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Universal Intelligence: A Definition of Machine Intelligence

Shane LeggMarcus Hutter

article2007Minds & MachinesSingularity Institute for Artificial Intelligence Prize

Develops a mathematically rigorous definition of intelligence that applies to any agent—biological or artificial—by formalizing the intuition that intelligence means succeeding at a wide variety of tasks weighted by their complexity.

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A fundamental challenge in artificial intelligence is defining and measuring intelligence for machines that may operate very differently from humans. Traditional approaches, like IQ tests for people or the Turing test for computers, often rely on human-like traits such as language or reasoning in familiar contexts. These fall short for diverse systems, from robots to algorithms, especially as technology advances and creates entities with novel capabilities. Without a clear, general definition, it's hard to gauge progress in AI or compare systems objectively, which hinders decisions on investment, development, and risks like unintended behaviors in superintelligent machines.

This paper sets out to create a formal, universal definition of machine intelligence by drawing from established ideas on human intelligence. The authors review psychological theories, tests, and expert definitions to identify core elements—such as adapting to new situations, learning from experience, and achieving goals—and translate them into a mathematical framework suitable for any machine.

The approach starts with a simple model: an agent interacts with an environment by taking actions, receiving observations and rewards, much like how animals learn through trial and feedback. To capture generality, the authors consider all possible computable environments—simulated scenarios drawn from an infinite but describable set—weighted by their simplicity, using a concept from information theory called Kolmogorov complexity (essentially, the shortest program needed to describe something). This favors testing against straightforward yet varied challenges, reflecting the principle that simpler explanations are often best, as seen in human IQ tests like pattern recognition. They then define universal intelligence as an agent's expected success (total rewards) across these environments, without assuming specific hardware, senses, or goals.

The key results include a precise equation for universal intelligence, denoted Υ, which ranks agents logically: a random actor scores near zero, basic learners that track patterns do better, specialized systems like a chess computer falter on unfamiliar tasks, and the theoretical optimal agent (AIXI) achieves the maximum. The paper also surveys other machine intelligence proposals, from Turing test variants to compression benchmarks, and compares them favorably against criteria like generality and objectivity—universal intelligence stands out for its breadth and lack of human bias. Notably, it aligns with proven optimal learning theories, showing that highly intelligent agents excel in prediction, planning, and adaptation across domains.

These findings imply a robust way to think about machine intelligence as the ability to thrive in diverse, unpredictable settings, impacting costs by guiding efficient AI design, reducing risks through better evaluation of adaptability, and informing policy on ethical AI deployment. Unlike narrower tests, this definition avoids cultural or species biases, evolving with technology rather than tying intelligence to human norms. It challenges views that equate intelligence with efficiency or consciousness, focusing instead on measurable performance.

Next steps should involve building practical tests to approximate Υ, such as sampling environments by generating short programs and running agent simulations, then weighting results by program length. Pilot these on existing AI systems to validate rankings against real-world utility. Trade-offs include balancing test scope (more environments mean higher accuracy but longer computation) with feasibility. Further analysis could explore how human cognition fits this scale.

Limitations include the definition's reliance on computable environments, which assumes the universe follows Turing-machine rules (no evidence against this yet, but unproven), and the uncomputability of exact complexity measures, requiring approximations that might introduce errors. Confidence is high in the theoretical foundation—rooted in decades of work on learning and complexity—but lower for immediate applications; readers should be cautious about over-relying on it without empirical validation through tests. Overall, this work provides a foundational tool for assessing AI's potential and progress.

