Basic Concepts: Definition of Randomness

When we talked about predictability and living things, I have noted that we need a more precise definition of randomness before diving into the details of complex living systems.

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.

Definition of randomness

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.