Universal Intelligence: A Definition of Machine Intelligence

Shane Legg; Marcus Hutter

Source-linked AI summary

Universal Intelligence: A Definition of Machine Intelligence

Shane Legg, Marcus Hutter

arXiv:0712.3329v1Dec 20, 2007cs.AI

TL;DR

Machine intelligence lacks a broadly applicable, absolute measure that avoids human-centered and relative testing problems. The paper formalises expert definitions into a general measure, relates it to universal optimal learning agents, and argues that practical tests estimating universal intelligence may be possible.

Problem
Existing intelligence tests can be relative, human-centered, or biased across systems with different perceptual and cognitive capacities, motivating an absolute machine measure.
Method
The paper extracts essential features from expert definitions, formalises them mathematically, and uses universal discounting to avoid a globally fixed effective horizon.
Results
The authors conclude that their mathematical definition captures machine intelligence broadly and relate it to universal agents whose optimality extends across many problem classes.
Takeaways & Limitations
Alternative tractable complexity measures could support practical tests that estimate universal intelligence.
Takeaways & Limitations
The formal framework must avoid discounting schemes that impose a fixed horizon or leave intelligence dependent on a free weighting parameter.

Abstract

from arXiv · show

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.

1 Introduction

The introduction frames intelligence as difficult to define and measure for arbitrary machines because systems may differ radically from humans in capabilities and environments. The paper therefore extracts general features from human-intelligence definitions, formalises them for machines, relates the measure to AIXI, and surveys alternative tests and definitions.

Motivation: Intelligence is intuitive in everyday life but difficult to define precisely, despite broad scientific consensus about human intelligence and its measurement.Standard intelligence tests are described as measuring approximately the same stable, individually varying construct when correctly applied and interpreted.
Motivation: Machine intelligence is harder to characterise because machines can differ profoundly from humans in physical form, sensors, actuators, communication, processing, and environments.The intended scope includes humans, animals, robots, chat-bots, expert systems, classification systems, and prediction algorithms.
Motivation: Human-oriented tasks such as memory, numerical computation, and chess may be trivial or uninformative for machines, so intelligence tests must avoid anthropocentric assumptions.The introduction argues that a general definition should not depend on particular senses, environments, goals, hardware, or time-specific principles.
Paper approach: The paper extracts broadly applicable ideas from established psychological theories and tests, mathematically formalises them as a machine-intelligence measure, and compares proposed machine tests and definitions.The paper’s stated structure includes an overview of psychological work, formal development, and a concluding survey comparison.
Paper approach: The work originates from Hutter’s universal optimal learning agent AIXI, whose agent-order relation motivates the universal intelligence measure.AIXI is described as provably optimal and maximal under that order relation; the measure is presented as a derivative of it.

2 Natural Intelligence

Natural-intelligence research moved from task-based tests toward broader accounts of intelligence that address cultural bias, changing reference groups, and the distinction between general and multiple abilities. The paper synthesizes these concerns into an abstract definition: intelligence is an agent’s ability to achieve goals across a wide range of environments.

Intelligence tests: Binet’s 1905 test assessed intelligence through 30 short tasks involving relatively complex, everyday problems rather than simple sensory or reaction-time measures.Earlier Galton tests emphasized reaction times, auditory discrimination, physical coordination, and related abilities.
Intelligence tests: Stanford-Binet and Wechsler testing combined verbal and non-verbal assessment, with Wechsler also advocating performance profiles across different abilities.Modern versions share a structure spanning multiple verbal and non-verbal areas, including reasoning, knowledge, quantitative reasoning, visual-spatial processing, and working memory.
Limits of human measures: IQ is relative to a reference group, making it problematic for machines whose performance may differ by many orders of magnitude and continually change with technological progress.The paper therefore considers an absolute measure more meaningful for machines.
Testing requirements: Intelligence tests should be repeatable and minimize cultural bias, including through multiple forms of assessment such as verbal and visual tests.Longer tests can reduce statistical variability, although they cost more to administer.
Theories of intelligence: Theories divide between intelligence as multiple components and intelligence as a general ability underlying positively correlated mental abilities.Thurstone proposed seven primary mental abilities, while Spearman emphasized a general mental ability.
General definition: The paper defines intelligence as an agent’s ability to achieve goals in a wide range of environments, treating it as the effect of capacities rather than any specific capacity set.This definition is broader than knowledge alone because information must be used effectively for a purpose, and it can include capacities humans may not have.

