I had this idea while taking a shower and felt that I had to share it. It most likely has flaws, so I would appreciate any feedback at info@solmaz.io. My hunch is that it could be a stepping stone towards something more fundamental.

As the world heads towards Artificial General Intelligence—AGI—people rush to define what it is. Marcus Hutter historically described it as

AI which is able to match or exceed human intelligence in a wide class of environments

(...)

hypothetical agent that can perform virtually all intellectual tasks as well as a typical human could

(see his most recently published book)

whereas OpenAI historically described it as

a highly autonomous system that outperforms humans at most economically valuable work

and more recently, according to a The Information report

an AI system that can generate at least $100 billion in profits for OpenAI

which apparently could be the threshold at which Microsoft loses access to OpenAI models, according to the legal agreement between OpenAI and Microsoft.

Acknowledging all of this and other possible definitions, I want to introduce a definition of AGI that relates to information theory and biology, which I think could make sense:

An AGI is an autonomous system that can generate out-of-distribution (i.e. novel) information, that can survive and spread in the broader environment, at a rate higher than a human can generate.

Here, "survival" can be thought of as mimetic survival, where an idea or invention keeps getting replicated or referenced instead of being deleted or forgotten. Some pieces of information, like blog posts auto-generated for SEO purposes, can quickly vanish, are ephemeral and so recently have started being called "AI slop". Others, such as scientific theories, math proofs, books such as Euclid's Elements, and so on, can persist across millennia because societies find them worth copying, citing, or archiving. They are Lindy.

In that way, it is possible to paraphrase the above definition as "an autonomous system that can generate novel and Lindy information at a rate higher than a human can do".

Like Hutter's definition, the concept of environment is crucial for this definition. Viruses thrive in biological systems because cells and organisms replicate them. Digital viruses exploit computers. Euclid's Elements thrives in a math-loving environment. In every case, the information's persistence depends not just on its content but also on whether its environment considers it worth keeping. This applies to AI outputs as well: if they provide correct or valuable solutions, they tend to be stored and re-used, whereas banal or incorrect results get deleted.

The lifetime of information

Mexican cultural tradition of Día de los Muertos and the anime One Piece have a similar concept on death:

When do you think people die? Is it when a bullet from a pistol pierces their heart? (...) No! It's when they are forgotten by others! (—Dr. Hiriluk, One Piece)

You could call this specific type of death "informational death". A specific information, a bytestream representing an idea, a theory, a proof, a book, a blog post, etc., is "dead" when its every last copy is erased from the universe, or cannot be retrieved in any way. Therefore, it is also possible to call a specific information "alive" when it is still being copied or referenced.

So, how could we formalize the survival of information? The answer is to use survival functions, a concept used in many fields, including biology, epidemiology, and economics.

Let us assume that we have an entity, an AI, that produces a sequence of information $x_1,x_2,\dots ,x_n$. For each piece of information $x_i$ produced by the AI, we define a random lifetime $T_i\ge 0$. $T_i$ is the time until $x_i$ is effectively forgotten, discarded, or overwritten in the environment.

We then describe the survival function as:

$$S_i\left(t\right)=P[T_i>t],$$

the probability that $x_i$ is still alive (stored, referenced, or used) at time $t$. This is independent of how many duplicates appear—we assume that at least one copy is enough to deem it alive.

In real life, survival depends on storage costs, attention spans, and the perceived value of the item. A short-lived text might disappear as soon as nobody refers to it. A revolutionary paper may endure for decades. Mathematical facts might be considered so fundamental that they become permanent fixtures of knowledge. When we speak of an AI that "naturally" produces persistent information, we are observing that correct or notable outputs often survive in their environment without the AI having to optimize explicitly for that outcome.

An expanding universe of information

In our definition above, we mention "out-of-distribution"ness, or novelty of information. This implies the existence of a distribution of information, i.e. a set of information containing all information that has ever been generated up to a certain time. We denote this set of cumulative information as $U$ for "universe", which grows with every new information $x_i$ produced by the AI. Let

$$U_0\text{be the initial "universe" (or data) before any}{x}_i\text{is introduced,}$$

and then

$$U_{i+1}=U_i\cup \left\{x_{i+1}\right\}\text{for}i=1,\dots ,N.$$

In other words, once $x_{i+1}$ is added, it becomes part of the universe. Given an existing state of $U_i$, we can define and calculate a "novelty score" for a new information $x_{i+1}$ relative to $U_i$. If $x_{i+1}$ is basically a duplicate of existing material, its novelty score will be close to zero. If it is genuinely out-of-distribution, it would be large. Therefore, when a novel information $x_{i+1}$ is added to $U$, any future copies of it will be considered in-distribution and not novel. We denote the novelty score of $x_{i+1}$ as $n_{i+1}$.

