Rise of the machines & the fallacy of exponential thinking

     Unless you have been hiding under a rock for the past little while, you must have surely heard about artificial intelligence and the risks it poses to mankind. In fact I wont fault you for hiding under a rock after listening to some of the absolutely bombastic eventualities as described by some rather esteemed people of our generation. Most of the arguments made against AI are based on two fundamental premises. The first being that AI is growing at an exponential pace or close to it and that the risk of something happening in the near term (5-10 years) is real and definite, unless we control it today. The second premise is that since AI has the potential to become more intelligent than us, we have no surefire way of predicting how it will behave and therefore cannot use past technological developments as much of a basis because we’ve never created anything that has the ability to, wittingly or unwittingly, outsmart us. Nevertheless there is a fundamental flaw in this thinking. That flaw is "the fallacy of exponential thinking".

     To truly understand exponential thinking, one has to understand linear progression, and by that extension linear thinking. In mathematics linear progression is when something changes or progresses in fixed increments from one stage to another, and has a starting point and an ending point. An example is counting from 1 to 10. In this example you start from 1 and increment by 1 every time until you reach 10. Another example is increments of 5 (5,10,10,20 . . .). We have been doing this since we were in primary school and we easily understand linear progression. Linear thinking therefore is a process of thought following known cycles or step-by-step progression where a response to a step must be elicited before another step is taken. This is so ingrained in us that we automatically believe that  the world around us operates linearly and that all growth follows a linear pattern. That is until we first hear about exponential progression.

    Exponential progression on the other hand is an increase in number or size, at a constantly growing rate (eg:- 1,2,4,8,16,32 . . .). Compound interest is a good example of exponential growth. It seems easy enough at first, but to quote Albert Alan Bartlet "The greatest shortcoming of the human race is our inability to understand the exponential function".

      Lets use an example to better understand exponents. Lets assume you were to eat the ever popular M&M candies but with a twist. If you were to eat M&M candies that increment linearly for a month, on day 30 you would have eaten 30 candies weighing 27.44 gms with a total caloric intake of 90 calories. At the same token if you were to eat them growing exponentially on day 30 you would have eaten 536,870,912 candies weighing 491 metric tons and a total caloric intake in excess of 1 billion. Most people would never have guessed that on Day 6 the number of candies on the exponential side crossed the count of candies on the linear side. This is the power of exponents.

	Linear				Exponential
	M&M's	Calories	Weight (gm)		M&M's	Calories	Weight (gm)
Day 1	1	3	0.91		1	3	0.91
Day 2	2	6	1.83		2	6	1.83
Day 3	3	9	2.74		4	12	3.66
Day 4	4	12	3.66		8	24	7.32
Day 5	5	15	4.57		16	48	14.64
Day 6	6	18	5.49		32	96	29.27
Day 7	7	21	6.40		64	192	58.54
Day 8	8	24	7.32		128	384	117.08
Day 9	9	27	8.23		256	768	234.16
Day 10	10	30	9.15		512	1536	468.33
Day 11	11	33	10.06		1024	3072	936.65
Day 12	12	36	10.98		2048	6144	1873.31
Day 13	13	39	11.89		4096	12288	3746.61
Day 14	14	42	12.81		8192	24576	7493.22
Day 15	15	45	13.72		16384	49152	14986.44
Day 16	16	48	14.64		32768	98304	29972.89
Day 17	17	51	15.55		65536	196608	59945.78
Day 18	18	54	16.46		131072	393216	119891.56
Day 19	19	57	17.38		262144	786432	239783.12
Day 20	20	60	18.29		524288	1572864	479566.23
Day 21	21	63	19.21		1048576	3145728	959132.47
Day 22	22	66	20.12		2097152	6291456	1918264.93
Day 23	23	69	21.04		4194304	12582912	3836529.87
Day 24	24	72	21.95		8388608	25165824	7673059.74
Day 25	25	75	22.87		16777216	50331648	15346119.48
Day 26	26	78	23.78		33554432	100663296	30692238.95
Day 27	27	81	24.70		67108864	201326592	61384477.90
Day 28	28	84	25.61		134217728	402653184	122768955.80
Day 29	29	87	26.53		268435456	805306368	245537911.60
Day 30	30	90	27.44		536870912	1610612736	491075823.21

