Randomness in Evolution: Valid and Invalid Usage

[lenny]

For some life forms, they evolved to try something new or different when the old stuff isn't working...

ummm, did you really mean that?

wouldn't you expect the vast majority if things to just go extinct if it turns out "the old stuff isn't working"? evolution on demand sounds, well, a bit miraculous.

 

[lenny]

2. What if you used exactly specified a set of floating point precision numbers. For some set of specified floating precision numbers you will get some rounding error in the insignificant digits when multiplying. Do you consider this random?

no, it is not random. and how does it help us to introduce "rounding error"?

a chaotic mathematical system simulated on a digital computer is just a many to one map on the integers. nothing random, and in "error" only if one mistakenly identifies the simulation with the thing simlutated (an exact calculation). as a mere simlulation it can still yield interesting robust insights.

do you disagree with jimbob's analog/digital distinction?

 

[Prometheus]

ummm, did you really mean that?

wouldn't you expect the vast majority if things to just go extinct if it turns out "the old stuff isn't working"? evolution on demand sounds, well, a bit miraculous.

Not evolution on demand, but evolved adaptability.

 

[articulett]

ummm, did you really mean that?

wouldn't you expect the vast majority if things to just go extinct if it turns out "the old stuff isn't working"? evolution on demand sounds, well, a bit miraculous.

Yes... I meant it, and it followed an article I quoted and I also explained what I meant. It's not "evolution on demand". It's evolution with a brain coded for "spontaneity"... read what I actually said and referenced before commenting if you want your opinions respected. Otherwise, you appear jerkish to me.

 

[articulett]

Not evolution on demand, but evolved adaptability.

Thanks.

 

[zosima]

no, it is not random. and how does it help us to introduce "rounding error"?

I wasn't intending it as an argument. Mitchell is new to this thread, and he introduced an example that I wanted to understand, and I wanted to understand how he felt about the whole random issue. Different people, on this thread might answer it different ways, and their answers tell me things. But issues of numerical stability are incredibly important.

Rounding error on a digital system is generally a deterministic process that is very difficult to model. So it tells me if Mitchell is a random=unpredictable, or a random='fundamentally random' fellow.

a chaotic mathematical system simulated on a digital computer is just a many to one map on the integers. nothing random, and in "error" only if one mistakenly identifies the simulation with the thing simlutated (an exact calculation). as a mere simlulation it can still yield interesting robust insights.

A chaotic system is:
1. Deterministic.
2. A purely mathematical construct.

I don't really want to argue about this, I've really talked to enough on this thread already. Just read my older posts, if you're curious about my position.

do you disagree with jimbob's analog/digital distinction?

Absolutely. It demonstrates jimbo's fundamental misunderstanding of what constitutes a chaotic system.

If you want an example try learning about the logistic map. It is an extremely common chaotic model that is completely closed under the integers. There are numerous other chaotic equations that are completely discrete as well.

My position is that any physical instantiation of a chaotic system is only going to fit the mathematical model ideal approximately. The only reason I can see that Jimbo is supporting this analog/digital distinction, is because it limits us to examples that are harder to reason about precisely, and thus makes it easier for him to wave his hands into his conclusion.

But seriously,I can't take the time to explain all this. If you are still confused I've laid everything out in detail in my previous posts. There is nothing more that I can say that hasn't already been said there.

 

[Dancing David]

When I say the organism "evolved to", most people understand that those with the propensity in the genes preferentially survived and reproduced. When we say duck's corkscrew shaped members evolved to fit in the reproductive tracts of females discouraging rape... then I trust that all people who actually understand evolution understand that these are the environmental forces which "naturally selected" the genes coding for these traits.

I don't expect a creationist to understand or agree with anything I say. They can't. It spoils what they want to believe about their self appointed expertise in whatever it is they imagine themselves to be experts in.

I understand that, and I was not faulting you, more the extreme fine point drawing of some people and avoiding the trap of determinism.

I agree with you and understand your arguments. I understand what you are saying but the others, wow...

 

[Dancing David]

I agree. But, it does depend on how you define 'random'.

Like 'dog' and 'tree' an idiomatic self referencing symbolic exchange that looses meaning when some people forget about the consensus of communication and stop trying to exchange ideas.

 

[mijopaalmc]

Absolutely. It demonstrates jimbo's fundamental misunderstanding of what constitutes a chaotic system. It sounds like both ya'll have a fuzzy, hand-wavy, pop-sci understanding of how these things work.

