# Defining randomness?

1. Feb 5, 2009

### SW VandeCarr

If I use a simple algorithm to generate all the numbers from 000,000,000 to 999,999,999, is there a rule for determining how many of these digit sequences are "random"? Slightly more complex algorithms generate the irrational numbers. Are these digit sequences random? For any finite length of the digit sequence of an irrational number, the next digit is determined, not random.

It would appear that a true random number generator must be based on something that can be safely said to be not determined by any algorithm, such as a quantum level physical process. Does this take the notion of a "random variable" from mathematics to physics?

2. Feb 5, 2009

### FiberOptix

It's my understanding that randomness is a pretty hard concept to nail down, and I believe there is still debate as to whether there even exists true randomness - it depends on your definition, which bring us back to your question.

I believe computers can only generate pseudo-randomness, and for the reasons you mention. You might take a que from matlab as to how to possibly quantify randomness, as the uniformity of the distribution of the numbers output from some function.

Pretty interesting stuff.

3. Feb 6, 2009

### HallsofIvy

Staff Emeritus
There is no such thing as a "random sequence". The fact that you have a specific set of numbers in a specific order means it is NOT random. You can have "randomly generated" sequences depending on how the order is generated. But it is quite possible that such a "randomly generated" sequence might turn out to be 1, 2, 3, 4, ...!

4. Feb 7, 2009

### SW VandeCarr

I agree. But how do you know a generator is truly a random generator? Is this a question of mathematics or physical theory (referring to my OP)? In other words, is there any way to mathematically prove a generator is truly a random generator?

Last edited: Feb 7, 2009
5. Feb 7, 2009

### HallsofIvy

Staff Emeritus
Hey, that wasn't part of the question! It's not my department!

6. Feb 7, 2009

### SW VandeCarr

You responded just as I was editing my last post. Can there be any mathematical proof that a given number sequence generator is a random number generator? Based on your response it appears the answer is 'no' and random number generators can only be defined in terms of physical theory. Is this correct?

7. Feb 7, 2009

### gel

There are definitions of randomness -- see http://en.wikipedia.org/wiki/Algorithmically_random_sequence" [Broken]. However, that's not really "true" randomness. You can also test statistically whether a sequence follows any particular random distribution. But you can't really define true randomness mathematically

Last edited by a moderator: May 4, 2017
8. Feb 7, 2009

### SW VandeCarr

I agree. I'm aware of algorithmic (Kolmogorov) randomness regarding non-compressible algorithms and statistical randomness based on probability distributions. If a physical process such as quantum process is needed to define a truly random (non-algorithmic) generator, how does this relate to the notion of a random variable? We can define a random variable in terms of a mapping from a probability space into an event space, but can we define how this function actually works?

I can specify the required parameters of a distribution and a random generator will output a series of 'n' values which will converge to these parameters as 'n' grows large. However, computer based random generators are really pseudo-random. I cannot know if each unit of output is statistically independent of any other unit of output.

To me this introduces an unusual feature to Probability Theory and other theories which utilize probability, in that it ultimately seems to depend on the undefined notion of randomness, or have missed something here?

Last edited by a moderator: May 4, 2017
9. Feb 7, 2009

### gel

I agree that ultimately you can't define what randomness is. You can mathematically define the notion of a probability space, and apply it to real-world problems. However, you can't prove at a fundamental level that it really does apply -- although you can perform statistical tests.

I don't think that this is an unusual feature of probability theory though. Maths deals with abstractions, and any applications rely on ultimately undefinable concepts as to what it actually means. But it does correspond with what we experience (hopefully).

10. Feb 7, 2009

### netometry

The only way a sequence could possibly be random is if it were part of an infinite set.
There is no such thing as random in a finite universe.

11. Feb 8, 2009

### SW VandeCarr

That's pretty radical. For one thing the digit sequences of the irrational numbers are infinite and some think they are random. I disagree since they are produced by an algorithm which will always repeat the exact same sequence up to any n every time it is invoked. In my view quantum level physical processes are our best, probably only, example of true (non- algorithmic) random number generators.

I don't know what this has to do with the size of the universe.