     Seeing a table like the one above re-enforces the fact that if AI is truly growing exponentially it poses huge risks for us. Fortunately for us, exponential growth works great on paper, but almost always runs into physical barriers in the real world. For example try folding a piece of paper in half more than eight times. Its starts to become very hard. the thickness of your average A4 size paper is roughly 0.1mm.  If you can fold it 10 times it will be approximately 2 inches high but if you somehow manage to fold it 20 times..... wait for it....  it will be 172 feet high and lastly if you fold it 30 times it will be roughly 53 kilometers high. You can see where this is going. The problem is that we just don't have the capability to fold paper that high or eat that many candies in a sitting. AI is going to hit hardware limitations and processing limitations and to get it to reach those next steps (the 20th fold of paper or the 20th day of eating candies) we will need to spend enormous resources (time, money, personnel, e.tc.). At some point we will ask ourselves why we are doing this at all. We will ask ourselves if it is truly worth diverting all these resources to achieve some utopian AI fantasy, when 80% of the worlds people still live on under 10$ a day.

      Let me make it clear. I am not against AI, especially "Specialized AI". It is what allows us to have self driving cars, enable businesses to predict online purchases and allow credit card companies to detect fraud. What irks me is when rich, famous people hijack the podium and use their money to tell us about the dangers of general AI as depicted in science fiction novels and popular movies. Humanity is far from the "general AI" of the HAL 9000 variety. Before we reach there humanity will come to the conclusion that its just not worth it. 

The Timelessness of Bulleh Shah

     Endless hours of YouTube binge watching over the weekend led me to a much forgotten, but loved treasure I has originally stumbled upon a few years back. It was a rendition of Bulleh Shah's "Ek Aliph" by Saieen Zahoor. It was as mesmerizing this time around as it was when I first listened to it. I also happened to hear it a few weeks into the violent death of Amjad Sabri, who was one of Pakistan’s most famous and respected musicians, celebrated for devotional songs from a centuries old offshoot of Islam with a focus on mysticism and asceticism called Sufism, of which both Shah and Zahoor hold lineage. The South Asian region in general, has somehow, over the years managed to stray away and eventually loose its rich roots of tolerance. An English translation of "Ek Aliph" will make all of us realize that not much has changed in the human condition between the 17th century and today. I just had to share it.

You read so many books

to know it all,

yet fail to ever read your

heart at all.

You rush to holy shrines to play a part,

Would you dare enter the shrine of your heart

You are quick to attack the evil one,

yet pride is a battle you have not won.

You grab for a star you can control,

yet fail to grasp the light in your soul.

Let the race end, my friend

Stop trying to be the one who knows,

for ‘God is One’ you need to know.

End the race, my friend

God is All we need! God is All!

Follow the wandering dervish!

If you deny the power of all that’s true,

God will not grant strength to you.

We are lost in this river of self,

no boat or streams are of any help.

End the race, my friend

Stop trying to know it all, my friend.

God is All we need! God is All!

Numbers with Character

     Number theory is an interesting branch of mathematics that deals with the study and analysis of integers and is primarily done for research or academic purposes. Recreational number theory on the other hand is mostly done by amateurs such as myself for fun and can involve finding patters and solving puzzles using numbers. One can slot number theory into three broad categories. The first being classic or elementary number theory, The second is algebraic number theory and lastly analytic number theory. The List of recreational number theory topics we will discuss herein purely deals with classic number theory.The dutch mathematician Hendrik Lenstra with tounge firmly in cheek stated that "Nowadays, when a Number Theorist applies for a grant, he says that Number Theory is used in cryptography, and so doing Number Theory is good for National Security. Back then, since it was before the discovery of America, they said Number Theory is used in music. But I won't comment on the progress of civilization since then:"

Perfect Numbers

They were most probably first discovered in the time period that spanned the 7th century BC and the 4th century AD by the then Greek speaking world. A perfect number can only be a positive number that equals the sum of its divisors, excluding itself. In the first examples listed the integer 6 is not only divisible by 1, 2 and 3 but the sum of 1,2 & 3 is also equal to 6.

