Characterizing Open-Ended Evolution Through Undecidability Mechanisms in Random Boolean Networks

TL;DR

This paper introduces a new metric, {}, to quantify open-ended evolution in discrete models like Random Boolean Networks, highlighting mechanisms that enable sustained novelty and proposing extensions for continuous systems.

Contribution

The paper presents a model-independent metric for open-ended evolution and compares classical and non-classical mechanisms in RBNs, identifying state-dependent undecidability-adjacent mechanisms as key to sustained novelty.

Findings

{} increases with persistent cyclic phenotypes.

Undecidability-adjacent mechanisms support sustained novelty.

Proposes extension of {} to continuous/hybrid spaces.

Abstract

Discrete dynamical models underpin systems biology, but we still lack substrate-agnostic diagnostics for when such models can sustain genuinely open-ended evolution (OEE): the continual production of novel phenotypes rather than eventual settling. We introduce a simple, model-independent metric, {\Omega}, that quantifies OEE as the residence-time-weighted average of attractor cycle lengths across the sequence of attractors realized over time. {\Omega} is zero for single-attractor dynamics and grows with the number and persistence of distinct cyclic phenotypes, separating enduring innovation from transient noise. Using Random Boolean Networks (RBNs) as a unifying testbed, we compare classical Boolean dynamics with biologically motivated non-classical mechanisms (probabilistic context switching, annealed rule mutation, paraconsistent logic, modal necessary/possible gating, and quantum-inspired superposition/paired-state coupling) under homogeneous and heterogeneous updating schemes. Our results support the view that undecidability-adjacent, state-dependent mechanisms -- implemented as probabilistic context switching, modal necessity/possibility gating, paraconsistent logic (controlled contradictions), or quantum-inspired superposition/paired-state coupling (correlated branching) -- are enabling conditions for sustained novelty. At the end of our manuscript we outline a practical extension of {\Omega} to continuous/hybrid state spaces, positioning {\Omega} as a portable benchmark for OEE in discrete biological modeling and a guide for engineering evolvable synthetic circuits.