So let’s analyze properties of symbol sequence generated by the logistic equation

So here I'll think about the symbol sequence generated from an orbit of the logistic equation

and this is with r equals 4.0 so this is a chaotic value, it’s aperiodic

and it has a sensitive dependence on initial conditions

and I’m going to imagine an experiment

and I’m going to do another experiment side by side

and I’m going to imagine tossing a fair coin many many times

so if I tossed a fair coin the outcomes are heads and tails

probability of heads is a half

probability of tails is a half

and the tosses are independent

meaning that the outcome of one toss has no influence

on the outcome of the next toss or any other toss.

symbol sequence I would get the orbit

and then I would convert that to L's and R's

and what I want to do is compare the statistical features of these L’s and R’s

to the statistical features of these H’s and T’s

and I’m going to imagine that we have an infinite amount of data

so we’ve done this infinitely long

and we’ve done this infinitely long.

Ok, that’s a little bit unrealistic.

But there is simplify the discussion and I’ll talk about again a little bit.

Ok, so the first thing we might do we have this infinitely long string sequence of L’s and R’s

so we could say what’s the frequency of L

what’s the frequency of R

and if we did that we would find that half of these symbols were L

and half were R

and we could do the same thing here.

I’ve tossed a fair coin a million and billion an infinite number of times

and we could ask the same question what’s the frequency of H?

what’s the frequency of R ?

and the frequency of H is a half

frequency of tails is a half

So all this says is the heads and the tails are equally likely

occur equally often and the L's and R's occur equally often

doesn’t mean the sequences are the same or have the same statistical features.

It just means that the statistics or the frequency of a single symbol are the same.

So now let’s do another experiment.

Let’s ask about the frequency of L L R L R L

LLR LLR and RR

I better not say that and write that instead, so

Ok, so I was trying to say the frequency of these outcomes

So if I scanned down this very long string and said

how often do I see or what fraction the time do I see an LL together

what fraction the time do I see an LR, RL and RR

and that would be a quarter for all of these

All the four possible pairs are equally likely

the same thing will happen to over here

so if I ask what’s the frequency of HH HT TT and TH

I would see a similar thing

so the frequency of all possible pairs

the frequencies are equal, they’re all equally likely

and the frequency of all possible pairs here are all equally likely.

and you can probably guess the next step

Let’s ask about the frequency of all possible triples LLL, LLR, LRL and so on

Ok, so for the symbolic dynamics, the symbolic sequence from the logistic equation

there are 8 possible outcomes of 3 consecutive symbols

LLL , LLR and so on and they all occur equally likely with a fraction of an eighth

And it’s the same story for the fair coin

there are 8 possible outcomes of 3 consecutive symbols

HHH, HTH and so on

and they all occur equally likely,

equally often their frequencies are the same 1 over 8

I could keep doing this experiment

I could do look at 4 consecutive symbols

5 consecutive symbols, 6 consecutive symbols

for both of these situations and I would see in all cases

all sequences occur equally often

so I considered all five, sequences of five possible point tosses out to see there

occur with equal frequency, same story with L's and R's over here.

So what this means is that the logistic equation

the particular symbolic dynamics from the logistic equation with r equals 4

are as random as a coin toss.

So that’s an important result that I want to think about for bit

So let me write that down.

So the logistic equation with r equals 4, is as random as a coin toss.

for coin toss all possible sequences of H's and T's are equally likely,

for the logistic equation all possible sequences of L's and R's are equally likely.

Here is another way to think about what this means

suppose I would do generate two symbolic sequences

and again that the names of the letters don’t matter

I could use H's and T's, 0's and 1's, the letters are arbitrary.

One of those very long symbol sequences I generated by tossing a coin

again and again and again and again

The other symbol sequence I would generate by iterating the logistic equation

again and again and again and again

and then writing it in symbolic form

and then I handed you those two sequences

and I ask you to tell me which was which

there would be no way

what this means is

there would be no way to tell those two sequences apart

one is generated by a fair coin

but we think of as sort of the most random process imaginable.