3 A Definition of Machine Intelligence · 3.1 Basic agent-environment framework

The paper formalizes intelligence as an agent’s ability to achieve goals across diverse environments. Its basic framework models agents and environments exchanging actions, observations, and rewards, with reward maximizing providing flexible goals.

3 A Definition of Machine Intelligence: The section formalizes the informal definition of intelligence mathematically for machines and examines properties of the resulting definition.
3.1 Basic agent-environment framework: Intelligence measures an agent’s ability to achieve goals in a wide range of environments.
3.1 Basic agent-environment framework: The framework has three essential components: an agent, environments, and goals.
3.1 Basic agent-environment framework: Agents and environments must interact through signals sent and received by both sides.
3.1 Basic agent-environment framework: Goals are necessary because intelligence without an objective would be unobservable and of no practical consequence.
3.1 Basic agent-environment framework: Reward signals let agents maximize their current situation while avoiding a fixed, single built-in goal.The environment signals how good the agent’s current situation is through reward.
3.1 Basic agent-environment framework: Agents must learn environmental structure and which actions produce reward in order to succeed.Rewards can encode flexible goals such as winning a game or solving a puzzle.
3.1 Basic agent-environment framework: The complete system treats reward as part of perception alongside observations, with the environment implicitly defining goals through reward generation.This setup is the framework used in reinforcement learning and also describes controller–plant systems in control theory.

3.1 Example. · 3.2 Formal agent-environment framework · 3.2 Formal agent-environment framework

The paper illustrates intelligence through an agent-environment coin-guessing game, then formalizes agents, environments, actions, perceptions, rewards, and interaction histories. The framework is deliberately general and does not constrain how agents or environments operate.

3.1 Example.: In the Two Coins Game, the player is the agent, coin system is the environment, guesses are actions, settled coins are observations, and rewards are 0 or 1.The agent guesses 0, 1, or 2 heads on each iteration.
3.2 Formal agent-environment framework: For unbiased coins, always guessing 1 is optimal; with significant bias, an intelligent agent can detect the bias and switch to guessing 0 or 2.The strategy depends on which outcome becomes more likely under the bias.
3.2 Formal agent-environment framework: Games, challenges, problems, and tests can generally be expressed in the agent-environment framework, which specifies roles without describing internal mechanisms.The framework is intended as a role-based description rather than an implementation theory.
3.2 Formal agent-environment framework: The paper formalizes agent-environment interaction together with the other components of its informal intelligence definition.This formalization begins with the interaction between agent and environment.
3.2 Formal agent-environment framework: The agent sends actions from finite action space A, while the environment sends perceptions from finite perception space P and rewards from R ⊆ [0, 1] ∩ Q.Rewards are restricted to the rational unit interval.
3.2 Formal agent-environment framework: Agent and environment alternate transmissions, beginning with the environment, producing histories such as o1r1a1o2r2a2o3r3a3.Actions, observations, and rewards are indexed by their order of occurrence.
3.2 Formal agent-environment framework: The environment µ assigns probabilities to each next observation-reward pair conditional on the preceding interaction history, with no further restrictions initially.In the Two Coins Game, P := {0, 1, 2}×{0, 1} and A := {0, 1, 2} represent outcomes, rewards, and guesses for fair coins.

3.2 Example.

The example contrasts a simple fixed strategy with an adaptive agent and shows that evaluating success depends on how near-term and distant rewards are weighted. Geometric discounting formalizes this tradeoff by reducing future rewards according to γ.

