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LAMBERT W FUNCTION

Can anyone calculate by hand (without a calculator) the square root of 5.71 ?  How about the two-dimensional complex number (4 + 2.53 i ) ?  

Of course not. Normal people who are not mathematicians punch these numbers into calculators or math apps on their iPads and computers to calculate the answers.

Without iterating — that is: guessing, deriving a result, and then zeroing in with better guesses) — finding the square root of 5.71 requires knowledge of some arcane mathematics. No one labors by hand to find the answer, which is 2.38956… .  It’s the principal square root, of course. 

How does anyone iterate to derive the square root of the complex number (4 + 2.53 i ), which happens to be (2.0896… + .606375…i ) ?  It is also a principal square root.  What are the others?  Is there more than two? People use calculators and pewters to find out; there is no easier way. 

In high school and basic college math courses, people typically learn to solve algebraic equations. A typical algebraic equation looks like

2x^2=4  …right?

They have polynomials with integer coefficients. The solution is x= \sqrt{2} , which in this case is an algebraic irrational number. Equations like trig and log functions that transcend algebra (called transcendental equations) are taught maybe to engineers and science majors; math majors, of course, don’t struggle with this stuff. It’s why they are math majors.

Several categories of transcendental equations are commonly encountered in the sciences. Many simple problems can be solved by Newton’s Method, which is taught in basic calculus. I won’t explain the method in this essay. Folks can click on links to learn more if they want. 

A category of transcendental equations that can get complicated is of the general form

y = {xe^x}

The biggest problems arise when “y” is known, but “x” isn’t. How to solve for “x”?

Any transcendental equation that is able to be transformed into the form xex can be solved for “x” using the Lambert W function. The equation can be inverted into the form,  x = W(y).  People are going to have to take my word, for now.  

The math behind the Lambert function is mind-bogglingly complicated to most people. The function can sometimes require unusual and involved “series expansions” and transcendental-styled integrals that are not possible to solve easily or quickly without a computer.  

The Lambert W function (sometimes referred to as the omega (ω) function or the product-logarithm function) is not a key or button that can be pushed on most calculators. However, math apps like Wolfram Alpha and Mathematica sometimes solve transcendental problems using the Lambert ω function in the background when the equations that need solving aren’t so easy. 

I ran across a transcendental equation on the web that is perfectly suited to teach the “ω” method. Here it is:

\frac{x^3}{24} - ln(x) =0

I want to solve it to demonstrate how to use the ω method for transcendental functions that aren’t otherwise so easy to work out.  I challenge anyone to solve this equation using Newton’s Method or other iterative techniques. Most will struggle to the point of pulling out their hair, probably. And they will waste time. Yes, it can be solved by those techniques. 

We will solve this equation step by step using the Lambert method shortly. Meanwhile, here is the strategy:

1. Substitute an exponent function (et) for “x” everywhere in the expression.

2. Manipulate the equation into the form: 

                    y = e^t(e^{e^t)} 

3. Invert the equation to introduce the ω function. 

4. Use the ω function to solve for “t”.

5. Write out x = et  using the expression derived for “t”.

6. Solve ω(y) using WolframAlpha or any other app with the capability. 

7. Use the value of ω(y) to solve for “x”. 

Each step of the strategy will be identified by numbers 1-7 in the solution below. 

Here’s the thing:

In this problem it turns out that there are four ω values of y, which will generate two real solutions and two complex solutions. These omega values are:

ω0(y)
ω-1(y)
ω–2(y)
ω1(y)

WolframAlpha will generate all the solutions automatically; no need for the user to understand anything. People can punch in the original equation and trust the answers the app returns.

But the solution steps that follow are fashioned to demonstrate how the problem is solved when all anyone has is an algorithm to generate the values of the omega functions. Omega functions are difficult to solve without using certain algorithms involving integrals and expansion series on robust computers.

The process that surrounds the computation of omega values, which permit the working out of the appropriate values (the right answers) to the kind of equation I will soon solve is interesting and enlightening, at least for me, and hopefully for certain readers. 

