Undecidability and Irreducibility Conditions for Open-Ended Evolution and Emergence

TL;DR

This paper demonstrates that undecidability is essential for open-ended evolution, showing that systems with ongoing complexity growth must be undecidable, which leads to unpredictable and irreducible evolutionary behaviors.

Contribution

It introduces robust computational definitions of open-ended evolution and proves that undecidability is a necessary condition for such systems to exhibit ongoing complexity growth.

Findings

Decidability limits the stable growth of complexity in dynamical systems.

Systems with strong open-ended evolution are necessarily undecidable.

Undecidability leads to unpredictable and irreducible evolutionary behaviors.

Abstract

Is undecidability a requirement for open-ended evolution (OEE)? Using methods derived from algorithmic complexity theory, we propose robust computational definitions of open-ended evolution and the adaptability of computable dynamical systems. Within this framework, we show that decidability imposes absolute limits to the stable growth of complexity in computable dynamical systems. Conversely, systems that exhibit (strong) open-ended evolution must be undecidable, establishing undecidability as a requirement for such systems. Complexity is assessed in terms of three measures: sophistication, coarse sophistication and busy beaver logical depth. These three complexity measures assign low complexity values to random (incompressible) objects. As time grows, the stated complexity measures allow for the existence of complex states during the evolution of a computable dynamical system. We show, however, that finding these states involves undecidable computations. We conjecture that for similar complexity measures that assign low complexity values, decidability imposes comparable limits to the stable growth of complexity, and that such behaviour is necessary for non-trivial evolutionary systems. We show that the undecidability of adapted states imposes novel and unpredictable behaviour on the individuals or populations being modelled. Such behaviour is irreducible. Finally, we offer an example of a system, first proposed by Chaitin, that exhibits strong OEE.