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OfflineRhizoid
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The concept of randomness
    #1702234 - 07/10/03 01:05 PM (14 years, 2 months ago)

This post is a rather long attempt to explain my understanding of randomness. All comments and questions are welcome of course.

What do the concepts random, free will, act of god, and unknown have in common? Answer: they are all labels for various situations where information is indeterministic. Are there any fundamental differences between these various types of indeterminism? I think the answer to that is "no", but before I get to that, let me first talk about randomness.

Randomness is the absence of patterns in data, and since the observation of patterns in time sequences is the foundation of all kinds of prediction, the absence of patterns is the reason why we say that a random event is unpredictable. So let's take a closer look at patterns. For the sake of discussion I'll limit myself to data that consists of sequences of 1's and 0's, like "1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 0 0 0 1 1 1" etc.

If someone gives you a piece of paper with a sequence like "0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 *" where * is some smudge that covers the last digit, you will probably guess that the last digit was "1". Why? Because you noted the obvious pattern in the preceding digits. Your guess could be wrong of course, but if your guess turns out to be correct, then you have made a successful prediction and you will be strengthened in your belief that there is some underlying causal reason why the 1's and 0's always alternate in this particular pattern.

The presence of the pattern makes it possible to write "0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1" as "01 repeated eleven times". The latter is an instruction for how to re-create the sequence. If it's a really long sequence, this can save a lot of space compared to just writing out the 1's and 0's. This is called data compression, and our computers do it all the time. There is a mathematical measure for how much a sequence can be compressed, called Kolmogorov complexity, which is defined as the minimum length of a program that can re-create the sequence. For sequences that have simple patterns the programs are short and the complexity is low, and for more complicated patterns the programs are longer and the complexity is high. In some cases no program exists that is shorter than the sequence itself because the pattern, if there is any, is way too complicated. Such sequences are said to be Kolmogorov-irreducible.

Kolmogorov-irreducibility is the best definition of randomness that mathematics has come up with. But there is a catch: it doesn't specify the capabilities of the computer that runs the program. For example, a computer that has a built-in algorithm for computing pi could compress the sequence "1 0 0 1 0 0 0 0 1 1 1 1 1 1 0 1 1" into the description "the first 17 binary digits of the fractional part of pi". Okay, that description wasn't actually shorter, but with more digits it would be. You get the picture. The important point here is that a computer that doesn't have a built-in algorithm for pi will either have to say that the sequence is irreducible, or it will have to include an algorithm for calculating pi in its description of the sequence. This works if the sequence is long, but if it's short enough then any description will be longer than the sequence itself (despite the presence of a pattern), so it will be irreducible. Seemingly random.

Okay, now apply this to free will. What kind of sequence do we get if a person exercises his free will and chooses a "1" or a "0" by his own free choice for each digit position in the sequence? Is this sequence Kolmogorov-irreducible or not? If it isn't then we have found a pattern, and the likely explanation is that the person either used some conscious rule for the decisions instead of choosing completely freely, or that some unconscious non-free-will activities influenced the results. So the choice was only completely free when there was no bias, and the Kolmogorov-complexity measure will show if this is the case.

When I apply all this to the real world, my conclusions are:

1. We sometimes see patterns in data.
2. Even if no pattern is seen, we can't prove that no pattern is possible.
3. The patterns are a function of both the observer and the observed.
4. We can create completely new patterns by choice.

And using Occam's razor I can't see any reason to say that any of the different types of indeterminism is fundamentally different from the free will case: previously unknown information is revealed, and the reason why it was unknown is either because it was part of an unseen pattern, or because it's completely new, in which case it may be the beginning of a new pattern, depending on what happens in the future.

Coincidences and "acts of God", by the way, are just the other side of the randomness coin. If randomness comes from causality where the patterns are hidden from view, then coincidences are patterns in plain sight but where the causal connection is hidden from view.


Edited by Rhizoid (07/10/03 01:16 PM)


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Offlinepattern
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Re: The concept of randomness [Re: Rhizoid]
    #1702370 - 07/10/03 01:59 PM (14 years, 2 months ago)

Great post!!