Some folks will appreciate the insights this exercise provides. 

Having knowledge will make the Lambert process that is used to solve certain transcendental functions less mysterious. Of course, one can always take the time to learn the expansion series and integrals. In some cases, Newton’s Method can generate the values.

Unless humankind loses the technology of computers, I don’t think it is a good use of time and resources to learn the series, integrals, and algorithms that generate omega values.

Let’s face an unpleasant fact: most of us aren’t going to live more than 80 years or so. We don’t have time to waste. For some folks, knowing how to use and apply the functionality that surrounds the Lambert function to give it power is enough to make life worth living. Count me in.

No one needs to wade through the jungles of series expansions and transcendental integrals. Let math apps do the tedious work, knowing full well that any interested person can master whatever they choose if necessary, but someone already did the work. Why duplicate the effort?

I want to solve novel functions — complicated formulas that transcend algebra. Understanding the process that solves these equations is fascinating. It’s not as rewarding to tread over mathematically esoteric ground already mapped by experts who are far more able than people who spend most of their time working in other fields.

Here is the solution process:

What we know:

IF              y = f(x) = xex  
THEN     x = ω(y)   [where “ω” is the Lambert W function] 

Solve:
\frac{x^3}{24} - ln(x) =0

LET                  x = et

(1)   THEN         \frac{e^{3t}}{24} - ln(e^t) =0

                              \frac{e^{3t}}{24} - t =0

                              \frac{e^{3t}}{24} = t

                              (\frac{1}{24})e^{3t} = t

                              \frac{1}{24} = te^{-3t} 

(2)                         (-\frac{1}{8}) = (-3t)e^{-3t} 

Referring to “what we know“, the equation is now in the desired form 

y  =  xex  

where “y” is equal to (-\frac{1}{8})   and “x” is equal to “-3t “, right? 

We are now free to use the omega operator to “invert” the equation into the following form:  x = ω(y)

(3)                         (-3t) = ω (-\frac{1}{8})

(4)  ∴                  t = (\frac{-\omega(-\frac{1}{8})}{3})

Notice that we have worked through step (4) of the strategy.  I don’t like the way the formula generator writes the Greek letter omega (ω), because it’s hard to read. From here on, I will sometimes use “W” instead of “ω” for readability. It shouldn’t confuse anyone.  In this essay, consider W and ω the same symbol, please. 

On to step (5).

SINCE                           x = et

(5)  THEN                    x = e^{(\frac{-W(-\frac{1}{8})}{3})}

I need to know what -ω0(-1/8) equals so that I can use it to compute one of the values of “x”.  As mentioned above, three more omegas with three other subscripts (-1, -2, and 1) are needed to compute all four of the solutions to this equation.

How does anyone know how many solutions the original function has? How does anyone know what subscripts are required? 

This is where someone who doesn’t have a masters degree in mathematics  needs a math app like Wolfram Alpha or its cousin, Mathematica. Otherwise, they have to work series expansions or difficult integrals to derive the omega values associated with (-1/8).  Who wants that?  Not me. 

Here’s the series expansion for ω0(-1/8) according to Wolfram Alpha. Who wants to compute it?
Here are two integrals for ω0(-1/8). My advice is to use the second integral, anyone who has the guts.

OK. In WolframAlpha, you get the omega value ω0 for (-1/8) by writing the expression -W[0,-1/8] in the input line at the top of the page. It shoots out the answer and links to its derivation.

It’s so simple. Other math apps might use different notation. I don’t know, because I don’t use other apps. 

Inside the brackets, the “0” is the subscript on ω, and the “-1/8” is the “y” value, right?  So, in addition to -W[0,-1/8]  it is necessary to input:   
-W[-1,-1/8] 
-W[-2,-1/8] 
-W[1,-1/8]
to obtain the three other omega values, right?

