Learning the Undecidable from Networked Systems

Abstract

This article presents a theoretical investigation of computation beyond the Turing barrier from emergent behavior in distributed systems. In particular, we present an algorithmic network that is a mathematical model of a networked population of randomly generated computable systems 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 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 emergently solving the halting problem in the minimum number of communication rounds. Furthermore, we introduce an algorithmic-informational measure of synergy for networked computable systems, which we call local algorithmic synergy. Then, we show that such algorithmic network can produce an arbitrarily large value of expected local algorithmic synergy.

Emergence and Algorithmic Information Dynamics of Systems and Observers

“…Instead of the communication protocol of plain diffusion, Abrahão et al [6] shows that a susceptible-infected-susceptible (SIS) contagion scheme [38,39] in algorithmic networks with a power-law degree distribution is also sufficient for triggering EEOE. In [4], it is shown that a slight modification in the communication protocol of plain diffusion from [7] is sufficient for enabling the whole algorithmic network to synergistically solve problems at a higher computational class than the computational class of its individual nodes.…”

Emergence of complex data from simple local rules in a network game

“…The model is defined in a general sense in order to allow future variations, to add specificities and to extend the model presented, while still being able to formally grasp a mathematical analysis of systemic features like the emergence of information and complexity along with its related phenomena and, this way, proving theorems. It was introduced in [16,18] and we have studied other variations of this first model with a static scale-free network topology in [17] and with a modified communication protocol to synergistically solve mathematical problems in [19]. In the present article we will focus on the model in [16].…”

“…The computation of each node may be seen in a combined point of view or taken as individuals. Respectively, nodes/programs may be computing using network's shared information to solve a common purpose [19]as the classical approach in distributed computing-or, for example, nodes may be "competing" with each other-as in a game-theoretical perspective, which we employ in this article (see Section 3). For the present purposes, we are interested in the average fitness (or payoff), and its related emergent complexity that may arise from a process that increases the average fitness.…”

“…That is, the algorithmic complexity of the networked population's output as a whole minus the algorithmic complexity of the isolated population's output as a whole. An initial step in the direction of tackling this problem is already mentioned in [19]. Analyzing this systemic property is not part of the scope of the present article and it will be a necessary future research, not only in the context of networked computable systems, but it is also an open problem for multivariate stochastic processes [27].…”

“…This mathematical phenomenon may be also fruitful for explaining synergistic behavior found in Nature and societies and why some network topological properties seem to be favored in biological networks. Indeed, algorithmic synergy was already shown to exist in networked resourceunbounded computable systems with a slight modification into the IFP [19]. Future research in this direction will be interesting for developing resource-bounded versions and, therefore, more realistic network-distributed computing models and architectures.…”