Biological arrow of time: Emergence of tangled information hierarchies and self-modelling dynamics

TL;DR

The paper reframes biological complexity and major evolutionary transitions as open-ended computational processes, proposing tangled hierarchies and self-modelling as core mechanisms. It integrates Gödel–Turing–Post recursion theory with dynamical systems and information theory to argue that open-ended meta-simulation resolves computational tensions by expanding problem-spaces, thereby producing novelty. The work highlights two types of tangled hierarchies (with and without self-modelling) and provides concrete biological and computational examples, such as genotype–phenotype mapping, stigmergic ant foraging, and the Game of Life, to illustrate undecidability and open-ended evolution. Overall, the authors offer a principled framework linking information processing, hierarchical organisation, and time's arrow, with broad implications for understanding evolutionary theory, genome evolution, and the nature of novelty generation.

Abstract

We study open-ended evolution by focusing on computational and information-processing dynamics underlying major evolutionary transitions. In doing so, we consider biological organisms as hierarchical dynamical systems that generate regularities in their phase-spaces through interactions with their environment. These emergent information patterns can then be encoded within the organism's components, leading to self-modelling "tangled hierarchies". Our main conjecture is that when macro-scale patterns are encoded within micro-scale components, it creates fundamental tensions (computational inconsistencies) between what is encodable at a particular evolutionary stage and what is potentially realisable in the environment. A resolution of these tensions triggers an evolutionary transition which expands the problem-space, at the cost of generating new tensions in the expanded space, in a continual process. We argue that biological complexification can be interpreted computation-theoretically, within the Gödel--Turing--Post recursion-theoretic framework, as open-ended generation of computational novelty. In general, this process can be viewed as a meta-simulation performed by higher-order systems that successively simulate the computation carried out by lower-order systems. This computation-theoretic argument provides a basis for hypothesising the biological arrow of time.

Figures (13)

• Figure 1: An example finite-state specification of a Turing Machine. States include A, B, C and halting state H. An arrow label (e.g., 0/P,R) specifies the tape symbol (e.g., symbol 0) that upon reading triggers a particular transition to another state, followed by the action, e.g., print (P) and move tape to the right (R) wiki-TM.
• Figure 2: Universal Turing Machine (UTM) simulating Turing Machine $M$ on input $I$. An encoding of program $P$ converts it into input data which is separated on the tape from the input $I$ by suitably chosen separator symbol(s) $\times$.
• Figure 3: Turing oracle. Wikimedia Commons, licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
• Figure 4: Open-ended sequence of $\alpha$-order oracle machines, with the corresponding Turing jump operations assigning successively harder decision problems $X, X', X"$,..., in an open-ended way.
• Figure 5: $n$-body simulation demonstrating unpredictability and undecidability of Newtonian dynamical systems, in this case comprised of $n=3$ stars of different masses (indicated by coloured trajectories) orbiting each other under the influence of gravity.

Theorems & Definitions (1)

• Lemma : Self-reference