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Author
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Topic: Defining Randomness
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Micah Sparacio
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Member # 6
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posted 03. April 2003 14:44
Defining Randomness
I like these threads that are exploring various ways of applying certain terminology. I'd like to explore the notion of randomness.
Here's a start:
According to Kolmogorov/Algorithmic Complexity, we identify randomness in strings by finding strings for which the smallest program that could produce the string as output is the same size as the string itself.
This is determining randomness after the fact. The implication is that a string lacks any sort of regularity (or pattern) that can be expressed computationally and therefore must be completely random.
The problem with this definition is that it always equates randomness with maximum complexity. There is no reason to think that this holds in all cases. We can reasonably expect for a random sequence of events to produce an object of high regularity. In addition, we can expect for some non-random sequence of events to produce an object with maximum complexity.
[ 03. April 2003, 14:58: Message edited by: Micah Sparacio ]
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Danpech
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Member # 163
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posted 03. April 2003 16:57
Trying to understand what our idea of 'randomness' is. Surely, when we think of this thing that we call 'randomness', there is something that we are actually thinking. It is not nothing. But, it seems to me that we can give some account of it only by way of reference to pattern (which would seem to make the term 'non-randomness' a derivative term). In other words, it seems to me that, in order to have this idea which we call 'randomness', we (I, anyway) presuppose order. 'Randomness' presupposes order.
But, do we presuppose non-order when we have the idea which we call 'order'? At the moment, in my foggy-ness, I'm not sure that we don't. If I posit that we do presuppose 'non-order' ('randomness') in thinking of 'order', and vise versa, then does this mean anything as to which of the two (if either) is fundamental?
I'm not sure that I have formulated that quite right. Perhaps the ideas of 'randomness' and 'order' may depend on nothing but what we are looking for. If we look for an example of non-order, and then we find it, are we not selecting something; are we not merely limiting the range of selection, so that anything outside the range is seen as non-order? Is 'randomness' nothing but a level of fine complexity that is beyond what we are used to?
At the moment, I can't think well. Hmm... There seems to be disorder for a system here: my brain. As far as the system of the body is concerned, deterioration of a joint is an example of disorder. But, I'm not sure that we say so because of logic alone, for it seems to me that a logically neutral position cannot identify disorder where one system (a joint) is being replaced with another system (say, calcium loss in micro-gravity). Firtile dirt is a system and, so, I think, is the process that turns a body into dirt.
Perhaps randomness is just the 'failure' of a given system to maintain itself. But, then, how did it arise in the very first place? Or, have all possible systems always existed, and nothing is new? This is the Oscilating Universe (OU) idea, like tides of the ocean. But, that is just no good, for I wish to live forever. How can such a system produce a being who not only feels, but who wishes to never end? As opposed to the OU, I prefer to believe in a foundation of reality that is fundamentally beyond my ability to fully define ("make sense" of). I think this is why 'religion' has such an appeal: it's more ideal than anything that we can fully define. If we are not the foundation of reality, then a metaphysics which implicitly asserts that we are equal to the foundation is not mentally ideal, since it would not be true. Presuppositions which are untrue can only mess up our thinking and, ultimately, our lives.
Can 'randomness' be fully defined in our minds? Perhaps only if order can be fully defined in our minds. ![[Smile]](smile.gif)
[ 03. April 2003, 17:30: Message edited by: Danpech ]
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yersinia
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Member # 324
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posted 03. April 2003 20:49
One decent definition of "random" that I think gets neglected in philosophical discussions is simply the statistical one of ~0 correlation to a particular variable.
I.e., if A is random w.r.t. B then knowledge of A gives you no additional predictive ability regarding B.
This is, for instance, the sense in which "random mutation" of "random mutation and natural selection" is meant. It has very little to do with the various metaphysics that people (on all sides of the evolution question) attach to the word.
To broaden into philosophy a bit, it may be that there is no coherant notion of "pure" randomness, things are only random, or not, with respect to specific other things.
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gedanken
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Member # 594
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posted 03. April 2003 21:44
Micah, your purported start is to understand “randomness”, but jumped immediately into complexity measure based on concepts of randomness.