Australian National UniversityCSIRO’s Data61Dalle Molle Institute for Artificial Intelligence Researchformal modelformal semanticsformal specificationuniversal intelligencemachine intelligenceintelligence measure
arXiv: 0712.3329
  • Book: Machine Super Intelligence, Shane Legg. Its formal treatment of universal intelligence, algorithmic probability, and AIXI supplies the theoretical framework this paper distills into a definition and measure.
Cover for Universal Intelligence: A Definition of Machine Intelligence

Abstract

A fundamental problem in artificial intelligence is that nobody really knows what intelligence is. The problem is especially acute when we need to consider artificial systems which are significantly different to humans. In this paper we approach this problem in the following way: We take a number of well known informal definitions of human intelligence that have been given by experts, and extract their essential features. These are then mathematically formalised to produce a general measure of intelligence for arbitrary machines. We believe that this equation formally captures the concept of machine intelligence in the broadest reasonable sense. We then show how this formal definition is related to the theory of universal optimal learning agents. Finally, we survey the many other tests and definitions of intelligence that have been proposed for machines.

Topics

formal modelformal semanticsformal specificationlearning agentscombinatorial optimizationcombinatorial problemsparallel machine

Keywords

universal intelligencemachine intelligenceintelligence measureuniversal optimal learning agentsintelligent agentPsychometric AIgeneral intelligencecrystallized intelligencefluid intelligence

Table of Contents

  • A Definition of Machine Intelligence
  • 2 Natural Intelligence
  • 2.1 Human intelligence tests
  • 2.2 Animal intelligence tests
  • 2.3 Desirable properties of an intelligence test
  • 2.4 Static vs. dynamic tests
  • 2.5 Theories of human intelligence
  • 2.6 Ten definitions of human intelligence
  • 2.7 More definitions of human intelligence
  • 3 A Definition of Machine Intelligence
  • 3.1 Basic agent-environment framework
  • 3.2 Formal agent-environment framework
  • 3.3 A formal definition of machine intelligence
  • 3.4 Universal intelligence of various agents
  • 3.5 Properties of universal intelligence
  • 4 Definitions and Tests of Machine Intelligence
  • 4.1 Informal definitions of machine intelligence
  • 4.2 Formal definitions and tests of machine intelligence
  • 4.3 Comparison of machine intelligence tests and definitions
  • 5 Discussion and Conclusions
  • 5.1 Constructing a test of universal intelligence
  • 5.2 Response to common criticisms
  • 5.3 Conclusion
  • Acknowledgements
  • References

Knowls

  1. Knowl 1 — Universal Intelligence Measure

    definition

    Universal intelligence is a formal mathematical measure of machine intelligence that quantifies an agent's expected performance across all well-defined environments, weighted inversely by their algorithmic complexity according to Occam's razor.

    Let E\mathcal{E} denote the set of all computable, reward-summable environmental measures with respect to a reference prefix universal Turing machine UU. Let K(μ)K(\mu) denote the Kolmogorov complexity of environment μ∈E\mu \in \mathcal{E}, and let VμπV_\mu^\pi denote the expected total reward earned by agent policy π\pi when interacting with environment μ\mu. The universal intelligence Υ(π)\Upsilon(\pi) of agent π\pi is defined as:

    Υ(π):=∑μ∈E2−K(μ)Vμπ\Upsilon(\pi) := \sum_{\mu \in \mathcal{E}} 2^{-K(\mu)} V_\mu^\pi

    In this formulation, the term 2−K(μ)2^{-K(\mu)} represents the algorithmic probability of environment μ\mu, ensuring that simpler environments contribute with greater weight than complex ones while assigning non-zero probability to all computable environments.

  2. Knowl 2 — Formal Agent-Environment Interaction Framework

    model/method

    The agent and environment interact sequentially over discrete time cycles k∈Nk \in \mathbb{N} via structured communication channels:

    1. Action space A\mathcal{A}: A finite set of symbols representing actions emitted by the agent to the environment.
    2. Perception space P=O×R\mathcal{P} = \mathcal{O} \times \mathcal{R}: A finite set of symbols generated by the environment and received by the agent, where O\mathcal{O} is a finite observation space and R⊂[0,1]∩Q\mathcal{R} \subset [0, 1] \cap \mathbb{Q} is the rational reward space.

