Learning the undecidable from networked systems

This article presents a theoretical investigation of hypercomputation from emergent behavior in distributed (or parallel) systems. In particular, we present an algorithmic network that is an abstract mathematical model of a networked population of randomly generated Turing machines with a fixed communication protocol. Then, in order to solve an undecidable problem, we study how nodes (i.e., Turing machines or computable systems) can harness the power of the metabiological selection of the fittest and the power of information sharing (i.e., communication) through the network. Formally, we show that there are pervasive network topological conditions that ensure the existence of a central node capable of solving the halting problem for every program with a length upper bounded by a logarithmic order of the population size. We also discuss the implications of such emergent phenomena on synergistic versus evolutionary paradigms in complex systems