Open-Endedness from a Computational Perspective

Open-endedness is the capacity of a system to keep generating novelty, complexity, and diversity without ever settling into a fixed endpoint. Biological evolution is the reference case: four billion years of continuous innovation with no convergence. The question that drives the field is whether artificial systems can do the same, and if so, what it would take. This document works through the definitions, the design choices that distinguish one open-ended system from another, a striking early physical example, and the philosophical stakes.

1. What "open" means

There is no single definition, but the useful ones cluster into a few families.

The most rigorous recent framing is observer-relative (Hughes et al., 2024). A system is open-ended when its outputs are both novel, meaning increasingly hard for an observer's model to predict over long horizons, and learnable, meaning more predictable once the observer has watched longer. Formally, for a sequence of artifacts, prediction error should rise as you look further ahead and fall as you accumulate history. The key move is putting the observer inside the definition, which makes open-endedness measurable through prediction error and ties it directly to whoever is judging novelty.

An information-theoretic framing describes open-endedness as unbounded growth in the complexity of what the system produces, so that the shortest description of its outputs keeps getting longer. This captures true algorithmic novelty rather than mere reshuffling of existing parts.

A process framing (Taylor et al., 2016) focuses on systems that never reach equilibrium and keep producing complex solutions, which requires an effectively unlimited space to explore, viable paths between solutions, and sustained pressure that never fully converges.

An emergence framing (Gershenson, 2023) emphasizes higher-order properties that are not present in the parts. Conway's Game of Life is the standard illustration: simple cell-update rules give rise to gliders, oscillators, and structures complex enough to compute, none of which is visible in the rules themselves.

2. Properties every open-ended system shares

Across domains, four properties recur.

Novelty generation is the universal one: the system keeps producing outputs that surprise even its designers. Complexity progression means the outputs get more sophisticated over time, often by building on earlier results as stepping stones. Self-organization means the system generates its own challenges and structure without being told to, step by step. Non-convergence means it never locks into a single fixed solution, whether because of competitive arms races, criticality, or continual exploration.

3. The design axes

Rather than a list of algorithms, it helps to see open-endedness as a space of choices. Real systems sit at particular points along these axes.

• Substrate: biological versus artificial. Biological evolution has material openness (energy flows, chemistry, physical embodiment) and operates at many nested levels at once. Most artificial systems run in discrete informational spaces at a single level, which is a large part of why they tend to plateau.
• Emergence type: syntactic versus semantic. Syntactic emergence is new structure from rules (Game of Life patterns). Semantic emergence is new meaning, which requires interpretation and grounding. Most computational systems reach the first and struggle with the second.
• Space: discrete versus continuous. Borges' Library of Babel is the discrete archetype: a finite alphabet yields an unbounded space of possible books. Fluid dynamics and neural weight spaces are continuous cases.
• Time course: perpetual (steady novelty), periodic (innovation then stabilization, as in punctuated equilibrium), or phase-transition (a sudden jump to a new regime, as when language or writing appeared).
• Growth pattern: cumulative (each discovery enables the next, as in science) versus explosive (rapid divergence from one breakthrough, as in the Cambrian explosion).
• Source of novelty: combinatorial (recombining existing parts) versus emergent (genuinely new properties from interaction).
• Drive: extrinsic (external pressure or objectives) versus intrinsic (curiosity, play, exploration for its own sake).
• Scale: single-agent versus collective, where interaction between agents produces system-level properties.
• Architecture: archive-based (keeping an explicit record of what has been found, as in Novelty Search and MAP-Elites) versus population-based (relying on competition among a population).
• Organization: hierarchical with cross-scale effects versus flat, and modular versus holistic.

4. How systems actually pursue it

Several method families put these choices into practice.