Why not start with more primitive understandings of randomness itself. What does it mean for an event to be “random”?
Let me start that discussion with some thoughts:
Randomness is a description of a certain type of lack of knowledge of an event. For example we might know that an event will be limited to certain alternatives (or purposely constraint the alternatives we will consider even though the possibilities may exceed that limited set in unusual circumstances like a coin landing on edge or being lost). In this sense “randomness” is usually not a complete lack of knowledge of events -- though some times the term “random” is applied in such cases. But I don’t think that is typical of the cases of interest relating to complexity or K-complexity, for example. So we are dealing with constrained alternatives among which we have limited knowledge.
There are important distinctions of viewpoint -- just as our language has forms to represent certain viewpoints. We are most likely speaking of a future event. Or possibly we are speaking in a future tense about a past event -- viewing it from the viewpoint of the past looking at the future in which the actual outcome was not known. I think this viewpoint orientation is very important to understanding randomness -- the viewpoint may specify what we do know, and what subset of all possible events to be considered, as an artificial and stylistic manner of speaking very specific to solving of technical problems. This special viewpoint language allows us to make use of the knowledge of constraints that we are specifically going to accept, while rejecting consideration of other aspects that are not part of what we want to consider in the specific case. This allows for “reductionism” to be applied to the problem being considered, to develop “probability” analyses of the event possibilities.
Now I want to emphasize the aspect of this viewpoint once more, as it may actually be the case that we don’t know what will happen because the actual event is only planned or something to be considered for the future. Or the event may have actually been in the past, and we are very specifically asking what could have happened if we “rewound the tape and played it over” with outcomes not determined beyond the constraints that we are going to consider. When we do this backward viewing of possible futures, we explicitly decide to exclude things we know from the present, such as what actually happened. (I flip a coin and it comes up heads. What is the probability that the coin just came up heads -- now that everyone knows what happened? Would you like to place a bet? I’ll give you 10:1 payment ratio or “odds” if you’ll bet on tails. See how the special viewpoint excludes information that may come from other sources, and usually is constrained to specifically view the coin toss from the past as though it were about to happen.)
Now consider a coin toss. But the tosser of the coin is not a mere human, but a humanoid appearing robot (or other device) that is mechanically constructed and programmed to be able to give the coin a very precise flip with controlled properties. One could use laws of physics to make such a robotic device that could cause a particular outcome with high probability. But to the viewers of the toss who do not know the specific programming, the coin could still be random and the outcome is not known. So the viewpoint depends on specific knowledge of the robot programming.
Further consider that said robot uses a tightly coded pseudo-random number generator to generate an apparently fair toss, but in which the outcome is predetermined by the robot’s pseudo-random generator. Without knowing the specific coding of the pseudo-random generator, one might analyze the physical coin tosses as random. (In K-complexity terms, one would not know how to compress the string significantly.) But once the pseudo-random generator was identified, then the generator program could be substituted for the string, and the string could be highly compressed.
One should at this point become aware of fractal compression software techniques which will highly compress many random event sequences. The compression may have absolutely no relationships to the processes used to cause the outcomes -- rather one simply looks for a generator function that simulates the events as observed and arbitrarily picks the shortest such generator. Actual fair coin toss sequences may sometimes be compressed highly by such a fractal compression algorithm, having absolutely nothing to do with whether the coin toss was fair. One should be aware that actual compression techniques will have a fairly high probability of performing a slight compression, just slightly under the actual string length. So there will be a high fraction of truly “random” strings that compress just slightly under their full string length. In fact a particular state machine can be defined in which exactly half of all truly random strings compress to shorter than their original length -- at the cost that the other half will require longer than original code representation. But this is a digression.
Now lets consider events in nature or at least in physical systems. For a moment, consider a bowling ball rolling toward pins. If the bowler has low skill, the outcome will be quite random, as the range of direction/speed/rotational aspects will be large in comparison to those which produce certain results that are scored highly in bowling. Because we will not know the path, there will be a fairly random distribution of such outcomes. This is almost equivalent to the die toss case -- because there is very little one could do in a fair die toss situation to control the throw sufficiently as to change the outcome. Yet, as the coin tosser mechanism has suggested, the actual physics of the bounces of the die in a reasonably well analyzed circumstance could predict the outcome early in the toss. It our specific lack of knowledge of those aspects that makes the “randomness” of the outcomes apparent to our viewpoint. As the outcomes are constrained, and by symmetric relationships that are unrelated to the tosser’s slight variations, the six outcomes can be classified as equal probability.