    At each cycle kk, the environment generates an observation ok∈Oo_k \in \mathcal{O} and a reward rk∈Rr_k \in \mathcal{R}, and the agent responds with an action ak∈Aa_k \in \mathcal{A}. This yields an alternating interaction sequence o1r1a1o2r2a2…o_1 r_1 a_1 o_2 r_2 a_2 \dots.

    • Agent π\pi: A probability distribution over actions conditioned on the complete preceding interaction history, π(ak∣o1r1a1…ok−1rk−1ak−1okrk)\pi(a_k \mid o_1 r_1 a_1 \dots o_{k-1} r_{k-1} a_{k-1} o_k r_k). A deterministic agent assigns probability 1 to a single action.
    • Environment μ\mu: A probability measure over observation-reward pairs conditioned on the past interaction history, μ(okrk∣o1r1a1…ok−1rk−1ak−1)\mu(o_k r_k \mid o_1 r_1 a_1 \dots o_{k-1} r_{k-1} a_{k-1}).

  3. Knowl 3 — Expected Total Reward Value Function for Reward-Summable Environments

    equation

    The expected performance of an agent π\pi interacting with an environment μ\mu is evaluated by the total expected reward accumulated over an infinite lifetime:

    Vμπ:=E[∑i=1∞ri]≤1V_\mu^\pi := \mathbb{E}\left[ \sum_{i=1}^\infty r_i \right] \le 1

    where ri∈[0,1]∩Qr_i \in [0, 1] \cap \mathbb{Q} is the reward received in cycle ii, and the expectation E\mathbb{E} is taken over all possible interaction sequences generated by the joint execution of agent π\pi and environment μ\mu.

    To ensure that VμπV_\mu^\pi is well-defined, finite, and bounded without requiring an external geometric discount factor γ∈(0,1)\gamma \in (0, 1) that would introduce an arbitrary horizon parameter 1/(1−γ)1/(1-\gamma), environments μ\mu are constrained to be reward-summable, meaning the total reward that the environment can ever emit satisfies ∑i=1∞ri≤1\sum_{i=1}^\infty r_i \le 1 along every realization.

  4. Knowl 4 — Algorithmic Probability Prior Over Computable Environments

    model/method

    The space of environments E\mathcal{E} is restricted to all computable environmental probability measures with bounded total reward. Because computable measures are enumerable as μ1,μ2,μ3,…\mu_1, \mu_2, \mu_3, \dots, each environment μi\mu_i can be encoded as a binary string ⟨i⟩\langle i \rangle.

    The Kolmogorov complexity of environment μi\mu_i is defined with respect to a prefix universal Turing machine UU as:

    K(μi):=K(⟨i⟩)=min⁡p{l(p):U(p)=⟨i⟩}K(\mu_i) := K(\langle i \rangle) = \min_p \{ l(p) : U(p) = \langle i \rangle \}

    where l(p)l(p) is the bit length of binary program pp.

    Occam's razor is formalized by assigning to each environment μ\mu an a priori algorithmic probability 2−K(μ)2^{-K(\mu)}. This universal prior guarantees that:

    1. Simple environments (those with short description lengths) receive high probability mass.
    2. Highly complex environments receive lower probability mass.
    3. The distribution is invariant up to a multiplicative constant under changes of the reference universal Turing machine UU.

  5. Knowl 5 — Maximal Intelligence of the Universal AIXI Agent

    theoretical result

    The theoretical upper bound on universal intelligence Υ\Upsilon is achieved by Hutter's universal algorithmic agent πAIXI\pi^{\text{AIXI}}:

    Υˉ:=max⁡πΥ(π)=Υ(πAIXI)\bar{\Upsilon} := \max_\pi \Upsilon(\pi) = \Upsilon(\pi^{\text{AIXI}})

    At each step, πAIXI\pi^{\text{AIXI}} updates the algorithmic posterior over all environments μ∈E\mu \in \mathcal{E} given the observed history and executes the action that maximizes the expected future total reward under the universal mixture distribution. Because πAIXI\pi^{\text{AIXI}} is provably Pareto-optimal and self-optimizing across all ergodic Markov decision processes, prediction problems, classification tasks, and bandit settings, agents with maximal universal intelligence are provably optimal universal learners.