Novelty Search, from Kenneth Stanley, throws away the fitness objective and rewards behavioral novelty instead, reaching solutions through indirect paths that objective-driven search would skip. Quality-Diversity algorithms such as MAP-Elites keep an archive of high-performing solutions across many behavioral niches, illuminating a whole space rather than optimizing a single point. POET co-evolves environments and the agents that solve them, generating an endless curriculum of challenges.

Foundation-model approaches use large models as engines of variation: LLMs as mutation operators over code, population-based training of diverse agents, and recursive self-modification. Reinforcement learning contributes intrinsic motivation, where curiosity or prediction error drives exploration, and automatic curriculum generation through environment design. Swarm and distributed methods produce coordination and collective intelligence with no central controller. Generative and procedural methods (grammars, L-systems, cellular automata, GAN variants) generate unbounded detail.

Working examples span games (AlphaZero discovering strategies that surprise experts through self-play), embodied agents (Voyager growing a skill library through code generation), swarms (large robot collectives with emergent group behavior), and scientific discovery (FunSearch finding new results by LLM-guided evolution with formal verification).

5. The measurement paradox

Measuring open-endedness runs into a self-referential problem. Stepney and Hickinbotham (2023) argue that any fixed yardstick eventually misses the point, because a genuinely open-ended system will, given time, exceed the framework built to evaluate it. This is the detection problem: no fixed measure can capture all forms of novelty, because the system outgrows the measure.

The observer-relative definition partly addresses this by making evaluation depend on the observer's model, so the yardstick can move. There are also hard computational limits underneath: the exact complexity measures involved are uncomputable in general, so real work uses approximations such as compression ratios and sampling. The paradox has a philosophical flip side worth stating: if we could perfectly predict and measure an innovation in advance, it is not clear it would count as innovation at all. Unpredictability may be a defining feature rather than a measurement failure.

6. A physical precedent: Pask's electrochemical devices

Long before modern artificial life, Gordon Pask built physical systems in the 1950s and 1960s that showed open-ended behavior in a way most software still cannot. His devices were electrodes suspended in acidic solutions of metallic salts, typically ferrous sulfate. By limiting the available current, he forced the growing structures to compete rather than expand freely.

Under current, metal ions were reduced at cathode sites and grew into iron threads along paths of highest current density. These threads had lower resistance than the surrounding solution, so each thread changed the electrical field that shaped the growth of the next. That feedback loop is what produced genuine self-organization.

The results were striking. In the most famous experiment, a device developed sensitivity to sound, effectively growing its own ear, without being designed as an audio sensor. Others developed sensitivity to magnetic fields or vibration. The systems constructed their own relevance criteria, deciding through their own dynamics which features of the environment mattered. Cut threads regrew their patterns, giving the systems a form of distributed structural memory. When conditions were ambiguous, threads bifurcated to handle the ambiguity, and retained the modification afterward, which is a concrete demonstration of learning.

Several principles come out of this work, and they generalize. Material embodiment matters: a physical substrate that can reorganize itself does things abstract models struggle to capture. Constraints enable creativity: scarce current forced competition that drove elaboration, so limitation produced abundance rather than restricting it. The observer is part of the system, since a device that redefines what counts as a relevant signal cannot be cleanly separated from whoever is watching. Emergence requires operating near instability, organized enough to persist but sensitive enough to be reshaped. And learning and self-organization turn out to be two views of the same underlying process.

7. Two grades of open-endedness: new points versus new dimensions

Not everything called open-ended is open-ended to the same degree, and the difference is the crux of the whole subject. It helps to separate three grades.

The weakest is complex emergence in a fixed space. Conway's Game of Life is the standard case: fixed rules, a fixed grid, a fixed set of possible states. It produces endless intricate patterns, but every one of them was always a point in the same unchanging space. Nothing genuinely new can enter; the system only rearranges what the rules already permitted. This is why the Game of Life, for all its visual richness, is a weak example of open-endedness. It never surprises the space it lives in, only the observer who has not yet enumerated it.