If you want an education, try learning about the logistic map. It is an extremely common chaotic model that is completely closed under the integers. There are numerous other chaotic equations that are completely discrete as well.

But seriously, I'm not gonna take the time to explain this to you. If you are still confused I've laid everything out in detail in my previous posts. There is nothing more that I can say that hasn't already been said there.

If you are going to criticize someone's knowledge of chaotic systems you might want to review your own. First off, the the logistic map may be closed under the integers (i.e., any integer input yields an integer output), but it not chaotic on the integers, because the integers do not form a dense set of perioduic orbits, a defining element of chaos.

 

[zosima]

If you are going to criticize someone's knowledge of chaotic systems you might want to review your own. First off, the the logistic map may be closed under the integers (i.e., any integer input yields an integer output), but it not chaotic on the integers, because the integers do not form a dense set of perioduic orbits, a defining element of chaos.

Yeah, I edited that language out of my post(before mijo reposted what I said) because I realized that I was a little annoyed and had used language that was too strong. So I apologize.

But my claim is correct. The logistic map is chaotic over the integers, you are incorrect. Write a program yourself, you'll see it cycling even in one dimension around 0. (Just to be sure I just tested it out with r=5 and x(0) = 7)

Moreover, there are plenty of other discrete chaotic systems. (Probably there are an infinite number) I would recommend this book if you are still confused:
http://www.amazon.com/exec/obidos/ASIN/1584880023/ref=nosim/weisstein-20
What's the book called? Discrete Chaos

ETA: Also here's an example from Wolfram's Mathworld: http://demonstrations.wolfram.com/DiscreteLogisticEquation/
The actual output is normalized between [0,1] for the purpose of plotting.(it is easier to see, 'cause don't need to rescale the plot every time you perform a step). The name of the example "Discrete Logistic Equation" Do you ever get sick of being wrong Mijo?

 

[mijopaalmc]

The logistic map is chaotic over the integers, you are incorrect. Write a program yourself, you'll see it cycling even in one dimension around 0. (Just to be sure I just tested it out with r=5 and x(0) = 7)

Moreover, there are plenty of other discrete chaotic systems. (Probably there are an infinite number) I would recommend this book if you are still confused:
http://www.amazon.com/exec/obidos/ASIN/1584880023/ref=nosim/weisstein-20
What's the book called? Discrete Chaos

ETA: Also here's an example from Wolfram's Mathworld: http://demonstrations.wolfram.com/DiscreteLogisticEquation/
The actual output is normalized between [0,1] for the purpose of plotting.(it is easier to see, 'cause don't need to rescale the plot every time you perform a step). The name of the example "Discrete Logistic Equation" Do you ever get sick of being wrong Mijo?

Actually, yet again it is you who is wrong. The logistic map on the integers contains only stable fixed points between 1 and 4 and unstable points for intergers greater than 4. It does not contain any periodic orbits, let alone are the periodic orbits dense in the integers. The logistic map on the integers is therefore not chaotic on the integers, at least according to Robert Devaney's definition of a chaotic dynamical system.

 

[Vorticity]

...
If you want an example try learning about the logistic map. It is an extremely common chaotic model that is completely closed under the integers. There are numerous other chaotic equations that are completely discrete as well.
...

I've been trying very hard to stay out of this futile thread, but I thought I'd jump in here with a quick note. The current disagreement over the logistic map seems to be resulting from some confusion over terms.

The logistic map is a discrete dynamical system which is usually written as:

[latex]$x_{n+1} = r x_n(1-x_n)$[/latex]

zosima, when you say that the logistic map is "closed under the integers", what do you mean. To me, for a map defined as

[latex]$x_{n+1} = f(x_n)[/latex]

to be "closed under the integers", that would mean that when [latex]$x_n$[/latex] is an integer, then [latex]$f(x_n)$[/latex] is such that [latex]$x_{n+1}$[/latex] is always an integer as well.

For arbitrary, real [latex]$r$[/latex], it is clear that the logistic map does not satisfy this criterion, although this is the case if [latex]$r$[/latex] is an integer.

However, in the next sentence, you say "There are numerous other chaotic equations that are completely discrete as well." (Bolding mine) This wording suggests to me that you are equating the property of being discrete (i.e. possessing a discrete independent variable) with the property of being closed under the integers. In fact, these are two different properties.

