Math, Thoughts

Growing Relations, Agents, and Intelligence

By analyzing a behavior, from its input to its output, using metrics we can write down a timeline mapping of the behavior.

Making a timeline mapping is the dumbest way to analyze a behavior, it’s also its first step.

Refining a relation

Through mathematics, and still beyond it, there exists many ways to compress a timeline mapping of inputs against outputs. This can start with a reduction of the time: as a parameter, a Bayesian causation, a stream for statistical analysis, etc.

Maybe your observed behavior is independent from time? Then, is it deterministic or non-deterministic? Is it symmetrical? Is it bijective? Is it linear?

I mean, really, the space of classification of relations between inputs and outputs is quite large and can become as abstract or specialized as goes the mathematicians imagination. When you first dig into those kind of domains, diversity exists… ad nauseam!

The compressed relation

So that’s cool, we already have a huge library of what functions can be.
Though, in practice, some methods can make parameterized functions give a good approximations; those play on the function parameters (gradient descent, Ziegler-Nichols,…) and the encoding space of the I/O (Fourier transform, Z transform,…).

Outside of the scope of perfect precision mathematics, real world technologies are designed with reasonable approximations over metrics. Expressing a behavior with a well compressed relation is always a success, but it never means we can conclude our relation perfectly fits the observed behavior. Even if precision is good enough, we can never consider that the behavior will always be predicted by the system.

But we do technology, so we’ll make approximations suiting our assumptions, to avoid being stuck in a philosophical idea of perfection.
It means we have to care about both the metrics and the fitting of the applied relation, up to a reasonable limit.

That way, no need to know everything, but there’s a trade for approximation; it’s specialization. So most of us assume it’s fair to consider there exists some generic rules that lead to intelligent behavior: linking general high and abstract point of views (e.g. fairness, love, round, blue, accent, etc) to specific low details point of views (e.g. logic circuit, pattern recognition, graph representation, design pattern, periodicity, causality assumption, arbitrary mapping, etc).

But is there really any simple set of rules?

There comes the Concept

From good-enough approximations of relations based on highly-customed parameters and well described spaces of inputs and outputs, we could theoretically mimic any behavior, or at least bruteforce it. There’s no objective test, meaning expressing measurable expectations, that we cannot put metrics on and figure out an approximate mapping of.

The issue remains; it’s monotask AI, not AGI.
But if this mimicking machine is capable of developing and selecting multiple ways to produce high-level outputs from its low-level inputs, those can start competing with each others, and a context orchestrator will have to apply a selective intelligence to make a conclusion out of the myriade of responding, or not, processes.

That’s the huge work of AGI: generating a proper focus to get the information related to context, and switch action and goals as you switch context.

In the opposite direction, the high-level concept should lead to low-level actions. So the high-level idea should be convertible into a low-level sequence of commands for the motors.

This dense mapping leading to a high-level instance, an integration of low-level sensor inputs and a derivation to low-level motor outputs is called the concept. It contains its input and output domains, its high-level states and its experiences over its processing: local relations (caused by memories), distribution (if statistics required), priors (if Bayes required), latent space (for generation and operations), etc.

The sigma machines

So a concept is this specific entanglement that allows us to grasp a general idea from its details: a character keeps its meaning despite the font, ambient light, inclination, style, color, etc. It stays the same concept.

And, as a concept corresponds to multiple I/O configurations, there might be multiple ways to implement a high-level concept to a low-level I/O.
For instance, I want to match the concept of “blue”, but I don’t have a specific I/O to fill that needs. Though, after observations, I noticed I have a low-level input responding to the concept of “dark blue”, and another  one to the concept of “light blue”.

The simplest turing machine I can think is just an OR operation on those 2 inputs to produce a good approximation of “blue” detection.
If my inputs were less convenient though, that problem could have become much harder to solve, and a conditional mix of relations could have been used. Each of these solutions, that can produce the expected response under approximations, are sigma machines.

Of course, they can vary in precision, resolution, relevancy, border errors, domain distribution, bandwidth, etc. Therefore getting the contextual information right is important to use the correct sigma machine for the job. But if one can be proven the most efficient to encompass a concept, it is the simplest turing machine of the concept.

Could there be a simplest Turing Machine of all the Concepts?

And that’s where I wanted to arrive; if a concept can describe a general view of anything; could there be an unknown concept of all the concepts?

Because, if such a thing exists, one of its sigma machine could lead us to get all the concepts right from the beginning. Some sort of all-knowing algorithm that should lead us to the god-like singularity, right?