Returning to the mechanical coin tosser, we could vary slightly the amount of angular momentum applied to the coin. As one varies this slightly for a given toss (parabolic) trajectory, the result will pretty smothly vary from heads to tails to heads, etc. Because this slight variation, due to the physics of the coin and its symetrical layout, evenly distributes the variation of outcome to the control parameter, we see an even distribution of outcomes to heads and tails over the analog parametric variations of angular momentum. And similarly vary the parabola so as to change the toss time, and once again the outcome symetrically oscilates between heads and tails as the toss time increases. Because of this, over the possible system inputs the outcome is distributed evenly among the outcome cases of “heads” and “tails”. Now consider the human tosser (coins, die) and we see that these slight variations cannot be controlled, yet because we have symetry of the physical aspects of the outcomes -- and thus can analyze the “chance” in terms of a probability or frequency measure.
(An interesting case is the robotic tosser with a quantum random generator, truly “random”, but with the toss physics deterministic after the quantum generator’s decision. In this case what we see as symmetry argument for even distribution is irrelevant, as it is the distribution of the quantum generator that is important.)
To put this in terms of the K-complexity -- we simply don’t have knowledge of the initial parameters, and thus cannot compress the string. If we had such knowledge, especially of the events that lead from case to case of the tosser (such as our mechanical coin tosser programmed with a desired outcome tape) then we may view the randomness differently. K complexity is simply an abstraction that is equivalent to a certain kind of generalized fractal compression -- simply find the shortest algorithm.
The K-complexity really has very little to do with the analysis of randomness itself. Rather it has to do with the dispersion of such random events among methods of coding the outcomes in terms of algorithms. Highly predictable outcome sequences can be written in terms of the generators of those sequences. For a given “Universal Turing Machine” there is a specific distribution of output strings that enumerated programs would output. For that specific UTM, there simply is a mapping of the programs to the outputs (with may programs not terminating or not known to terminate and thus not having a defined output mapping). But we simply identify the fact that short programs can generate long output strings, and examine the density of output strings that can be generated by such programs. For a given UTM, there is simply a mapping of input (program) strings to output strings, and these can be enumerated by enumerating the input programs. For a given length, we can simply classify the particular UTM’s responses in terms of how long the output strings are. Since the UTM simply defines a mapping of strings to strings, the UTM itself is irrelevant. Nothing has so far directly related K-complexity to actual reality.
We can make such a relationship when we can identify the “programs” for particular UTM as corresponding in some way to realistic simulations of actual physical systems. To put this in terms of the mechanical coin tosser case, we simply are identifying the UTM with a knowledge of the specifics of the coin tosser mechanism and its internal programming. To the extent that it produces a deterministic output, and we can identify such programming, we can compress the string at least as much as that programming is compressed below the length of the output to be generated (as in the pseudo-random generator that has an actual highly compressed representation of the output which is deterministically produced by the mechanical coin tosser mechanism.) In other cases we may be able to compress the knowledge of the physical world’s constraints, and thus identify the reduced structure of the physical world beyond the randomness that comes of the expansion of sensitivity of outcomes to parametric variations of systems involved.
This is the sort of case in which K-complexity actually becomes meaningful in noting the randomness. Consider the “Caputo” example. There is a reason for someone to cheat in the choice by a corrupt politician. One can compress the output due to such knowledge, and identify a cause by such a mechanism because the mechanism (desired outcome by a politician) has a simpler description than the random outcome of a random decision mechanism.
But I would like to return to the notion introduced above of the lack of knowledge. If one had a knowledge of the physical interactions that cause the events, then one can predict the outcomes to a greater extent. It is the knowledge that counteracts the widening of choices. We can have that knowledge in varying ways. So randomness is essentially related to partial lack of knowledge.