Quote:

Rhizoid said:
Okay, now apply this to free will. What kind of sequence do we get if a person exercises his free will and chooses a "1" or a "0" by his own free choice for each digit position in the sequence? Is this sequence Kolmogorov-irreducible or not?
If it isn't then we have found a pattern, and the likely explanation is that the person either used some conscious rule for the decisions instead of choosing completely freely, or that some unconscious non-free-will activities influenced the results. So the choice was only completely free when there was no bias, and the Kolmogorov-complexity measure will show if this is the case.




If all choices must be purely random to be free, then that is a limit on free will. There must also be free will for patterns to choose to continue being patterns. I would like my hands to continue to have five fingers: I am not a mindless slave to this notion, I could choose to chop off a finger, but I don't. Not because there is no randomness to be able to imagine and take that option, but because I continually exercise the same boring nonaction according to my will. Similarly I can choose 0 many times in a row, even if it is predictable, it is still a free choice to be predictable.

Even when a sequence is Kolmogorov-irreducible, there can still be a formula that always shows us what number will come next in the sequence. We have a formula to tell us the nth digit in pi.



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OfflineRhizoid
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Re: The concept of randomness [Re: pattern]
    #1702481 - 07/10/03 02:27 PM (14 years, 2 months ago)

Good point. But I don't consider it to be an exercise of free will when I continuously keep avoiding chopping off a finger. I consider this to be the result of some earlier choice to have five fingers. The continual non-chopping-off that lets me keep five fingers is a simple question of cause and effect that originated in some earlier decision. That's why it's a pattern. In fact, I am a pattern. The question is, does the potential ability to reverse that decision count as free will also?

And yes, there might always be a formula for the next digit in a Kolmogorov-irreducible sequence. The point is that whether it's irreducible or not depends on what kind of computer you use for the calculation. If you use a large enough computer, any sequence will be reducible and non-random, because you could always include the sequence in the computer's own database and give it a short label.


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OfflineMAIA
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Re: The concept of randomness [Re: pattern]
    #1702624 - 07/10/03 03:00 PM (14 years, 2 months ago)

Probability and quantitative methods tells us that even a universe created with random elements have a determinate behavior, the mode and median value are examples. That way we can measure the central tendency of the distribution using the arithmetic mean.

median=L+i[(n/2-F)/f]

where
L is the lower boundary of the median class
i is the width of the median class
F is the cumulative frequency up to the median class
f is the frequency within the median class
n is the sample size

In life, the distribution of thoughts and actions (elements) have a certain frequency, they can be considered as random but by knowing something like the median you can be able to draw a line representing your personality and the sum of all actions. The problem is that you would have to collect every data of your everyday life, i find it unthinkable.
Here where i live, when somebody wants to make something out of his life, achieve some objective, we say "i have to draw a line for my life".

MAIA


--------------------
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OfflineStrumpling
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Re: The concept of randomness [Re: Rhizoid]
    #1703027 - 07/10/03 04:49 PM (14 years, 2 months ago)

good stuff, man :smile:

"What kind of sequence do we get if a person exercises his free will and chooses a "1" or a "0" by his own free choice for each digit position in the sequence? Is this sequence Kolmogorov-irreducible or not? If it isn't then we have found a pattern, and the likely explanation is that the person either used some conscious rule for the decisions instead of choosing completely freely, or that some unconscious non-free-will activities influenced the results."

are you implying that unconscious non-free-will activities are all totally "random?"

I wanted to point out the idea that anything seeming random is only "unpredictable" because we haven't figured out how to predict it yet.


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Offlinepattern
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Re: The concept of randomness [Re: Strumpling]
    #1703066 - 07/10/03 05:00 PM (14 years, 2 months ago)

Would you agree with this statement?

