Stop arguing about what intelligence is. Write down one equation.
Every debate about machine intelligence stalls on the same missing step: nobody agrees what the word means. Legg refuses to argue. He starts from one plain English sentence the psychologists roughly agree on — intelligence measures an agent's ability to achieve goals in a wide range of environments — and turns it, piece by piece, into a single formula.
The formula scores an agent by its average reward across every computable environment, with simpler environments weighted more heavily. That one weighting choice — Occam's razor, made mathematical — is the whole trick. It makes a single number, the Υ (“upsilon”) score, that ranks a dice-roller, a chess machine, a human, and the theoretical perfect agent on one scale. The perfect agent at the top has a name — AIXI — and a catch: it can be defined exactly but can never be run.
Intelligence is not a task you pass. It is a weighted average over all the tasks there could ever be — with the simple ones counting most.
The question
What could a definition of intelligence even look like?
Step 1 · One sentence, three pieces
An agent, an environment, and a single reward signal between them.
Legg's whole construction grows from one plain sentence that psychologists and AI researchers loosely agree on:
“Intelligence measures an agent's ability to achieve goals in a wide range of environments.”
Pull it apart and there are exactly three pieces: an agent, the environments it acts in, and a goal. To make the goal something we can score without speaking the agent's language, we hand it over a single wire — a reward, one number per turn that says “how good is your situation right now.” The agent and the environment then take turns, forever:
One clean trick keeps the math finite. Instead of discounting far-future rewards with extra knobs, Legg only allows reward-summable environments: ones whose total reward over all time is bounded — by 1, without loss of generality. So an agent's score in environment μ is just the expected sum of rewards, a number between 0 and 1: Vπμ = E(Σ rᵢ) ≤ 1.
To test an agent in any way at all, it is enough to fully specify the environment. The goal hides inside the environment, in when it chooses to pay out reward. So “a wide range of goals” and “a wide range of environments” become the same thing.
Thesis §4.1: the agent π is a distribution over actions given history; the environment μ a distribution over perceptions given history. Restricting to reward-summable μ ∈ E removes the discount parameters and bounds Vπμ ≤ 1.
Step 2 · Our running example
One tiny world we'll carry all the way through: a playground slide.
You have the loop — agent acts, environment pays reward. To feel what “intelligence” means in this framework, we need one concrete environment to test agents in. We'll use a single example from the thesis (Figure 4.1) for the whole page: a child's playground slide.
The rules are deliberately trivial. Each turn k the agent picks one of two actions:
Now the trap. Compare just the next reward: resting pays 2⁻ᵏ⁻⁴, climbing pays 0. A short-sighted agent — one that only maximises the very next reward — will rest at the bottom forever and never discover the slide. Yet the climb-then-slide payoff 2⁻ᵏ is 16× larger (since 2⁻ᵏ⁻⁴ = 2⁻ᵏ⁄16). The smart move is to give up one small reward now to win a big one next turn.
That single gap — between agents that grab the immediate crumb and agents that look one step ahead — is exactly the kind of difference a real measure of intelligence must detect. We'll watch several agents face this slide, and score them.
The slide is one of the simplest environments where greedy fails. Keep it in mind: a good intelligence measure should reward the agent that plans ahead here — and, as we'll see, reward it for solving simple worlds like this most of all.
Thesis Example (µ-slide), Fig 4.1: A = {rest, climb}; resting pays 2⁻ᵏ⁻⁴, climbing pays 0, and from the top the agent always slides down for 2⁻ᵏ. Both π-basic and a one-step-greedy agent get stuck resting.
The measure
From one environment to all of them · Occam's razor · the equation · AIXI · the catch
Step 3 · Run the agents (interactive)
“Performance in an environment” is just the reward an agent collects. Let's collect some.
You have the slide and you can see the trap. Before we score intelligence across all environments, let's nail down what a score in one environment is — by actually running two agents on the slide and adding up their reward.