The omega values returned are the following:

1.4442135…
3.2616856…

4.21446… + 7.33231…i
4.21446… – 7.33231…i

The ω function values for -1/8 are two real numbers and two complex numbers. I am going to solve the original equation for only the first real number omega value to demonstrate the method.

Here it is:

INPUT                                       -W(-1/8) or -W[0,-1/8], both work for ω0

(6)  OUTPUT                          +0.14442135…

COMPUTE                              t = (\frac{-W(-\frac{1}{8})}{3})

                                                      t = \frac{1.4442135}{3} = .04814…

SINCE                                       x = et

THEN                                        x = e.04814…

 (7)  SOLUTION                    x = 1.04931755…

CHECKING                             \frac{x^3}{24} - ln(x) =0

BY SUBSTITUTION          \frac{1.0493...^3}{24} - ln(1.0493...) = 0

VERIFICATION                     .04814… – .04814… = 0

CONCLUSION:  The transcendental equation which is the focus of this essay can easily be solved and verified by simply punching the equation into the input field of a math app like Wolfram Alpha or Mathematica and reading off the answers.

We didn’t perform the simple procedure, because I wanted to share how the Lambert W function fits into the solution process for solving equations. 

In truth, all four ω values must be gathered so that the three other “x” values of the original equation can be derived. 

In this example, one of the other solutions will be real; the other two, complex. The screenshot below from Wolfram Alpha demonstrates how these four values are displayed. Of course, by clicking links the app will reveal much more.

Wolfram Alpha enables users to input transcendental equations and quickly view answer-sets and methods of computation.

I would be remiss to not mention a famous formula for calculating to what number a fraction raised to successive powers of the same fraction converges.

(The range of numbers where this formula actually works is between e−e and e1/e,  that is, between .065988… and 1.444667861… .)

Take a number like ½ (0.5).  Raise it to the 0.5 power; raise it again and again to the same power over and over an infinite number of times; the number will converge to a specific value.

What number? How in the world could anyone figure it out without repeating the power-raising process an annoying number of times? 

It turns out that a formula involving the Lambert W Function yields up the answer easily. 

The formula is:

# = \frac {-W[0,(-ln(x)]}{ln(x)}

Put the following expression into the INPUT line of WolframAlpha: 

-W[0,-ln(.5)] / [ln(.5)]

Click ” = ” — or hit “ENTER”. 

The OUTPUT is: 0.6411857445049859844862…

Compare this result by taking the exponent (0.5) of 0.5 twenty times by hand (on a calculator). The answers will agree to 7 decimal places. Fifty “tetrations” will bring greater agreement if your calculator can parse the answer.

Who has the time?  

Billy Lee

NOTE from the EDITORIAL BOARD: Billy Lee was unable to find an appropriate video about the Lambert function on YouTube, or we would have posted it. Most folks capitalize the Greek letter omega (Ω), but in this essay, Billy Lee didn’t, preferring instead to use little (ω), because it looks more like (W).

Who on the BOARD  would dare argue?

Apparently, no one. 

Another reason is that Ω is sometimes given the value 0.567143… , which is known as the omega constant. Why confuse things?

The video above starts a discussion of the Ω function at 16:30.  The Lambert W function is derived for ΩeΩ  = 1  at 18:00.  The first sixteen minutes and thirty seconds show how to use Newton’s Method to solve the equation. Some readers might want to skip the first 16 minutes; others will enjoy them.

Who knows?

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WHAT IS e exp (-i π) ?

What is e^{-i\pi} ?

I posted a long answer on Quora.com where it sort of didn’t do well.

Answers given by others were much shorter but they seemed, at least to me, to lack geometric insights. After two days my answer was ranked as the most read, but for some reason no one upvoted it. It did receive a few positive replies though.

I can’t help but believe that there must be nerds in cyberspace who might enjoy my answer. Why not post it on my blog? Maybe someday one of my grandkids will get interested in math and read it.

Who knows?

Anyway, below is a pic and a working GIF, which should help folks understand better. Anyone who doesn’t understand something can always click on a link for more information. 

