The concept of randomness seems obvious at first glance. Random and Randomness are so popular in our daily talks which we never think about their exact definition.

Let me start with a simple question. Which of the following series of the numbers can be called random:

1  0  0  1  0  0  1  0  0  1  0  0 1  0  0  1  0  0  1  0  0  1  0  0 1  0  0  1  0  0  1  0  0  1  0  0

1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1

1  0  1  1  1  1  0  1  1  0  1  0  0  0  0  1  0  1  1  1  0  1  1  0  0  0  1  0  1  0  0  0  1  1  0

Most of us consider the third-row as a random series of bits. The second-row doesn’t seem a random serie and the first-row is something in the middle. Not as random as the third-row and not as ordered as the second row. But sure it has a pattern inside. It’s just a ‘1  0  0’ repeated 12 times.

Now let me tell you that all these 3 rows are the result of consecutive tossing a coin. Anyone familiar with the basics of probability knows that if you toss a coin for 36 times all of the above series have exactly the same chance of appearance. Actually the chance is 0.000000000014551915. So they all can be called a random series of numbers somehow.

In theory, it’s understandable but we all feel that there’s a difference between these 3 series. The first and the second series have some pattern, but the third one seems absolutely patternless.

Here we can define two different concepts of randomness: Product Randomness and Process Randomness. As far as I know, this distinction between the product randomness and the process randomness was done by Gregory Chaitin. Although he didn’t use the same names, but sure he distinguishes the two different approaches to the randomness.

So let’s make a brief definition of the both terms:

Product Randomness is an attribute of a series of events with no visible pattern.  So any patternless series of events can be called as a sample of product randomness.

Process Randomness is an attribute of a series of events resulting from a process with two or more products which all have the same probability of happening.

So considering my question at the start of this article, all three rows can be called random if we consider the process definition of randomness, but only the third one can be called random if we consider the product definition of randomness.

Gregory Chaitin has an amazing definition of product randomness: Any series of numbers can be considered as random if there is no shorter way to communicate them with someone else (or with a computer) than copying the whole series of numbers itself.

So in his point of view the second row is not a random row as you can describe it for a computer in this way:

Print ‘1’ for 36 times

Even the first raw is not random because you can write:

Print ‘1  0  0’ for 12 times

But the third one can be considered random as there’s no way to compress it to a shorter message.

But what if our computer understands other commands too? Let’s suppose that our computer understands DecToBin command for converting decimal numbers to binaries.

then all of the three rows are not random anymore (or they are at the equal randomness level):

First row: DecToBin(39268272420)

Second row: DecToBin(68719476735)

Third row: DecToBin(25451802950)

Here Chaitin has a very simple answer: as far you can make it shorter, just do it. when it’s not absolutely possible then you have a random series in hand!

So this is an understandable yet non-measurable definition of product randomness. Anyway, he is a mathematician and for the mathematicians the most important concern is to prove that there exists an answer. To know the actual answer is not the first priority.

But in the real world we need an exactly quantifiable definition. The whole existing world is a series of events. So considering it as a sample of process randomness means that this nice pattern-full world has the same value as any other pattern-less world which could happen.  On the other hand considering the whole world as a result of a process randomness has its own implications and complications.

The most serious challenge in understanding the world is limitation of our brain as it is hardwired to look for pattern and meaning even in pattern-less and meaningless things.

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After you play too much with the spring something extraordinary happens. spring moves away from you with a worm-like movement. The spring pushes himself forward by stretching and contracting its body and hides somewhere behind a book.

Seems it was not a real spring. It was a spring-like living creature! This is a simplified statement of my idea behind the concept of predictability and connection of this concept with being alive.

Taking a look at the literature on mind and free will, you see the most examples and metaphors about predictability  revolves around predictability of physical movements. Same holds true for mine. But soon we will extend it to the other aspects of predictability.

When an object becomes absolutely predictable for us, it’s not considered as a living entity anymore. Physical movements of water are predictable. So it’s not considered a living thing. Movements of a fish are not predictable. So it’s kind of living entity!

The idea of predictability of behavior in living things is easy to grasp yet hard to define. As Dale states four decades ago:

An important feature of the response made to stimuli by living things is that in general the magnitude of the response has no obvious relation to the intensity of the stimulus.

Taking a look at the general literature, seems that without being seriously criticized, we can consider unpredictability and randomness as similar or same concepts. As Patrick Suppes states in his book called Probabilistic Metaphysics:

Phenomena that we cannot predict must be judged random.

In our daily talks, such a definition seems a bit radical. We are not able to predict stock market movement, however, no-one would be happy if you call the market movement as a random walk. Malkiel‘s book Random Walk Down Wall Street, caught the attention of the great stock market players and individual investors, but investors are still asking expensive consultants to help them with selecting an above average portfolio based on past performance of them.

The same story holds true for destiny of a marriage or result of a surgery and many other life incidents.

But taking a second look at the randomness definition shows that we have missed the context of the statement.

Are we trying to predict a single behavior or the pattern of behavior over time? If we flip a coin for 1000 times would be easy to predict that it will land head-side-up for about 500 times. But can you have any prediction with the same certainty if I have just one chance to toss the coin? Sure not.

Therefore, we must clearly define our scope of the study and boundary of the system before making any judgment about predictability.

So, let me state an oversimplified definition of unpredictability at this stage:

An entity can be considered unpredictable if giving the same history of states and unlimited time to experiment, still be impossible to predict the response to the same stimulus  within a predefined tolerance.

This is the place where the real story begins. There are some obvious and many hidden assumptions in the above definition. Not only about scope and the boundary, but also about observer and stimuli. Back to Wittgenstein’s ruler metaphor mentioned before, here’s the challenge. We are unreliable rulers trying to measure the table. But let’s try to do so.

I have continued this topic with a brief look at the concept of randomness.

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