Using algorithmic complexity theory methods, we propose a robust computational definitions for open-ended evolution (OEE) and adaptability of computable dynamical systems. With this framework, we show that decidability imposes absolute limits to the growth of complexity on computable dynamical systems up to a logarithm of a logarithmic term. Conversely, systems that exhibit open-ended evolution must be undecidable and have irreducible behaviour through the evolution of the system. Complexity is assessed in terms of three measures: sophistication, coarse sophistication and busy beaver logical depth.

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