6 = 1+2+3

28 = 1+2+4+7+14

496 = 1+2+4+8+16+31+62+124+248

Narcissistic Numbers

Greek mythology states that Narcissus who was the son of river god Cephissus and nymph Liriope was a person of exceptional beauty. Narcissus ended up dying because he was unable to leave a pool after falling in love with his own reflection. we also happen to have numbers that can’t get enough of themselves. They happen to be numbers that are the sum of its own digits when each digit is raised to the power of the number of digits. By that definition all one digit numbers are narcissistic. There are no two digit narcissistic numbers but there are four three-digit narcissistic numbers.

370 = 3^3 + 7^3 + 0^3

1^3 + 5^3 + 3^3

1^3 + 5^3 + 3^3

Mersenne Primes

Marin Mersenne who "Mersenne Primes" were named after is a very interesting individual. Mersenne was an ordained priest and a friar who while in Paris at the convent of the annunciation met other greats like René Descartes and Étienne Pascal (father of Blaise Pascal) who influenced his work. Mersenne also seemed to be a man of strong convictions who was an ardent supporter of Galileo during the geocentric vs heliocentric debates even though he was a Catholic. Mersenne primes are prime numbers that fit the pattern 2^p-1, where p is any prime number. An example is where p=2, therefore 2^2-1 = 3. Three happens to be a prime hence it is a Mersenne prime. Other Mersenne primes include

p = 3, therefore 2^3-1 = 7

p = 5, therefore 2^5-1 = 31

p = 7, therefore 2^7-1 = 127

p = 11, therefore 2^11-1 = 2047

Dudeney Primes

Henry Dudeney was an English civil servant who at a very young age was much inspired by his grandfather who was also a mathematician. He was considered one of Englands premiere recreational number theorists and puzzle maker. Dudeney noticed that some  numbers exhibit a behavior where the numbers individual digits sum add up to their cube root. There are only six known Dudeney numbers until someone proves this wrong.

1) number 1  where, 1 = 1, and 1*1*1 = 1

2) number 512  where, 5+1+2 = 8, and 8*8*8 = 512

3) number 4913 where 4+9+1+3 = 17 and 17*17*17 = 4913

4) number 5832 where 5+8+3+2 = 18 and 18*18*18 = 5832

5) number 17576 where 1+7+5+7+6 = 26 and 26*26*26 = 17576

6) number 19683 where 1+9+6+8+3 = 27 and 27*27*27 = 19683

Twin, Cousin & Sexy Primes

I first heard about twin, cousin and sexy primes while watching an episode of the "Colbert Report" when they had on Terence Tao the mathematical genius from Caltech. Twin primes are when the difference between two consecutive primes is equal to two. Examples of twin primes are:

(3,5) (5,7) (11,13) (17,19) (29,31)

Cousin primes are when the difference between two consecutive primes is equal to four. Examples of cousin primes are:

(3, 7), (7, 11), (13, 17), (19, 23), (37, 41)

Finally Sexy primes are when the difference between two primes is equal to (you guessed it) six:

(5,11), (7,13), (11,17), (13,19), (17,23)

     The recreational number theory topics listed above are the ones that intrigued me and more importantly the ones that I could understand. If this is something you like, you can lookup a fairly exhaustive list of topics listed on wiki which can be found here (https://goo.gl/OVVH0m). 

The Magic of Primes in PKI

     Almost all of us that live in an internet enabled world have used PKI (Public Key Infrastructure) in some form or fashion. If you work in the information technology field, more specifically on the security side of the house, you would have definitely heard of PKI. Every time we use email, do our online banking or connect to our offices networks using VPN we could possibly be using PKI. The PKI framework itself is so pervasive and robust that it is abstracted from the general user community. In the bargain we have forgotten that it evens exists or bother to think about how it works.