Two contestants. π-greedy only ever maximises the very next reward. π-lookahead looks one turn into the future before acting — exactly the π-2forward agent from the thesis. Step them turn by turn and watch the reward tape fill:
Pick an agent, then step or play — watch where it goes and what it earns
π-greedy chosen. Step it and watch it settle for the crumb.
Watch the totals diverge. π-greedy rests every turn and pockets only crumbs (2⁻⁵, 2⁻⁶, …). π-lookahead climbs, slides, and banks the big 2⁻ᵏ each cycle — far more total reward in the same world. Same environment, different Vπμ.
So a score in one environment is concrete: just the expected total reward, Vπμ. π-lookahead beats π-greedy here. But beating one hand-picked world proves nothing — a chess machine beats everyone at chess and is still an idiot at everything else. The real question is the next part: how do you combine performance across every environment into one honest number?
One environment gives one value Vπμ. Intelligence is “a wide range of environments,” so we'll need to average V over all of them — and the entire difficulty is deciding how much each environment should count. That weighting is Part B's big idea.
π-greedy ≈ the thesis's π-basic / π-2back (maximise next reward, 0.9 of the time). π-lookahead = π-2forward: it maximises r̂ₖ + r̂ₖ₊₁, so it sacrifices the crumb to win the slide. By construction Υ(π-2forward) > Υ(π-2back).
Step 4 · The weighting problem
There are infinitely many environments. How much should each one count?
One environment gives one value. Intelligence is performance across a wide range of them — and Legg takes “wide range” to its limit: the space of all computable environments (any world a program could simulate). There are infinitely many. You can't average them evenly, so some must count more than others. Which?
The answer is the oldest rule in science, made mathematical — Occam's razor: prefer the simplest explanation. We weight each environment by how simple it is. And “simple” gets a precise meaning: the length of the shortest program that produces it — its Kolmogorov complexity K(μ).
Here's the intuition the thesis uses. Picture programs that print infinite sequences:
very simple world
while(1) print("1");
K small → weight HIGH
A few bytes of program. Highly probable, so it carries lots of weight in the sum.
moderate world
print digits of π …
K medium → weight medium
Ten-plus times longer to write, so exponentially less weight than the all-ones world.
complex world
a random-looking stream
K large → weight tiny
No short program. Its weight 2⁻ᴷ is vanishingly small — it barely enters the sum.
Weight by complexity, not by difficulty. A simple world and a complex world can be equally easy to win in — but the simple one is a priori more likely to be the world you're actually in, so an intelligent agent should be rewarded most for handling simple worlds well. That's Occam's razor, applied to environments.
Thesis §2.4 & §4.1: K(μ) = length of the shortest program for μ. The Solomonoff weight 2⁻ᴷ⁽µ⁾ is, up to a constant, the probability a random program produces μ. A C-loop printing “1” forever is near-minimal; printing π needs a program ~10× longer, hence far less prior weight.
Step 5 · The equation (interactive centerpiece)
Add up an agent's reward over every world, weighting simple worlds most. That sum is its intelligence.
You have all the pieces now: a per-environment score Vπμ (Step 3) and a weight 2⁻ᴷ⁽µ⁾ for each environment (Step 4). The universal intelligence of an agent is simply the weighted sum of the two, over every computable environment:
Υ(π)= Σ 2−K(μ)·Vπμ
sum over all computable, reward-summable environments μ · weight 2⁻ᴷ⁽µ⁾ (simplicity) · score Vπμ (performance) · the result Υ is one real number
Read it left to right and it's just bookkeeping: for every possible world, take how well the agent does there, multiply by how much that world counts, and total it all up. Let's build that total by hand for a few agents. Pick one and watch its score accumulate, world by world:
Choose an agent — see how much it scores in each world, weighted by simplicity, summed into one Υ
π-random chosen. It exploits nothing, so it scores near zero almost everywhere — and Υ is tiny.