  6. Knowl 6 — Hierarchy and Separation of Agent Architectures Under Universal Intelligence

    theoretical result

    Universal intelligence induces a strict, intuitive ordering over standard classes of learning and non-learning agents:

    Υ(πrand)<Υ(πbasic)<Υ(π2back)<Υ(π2forward)≤⋯≤Υ(πAIXI)\Upsilon(\pi^{\text{rand}}) < \Upsilon(\pi^{\text{basic}}) < \Upsilon(\pi^{\text{2back}}) < \Upsilon(\pi^{\text{2forward}}) \le \dots \le \Upsilon(\pi^{\text{AIXI}})

    where:

    • πrand\pi^{\text{rand}} chooses actions uniformly at random.
    • πbasic\pi^{\text{basic}} maintains empirical statistics over single observation-action pairs.
    • π2back\pi^{\text{2back}} conditions reward predictions on histories of length 2, allowing it to adapt to temporal patterns.
    • π2forward\pi^{\text{2forward}} plans two steps ahead to maximize estimated multi-step cumulative reward, resolving delayed gratification tasks.

    In addition, narrow expert systems (such as chess computers πdblue\pi^{\text{dblue}}) obtain very low universal intelligence Υ(πdblue)\Upsilon(\pi^{\text{dblue}}). While they achieve high value VμchessπdblueV_{\mu_{\text{chess}}}^{\pi^{\text{dblue}}} in a specific complex environment μchess\mu_{\text{chess}}, its weight 2−K(μchess)2^{-K(\mu_{\text{chess}})} is small, and the agent fails to extract rewards across the vast majority of other simple and structured environments μ≠μchess\mu \ne \mu_{\text{chess}}, resulting in Υ(πbasic)>Υ(πdblue)\Upsilon(\pi^{\text{basic}}) > \Upsilon(\pi^{\text{dblue}}).

  7. Knowl 7 — Inapplicability of the No-Free-Lunch Theorem to Universal Intelligence

    theoretical result

    Wolpert and Macready's No-Free-Lunch (NFL) theorems do not limit or invalidate universal intelligence.

    The NFL theorems establish that all optimization or learning algorithms have identical average performance when evaluated over a uniform distribution across all possible problem instances. In universal intelligence, the distribution over environments is explicitly non-uniform: environments are weighted by their algorithmic probability 2−K(μ)2^{-K(\mu)}. Because simpler environments are exponentially favored and structured regularities dominate the prior probability mass, universal learning agents can strictly outperform random or narrow agents on average.

  8. Knowl 8 — Comparative Evaluation of Machine Intelligence Tests and Definitions

    data/table

    The proposed universal intelligence measure is evaluated alongside ten existing tests and definitions of machine intelligence against twelve criteria:

    Intelligence Test / Definition Valid Informative Wide Range General Dynamic Unbiased Fundamental Formal Objective Fully Defined Universal Practical
    Turing Test debatable no no no yes no no no no yes no debatable
    Total Turing Test debatable no no no yes no no no no yes no no
    Inverted Turing Test debatable debatable no no yes no no no no yes no debatable
    Toddler Turing Test debatable no no no yes no no no no no no debatable
    Linguistic Complexity debatable yes debatable no no no no yes yes no no yes
    Text Compression Test debatable yes debatable no no debatable debatable yes yes yes no yes
    Turing Ratio debatable yes debatable debatable ? ? ? ? ? no ? ?
    Psychometric AI debatable yes debatable debatable ? debatable no yes yes yes no debatable
    Smith's Test debatable yes debatable debatable no ? no ? yes no ? yes
    C-Test debatable yes debatable debatable no yes yes yes yes yes yes no
    Universal Intelligence yes yes yes yes yes yes yes yes yes yes yes no

    Universal intelligence satisfies all formal, theoretical, and conceptual requirements (validity, informativeness, wide range, generality, dynamic adaptation, unbiasedness, fundamentality, formality, objectivity, full definition, universality) while lacking only direct practicality due to incomputability.