     From a high level perspective it is fairly easy to understand how PKI functions. It is essentially a two key (asymmetric) encryption system that provides authentication and confidentiality. Lets assume two users (Andy and Bob) want to communicate securely over a public network where transmitted data can by spied upon easily. To prevent the spying Andy can encrypt the message he sends to Bob using key X. The issue we now have is that if Andy encrypted the data using key X, how does he send the key to Bob to decrypt it without it being compromised. This problem is what the PKI framework solves. It does this by generating two keys per user (a public key and a private key) that are mathematically linked to each other. The public key as its name implies is public and can be shared with others. Now if Andy needs to communicate with Bob he requests the public key for Bob. Once Andy receives the key he encrypts the message and sends it to Bob. This message can only be decrypted using Bob's private key which never leaves his person. A hacker that compromised the encrypted message and Bob's public key will never be able to decrypt Andy's message leaving him rather perplexed.

     Lets take a look at the actual math behind PKI, because that is where it gets really interesting. Before we do that, we have to understand that PKI is a framework and not a technology implementation, therefore I am going to show you how the RSA algorithm that implements the PKI framework functions. The public key is actually made up of two parts. One is a composite of two primes (two primes multiplied by each other) which you will see below referenced as k1 and the second is an exponent which will be referenced as e. This is a special exponent that needs to fulfill the following requirements. The exponent needs to be an integer. The exponent cannot be a factor of k1. In addition it needs to be greater than 1 but less than φ(k1). What exactly is φ(k1) you ask ? It is all the numbers that are less than k1 but greater than 1 and do not share a common factor with k1.  It all sounds too complicated but lets take a look at φ(9) as an example.

1. Lets list out all the numbers that are lesser or equal to 9 = (1,2,3,4,5,6,7,8)
2. Now we will mark in red all the numbers that share a factor with 9 except for the number 1 and erase them (1,2,3,4,5,6,7,8)
3. We are now left with 6 numbers (1,2,4,5,7,8 ) 
4. So the φ(9) =  6

Step 1 - Generate Public Key

To generate a public key we pick two large primes and multiply them together to form a composite. For this example we will pick small primes so that it becomes easier to work with.

• k1 = PrA x PrB
• k1 = 23 x 29
• k1 = 667

Lets also pick an exponent that meets the requirements such that 1 < e < φ(k1). For this example I am going to pick  the number 3. What we are left with is two numbers k1 = 667 and e = 3. These two numbers together make up the public key.

Step 2 - Generate Private Key

• k2 = 2(φ(k1))+1 / e
• k2 = 2(φ(667))+1 / 3
• k2 = 2(616)+1 / 3
• k2 = 1232+1 / 3
• k2 = 1233 / 3
• k2 = 411

Remember that k2 is the private key that is never distributed to anyone, It is only used to decrypt messages that are sent to you which is encrypted by k1 and e which together make up your public key.

Step 3 - Data Encryption

Now that we have both a public and private key we can begin to encrypt data. We will attempt to encrypt the word "bag" as an example. Firstly we need to convert the word "bag" to some sort of numerical value so that we can do calculations against it. The easiest way to do this is to find the numerical place value of the letters. For example "b" will have a value of 2 as it is the second letter in the English alphabet. By that logic the word bag = 217. Now we can begin the encryption process.

• ed = data^e mod k1
• ed = 217^3 mod 667
• ed = 10218313 mod 667
• ed = 540

Step 4 - Data Decryption

If you are attempting to do this on a calculator even with the fairly small numbers that I have shown here the calculations are going to be huge. I am including a link for a big integer calculator here which will help you make these calculations (http://goo.gl/GXSdCv)

• dd = ed^k2 mod k1
• dd = 540^411 mod 667
• dd= 10323697896242668219954362115926597275247250862687291172394408598235646340560963277503503134090563294398280979796545134818573213064741511000526138653105369880225616050484894381599128674424417000626405170045975711797996349495340567543681315349755878142749780282862273187818683269529600679603730315376728985279076969364683520463773650596665685647638324598792650556617337357191019481317184045352277937331907517838610430973762375649970512352333359870591860750640525281545205463597201920722441526847683758018612285502996206303677486482364237501952886563083240964158025078640998912337940214285221812259921170151592015645065351114155483987914299497389946764598690494890769755112018882602072423747515993913989410381103104000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 mod 667
• dd = 217

Our decrypted message is 217. If we pass his through our letter place value calculation we will come up with the word "bag". 