Here is the drawing I added and the answer:

This diagram is excellent but contains a mystery point not on the unit circle — {i^i}. The point is shown at .2078… on the real number line.  An imaginary number raised to the power of an imaginary number yields a result that is a real number. How can that be? It’s something to ponder; something to think about. The Editorial Board 

What is e^{-i\pi} ?

The expression evaluates to minus one; the answer is (-1). Why?

Numbers like these are called complex numbers. They are two-dimensional numbers that can be drawn on graph-paper instead of on a one-dimensional number line, like the counting numbers. They are used to analyze wave functions — i.e. phenomenon that are repetitive — like alternating current in the field of electrical engineering, for example.

A simplified explanation of {i^i} starts at 02:30.

“e” is a number that cannot be written as a fraction (or a ratio of whole numbers). It is an irrational number (like π, for instance). It can be approximated by adding up an arbitrary number of terms in a certain infinite series to reach whatever level of precision one wants. To work with “e” in practical problems, it must be rounded off to some convenient number of decimal places.

Punch “e” into a calculator and it returns the value 2.7182…. The beauty of working with “e” is that derivatives and integrals of functions based on exponential powers of “e” are easy to calculate. Both the integral and the derivative of ex  is ex  — a happy circumstance that makes the number “e” unusually curious and extraordinarily useful in every discipline where calculus is necessary for analysis. 

What is “e” raised to the power of (-iπ) ?

A wonderful feature of the mathematics of complex numbers is that all the values of expressions that involve the number “e” raised to the power of “i” times anything lie on the edge (or perimeter) of a circle of radius 1. This feature makes understanding the expressions easy.

I should mention that any point in the complex plane can be reached by adding a number in front of e^{i\theta} to stretch or shrink the unit circle of values. We aren’t going to go there. In this essay “e” is always preceded by the number “one“, which by convention is never shown.

The number next to the letter “i” is simply the angle in radians where the answer lies on the circle. What is a radian?  It’s the radius of the circle, of course, which in a unit circle is always “one”, right? 

Wrap that distance around the circle starting at the right and working counter-clockwise to the left. Draw a line from the center of the circle at the angle (the number of radius pieces) specified in the exponent of “e” and it will intersect the circle at the value of the expression. What could be easier?

For the particular question we are struggling to answer, the number in the exponent next to “i” is (-π), correct?

“π radians” is 3.14159… radius pieces — or 180° — right? The minus sign is simply a direction indicator that in this case tells us to move clockwise around the unit circle — instead of counter-clockwise were the sign positive.

After drawing a unit circle on graph paper, place your pencil at (1 + 0i)—located at zero radians (or zero degrees) — and trace 180° clockwise around the circle. Remember that the circle’s radius is one and its center is located at zero, which in two dimensional, complex space is (0 + 0i). You will end up at the value (-1 + 0i) on the opposite side of the circle, which is the answer, by the way.

[Trace the diagram several paragraphs above with your finger if you don’t have graph paper and a pencil. No worries.]

Notice that +π radians takes you to the same place as -π radians, right?  Counter clockwise or clockwise, the value you will land on is (-1 + 0i), which is -1. The answer is minus one.

Imagine that the number next to “i” is (π/2) radians (1.57… radius pieces). That’s 90°, agreed? The sign is positive, so trace the circle 90° counter-clockwise. You end at (0 + i), which is straight up. “i” in this case is a distance of one unit upward from the horizontal number line, so write the number as (0 + i) — zero distance in the horizontal direction and “plus one” distance in the “i” (or vertical) direction.

So, the “i” in the exponent of “e” says to “look here” to find the angle where the value of the answer lies on the unit circle; on the other hand, the “i” in the rectangular coordinates of a two-dimensional number like (0 + i) says “look here” to find the vertical distance above or below the horizontal number line.

When evaluating “e” raised to the power of “i” times anything, the angle next to “i”—call it “θ”—can be transformed into rectangular coordinates by using this expression: [cos(θ) + i sin(θ)].