Example: The simple agent always guesses that one coin toss will result in heads, while a more complex agent can count heads and adapt if the coins appear biased.The adaptive strategy changes when observed outcomes suggest the coins are sufficiently biased.
Measure of success: An agent’s success cannot be judged without specifying how much near-future reward matters relative to more distant reward.One agent may achieve high rewards quickly, whereas another may explore first and reach higher eventual performance.
Measure of success: γ = 0.95 discounts a reward of 0.7 by 10 time steps to approximately 0.42 and by 100 time steps to just over 0.004.Rewards decay geometrically into the future, and increasing γ toward 1 places greater weight on long-term rewards.

3.2 Formal agent-environment framework

The framework evaluates agents through expected future rewards over agent–environment interactions, while addressing how discounting should balance normalization and temporal preference. It seeks broad but effective universality by constraining reward sequences and requiring computable environmental measures.

Future value: Expected future value sums discounted rewards into the infinite future and takes the expectation over all interaction sequences between agent π and environment µ.The reward in cycle i is r_i, γ is the discount rate, γ^i discounts the ith future reward, and Γ := P∞ is the normalizing constant.
Discounting: Geometric discounting gives limited farsightedness: γ = 0.95 plans about 20 cycles ahead, while every finite horizon excludes tasks requiring longer planning.A horizon of 5 suffices for tic-tac-toe but is insufficient for chess, so geometric discounting leaves temporal weighting as a parameter.
Discounting: Near-harmonic discounting replaces γ^i with 1/i^2, making the agent’s forward planning proportional to its current age without relying on a fixed geometric horizon.This is presented as a candidate for universal discounting because people of age k years may plan roughly the next k years.
Reward structure: An alternative removes external discount parameters by requiring rewards to incorporate temporal preference while ensuring their total sum remains bounded.This condition replaces the prior requirement that each reward lie in [0, 1]∩Q with an additional bounded-sum constraint on reward signals.
Space of environments: To keep the environment space broad yet make intelligence effectively measurable, the framework requires environmental probability measures to be computable.The resulting class remains infinite and permits unbounded maximal complexity, while only the measure describing the environment must be computable.

3.3 A formal definition of machine intelligence · 3.3 A formal definition of machine intelligence

The formal definition must combine an agent’s performance across infinitely many environments by weighting environments rather than assigning them a uniform distribution. Because observations leave many hypotheses consistent with the data, the agent must prefer among them using Occam’s razor, selecting simpler explanations as the rational and intelligent choice.

3.3 A formal definition of machine intelligence: An overall performance measure must combine an agent’s performance across many different environments.The paper identifies aggregation across environments as necessary for defining a single measure.
3.3 A formal definition of machine intelligence: Because there are infinitely many environments, they cannot receive a uniform distribution and must instead be weighted unequally.The passage poses the unresolved question of how those weights should be assigned.
3.3 A formal definition of machine intelligence: The true environment is described by an unknown probability measure, while the agent has only past observations available.The agent uses those observations to construct probability measures consistent with its experience.
3.3 A formal definition of machine intelligence: The agent constructs hypotheses as potential explanations of the true environment that are consistent with its observations.The passage states that the set of hypotheses changes as the number of observations increases.
3.3 A formal definition of machine intelligence: A given situation can leave a large number of hypotheses consistent with the current observations.Observational information alone therefore does not uniquely identify the correct hypothesis.
3.3 A formal definition of machine intelligence: To predict which hypotheses are most likely correct, the agent must use information beyond its observations.The passage frames this as a recurring problem in inductive inference.
3.3 A formal definition of machine intelligence: Occam’s razor recommends preferring the simplest hypothesis among multiple hypotheses consistent with the data.This principle is presented as the standard response to the inductive-inference problem.
3.3 A formal definition of machine intelligence: The paper characterizes applying Occam’s razor as rational and intelligent, noting that IQ tests often implicitly assess this ability.The IQ-test connection is attributed to the cited discussion in Subsection 2.1.