While for certain initial conditions and certain values of [latex]$r$[/latex] (e.g. for [latex]$r = 1$[/latex]), it is undeniably the case that the logistic map is chaotic, it is certainly not the case for integer-valued initial conditions.

 

[Vorticity]

But my claim is correct. The logistic map is chaotic over the integers, you are incorrect. Write a program yourself, you'll see it cycling even in one dimension around 0. (Just to be sure I just tested it out with r=5 and x(0) = 7)

Unless we're having some huge confusion over definitions, then I'm baffled.

I just wrote a program and tried it with these values. The logistic map does not display chaos for these values of r and x(0). It shoots off to negative infinity.

ETA: Also here's an example from Wolfram's Mathworld: http://demonstrations.wolfram.com/DiscreteLogisticEquation/
The actual output is normalized between [0,1] for the purpose of plotting.(it is easier to see, 'cause don't need to rescale the plot every time you perform a step).

It is not normalized. It wasn't necessary.

For 0 <= r <= 4, the interval [0,1] is closed under the standard logistic map.

That is why they constrain their initial condition to be between 0 and 1, and also they constrain their parameter lambda (which is equivalent to r) to be between 0 and 4.

The name of the example "Discrete Logistic Equation" Do you ever get sick of being wrong Mijo?

Careful.

Although this is to a certain extent a confusion over nomenclature, as far as I can tell, mijo is much closer to being right than you are: The logistic map with integer r (and so closed under the integers) is, in fact, NOT chaotic with integer initial conditions.


By the way, an interesting side note, since we're speaking tangentially about probability:

For r = 4, the logistic map has an invariant measure p(x) on [0,1], where
[latex]$p(x) = \frac{1}{\pi\sqrt{x(1-x)}}$[/latex]

 

[mijopaalmc]

zosima, when you say that the logistic map is "closed under the integers", what do you mean. To me, for a map defined as

[latex]$x_{n+1} = f(x_n)[/latex]

to be "closed under the integers", that would mean that when [latex]$x_n$[/latex] is an integer, then [latex]$f(x_n)$[/latex] is such that [latex]$x_{n+1}$[/latex] is always an integer as well.

For arbitrary, real [latex]$r$[/latex], it is clear that the logistic map does not satisfy this criterion, although this is the case if [latex]$r$[/latex] is an integer.

However, in the next sentence, you say "There are numerous other chaotic equations that are completely discrete as well." (Bolding mine) This wording suggests to me that you are equating the property of being discrete (i.e. possessing a discrete independent variable) with the property of being closed under the integers. In fact, these are two different properties.

Yeah, this was something that also confused me about zosima's post. I interpreted the property of being "closed under the integers" as being the property of algebraic closure you described above. As you also noted this propert say nothing about the discreteness or continuity of the dynmical system, which, as far as I understand, pertains to the "number" of iterations (i.e., compositions of the time evolution function). This is in generally more rigorously explained as the group action of the integers (in the discrete case) or the reals (in the continuous case) on a manifold, and, as far as I understand, is roughly analogous to indexing random variables to a competely ordered set for stochastic processes.

 

[jimbob]

For some life forms, they evolved to try something new or different when the old stuff isn't working...

I take issue with virtually every word in that post.

Firstly:

Organisms do not evolve "to" do anything. Nor do they "try" things.

Secondly:

If the "old stuff" wasn't working, they would become extinct.

A random variation increased the reproductive success of holders of this variation, and so the new variation spread. If the holders of the variation (mutation) are still competing with organisms without this mutation, then their success will reduce the reproductive success of organisms without the mutation.

I agree many organisism do not evolve 'to do things'.
I take exception with many of your words.

Um variation need not be random, but please continue to abuse the term variation. There are means of variation that are not 'random', but whatever.

Dancing David,

It wasn't just the Evolved to... part. It was what they were evolving to do. It would still be wrong to say.

For some life forms, they evolved to try and tried something new or different when the old stuff isn't working...​

Lets change it again to make it less wrong:

For some life forms, they evolved to try and "tried" something new or different reproduced better than the old stuff (which initially still worked sufficiently to produce reproducing offspring otherwise there wouldn't be any descendents) when the old stuff isn't working...​

The original statement is almost 100% wrong. It is also a disagreement I had with articulet when she equated aspects of technical development to evolution.

Humans can, and do learn from mistakes. Evolution can only "learn" from success.

The post above articulett, which I suspect she was responding to concerned a single populationof e.coli that evolved citrate metabolism.