The other sequence is generated by iterating a function about as deterministic as it gets.

Nevertheless their outcomes are indistinguishable.

Now I should mention that of course if we were to do this experiment in real life

We can’t have an infinite sequence of fair coin tosses or iterates.

So what we'll have a very long one

these wouldn’t be exactly an eighth

these all wouldn’t be exactly an eight

some would be a little bit more

a little bit less but it wouldn’t effect the overall argument

that in the limit that the sequences

get longer and longer and longer

all of these frequencies become the same

and all of these frequencies become the same

and the two that two sequences are indistinguishable.

So we just saw an example of a deterministic function

producing a random outcome

and not just a random outcome but an outcome that I said is

in a sense as random as a fair coin.

Deterministically iterating a function produces something

that’s indistinguishable from a random coin toss.

So that seems really weird and we’re going to talk about that in a second.

But first I want to point out that this discovery

that iterating the logistic equation turning in a symbolic dynamics

make something that looks like a fair coin

is actually a tremendously useful result

because randomness is actually really useful resource

in science and in our lives.

Having access to randomness is in a sense a good thing

So let me explain what I mean by this.

In many numerical algorithms when does what’s called Monte Carlo simulation

and that requires access to a good source of random numbers

because it requires some sort of random sampling of some sort of a space.

In statistics it’s really important to do, to work with what’s called an unbiased sample

you want to get a random sample, so you have really big population

a million people in a country or state or something

and you want to figure out their statistical properties

some average income who knows

you want to make sure that

Ok, you could ask all million people, that probably takes too long

so instead, you ask some smaller sample

But you want that smaller sample of your larger population

to be representative of a whole.

and the best way to do that is to do a random sample

then I was easy and clear to know how to do that

but having access to good, some sort of good randomness algorithm

if that even makes sense, it does, as we’ll see is a very useful thing.

In other activities, randomness is also useful resource

So if I am engaged in some strategic interaction

with some opponent and that could be bargaining,

that could be, I don’t know, running away from somebody that’s trying to catch me,

evading somebody that’s trying to do something bad to me

I want my actions to be as unpredictable as possible

If the regularities and what I am doing

then my opponent can exploit that regularity

and that could be in a game like prisoner’s dilemma

or rock paper scissors or in some other sort of physical situations

where I’m running away from somebody

and I want to move left and right with equal, equal probabilities.

So in strategic interaction if you’re random

then there is no regularities that an opponent can exploit.

very similar story in cryptography I want to use,

there is a lots of different ways to do cryptography,

but one way is you need some symbol sequence, some source

yes, some symbol sequence

that you use to encode the message you want to keep secret

and the sequences that I’m using as part of that encoding scheme

needs to be as random or as close to random as possible

Because if there are regularities in that encoding sequence

then an opponent can exploit that

and could maybe figure out my secret messages

so anyway the point of all this is that may be we think of randomness

something unwanted and often it is unwanted.

I don’t want my car to like randomly starting in the morning or not

I’d like to know that even when it’s really cold out it’s going to start

So often we think of randomness something to avoid

but randomness actually is a really useful resource

so the logistic equation with r equals 4 can generate numbers

or bits zeros and ones

that can be used as a random number generator

which is really useful in all sort of applications

now I’m not an expert in random number generators

and I don’t but I don’t think that the logistic equation

in this exact form is used in random number generator

however it’s the same basic idea

one is iterating one the computer iterates a deterministic function

that is chaotic and so produces this random outcome

and then that serves as a source of randomness

for whatever is your computer is trying to do

so randomness is actually a useful resource in many circumstances

in any event in the next lecture we’ll dig deeper into randomness

because we clearly have to think a little more about what randomness means

given that we have a deterministic system producing a random result.