3.3 Example. · 3.4 Example. · 3.5 Example.

The examples distinguish intelligence from mere success: intelligent agents choose actions expected to maximize success, using prior assumptions such as Occam’s razor when experience is insufficient. They also characterize problem difficulty by the complexity of the simplest good solution, motivating Kolmogorov complexity as a measure of environmental structure.

3.3 Example.: The sequence 2, 4, 6, 8 illustrates that intelligent prediction identifies the simple rule 2k and predicts 10, despite multiple polynomials fitting the data.The example motivates preferring simpler hypotheses when extrapolating from limited observations.
3.3 Example.: Occam’s razor can be a more important demonstration of intelligence than achieving a successful outcome.The paper introduces a game in which the apparently successful answer may not be the intelligent choice.
3.4 Example.: By the 13th round, answering “Yes” is the most intelligent choice given the available knowledge, even though answering “No” can produce a successful outcome through dumb luck.The example contrasts selecting the action expected to succeed with accidentally obtaining a better result.
3.4 Example.: Intelligence concerns taking actions expected to be most likely to lead to success, rather than succeeding in every individual situation.Adequate experience may support this judgment, but agents often need prior assumptions about the world.
3.4 Example.: Tests should, at least on average, reward correctly applying Occam’s razor even when that application sometimes leads to failure.This criterion evaluates the quality of the agent’s decision process rather than isolated outcomes.
3.4 Example.: Environmental complexity does not by itself determine difficulty: a highly complex environment can be easy when every policy is optimal because rewards are always maximal.The example separates complexity of observations and relationships from the difficulty of achieving reward.
3.5 Example.: A problem is difficult when its simplest good solution is complex, whereas easy problems have simple solutions.This provides a solution-based account of environmental difficulty.
3.5 Example.: Kolmogorov complexity measures a binary string by the length of its shortest program on a prefix universal Turing machine, assigning low complexity to simple patterns such as a trillion zeros.The definition uses program length l(p) and a reference machine U; long irregular random strings are contrasted with structured strings.

3.3 A formal definition of machine intelligence

The paper defines universal intelligence as an agent’s expected performance across all computable reward-summable environments, weighted by their Kolmogorov complexity. The definition is general over communication channels and imposes no restrictions on an agent’s internal workings.

Formal definition: Universal intelligence is the expected performance of agent π under the universal distribution 2^-K(µ) across the environment space E.E comprises all computable reward-summable environmental measures relative to reference machine U, and K is Kolmogorov complexity.
Interpretation: The equation represents an agent’s ability to achieve goals through its value function V^π_µ across a wide range of environments.The environments are characterized as computable measures in E, with goals implicit in the environment.
Interpretation: The weighting term 2^-K(µ) formalizes Occam’s razor by weighting performance inversely with environment complexity.Simpler environments receive greater weight than more complex environments.
Generality: The definition applies to arbitrary systems because agent–environment information exchange uses very general communication channels and the agent’s internal workings are unrestricted.Its generality covers different possible sensors and actuators.

3.4 Universal intelligence of various agents

The section compares universal intelligence across random, highly specialised, and simple general-learning agents. Random and specialised agents receive low universal-intelligence assessments, while the simple learning agent exceeds the random baseline.

Agent comparisons: Universal intelligence is illustrated by comparing a range of agents and their relative degrees of universal intelligence.The subsection uses contrasting agent types to build intuition for the definition.
A random agent: A uniformly random agent, πrand, typically achieves low values across environments because it fails to exploit environmental regularities.Some environments may yield high reward regardless of action, but its value is generally low compared with other agents.
A very specialised agent: Deep Blue can perform extremely well in chess yet have very low universal intelligence because it is inflexible and narrow.Its high chess performance is offset by low performance outside chess, where it only plays chess.
A general but simple agent: The simple learning agent πbasic exceeds the random agent’s universal-intelligence value by selecting high-reward actions with 90% probability and random actions with 10%.It builds observation–action statistics and is described as more intelligent than πrand despite being simplistic.

3.4 Universal intelligence of various agents

The examples show that universal intelligence increases when agents exploit longer temporal histories, plan ahead, and adapt across a wider range of environments. More flexible and powerful agents therefore receive higher Υ, although comparisons become difficult for complex agents.

