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

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, Ω, that quantifies OEE as the residence-time-weighted average of attractor cycle lengths across the sequence of attractors realized over time. Ω 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 Ω to continuous/hybrid state spaces, positioning Ω as a portable benchmark for OEE in discrete biological modeling and a guide for engineering evolvable synthetic circuits.

Figures (2)

• Figure 1: Best open-endedness curves across mechanisms - Homogeneous Case. Each curve reports the across-network mean $\Omega(K)$ for the parameter combination within a logic that maximized the area under the curve (AUC) on the $K$ grid ($1.1$ to $4.5$ in steps of $0.2$). Homogeneity comprises Poisson in-degree, synchronous updates, and fixed bias ($0.5$). For every $K$, we averaged over $1,000$ independently generated networks of $N=100$ nodes; each simulation ran for $T=10^6$ time steps from a random initial state. The legend shows the selected (AUC-maximizing) parameters: Classical (deterministic baseline); PBN “(2 ctx, $\sigma=0.1$)” indicates two deterministic contexts with switching probability $\sigma=0.1$ per epoch; ARM “$(\mu=0.4)$” flips each LUT output with probability $0.4$ when read; Paraconsistent “$(c=0.1)$” replaces LUT entries by contradictory tokens with probability $0.1$; Modal “$(a=1, p_p=0.5, p_n=0.5)$” uses accessibility degree $a=1$, with probabilities $p_p=0.5$ and $p_n=0.5$ of writing possible and necessary, respectively; Quantum-inspired “$(sp=0.1, e=6)$” writes superposed with probability $sp=0.1$ and applies paired-state coupling of size $e=6$ (a logic-inspired correlation mechanism; not unitary quantum dynamics). Note that the Classical curve collapses to zero, so it is not visible in the plot. Shaded confidence bands are omitted here to emphasize the comparative shapes; full parameter sweeps are provided in the Supplementary Material: Figures S3, S4, S7, S8, S11, S12, S15, S16, S17, S20, and Table S1.
• Figure 2: Best open-endedness curves across mechanisms - Heterogeneous Case. Same plotting and selection protocol as Fig. \ref{['fig:homo']}, but for structurally and temporally heterogeneous networks (Exponential in-degree; stochastic asynchronous updates). For each $K$, results average $1,000$ networks of $N=100$ nodes simulated for $T=10^6$ steps. The AUC-maximizing parameter sets (legend) are: Classical (deterministic baseline); PBN “(2 ctx, $\sigma=0.1$)”; ARM “$(\mu=0.4)$”; Paraconsistent “$(c=0.1)$”; Modal “$(a=1, p_p=0.5, p_n=0.1)$”; and Quantum-inspired “$(sp=0.1, e=32)$”. As in Fig. \ref{['fig:homo']}, $\Omega$ values are residence-time weighted cycle-lengths normalized by $T^2$ (see Methods). Note that both Classical and PBN curves collapse to zero, so they are not visible in the plot. All intermediate sweeps, robustness checks (e.g., alternative $K$ grids and $T$ values), and per-logic ablation plots appear in the Supplementary Material: Figures S5, S6, S9, S10, S13, S14, S18, S19, S21, and Table S2.