Undecidability and Irreducibility Conditions for Open-Ended Evolution and Emergence

Area of Science:

• Theoretical Computer Science
• Algorithmic Information Theory
• Evolutionary Systems

Background:

• Open-ended evolution (OEE) describes systems capable of unbounded novelty and complexity.
• Algorithmic complexity theory provides tools to define and measure complexity computationally.

Purpose of the Study:

• To computationally define open-ended evolution and adaptability.
• To investigate the role of decidability and undecidability in the stable growth of complexity within computable dynamical systems.

Main Methods:

• Utilizing methods from algorithmic complexity theory.
• Defining and measuring complexity using sophistication, coarse sophistication, and busy beaver logical depth.
• Analyzing computable dynamical systems.

Main Results:

• Decidability imposes absolute limits on complexity growth in computable systems.
• Systems exhibiting strong open-ended evolution must be undecidable.
• Undecidable states lead to novel, unpredictable, and irreducible behavior in evolutionary systems.

Conclusions:

• Undecidability is a necessary requirement for strong open-ended evolution.
• The stable growth of complexity in evolutionary systems is fundamentally linked to undecidability.
• This framework offers insights into the limits and possibilities of complex adaptive systems.