Sieve of Eratosthenes (Part 2)

Eratosthenes

    Eratosthenes was by all accounts a polymath and probably the first recorded one in history. He was a close friend of Archimedes (of the Eureka fame) who also shared his love of Mathematics. Eratosthenes list of achievements is truly impressive. He was a mathematician, geographer, poet, astronomer, music theorist, the inventor of geography and the chief librarian at the ancient Library of Alexandria. While Eratosthenes worked at the library, a more modern theoretical physicist named Albert Einstein worked at the patent office. This allowed them to be exposed to many different ideas from various perspectives. This I believe is one of the critical reasons as to why both Einstein and Eratosthenes were so far ahead of their times.

    I am just going to focus on one of his achievements, which is a sieve used to find consecutive prime numbers that was named after him. Eratosthenes was a Greek citizen born in 276 BC in Shahhat, Libya and eventually died in 194 BC in his much beloved Alexandria, Egypt. After loosing his vision he suffered a bout of depression, leading him to voluntarily starve himself to death at the age of 82.  The reason it is important to keep these dates in perspective is because the next sieve to find primes was invented in 1934 by S. P. Sundaram, an Indian student 2128 years later. The one after that was invented in 2004 by Arthur Oliver Lonsdale Atkin a British mathematician 70 years later. Given the time that Eratosthenes lived in and the resources available to him, you can appreciate the sheer magnitude of his discovery. Although now there are so many faster ways to find large primes like "Fermat little theorem" or the "Sieve of Atken", Eratosthenes's sieve was used well into the 21st century, for cryptography and other purposes. 

    My simple python script to find primes turned into a history lesson which I thought is important to share. Learning is the act of acquiring by synthesizing different types of information. Not only were Einstein and Eratosthenes exposed to different ideas,  they wanted to learn by exploring the unknown. Today with the democratization of technology, and the penetration of the internet almost all information is a google search away. You are only bound by your willingness to learn. Keep your eyes open, find what you love to do and explore away. 

Sieve of Eratosthenes (Part 1)

    In an attempt to grasp the fundamentals of python, I decided to give myself a challenge. The challenge was to find prime numbers between 1 and x. The idea here was that while attempting to find these primes, I would end up learning about loops, conditional statements, breaks, e.t.c. I finished the challenge, but ended up learning a lot more than just the fundamentals of python.

    I started off with the brute force method. This method looped through a set of numbers one at a time, and then made sure that only 1 mod the current number and the current number mod itself had a remainder of 0. If so it was a prime number. Imagine looping though a set of numbers 1 to x, where x was greater than or equal to 1,000,000. You will quickly realise that although this method worked it had obvious flaws. It was inefficient and time consuming. To speed up the process I started storing the primes in memory as I kept finding them. Then I started looping through the found primes and doing a modulus with the current number. This reduced the number of modulo operations I had to do, but it was still fairly time consuming.

    I needed to find yet a faster method to find primes and came across a simple but elegant method called the "Sieve of Eratosthenes". Eratosthenes used a process of elimination until you were only left with primes. His method is summed up below. 

1. Create a list of consecutive numbers from 1 to x and cross out 1 because it is not a prime number.
2. Start with the next number p which is 2 in this case and circle it. Next eliminate all multiples of p (eg:- 4,6,8,e.t.c.)
3. p takes on the value of the next number in the list and repeat step 2
4. Keep going until p multiplied by p is greater than or equal to x

Sieve of Eratosthenes

    This was much quicker and could be implemented writing a lot less code. This got me interested in Eratosthenes and I wanted to know more about him. (Continued in Part 2 .... )