The vast majority of organisms reproduce asexually most of the time. Variation in such a situation (ignoring lateral gene transfer) is due to mutation.

 

[zosima]

Unless we're having some huge confusion over definitions, then I'm baffled.

I just wrote a program and tried it with these values. The logistic map does not display chaos for these values of r and x(0). It shoots off to negative infinity.

Well my initial claim was with respect to whether digital systems can be chaotic. So it does shoot off to negative infinity. But then it rolls around and oscillates chaotically, so technically, I'm describing it over a finite field.

Also, I am making the claim that r must also be an integer. So I'm not just talking about discrete time, I'm talking about a completely discrete system. If R is rational it is trivially not closed under the integers. (Or a finite field)

It is not normalized. It wasn't necessary.

For 0 <= r <= 4, the interval [0,1] is closed under the standard logistic map.

That is why they constrain their initial condition to be between 0 and 1, and also they constrain their parameter lambda (which is equivalent to r) to be between 0 and 4.

You are right about that. That sort of normalized plotting is a technique applied elsewhere is NKS I'd figured thats what they were doing there, so I stand corrected.

But even that being the case a normalized plot(dividing by some reasonable guarantee of an upper bound, or moding the output), will demonstrate the chaotic behavior of the map with large x. If you want to say that I've applied a modulus to plot the output,thats fine. Its also seems to be 'tangential' at best, as you would put it. It fits the definition of chaotic insofar as it has exponential error growth from neighboring starting conditions( the fundamental feature in the discussion of evolution.). Really, the discussion is about the broader question of unpredictability in systems, and often chaotic systems are being used to stand in for unpredictable systems. Constraining a system to fall within some bounded region seems to me to be an irrelevant constraint to this discussion.

But what I'm most concerned with is proving that a digital system can constitute a chaotic system just as well as an analog system of equivalent precision. I'm certainly not saying that any of the chaotic properties of the function follow from the function being defined over the integers, only that they are not inconsistent.

If you take issue with the logistic map there are plenty of other systems that would prove my point. Moreover rational numbers within a digital system would be a sufficient field to make proofs about chaos in digital systems

 

[mijopaalmc]

Also, I am making the claim that r must also be an integer. So I'm not just talking about discrete time, I'm talking about a completely discrete system. If R is rational it is trivially not closed under the integers. (Or a finite field)

But when r is restricted to the integers, the dynamical system is not chaotic, as there are no periodic orbits and periodic orbits are an essential element of chaos.

Constraining a system to fall within some bounded region seems to me to be an irrelevant constraint to this discussion.

As far as I have read in textbooks about chaos, chaotic maps are maps of sets in to themselves. Outside of some bounded interval ([0,1] in the case of the logistic, any intial interval will eventually not map into itself.

If you take issue with the logistic map there are plenty of other systems that would prove my point. Moreover rational numbers within a digital system would be a sufficient field to make proofs about chaos in digital systems

Nice assertions. Would you care to back them up with some actual examples?

 

[zosima]

But when r is restricted to the integers, the dynamical system is not chaotic, as there are no periodic orbits and periodic orbits are an essential element of chaos.

That is an excellent repetition of your claim, it doesn't seem like you've included any new content. The definition that I've been using insofar as evolution is concerned is exponential error growth. Restricting to periodic orbits necessarily excludes numerous systems that need to be part of the evolution discussion. For example, the pool(snooker) balls, would be chaotic in the sense of exponential error growth, but not in the sense that they ever have periodic orbits. Moreover, if we use your definition, then it is guaranteed that evolution has nothing to do with chaos, as evolution doesn't have periodic orbits. So the definition you've provided, while perhaps useful in limiting certain mathematical investigations, is self-defeating when applied to the topic at hand, whereas exponential error growth is a perfectly sufficient definition. Why should we restrict chaotic systems to periodic orbits? Is there any reasoning behind this claim?

As far as I have read in textbooks about chaos, chaotic maps are maps of sets in to themselves.

Okay, if that is the constraint you are placing, then closure under the integers was sufficient in the first place.

Outside of some bounded interval ([0,1] in the case of the logistic, any intial interval will eventually not map into itself.

You're wrong. r = .1, x = 10, will map into itself an infinite number of times, on the bounded region [-10,10], although this particular solution will not be chaotic. There actually is a set of solutions that have this property. r = .01, x = 100 [-100,100] r = .001,x = 1000,[-1000,1000] so, technically, there are an infinite number of bounded solutions outside of the interval [0,1] that map onto themselves indefinitely yet do not converge to 0(or ever enter the region [0,1])

Also I've already mentioned this but you seem to have missed it, if we're working with a finite set of integers or we do a normalized plot, it shows just as much chaotic oscillation as any other system. It is just a question of how you visualize the erratic and unpredictable behavior of the output.