So how could we calculate this novelty score? One way to calculate it is to use conditional Kolmogorov complexity:

$$n_{i+1}=K\left(x_{i+1}|U_i\right)$$

where

$$K\left(x|U\right)=\underset{p}{min}\{\left|p\right|:M(p,U)=x\}.$$

is the length (in bits) of the shortest program that can generate $x$, when the set $U$ is given as as a free side input, and $M$ is the universal Turing machine.

How does this relate to novelty?

Low novelty: If $x$ can be produced very easily by simply reading or slightly manipulating $U$, then the program $p$ (which transforms $U$ into $x$) is small, making $K(x\mid U)$ and hence the novelty score is low. We would say that $x$ is almost already in $U$, or is obviously derivable from $U$.

High novelty: If $x$ shares no meaningful pattern with $U$, or can't easily be derived from $U$, the program $p$ must be large. In other words, no short set of instructions that references $U$ is enough to produce $x$—it must encode substantial new information not present in $U$. That means $K(x\mid U)$ and hence the novelty score is high.

Informational fitness

We can now combine survival and novelty to formalize our informal definition of AGI-ness above. We integrate the survival function over time to the expected lifetime of information $x_i$:

$$L_i={\int }_0^{\infty }{S}_i\left(t\right)dt=E\left[T_i\right].$$

Therefore, for an entity which generates information $\{x_1,x_2,\dots ,x_n\}$ over its entire service lifetime, we can compute a measure of "informational fitness" by multiplying the novelty score $n_i$ by the expected lifetime $L_i$ over all generated information:

$$\text{IF}=\sum _{i=1}^n{w}_i{L}_i.$$

This quantity tracks the total sum of both how novel each new piece of information an entity generates, and how long it remains in circulation.

My main idea is that a higher Informational Fitness would point to a higher ability to generalize, and hence a higher level of AGI-ness.

Because each subsequent item's novelty is always measured with respect to the updated universe that includes all prior items, any repeated item gets a small or zero novelty score. Thus, it doesn't inflate the overall Informational Fitness measure.

Why worry about novelty at all? My concern came from viruses, which are entities that copy themselves and spread, and therefore could be considered as intelligent if we simply valued how many times an information is copied. But viruses are obviously not intelligent—they mutate randomly and any novelty comes from selection by the environment. Therefore, a virus itself does not have a high IF in this model. However, an AI that can generate many new and successful viruses would indeed have a high IF.

Information's relevance

Tying AGI-ness to survival of information renders the perception of generalization ability highly dependent on the environment, or in other words, state of the art at the time of an AI's evaluation. Human societies (and presumably future AI societies) advance, and the window of what information is worth keeping drifts over time, erasing the information of the past. So whereas an AI of 2030 would have a high IF during the years it is in service, the same system (same architecture, training data, weights) would likely have a lower IF in 3030, due to being "out of date". Sci-fi author qntm has named this "context drift" in his short story about digitalized consciousness.

Comparing AI with humans

Humans perish with an expected lifetime of 80 years, whereas AI is a digital entity that could survive indefinitely. Moreover, if you consider an AI's performance depends on the hardware it runs on, you realize that IF should be derived from the maximum total throughput of all the copies of the AI that are running at a time. Basically, all the information that is generated by that specific version of the AI in the entire universe counts towards its IF.

Given this different nature of AI and humans, how fair would it be to compare a human's informational fitness with an AI's? After all, we cannot digitize and emulate a human's brain with 100% fidelity with our current technology, and a fair comparison would require exactly that. We then quickly realize that we need to make assumptions and use thought experiments, like hypothetically scanning the brain of Albert Einstein (excuse the cliché) and running it at the same bitrate and level of parallelism as e.g. OpenAI's most advanced model at the time. Or we could consider the entire thinking power of the human society as a whole and try to back-of-the-envelope-calculate that from the number of Universities and academics. But given that a lot of these people already use AI assistants, how much of their thinking would be 100% human?

The original OpenAI definition "a highly autonomous system that outperforms humans at most economically valuable work" is a victim of this as well. Humans are using AI now and are becoming more dependent on it, and smarter at the same time. Until we see an AI system that is entirely independent of human input, it will be hard to draw the line in between human and AI intelligence.

Thank you for reading up to this point. I think there might be a point in combining evolutionary biology with information theory. I tried to keep it simple and not include an information's copy-count in the formulation, but it might be a good next step. If you think this post is good or just dumb, you can let me know at info@solmaz.io.