This article presents a theoretical investigation of computation beyond the Turing barrier 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 computable systems with a fixed communication protocol. Then, in order to solve an undecidable, we study how nodes (i.e., Turing machines or computable systems) can harness the power of the metabiological selection and the power of information sharing (i.e., communication) through the network. Formally, we show that there is a pervasive network topological condition, in particular, the small-diameter phenomenon, that ensures that every node becomes capable of solving the halting problem for every program with a length upper bounded by a logarithmic order of the population size. In addition, we show that this result implies the existence of a central node capable of solving the halting problem in the minimum number of communication rounds. We also discuss the implications of such emergent phenomena on synergistic versus evolutionary paradigms in complex systems by showing that such algorithmic network can produce an arbitrarily large value of expected local algorithmic synergy