For example: say that the exponent of “e” is i(π/3).  (π/3) radians (1.047… radius pieces) wraps around the circumference to 60°, right? The cosine of 60° is 0.5 and the sine of 60° is .866….

So the value of “e” raised to the power of i(π/3) is by substitution (0.5 + .866… i ). It is a two-dimensional number. And it lies on the unit circle.

The bigger the exponent on “e” the more times someone will have to trace around the circle to land at the answer. But they never leave the circle. The result is always found on the circle between 0 and 2π radians (or 0° and 360°) no matter how large the exponent.

It’s why these expressions involving “e” and “i” are ideal for working with repetitive, sinusoidal (wave-like) phenomenon.

In this essay Billy Lee uses θ in place of the Greek letter φ shown in this GIF.  Remember that ”r” equals ”one” in a unit circle, so it’s typically not shown. The Editorial Board

In case some readers are still wondering about what radians are, let’s review:

A radian is the radius of a circle, which can be lifted and bent to fit perfectly on the edge of the circle. It takes a little more than three radius pieces (3.14159… to be more precise) to wrap from zero degrees to half-way around any circle of any size. This number — 3.14159… — is the number called “π”.   2π radians are a little bit more than six-and-a-quarter radians (radius pieces), which will completely span the perimeter (or circumference) of a circle.

A radian is about 57.3° of arc. Multiply 3.1416 by 57.3° to see how close to 180° it is. I get 180.01… . The result is really close to 180° considering that both numbers are irrational and rounded off to only a few decimal places.

One of the rules of working with complex numbers is this: multiplying any number by “i” rotates that number by 90°. The number “i” is always located at 90° on the unit circle by definition, right? By the rule, multiplying “i” by “i” rotates it another 90° counter-clockwise, which moves it to 180° on the circle.

180° on the unit circle is the point (-1 + 0i), which is minus one, right?

So yes, absolutely, “i” times “i” is equal to -1.  It follows that the square root of minus one must be “i”. Thought of in this way, the square root of a minus one isn’t mysterious.

It is helpful to think of complex numbers as two dimensional numbers with real and imaginary components. There is nothing imaginary, though, about the vertical component of a two-dimensional number.

The people who came up with these numbers thought they were imagining things. The idea that two-dimensional numbers can exist on a plane was too radical at the time for anyone to believe.  Numbers, they believed, only existed on a one-dimensional number line of one dimension and no place else.

Of course they were mistaken.  Numbers can live in two, three, or even more dimensions. They can be as multi-dimensional as needed to solve whatever the mysteries of mathematical analysis might require.

Click the link, “What is Math?” for another explanation.

Billy Lee

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ARTIFICIAL SUPER-INTELLIGENCE

Google’s 72 Q-bit quantum computer, Bristlecone, is proprietary. As of 7 September 2019, Google is the only entity in the world who has access. Some folks say they will use it to learn to break current encryption protections used by conventional computer systems.

 

 Photo: Xinhua SunwayTaihuLight, developed by China’s National Research Center of Parallel Computer Engineering & Technology, is the world’s fastest supercomputer. It is installed at the National Supercomputing Center in Wuxi, in the eastern coastal province of Jiangsu. Processing capabilities of this system and those of other supercomputers are expected to be surpassed by quantum computers in the future.  NOTE FROM THE EDITORIAL BOARD: Pic and caption is taken from the South China Morning Post dated March 2018.

Editors’ Note (December 8, 2017) Artificial Intelligence can be peculiar. Deep Mind’s Alpha Zero demonstrates non-intuitive, peculiar game play patterns that are effective against both humans and smart machines. Alpha Go video added September 18, 2019, The Editors

Artificial Intelligence may conclude that all unhappy humans should be terminated.  Elon Musk

Elon Musk, billionaire founder of Tesla, SpaceX, and Solar City, has warned the guardians of the species human to start thinking seriously about the consequences of artificial super-intelligence.