Comparing agents: πdblue fails to exploit trivial regularities that πbasic can use, so Υ(πbasic) > Υ(πdblue).This favors agents that exploit basic regularities across many environments over specialised machines limited to narrow domains.
Longer history: πbasic cannot correlate its current action with its previous action, so it misses the optimal alternating strategy in µalt.The environment rewards switching actions and gives zero reward when the same action is repeated.
Longer history: π2back exploits µalt and generalises πbasic, yielding Υ(π2back) > Υ(πbasic).Extending the history used in the internal table enables adaptation to longer temporal structures and produces still higher machine intelligence.
Forward planning: π2forward improves on π2back by evaluating the next two rewards, allowing it to climb the slide for greater future reward.The agent sacrifices the immediate resting reward of 2^−k−4 to obtain 2^−k while sliding down in the following cycle.
Comparing agents: Increasing agent complexity and adaptability generally raises universal intelligence, though proving comparisons becomes difficult for complex agents.The simple examples establish that more flexible and powerful agents have higher universal intelligence.

3.5 Properties of universal intelligence

This section presents AIXI as an elegant theoretical model of perfect artificial intelligence, while noting that its incomputability and computational infeasibility prevent practical implementation. It also questions how much universal machine intelligence humans possess because human cognition is specialized for sensory processing.

AIXI: AIXI is incomputable because K is incomputable, and computing the expectations needed to maximize future expected rewards would be practically infeasible even if K were computable.Despite these limitations, AIXI remains theoretically significant.
AIXI: AIXI defines an elegant theoretical conception of perfect artificial intelligence and has been shown to possess strong optimality properties.The passage states that one such result is that AIXI converges to “opti,” but the supplied text is truncated.
Humans: Human universal machine intelligence may be limited because much of the human brain is specialized to process information from human sense organs, making performance on complex environments difficult to assess.In extremely simple environments, humans should be able to identify structure and exploit it to maximize reward.

3.5 Properties of universal intelligence

Universal intelligence is presented as a broad formal definition whose validity, meaningfulness, informativeness, and range follow from its formalisation and agent-ordering properties, although its value is not computable and practical approximation remains future work.

Limitations: Υ is not computable because the definition uses Kolmogorov complexity, so practical tests must approximate the ideal and may introduce weaknesses absent from the definition.The definition evaluates general performance across all computable environments with bounded reward sum, whereas a practical test can only cover a limited set.
Valid: Universal intelligence is a valid formal definition because it formalises a reasonable mainstream account of intelligence through a step-by-step argument.Its validity depends on the reasonableness of the informal definition and the soundness of the formalisation.
Meaningful: Universal intelligence is meaningful because it naturally orders simple agents by power and adaptability, while high Υ implies strong performance in most simple and moderately complex environments.Such agents would be powerful and flexible technologies with many potential uses.
Wide range: Universal intelligence spans basic agents such as πrand, πbasic, π2back, and π2forward through AIXI, whose Υ value is maximal.The paper presents this as the widest possible range for a machine-intelligence measure.

3.5 Properties of universal intelligence

Universal intelligence is presented as a broad, formal, objective, non-anthropocentric measure grounded in computation, information, and complexity. Its current form is theoretically useful but not directly testable because Kolmogorov complexity is uncomputable, motivating approximation through sampled environments.