Well, to get to the singularity, we need to have at least the concept encompassing all the concepts; a set of rules applied to all those concepts existence. So could we plot a space of concepts regarding something higher than I/O? (a sort of logic applied to each concept)

I thought I could prove easily there’s no such thing as the concept that encompasses the space of concepts, though I cannot prove or disprove this at this point… I guess, I will have to come back later on that topic.

Though, really, if you have suggestions; please leave them in the comments to open the debate; I’d be curious to see if others think they can prove this.

 

And, as a bonus, here’s the sentence that inspired me this article:

The conceptual space is the space containing all the concepts. Therefore, if there exists a concept that is not part of it ; and if the conceptual space would have encountered that element, it’d start by wondering if this element is a concept. As per the premise, it’s already identified as a concept, so it’s already part of the conceptual space. Therefore, the conceptual space encompasses all the concepts.

Math, Thoughts

My Learning grows like Math

I don’t think it’s innate, but I discern a pattern through the way I learn stuff in a unkown environment.
This is not a discrete process but more some fuzzy steps adding to a global context. So here’s the way I get the grip on this unknown environment :

  • When I don’t know, I’m just following orders and establishing rules based on induction and questioning
  • When I start figuring out the rules, I’m aiming for the limits implied by the rules and I observe what constrains are behind
  • When the constraints are established, I observe their reasons by the way I could go around them and overpass the limits
  • Finally, figuring out constraints and limits, I established a structure of the topic, I made the first tools to think on this unknown environment; seeding a knowledge I could grow the now-known environment by finding other ways around in different cases

What it seems to me is; the first two steps are hard to acquire. The two last parts can be iterated to populate an environment. When you know the limits, you get generally stuck. But, adding a hypothesis, there might be a work around. The nature of this assumption makes a specific context of your environment.

I like math, and I could picture it the same way it develops, in a sense.
When you end with something impossible. For instance, you haven’t heard about irrational number and you try to measure the diagonal of a one unit square.

You have a ruler of one unit to make the square, but the diagonal is a bit longer than your ruler. So you decide to put a mark after one unit and break the ruler into 10 pieces. Then you put pieces after your mark and you can fit a little more than 4 of them.
So you break a piece into 10 more pieces (it was a really huge ruler. Like skyscraper tall) and put a piece while still having to break another into 10. You figure out it can go for pretty long before you reach something.

Damn that’s not cool ! My goal was to find the way to measure the diagonal but there’s no limit to this breaking system.
As an engineer (yeaaaah! I finally got the title ffs!), I will have tools to make it (my ruler) but that have a given limit of “breakability”. So I assume I’m making an error.

You broke  your ruler 2 times into 10 pieces. If you had done that for every piece, you would have 10^2=100 of them. Also you calculated there is 1 unit and 4 subunits and 1 subsubunit and some subsubsubunits undefined. That gets a value of 141/100 but it’s not quite acurate.
We’ll give a margin to this value, it will account for the part we haven’t measure due to the precision. This part is less than a subsubunit, so we define a 1/100 error margin.

So I could express my diagonal as 1.41 (+/- 0.01) units. That’s cool and accurate enough for an engineer. Things are defined in a context; it’s somehow a local approximation of larger possibilities. I don’t need more accuracy to make it works.

But the mathematician won’t accept the engineer approximation. He doesn’t work in the real world, with its fuzzy noisiness, but in this thought world made of pure ideas where everything is perfectly defined and is also abstraction. Therefore, he won’t be satisfied with this idea and will try to find a logical way to express it.

You seem now desperate to measure the diagonal of a one unit square (you got some silly hobby though). To get a grip on this idea, you decide to make the context evolve by adding some stuff.
If you use this diagonal with others of the same length to make a new square, you observe its diagonal fitting 2 rulers exactly. That is an interesting property. So you decide to define this length as a square root of 2. By showing you can use your unit back and forth on this property, you opened crazy ideas that’ll lead to years of headaches. Now people will extend this “square root of” idea to crazy extreme like making a square root of 3, or even defining the 2 unit square diagonal as two times the square root of 2. But who really has an exact square root of 2 ruler ?

Well, that’s the idea and the generalisation. It can grow on different context while the rule still apply. Even better as there are some surprising square roots like 9 going back to 3 units, but you can navigate this acquired knowledge with this handy tool.
By adding other ideas, like negative numbers, you end up with new limits. The reason being; length are always positive. If you have a negative, it’s not a length but a distance from some point, and a positive direction have to be given. (leading to imaginary numbers and complex numbers allowing rotation in the plane)
That’s really something you can grow in tremendous way to establish a richer context; the spatial geometry, vectors, matrices and so on.

 

While my point here is reached on the comparison between how to grow math and how I think I approach unknown environment, I just thought of something really interesting !