Turning the informal definition of intelligence into a precise mathematical equation.
We have an informal definition of intelligence: an agent's ability to achieve goals in a wide range of environments. We have AIXI, a theoretical model of optimal intelligence. Now we flip the idea on its head: instead of using universal AI theory to build intelligent agents, we use it to measure intelligence.
Our informal definition has three pieces: agents, environments, and goals. Let's formalise each one.
Agents and environments communicate through the agent-environment model from Chapter 2. The agent sends actions, the environment returns observations and rewards. The reward signal implicitly defines the goal — the agent tries to maximise cumulative reward.
"Wide range of environments" means we consider the space of all computable environments with bounded reward sum (the set E). We require environments to be computable because incomputable environments cannot be simulated or tested. The bounded reward sum condition ensures that every possible temporal preference is represented.
"Ability to achieve" means expected performance: the value function Vμπ.
The remaining question: how do we combine performance across infinitely many environments into a single number? We cannot use a uniform distribution (it doesn't exist over infinite sets). Instead, we weight each environment by 2-K(μ) — its Kolmogorov complexity. Simple environments count more.
Bringing everything together:
The final equality is remarkable: universal intelligence equals the agent's expected performance under the universal mixture ξ. It is literally AIXI's value function. This means universal intelligence of an agent is simply its expected performance with respect to the universal distribution.
Let's unpack what each part captures from our informal definition:
| Informal | Formal |
|---|---|
| "Agent" | π — any function from histories to actions |
| "Environments" | μ ∈ E — all computable reward-summable environments |
| "Goals" | Implicit in the reward structure of each μ |
| "Ability to achieve" | Vμπ — expected total reward |
| "Wide range" | ∑ 2-K(μ) — weighted sum over all environments |
A random agent πrand chooses uniformly random actions. In most environments, it will fail to exploit any regularities, so Vμπrand will be low compared to other agents. Therefore Υ(πrand) is low.
But wait — some environments give high reward no matter what the agent does (imagine an environment that always gives reward 1 regardless of actions). For these, even the random agent scores well. However, such trivial environments are simple (short programs), so while 2-K(μ) is relatively large, the random agent's performance is no better than any other agent's. It gets no advantage.
IBM's Deep Blue plays chess at superhuman level. Its value function Vμchessπdblue is extremely high. But 2-K(μchess) is small (chess is complex), and for all other environments V is very low. So Υ(πdblue) is very low.
This tells us something profound about current AI: by focusing on increasingly specialised systems, we have in some sense been going backwards in terms of universal intelligence. A system that handles basic pattern recognition across many domains is more intelligent than one that dominates a single complex domain.
A general but simple agent πbasic builds a table of observation-action pairs and keeps statistics. It takes the best known action 90% of the time, explores 10%. For most environments it will find some structure to exploit, so Vμπbasic > Vμπrand almost everywhere. Thus Υ(πbasic) > Υ(πrand).
Extending πbasic to use more history improves it further. An agent π2back that conditions on the last two observations finds patterns that πbasic misses, like the alternating-action environment.
An agent π2forward that looks one step into the future (not just maximising immediate reward but also next-step reward) is even more powerful. It can see that climbing a hill (zero immediate reward) leads to sliding down (high reward next step), a pattern that greedy agents miss.
By construction, AIXI maximises Υ. No agent can have higher universal intelligence. This gives us the upper bound on intelligence:
This upper bounds the intelligence of all future machines, no matter how powerful their hardware and algorithms. Of course, AIXI is not computable, so no real machine can achieve this bound. But it tells us the theoretical ceiling.
Where would a human fall? For simple environments, a human should identify structure and exploit it. For complex environments (say, one that involves processing sensory data in formats the brain was not designed for), a human might perform poorly compared to a specialised algorithm. Perhaps the universal intelligence of a human is not that high compared to some machine learning algorithms? We genuinely don't know.
How does universal intelligence compare to the desirable properties of an intelligence measure?
| Property | Status |
|---|---|
| Valid | Yes — derived step by step from mainstream definitions of intelligence |
| Informative | Yes — assigns a real number, enabling comparison of any two agents |
| Wide range | Yes — spans from πrand to AIXI |
| General | Yes — hard to imagine a broader metric without contradicting Church-Turing |
| Dynamic | Yes — measures learning and adaptation over time, not one-shot problems |
| Unbiased | Yes — grounded in universal Turing computation, not any particular culture |
| Fundamental | Yes — based on computation and complexity, unlikely to change with technology |
| Formal | Yes — a mathematical equation |
| Practical | No — Kolmogorov complexity is not computable |
The one weakness: impracticality. But this mirrors the definition of randomness — incomputable to verify, yet theoretically fundamental. Future work aims to approximate Υ using computable complexity measures like Levin's Kt complexity.
Legg addresses common criticisms head-on:
"It's just a few equations." Yes, but so is E=mc2. The work is in showing that the equation correctly captures the concept. That required surveying 70+ definitions, building the agent-environment framework, and connecting it to universal AI theory.
"It's just reinforcement learning." The equation goes far beyond RL. It uses universal Occam-weighted priors, considers all computable environments, and produces an absolute measure. Simply writing down the RL framework does not give you universal intelligence.
"The universe might not be computable." There is no evidence of incomputable physical processes. Even if some exist, computable approximations would still work extremely well given that all known physics is computable.
"What about consciousness/creativity/soul?" These matter only insofar as they measurably affect performance. If understanding has a measurable impact on an agent's performance, then Υ is partly a measure of understanding. If not, it is irrelevant to intelligence in any practical sense.
"No Free Lunch theorem makes this impossible." NFL applies to uniform distributions over problems. Universal intelligence uses a highly non-uniform distribution (Occam's razor). The NFL theorem does not apply.
This equation is the central contribution of the thesis. It turns the age-old question "what is intelligence?" into a mathematical statement. It correctly ranks agents from random to optimal, emphasises generality over specialisation, and connects to the deepest ideas in theoretical computer science.
The next challenge: can we approximate this measure? Chapter 5 will show that fundamental limits on computation constrain how closely any real algorithm can approach AIXI.