"randomness is the incomprehensible imprint of an infinitely complex pattern"


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OfflineMAIA
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Re: The concept of randomness [Re: pattern]
    #1703254 - 07/10/03 05:55 PM (14 years, 2 months ago)

I found an interesting link about this same subject.
Randomness and Mathematical Proof
Scientific American 232, No. 5 (May 1975), pp. 47-52
Gregersen, From Complexity to Life, Oxford University Press, 2003, pp. 19-33
by Gregory J. Chaitin
Although randomness can be precisely defined and can even be measured, a given number cannot be proved to be random. This enigma establishes a limit to what is possible in mathematics.
http://www.cs.auckland.ac.nz/CDMTCS/chaitin/sciamer.html

I've found this part explaining some of the things i wanted to point out

Quote:

Properties of Random Numbers
The methods of the algorithmic theory of probability can illuminate many of the properties of both random and nonrandom numbers. The frequency distribution of digits in a series, for example, can be shown to have an important influence on the randomness of the series. Simple inspection suggests that a series consisting entirely of either 0's or 1's is far from random, and the algorithmic approach confirms that conclusion. If such a series is n digits long, its complexity is approximately equal to the logarithm to the base 2 of n. (The exact value depends on the machine language employed.) The series can be produced by a simple algorithm such as ``Print 0 n times,'' in which virtually all the information needed is contained in the binary numeral for n. The size of this number is about log2 n bits. Since for even a moderately long series the logarithm of n is much smaller than n itself, such numbers are of low complexity; their intuitively perceived pattern is mathematically confirmed.
.
Another binary series that can be profitably analyzed in this way is one where 0's and 1's are present with relative frequencies of three-fourths and one-fourth. If the series is of size n, it can be demonstrated that its complexity is no greater than four-fifths n, that is, a program that will produce the series can be written in 4n/5 bits. This maximum applies regardless of the sequence of the digits, so that no series with such a frequency distribution can be considered very random. In fact, it can be proved that in any long binary series that is random the relative frequencies of 0's and 1's must be very close to one-half. (In a random decimal series the relative frequency of each digit is, of course, one-tenth.)
.
Numbers having a nonrandom frequency distribution are exceptional. Of all the possible n-digit binary numbers there is only one, for example, that consists entirely of 0's and only one that is all 1's. All the rest are less orderly, and the great majority must, by any reasonable standard, be called random. To choose an arbitrary limit, we can calculate the fraction of all n-digit binary numbers that have a complexity of less than n-10. There are 21 programs one digit long that might generate an n-digit series; there are 22 programs two digits long that could yield such a series, 23 programs three digits long and so forth, up to the longest programs permitted within the allowed complexity; of these there are 2n-11. The sum of this series (21 + 22 + ... + 2n-11) is equal to 2n-10-2. Hence there are fewer than 2n-10 programs of size less than n-10, and since each of these programs can specify no more than one series of digits, fewer than 2n-10 of the 2n numbers have a complexity less than n-10. Since 2n-10 / 2n = 1/1,024, it follows that of all the n-digit binary numbers only about one in 1,000 have a complexity less than n-10. In other words, only about one series in 1,000 can be compressed into a computer program more than 10 digits smaller than itself.
.
A necessary corollary of this calculation is that more than 999 of every 1,000 n-digit binary numbers have a complexity equal to or greater than n-10. If that degree of complexity can be taken as an appropriate test of randomness, then almost all n-digit numbers are in fact random. If a fair coin is tossed n times, the probability is greater than .999 that the result will be random to this extent. It would therefore seem easy to exhibit a specimen of a long series of random digits; actually it is impossible to do so.




MAIA


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OfflinetrendalM
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Re: The concept of randomness [Re: Rhizoid]
    #1703255 - 07/10/03 05:56 PM (14 years, 2 months ago)

Bring chaos theory into the mix and you can find patterns that did not exist before  :smile:

Fractals are one example of this. Repetition exists even in a seemingly random system.


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Re: The concept of randomness [Re: trendal]
    #1703415 - 07/10/03 06:50 PM (14 years, 2 months ago)

A point of interest: computers that produce a random variable through a random output generator do not actually produce a truely random output, but something that is called pseudo-random.

Computers will always output a "perfectly" random number, meaning if a computer is to randomly choose between n possibilities, each possibility will have EXACTLY the same probability of appearing.

In nature this is NEVER the case. "Real" systems are vastly more complex than "virtual" systems, and even the tiniest part of the system will effect the output. This causes some possibilities to show up more often than others.

