# Learning the undecidable from networked systems

## TL;DR

This paper explores how networked systems can collectively solve undecidable problems like the halting problem through emergent behavior, communication, and selection mechanisms, surpassing traditional computational limits.

## Contribution

It introduces a mathematical model of networked computable systems, demonstrating conditions under which nodes can solve undecidable problems and defining a new measure of informational synergy.

## Key findings

- Nodes can solve the halting problem within logarithmic communication rounds.
- A central node can emergently solve the halting problem efficiently.
- The network can generate arbitrarily large local algorithmic synergy.

## 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.