138 7 A Formal Model of Intelligent Agents
7.3.3 Pragmatic General Intelligence
The above concept of biased universal intelligence is perfectly adequate for many purposes, but it is also interesting to explicitly introduce the notion of a goal into the calculation. This allows us to formally capture the notion presented in [Goe93a] of intelligence as “the ability to achieve complex goals in complex environments.”
If the agent is acting in environment jz, and is provided with g, corresponding to g at the start and the end of the time-interval T = {i € (s,...,¢)}, then the expected goal-achievement of the agent, relative to g, during the interval is the expectation
t ViiaT = OD T(Ig,s,i)) 1=s where the expectation is taken over all interaction sequences J, ., drawn according to pp. We then propose
Definition 5 The pragmatic general intelligence of an agent 7, relative to the distribution vy over environments and the distribution y over goals, is its expected performance with respect to goals drawn from y in environments drawn from v, over the time-scales natural to the goals; that is,
HEE GEG,T
(in those cases where this sum is convergent).
This definition formally captures the notion that “intelligence is achieving complex goals in complex environments,” where “complexity” is gauged by the assumed measures v and 4.
If v is taken to be the universal distribution, and + is defined to weight goals according to the universal distribution, then pragmatic general intelligence reduces to universal intelligence.
Furthermore, it is clear that a universal algorithmic agent like ATXT [Hut05] would also have a high pragmatic general intelligence, under fairly broad conditions. As the interaction history grows longer, the pragmatic general intelligence of AIXI would approach the theoretical maximum; as AIXI would implicitly infer the relevant distributions via experience. However, if significant reward discounting is involved, so that near-term rewards are weighted much higher than long-term rewards, then AIXI might compare very unfavorably in pragmatic general intelligence, to other agents designed with prior knowledge of v, y and 7 in mind.
The most interesting case to consider is where v and ¥ are taken to embody some particular bias in a real-world space of environments and goals, and this bias is appropriately reflected in the internal structure of an intelligent agent. Note that an agent needs not lack universal intelligence in order to possess pragmatic general intelligence with respect to some non-universal distribution over goals and environments. However, in general, given limited resources, there may be a tradeoff between universal intelligence and pragmatic intelligence. Which leads to the next point: how to encompass resource limitations into the definition.
One might argue that the definition of Pragmatic General Intelligence is already encompassed by Legg and Hutter’s definition because one may bias the distribution of environments within the latter by considering different Turing machines underlying the Kolmogorov complexity. However this is not a general equivalence because the Solomonoff-Levin measure intrinsically
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