If you have ever used PGP encryption you will know what I'm talking about. When you create your random starting point for a key pair, you are asked to hit keys on the keyboard and move your mouse around. These TRUELY random inputs (they are "natural") help to offset the pseudo-randomness of the computer system. This is required because in a pseudo-random system the probability of each possibility can be known and fairly easily calculated. This gives the system a weakness, in that if you have a probability model to use you can theoretically break the encryption. The addition of "true" random variables (from keystrokes and mouse movement) removes the ability to predict the output in any accuate way.


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OfflineStrumpling
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Re: The concept of randomness [Re: trendal]
    #1703670 - 07/10/03 08:25 PM (14 years, 2 months ago)

"'true' random variables (from keystrokes and mouse movement)"

so you're telling me that I can type "truely" random text, and make "truely" random mouse movements? That doesn't make any sense to me.

And Pattern, I can't agree with that because I don't understand what you're saying.


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Re: The concept of randomness [Re: Strumpling]
    #1703678 - 07/10/03 08:33 PM (14 years, 2 months ago)

No I think you misunderstood me, or my message wasn't exact enough  :smirk:

What I meant was that the "random" numbers generated by a computer are pseudo-random in the way I described...but input from a mouse by a HUMAN is not pseudo-random at all, so this type of input can be used to create a MUCH stronger starting point for encryption  :wink:

But yes, you can create mouse movements that are much more "random" than any simulated movement a computer could make.


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OfflineRhizoid
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Re: The concept of randomness [Re: Strumpling]
    #1705184 - 07/11/03 08:27 AM (14 years, 2 months ago)

Quote:

are you implying that unconscious non-free-will activities are all totally "random?"




No, the opposite, that they are sources of patterns. Let me give an example: suppose I ask you to produce, out of your head, a Kolmogorov-irreducible sequence of 1's and 0's. You try to avoid producing patterns, but it's hard for a human to do that consistently, so suppose you fail: patterns will be detected. Maybe just some subtle thing like a higher frequency of 1's than 0's. What I'm saying is that these patterns must have come from unconscious non-free-will influences, since you tried to use your free will to avoid all patterns.

#Edit: I mean subconscious, not unconscious...

My main point is this: The dichotomy of pattern/random is a function of the capabilities of the observer. Placed in a large enough context, anything that we call randomness is part of a pattern. And for a sufficiently limited observer, anything that we call pattern seems completely random.

In my eyes this takes away the paradox that a completely unbiased exercise of free will can be both random (unpredictable, Kolmogorov-irreducible) and part of a larger unseen pattern at the same time.

Take a very simple pattern: "1 0 1 0". Now choose a new digit to continue the sequence. If you choose 1, you have "1 0 1 0 1" which is still a pattern of alternating 0's and 1's. If you choose 0, you have "1 0 1 0 0" which is the first 5 digits of the pattern "1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 ..." (among others). Even if the choice was "random", you're still creating a pattern.


Edited by Rhizoid (07/11/03 08:40 AM)


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OfflineRhizoid
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Re: The concept of randomness [Re: trendal]
    #1705193 - 07/11/03 08:37 AM (14 years, 2 months ago)

Quote:

But yes, you can create mouse movements that are much more "random" than any simulated movement a computer could make.




The essential factor here is entropy. Not thermodynamic entropy, but a related and more general type of entropy called Shannon entropy. Every pseudo-random number generator starts with an initial "seed", and the number of unknown or "secret" bits in this seed is the amount of entropy. The more entropy, the more difficult it will be to second-guess the pseudo-random number sequence. And the only way to get entropy into a deterministic algorithm is to read random data bits from the outside world, like reading bits from the clock, or reading mouse movements.


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Offlinejohnnyfive
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Re: The concept of randomness [Re: Rhizoid]
    #1705706 - 07/11/03 01:52 PM (14 years, 2 months ago)

Order from chaos!


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OfflineStrumpling
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Re: The concept of randomness [Re: johnnyfive]
    #1706872 - 07/11/03 09:01 PM (14 years, 2 months ago)

novelty theory ties in very well with chaos in my opinion.


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