
Active and inactive edges are not to be confused with active and passive mobility. An edge is said to be active if its state is 1 and it is said to be inactive if its state is 0.
- 4.
Essentially, a predicate in this type of models is called symmetric (or commutative) if permuting the input symbols does not affect the predicate’s outcome.
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Michail, O., Spirakis, P.G. (2017). Connectivity Preserving Network Transformers. In: Adamatzky, A. (eds) Emergent Computation . Emergence, Complexity and Computation, vol 24. Springer, Cham. https://doi.org/10.1007/978-3-319-46376-6_15
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