The simple worlds dominate the sum. Doing well at chess adds almost
nothing to Υ (its weight is ~.002), while handling the all-zeros and slide
worlds — trivially simple, heavily weighted — is what moves the score. An
agent that's brilliant at chess but can't predict 0000… ranks
below a humble agent that can. Numbers are illustrative of the trend.
Generality, not narrow skill, is what gets rewarded. Because simple environments carry the most weight, an agent must solve the easy worlds to score well — exactly the broad adaptability we mean by intelligence.
Definition 4.1.3: Υ(π) := Σ_{μ∈E} 2⁻ᴷ⁽µ⁾ Vπμ = Vπξ, the agent's expected value under the universal distribution ξ. The four worlds shown are a tiny illustrative slice of an infinite sum; weights add to a small fraction of 1.
Step 6 · The ceiling, and the catch
Maximise Υ perfectly and you get AIXI — the optimal agent that can never be run.
The Υ score ranks agents. Natural next question: who sits at the very top? What would an agent look like that maximises this sum?
It would, in every world, weigh how likely that world is given its history and its simplicity prior 2⁻ᴷ⁽µ⁾, then pick the action with the greatest expected future reward. That agent already has a name. It's AIXI (Hutter, 2005) — the dual of the measure. AIXI takes the perfect agent for a known world and swaps the unknown true environment for the simplicity-weighted mixture of all worlds at once. By construction it sits at the top of the Υ scale — the ceiling on the intelligence of any future machine, no matter its hardware.
The weight 2⁻ᴷ⁽µ⁾ depends on Kolmogorov complexity K, and K is not computable — finding the shortest program is undecidable (it runs into the halting problem). So Υ can't be calculated exactly, and AIXI can be defined precisely but never run. The perfect agent is a mathematical North Star, not a program. The gap between AIXI and anything you can build is the entire point.
Thesis §2.10 & §4.2: AIXI = πξ, the optimal agent with the unknown μ replaced by the universal mixture ξ. Ῡ := max_π Υ(π) = Υ(πξ). K(μ) and Solomonoff induction are uncomputable; AIXI can only ever be approximated.
The ladder & the legacy
Does it rank agents the way we'd expect? · the stakes · and can you re-explain it?
Step 7 · The sanity check (and a shock)
It ranks agents the way we'd expect — except for one jolt worth understanding.
A definition is only as good as the orderings it produces. So Legg walks the measure up a ladder of agents and checks: does Υ rank them the way our gut says it should?
Mostly, yes. A dice-rolling π-random scores lowest. A table-keeping π-basic beats it. An agent that remembers more history (π-2back) beats that, and a one-step planner (π-lookahead) beats that — each is a strict generalization of the last, so its V is at least as high in every world. The ordering is clean and matches intuition. Then comes the jolt:
A ranking that surprises — and teaches
π-chess · Deep Blue — world-class, but only at chess
π-simple · only predicts 000… and 111…, nothing else
0000… ranks below one that only handles trivial patterns. Υ(π-chess) < Υ(π-simple).
The measure strongly emphasizes solving simple problems. Narrow
brilliance buys almost nothing; broad competence on the easy, common worlds is
what counts. (Any human chess player can also predict 0000… — only
artificial, hyper-specialized systems fall into this trap.) Viewed this way,
decades of ever-narrower AI actually moved away from general intelligence.
Thesis §4.2: Υ(π-rand) < Υ(π-basic) < Υ(π-2back) < Υ(π-2forward), each a generalization of the last. Yet Υ(π-dblue) < Υ(π-simple): weighting by complexity makes a chess supercomputer rank below a trivial sequence predictor.
Step 8 · Why a definition is worth the trouble
If you can rank intelligence, you can imagine — and worry about — what sits above us.
You've climbed the ladder from a dice-roller to the incomputable ceiling. Now the “so what”: why does pinning down a definition of intelligence matter beyond philosophy?