Stronger is open-ended exploration of a fixed space. Novelty search, which abandons the objective and rewards reaching new behaviors, keeps discovering points that objective-driven optimization skips, collecting stepping stones and escaping deception. This is a real and useful form of open-endedness. But the space of behaviors it searches is still fixed in advance. It finds new points in a given landscape; it does not change the landscape.

Strongest is a system that expands its own space by adding new dimensions. This is what Pask's electrochemical devices did when they grew a sensitivity to sound. Before that growth, sound was not even a variable the system could respond to; afterward it was. The device did not find a new point in an existing space of behaviors, it added a new axis to that space, a new way of being affected by the world. Biological evolution does this repeatedly through major transitions: photoreception, nervous systems, and language each added dimensions of variation that did not previously exist. Each one changed not just what the system did but what the system could vary at all.

The distinction is easy to blur, because the first two grades are so much easier to build and both feel open-ended from the outside. A fixed rule set that generates surprising patterns looks open-ended, and a search that keeps finding new behaviors looks open-ended. But the deepest and least solved form is the third: a system that keeps enlarging the very space of things it can do. That is the target Pask's work points at, and it is precisely the thing that cellular automata and novelty search, for all their strengths, do not reach. Most current artificial systems produce novelty within a space someone already defined. The open problem is building systems that define new spaces for themselves.

8. Constraint and creativity

A theme runs through every example above and deserves to be pulled out on its own: constraint is not the enemy of open-endedness. Borges' finite alphabet, Conway's handful of rules, and Pask's limited current all show the same thing. Without structure there is nothing to build on, and without scarcity there is no competition to drive elaboration. The constraint is what channels an unbounded space into productive exploration rather than noise. Biological evolution runs under thermodynamic limits for the same reason: they prevent runaway growth while still permitting endless structural refinement.

9. Why it matters

Open-endedness marks a shift from optimization, which drives toward a fixed best answer, to exploration, which keeps opening new possibilities. That shift has practical weight. It may be a route toward systems that keep improving past their initial objectives, and the stepping-stone pattern suggests that indirect paths often outperform direct optimization.

It also carries real risk, which is exactly why the observer-relative framing is attractive: building the human evaluator into the system's dynamics keeps it tractable and gives a handle on control. A system designed to surprise its creators is useful and, for the same reason, needs to stay aligned with human values as its capabilities move beyond its original training. The convergence of quality-diversity methods, foundation models, evolutionary approaches, and self-organizing materials is making genuinely open-ended systems more reachable, which puts both the opportunity and the safety question in front of us at once.

Key references

• Hughes et al. (2024). Open-endedness is essential for artificial superhuman intelligence. arXiv:2406.04268.
• Taylor et al. (2016). Open-ended evolution: perspectives from the OEE workshop in York. Artificial Life, 22(3), 408-423.
• Stepney and Hickinbotham (2023). On the open-endedness of detecting open-endedness. Artificial Life, 30(3), 390-404.
• Lehman and Stanley (2011). Abandoning objectives: evolution through the search for novelty alone. Evolutionary Computation, 19(2), 189-223.
• Mouret and Clune (2015). Illuminating search spaces by mapping elites. arXiv:1504.04909.
• Wang, Lehman, Clune, and Stanley (2019). POET: open-ended coevolution of environments and their optimized solutions. GECCO.
• Stanley and Lehman (2015). Why Greatness Cannot Be Planned: The Myth of the Objective. Springer.
• Gershenson (2023). Emergence in artificial life. Artificial Life, 29(2), 153-167.
• Conway (1970). The game of life. Scientific American, 223(4), 4-10.
• Cariani (1993). To evolve an ear: epistemological implications of Gordon Pask's electrochemical devices. Systems Research, 10(3), 19-33.
• Pask (1975). Conversation, Cognition and Learning. Elsevier.
• Romera-Paredes et al. (2024). Mathematical discoveries from program search with large language models. Nature, 625(7995), 468-475.
• Borges (1962). The library of babel. In Labyrinths.