I think that an important aspect should be considered. Due to supersensitivity of many outcomes to initial conditions, and also to lack of knowledge inherent in quantum indeterminacy, there are a great many cases in which the outcome has fairly predictable pathway aspects, yet becomes “random” within the problem constraints due to the lack of knowledge (even in the presence of fairly complete understanding of a system’s performance).
Consider a case like RMNS: Random mutations cause widely changing outcomes. Yet natural selection constrains those outcomes long term effects. Only working systems will result. So NS constrains the random outcomes of the system, which is fairly unconstrained by RM. Thus progress of populations undergoing mutational change (or other genetic change and descent with modification processes) can be predicted in many cases in which there is sufficient knowledge. This is simply the case of knowledge changing the probability distribution that we see. Remember that randomness is always a result of lack of knowledge, and that with sufficient knowledge all randomness disappears. (An omniscient being for example would not see “randomness” in the sense that the being would already know the outcomes, due to access to knowledge that mere humans do not possess.)
[ 04. April 2003, 12:47: Message edited by: gedanken ]
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Danpech
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posted 06. April 2003 19:17
In hope it will help in this question about randomness: Gedanken mentioned coin-tossing as a seeming 'random' event. If you flip coins enough while trying to keep track of the rythm, you can learn to make any coin land as heads or tails, at will. This is just a form of juggling.
What do we mean by randomness in regard to this, then? The least we can say here is that the coin 'toss' would not be random (in any normal sense), even though it can be thought to be random by someone who does not know that this person flipping the coin is actually an expert coin juggler.
But, does this imply that randomness is an illusion based on ignorance? Why should randomness and knowledge-of-the-outcome be mutually exclusive? The non-pattern idea of randomness still looks intact to me (for what little I know): A string that, by definition, has no unifying pattern. Is that possible, on some level of physics anyway? If it is possible, then randomness has an objective reality.
But, suppose it is impossible to have a string with no unifying pattern. Does this then do away with randomness as nothing but a lack of system- knowledge? If it does away with it as nothing but illusion, yet there are still two questions: 1)whether the system is finite or infinite, either now or successively; 2)whether the system can be fully known, either from inside it by conscious agents, or outside it.
If the system cannot be fully known from inside it (because the knowledge would be equal to the system), and if there is no outside, then how can we say that a given point in the system is determined, especially if the system is changing (changing perhaps because there are pattern-input agencies within it)? But, suppose we allow that there is an outside, and that the outside is a pattern-input agency? Then, the system can still be said to be undetermined except as this outside agency decides otherwise.
Or, perhaps we can do away with randomness? Why do we need randomness as an objective reality? Instead, what we might have are one or more pattern-input agencies, some or all of which could be partially or wholely independent. I seem to be able to input patterns into all kinds of things, and with what seems to me a good deal of independence from other agents.
Or, perhaps we have the objective randomness, plus one or more independent and, or, semi-independent pattern-input agencies. I like this idea the best, for variety, although I'm not sure what a pattern-input agency is.
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Micah Sparacio
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posted 06. April 2003 19:37
1. A phenomenon is random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.
2. Apparent lack of purpose or cause.
[ 06. April 2003, 19:42: Message edited by: Micah Sparacio ]
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Regvi
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Member # 586
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posted 07. April 2003 08:05
Context seems to be of crucial importance when discussing randomness and patterns.
Example: My adviser of studies' department phone number used to be 9905962, the same as my university matriculation number. To the two of us, 9905962 is not just any random number, it has (or at least had) significance as a pattern.
I assume that the number 9905962 meant nothing in particular to most readers before this posting.
In general, the more we know, the more significant patterns we can recognize.
Also, we cannot rule out for any random sequence of numbers that it has some meaning for some intelligence somewhere (eg God, angels or aliens).
Another example: most people would say that the sequence 00000000001111111111 is non-random. Why? Because it is completely symmetrical, and looks ordered (it has a psychological effect on us).
It is well known that babies prefer looking at face-like shapes than at non-face like shapes, and even that they prefer pretty faces to ugly ones.
Rógvi
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