Because the same construction that defines the ceiling defines a direction to climb. Once intelligence is one scalar that machines can be ranked on, “a machine more intelligent than any human” is no longer hand-waving — it's a point higher on the Υ scale, approached by ever-better approximations of AIXI. Legg leans on I. J. Good's 1965 warning:
“The first ultraintelligent machine is the last invention that man need ever make” — since it could design even better machines, triggering an intelligence explosion. — I. J. Good, 1965, quoted in the thesis
That is the thesis's closing stake. Practical, tractable approximations of these optimal agents could, in principle, set off such an explosion — so the ethics of building minds that exceed our own deserve serious thought now, not after the fact. The title's “super intelligence” is the literal top of the very scale you just built.
A formal measure turns a vague fear into a research target: it says what we'd be building toward, and gives a yardstick for how far along we are. The author would put that conviction into practice — two years later he co-founded DeepMind.
Thesis §7: an intelligence explosion is “not obviously” impossible; if even slightly likely, preparing early is prudent. Legg co-founded DeepMind in 2010, whose stated mission — “solve intelligence” — echoes this thesis directly.
Step 9 · What it claims & recap
A scorecard for the definition — and can you now re-explain it?
You've built the equation, run agents through it, found the ceiling, and met the catch. Final pass: how good a definition is this, and have you actually got it?
Report card · how the measure scores as a definition
Range · agents it can rank
Formal & objective · one equation, no subjectivity
Universal · not tied to humans or culture
as a practical test · can you run it?
Now you can explain it. Five questions — answer each out loud before opening it. If all five come easily, you've genuinely got this thesis.
State the Universal Intelligence Measure in one sentence.
An agent's intelligence is its expected reward summed over every computable environment, with each environment weighted by 2⁻ᴷ⁽µ⁾ — that is, by its simplicity. One number, Υ(π).
Why weight by simplicity (Occam's razor)?
Simpler environments have shorter programs, so they're a priori more likely to be the world you're in. Rewarding agents most for handling simple worlds is just applying Occam's razor to which environments matter.
What is AIXI, and why can't it be run?
AIXI is the agent that maximises Υ — the optimal agent over the simplicity-weighted mixture of all worlds, the top of the scale. It can't be run because the weight depends on Kolmogorov complexity K, which is uncomputable; AIXI can only be approximated.
Why does a chess supercomputer score so low?
Chess is a complex world with a long program, so its weight 2⁻ᴷ is tiny; an agent that only plays chess fails the heavily-weighted simple worlds (like predicting 0000…). Generality, not narrow skill, is what Υ rewards.
Why bother with a definition you can't compute?
A clean, strong definition is worth more than a convenient but flawed test. Like Martin-Löf randomness, you define the ideal precisely and then accept that real measurement only approximates it — better tests are the follow-up work, not a reason to weaken the definition.
What happened next — this thesis put a number on “general intelligence” and an explicit ceiling above us. Two years later its author co-founded DeepMind to chase that ceiling in practice, and the framing — reward, environments, generality, an optimal agent we approximate — became the vocabulary of modern reinforcement learning and AGI safety.
“Intelligence measures an agent's ability to achieve goals in a wide range of environments.” — Shane Legg & Marcus Hutter, the informal definition this thesis formalizes
It gave “intelligence” a definition the field could actually argue with.
You can disagree with the Universal Intelligence Measure — but you have to do it in equations now, not adjectives. By making intelligence a single, falsifiable quantity, the thesis handed a generation of researchers a target, a vocabulary, and a warning.
DeepMind
Two years after the thesis, Legg co-founded DeepMind, whose mission to “solve intelligence” is this formal stance turned into an engineering program.
Agents & AGI
Reward over environments, generality over narrow skill, an optimal agent we can only approximate — this is now the default framing for RL and AGI research.
Safety & the ceiling
The explicit upper bound above human ability, and the intelligence-explosion stake, anticipated the AI-safety conversation now central to frontier labs.
A fuzzy word became one equation — and that equation quietly sits under a decade of frontier AI that followed.