Nice assertions. Would you care to back them up with some actual examples?

y = 2*x mod z over the integers is a classical example.(z prime != 2)
Over the rationals: y = 2*x mod 1.0

Both digital and analog systems will make numerical approximations from the mathematical ideal, which has been my contention from the start, and thus they are both equivalent approximations.

ETA: Moreover, if we restrict the discussion to only systems that have dense periodic orbits, it essentially concedes the issue of randomness in evolution. Chaotic systems under your definition are generally subject to noise induced order. This eliminates the possibility of any large scale randomness in evolution.

 

[jimbob]

Both digital and analog systems will make numerical approximations from the mathematical ideal, which has been my contention from the start, and thus they are both equivalent approximations.

No they are not.

An analogue computer does not make numerical approximations. It is a physical model of a system, with one type of physical quantity replacing another, for example using an LCR circuit to model the oscillations of a mass on a spring. If you made two analogue computer simulations of chaotic system, they will diverge, by virtue of them not having identical starting conditions.

If you made two digital computer simulations of chaotic system, they will not dverge, if you set the initial numbers and precision to be the same.

The computer is a physical system, but a computer simulation is not a physical chaotic system.

Physically, assuming that you are using a computer with CMOS logic, the computer simulation consists of a set of tranistors with their gate voltages either "high" or "low" and their drain voltages either "high" or "low". (The computer simulaiton could also be made on a babbage-engine, or on a valve computer, or any other type of computer that could be made). In all these systems, as you say, there is a numerical approximation to solving the chaotic equations. And a "mapping" of these results to some form of output system.

An analogue system aims to be a physical analogue of the system that is modelled, so there is no numerical approximation. In other words, a chaotic analogue simulation will be chaotic, but obviously its behaviour will diverge from that of the system it is modelling because it is a different chaotic system.

 

[zosima]

No they are not.

An analogue computer does not make numerical approximations. It is a physical model of a system, with one type of physical quantity replacing another, for example using an LCR circuit to model the oscillations of a mass on a spring. If you made two analogue computer simulations of chaotic system, they will diverge, by virtue of them not having identical starting conditions.

The computer is a physical system, but a computer simulation is not a physical chaotic system.

Physically, assuming that you are using a computer with CMOS logic, the computer simulation consists of a set of tranistors with their gate voltages either "high" or "low" and their drain voltages either "high" or "low". (The computer simulaiton could also be made on a babbage-engine, or on a valve computer, or any other type of computer that could be made). In all these systems, as you say, there is a numerical approximation to solving the chaotic equations. And a "mapping" of these results to some form of output system.

An analogue system aims to be a physical analogue of the system that is modelled, so there is no numerical approximation. In other words, a chaotic analogue simulation will be chaotic, but obviously its behaviour will diverge from that of the system it is modelling because it is a different chaotic system.

Your claim that the behavior of the computer itself is not 'truly chaotic' is demonstrably false. See: http://mathworld.wolfram.com/ShadowingTheorem.html
QED: Computers can be chaotic.

But really, all you are really saying is that if a computer is a physical simulation of something else, then it can only approximate that thing. It is trivially true that in a simulation the simulator is not the simulatee. If I use an RC oscillator to simulate a spring(in analog) it will also only be an approximate simulation of the spring. Moreover there is numerical approximation in any analog system. It has finite resolution limited by the size of its components(although this might be smaller than computer components), moreover its resolution is limited practically by noise and the accuracy with which we can measure its output. The reason we use computers to do simulations and not analog systems is because they are more accurate not less.

If you made two digital computer simulations of chaotic system, they will not dverge, if you set the initial numbers and precision to be the same.

Technically, since chaotic systems are deterministic, you are now making the case that digital systems are better simulations of chaotic systems than analog systems. I think it is your confusion over this point that has leading you astray since the beginning.

@Mijo: If you are looking for more periodic chaotic systems over integers here are some more examples.

x(0)=-2
y(0)= 4

x(n+1) = 1 - y + |x|
y(n+1) = x

And:

x(0)=1
y(0)=-2

x(n+1) = 1 -x^2 + y
y(n+1) = x