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

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 contribution of each attractor’s cycle length 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/entanglement) under homogeneous and heterogeneous updating schemes. Our results support the view that undecidability-adjacent, state-dependent mechanisms – implemented as contextual switching, conditional necessity/possibility, controlled contradictions, or 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.

💡 Research Summary

The paper tackles the longstanding problem of how to recognize and quantify open‑ended evolution (OEE) in discrete dynamical models, especially those used to represent gene‑regulatory networks. The authors introduce a novel, model‑agnostic metric Ω that captures the “open‑endedness” of a system by weighting each attractor’s cycle length with the amount of time the system spends in that attractor. Formally, Ω = Σ_i (τ_i·L_i) / T, where L_i is the length of the i‑th attractor’s cycle, τ_i is the residence time on that attractor, and T is the total observation time. Ω equals zero for a single fixed point or a short, isolated cycle, and grows as more distinct, long‑lived cyclic phenotypes appear, thereby separating genuine sustained novelty from transient fluctuations.

To test Ω, the authors use Random Boolean Networks (RBNs) as a unifying testbed. They study both homogeneous RBNs (identical indegree K, synchronous updates, uniform Boolean functions) and heterogeneous RBNs (Poisson or exponential indegree distributions, asynchronous updating via the CUBEWALKERS library). Networks of size N = 100–500 and connectivity K = 2–4 are simulated for up to 10⁵ steps.

The core contribution lies in comparing classical Boolean dynamics with five “non‑classical” mechanisms that introduce undecidability‑adjacent, state‑dependent behavior:

1. Probabilistic Boolean Networks (PBNs) – multiple Boolean contexts are pre‑generated; at each context‑switch (probability c_l) the whole rule set changes. This mimics environmental contingency and produces abrupt creation of new attractors, leading to large spikes in Ω.

2. Annealed Rule Mutation (ARM) – each entry of a node’s lookup table is flipped with probability p_ARM at every read. This implements a continuously mutating rule set, slowly deforming existing attractors and generating a steady, moderate increase in Ω.

3. Paraconsistent Logic – LUT entries may output an “inconsistent” symbol (×) with probability π. A resolver ρ applies a majority‑vote or priority rule to collapse contradictions. Local contradictions cause brief instability that often triggers transitions to new attractors, producing occasional Ω explosions.

4. Modal Necessity/Possibility Gating – nodes carry □ (necessary) or ◇ (possible) flags that enable a rule only when a global condition holds. Conditional activation creates multiple logical pathways for the same input, expanding the attractor landscape and yielding a sustained, intermediate Ω.

5. Quantum‑Inspired Superposition/Entanglement – node states can be 0, 1, or a superposed (0+1) state; entangling operations couple several nodes simultaneously. Because deterministic Boolean updates are impossible, the system is sampled probabilistically at “measurement” steps. This produces a highly non‑deterministic dynamics where many virtual attractors coexist, driving Ω to the highest values observed.

Experimental results show: (i) pure homogeneous Boolean RBNs keep Ω≈0, i.e., no OEE; (ii) structural and temporal heterogeneity alone yields only modest Ω growth; (iii) each non‑classical mechanism markedly raises Ω, with PBNs and paraconsistent logic giving the largest spikes, ARM providing a long‑term baseline, modal gating offering stable intermediate novelty, and quantum‑inspired dynamics generating the most extreme open‑endedness. The authors argue that these mechanisms embody undecidability: the system’s future cannot be predicted by any single algorithm, mirroring the halting problem. Consequently, continual generation of novel phenotypes becomes inevitable.

Finally, the authors outline how Ω can be extended to continuous or hybrid state spaces. By discretizing continuous variables into bins and defining “virtual cycles” for flows, one can compute a weighted residence‑time sum analogous to Ω, enabling its application to metabolic networks, signaling pathways, or even agent‑based models with continuous traits.

In sum, the paper delivers a robust, quantitative OEE metric and demonstrates that undecidability‑adjacent, state‑dependent mechanisms are key enablers of sustained evolutionary novelty. This work provides a practical benchmark for assessing open‑endedness across biological modeling platforms and offers concrete guidance for engineering synthetic circuits that remain evolvable over long time scales.