  9. Knowl 9 — Incomputability of Universal Intelligence and Approximations via Time-Bounded Complexity

    limitation

    Universal intelligence Υ(π)\Upsilon(\pi) is theoretically uncomputable due to the incomputability of the Kolmogorov complexity function K(μ)K(\mu) and the infinite summation over all computable environments E\mathcal{E}.

    To derive a workable, practical intelligence test from the theoretical definition, approximations must replace K(μ)K(\mu) with computable time-bounded simplicity priors, such as:

    1. Levin's KtKt complexity: Defines complexity by jointly penalizing description length and execution time.
    2. Schmidhuber's Speed Prior: Weights environments based on both code length and the computational time required to generate outputs.
    3. Monte Carlo sampling: Approximates the infinite series by sampling a finite set of simulated environments from the computable prior.

  10. Knowl 10 — Functionalist Behavioral Stance on Machine Intelligence

    assumption

    The universal intelligence framework adopts an explicitly functionalist, black-box behavioral perspective: intelligence is defined entirely by an agent's external input-output interactions (actions chosen given histories of observations and rewards) across the spectrum of environments, without regard to internal architecture, internal data structures, or computational efficiency.

    Under this premise:

    1. Block's 'Blockhead' lookup-table argument: If an agent utilizes an enormous precomputed lookup table to achieve high reward across a vast variety of computable environments, it is considered intelligent because it achieves the goal of general performance across environments.
    2. Searle's 'Chinese Room' argument: Internal 'understanding' or consciousness is relevant if and only if it produces a measurable impact on an agent's performance in well-defined environments. If there is a measurable difference, universal intelligence measures it; if there is no measurable difference, the internal property is deemed scientifically inconsequential for the definition of intelligence.

Coverage note — Omitted the introductory historical overview of human IQ tests (Binet, Wechsler, Raven), animal intelligence testing, and specific toy problem step-by-step arithmetic (the Two-Coins game and slide example) as these serve solely as background motivation and pedagogical illustrations for the formal agent-environment framework.

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Citation

MLA
Legg, S., and M. Hutter. “Universal Intelligence: A Definition of Machine Intelligence”. Minds & Machines, 17:4 (2007) Pages 391-444, 2007, http://arxiv.org/abs/0712.3329v1.
APA
Legg, S., & Hutter, M. (2007). Universal Intelligence: A Definition of Machine Intelligence. Minds & Machines, 17:4 (2007) Pages 391-444. http://arxiv.org/abs/0712.3329v1
Chicago
Legg, S., and M. Hutter. 2007. “Universal Intelligence: A Definition of Machine Intelligence”. Minds & Machines, 17:4 (2007) Pages 391-444. http://arxiv.org/abs/0712.3329v1.
Harvard
Legg, S. and Hutter, M. (2007) “Universal Intelligence: A Definition of Machine Intelligence”, Minds & Machines, 17:4 (2007) pages 391-444 [Preprint]. Available at: http://arxiv.org/abs/0712.3329v1.
Vancouver
1. Legg S, Hutter M (2007) Universal Intelligence: A Definition of Machine Intelligence. Minds & Machines, 17:4 (2007) pages 391-444

BibTeX

@article{legg2007universal,
  title = {Universal Intelligence: A Definition of Machine Intelligence},
  author = {Legg, Shane and Hutter, Marcus},
  year = {2007},
  journal = {Minds & Machines, 17:4 (2007) pages 391-444},
  url = {http://arxiv.org/abs/0712.3329v1},
  eprint = {0712.3329}
}
Metadata:arXiv

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