General: Universal intelligence aggregates an agent’s performance across all well-defined environments, making a broader computable performance metric difficult to conceive.A broader measure would appear to contradict the Church-Turing thesis by measuring performance outside computable measures.
Unbiased: The measure is based on universal Turing computation and weights environments according to their Kolmogorov complexity rather than culture-specific problem choices.The passage motivates computation as a general, principled basis because proposed computational models have equivalent expressive power.
Limitations: The environmental distribution 2^-K(µ) is invariant up to a multiplicative constant under reference-machine changes, but relative agent intelligence can still change.Limiting the reference machine’s complexity is proposed as one possible mitigation.
Properties: Universal intelligence is formal, objective, universal, and grounded in computation, information, and complexity.These foundations are presented as fundamental concepts, while the mathematical equation leaves little room for definitional ambiguity and avoids subjective or anthropocentric criteria.
Practical: Universal intelligence cannot currently be directly converted into an intelligence test because Kolmogorov complexity is not computable.In pure form, it can instead analyze intelligence’s nature and theoretically examine mathematically defined learning algorithms.
Practical: A workable approximation could randomly generate programs describing environmental probability measures and test agent performance across sampled environments.This follows from viewing universal intelligence as a weighted sum of performance over all environments.

4 Definitions and Tests of Machine Intelligence

This section surveys informal and formal definitions and tests of machine intelligence, treating them as a spectrum rather than strictly separate categories. It highlights diverse goal-, environment-, and performance-based perspectives while examining limitations of established tests and definitional restrictions.

Definitions and Tests: The survey treats formal definitions and tests as a spectrum because proposals often combine both roles.The authors explicitly avoid artificially dividing proposals into tests or formal definitions.
Survey Scope: The authors present the section as a general survey intended to broaden attention beyond the Turing test.They describe it as, to their knowledge, the only general survey of machine-intelligence tests and definitions, noting that textbooks usually say little beyond mentioning the Turing test.
Informal Definitions: Informal definitions commonly characterize intelligence through successful goal achievement, adaptive behavior, broad task performance, or action in uncertain environments.The sampled definitions connect intelligence with sustaining successful life, achieving goals, adapting across environments, and maximizing success under uncertainty.
Definitional Restrictions: The authors reject making resource limits or membership in P part of intelligence’s definition, because future computational breakthroughs should not make more capable agents less intelligent.They regard resource additions as superfluous or wrong and similarly question restricting intelligence to problems in P rather than allowing possible solutions in NP.
Tests of Intelligence: The Turing test is criticized as neither sufficient nor necessary for intelligence and as currently offering limited practical guidance.Criticisms include logically possible answer-table systems, its emphasis on human knowledge and behavior, the distance of current systems from unrestricted passage, and unreliable classifications.
Limitations of Existing Tests: Human-oriented tests and compression tests may be inadequate for machines because they are anthropocentric or do not clearly establish general-purpose intelligence.The authors argue that standard human tests assume human-like subjects, while compression may not transfer to general intelligence and elementary child knowledge may not compress abstract adult knowledge effectively.

5 Discussion and Conclusions

The discussion defends universal intelligence as a precise, broad measure while identifying computability, testing, and philosophical objections as challenges for future work. It argues that the definition supports more rigorous scientific debate about machine intelligence.

Future work: The proposed next step is a workable test that evaluates agents across many simulated environments and combines performance into an overall intelligence value.The test structure follows directly from the equation for Υ.
Future work: A central theoretical challenge is replacing incomputable Kolmogorov complexity with computable measures incorporating description length and running time.The passage identifies Levin’s Kt complexity and Schmidhuber’s Speed prior as possible replacements.
Responses to criticism: The authors defend Υ by linking it to expert-based informal definitions, demonstrating consistency with properties of intelligence, and relating it to existing ideas about artificial intelligence.They present these as three main responses to criticisms, with the passage specifying the first two.
Responses to criticism: Υ evaluates intelligence by successful operation across a wide range of environments, regardless of whether performance comes from an efficient algorithm or an impractically large lookup table.This directly addresses objections based on lookup-table machines and emphasizes measurable problem-solving ability.
Theoretical limits: The standard No Free Lunch theorem does not apply because universal intelligence does not assume a uniform distribution over environments, though a broader limitation theorem remains conceivable.The authors report that no such broader result is currently known and that it would apply only to computable agents.
Conclusions: The paper generalizes and formalizes expert-inspired intelligence definitions to provide a direct alternative to imitation-based tests and promote greater scientific rigor.The authors emphasize that precise mathematical formulation makes criticisms clearer and more precise.