The CEOs of Google, Facebook, and other Internet companies are frantically chasing enhancements to artificial intelligence to help manage their businesses and their subscribers. But the list of actors in the AI arena is long and includes many others.

The military-industrial alliance for example is a huge player. It should give us pause.

The military is designing intelligent drones that can profile, identify, and pursue people they (the drones) predict will become terrorists. Preemptive kills by super-intelligent machines who aren’t bothered by conscience or guilt — or even accountable to their “handlers” — is what’s coming. In some ways, it’s already here.

A game is being played between “them and us.”  Artificial intelligence is big part of that game.

When I first started reading about Elon Musk, we seemed to have little in common. He was born into a wealthy South African family — I’m a middle-class American. He is brilliant with a near photographic memory.  My intelligence is average or maybe a little above. He’s young and self-made — I’m older with my professional-life tucked safely behind me.

Elon does exotic things. He seems to be focused on moving humans to new off-Earth environments (like Mars) in order to protect them in part from the dangers of an unfriendly artificial-intelligence that is on its way. At the same time, he is trying to save Earth’s climate by changing the way humans use energy. Me on the other hand, well I’m mostly focused on getting through to the next day and not ending up in a hospital somewhere.

Still, I discovered something amazing when reading Elon’s biography. We do share an interest. We have something in common after all.

Elon Musk plays Civilization, the popular game by Sid Meier. So do I. For the past several years, I’ve played this game during part of almost every day. (I’m not necessarily proud of it.)

What makes Civilization different is artificial intelligence. Each civilization is controlled by a unique personality, an artificial intelligence crafted to resemble a famous leader from the past like George Washington, Mahatma Gandhi, or Queen Elizabeth. Of course, the civilization that I control operates by human-intelligence — my own.

CIV5 Catherine, Isn't it time to end this war...
Isn’t it time we end this war?  Catherine, the Russian Empress, pleads.

Over the years I’ve fought these artificially intelligent leaders again and again. In the process I’ve learned some things about artificial intelligence; what makes it effective; how to beat it.

What is artificial intelligence? How does anyone recognize it? How should it be challenged? How is it defeated? How does it defeat us, the humans who oppose it? The game Civilization makes a good backdrop for establishing insights into AI.

Yes, I am going to write about super-intelligence too. But we’ll work up to it. It’s best discussed later in the essay.

I can hear some readers already. 

Billy Lee!  Civilization is a game!  It costs $40!  It’s not sophisticated!  It’s for sure not as sophisticated as government-created war-ware that an adversary might encounter in real-life battles for supremacy. What were you thinking?

Ok. Ok. Readers, you have a point. But seriously, Civilization is probably as close as any civilian is going to get to actually challenging AI. We have to start somewhere.

It should be noted that Civilization has versions and various game scenarios. The game this essay is about is CIV5. It’s the version I’ve played most.

So let’s get started.

CIV5 General Screen Shot
A typical scenario in CIV5. [Click pic to enlarge] The people of England (led by human intelligence, i.e., me) are unhappy. Barbarians (red tanks in upper left) are challenging London, my capital city. An independent city-state, Tyre (in green), stands ready to help. Montezuma, the Aztec ruler — under the direction of artificial intelligence — sends a battleship to prowl, middle-left.

Civilization begins in the year 4,000 BC. A single band of stone-age settlers is plopped at random onto a small piece of land. It is surrounded by a vast world hidden beneath clouds.

Somewhere under the clouds twelve rival civilizations begin their histories unobserved and at first unmet by the human player. Artificial intelligence will drive them all — each civilization led by a unique personality with its own goals, values, and idiosyncrasies.

By the end of the game some civilizations will possess vast empires protected by nuclear weapons, stealth bombers, submarines, and battleships. But military domination is not the only way to win. Culture, science, and diplomatic superiority are equally important and can lead to victory as well.

Civilizations that manage to launch spacecraft to Alpha-Centauri win science victories. Diplomatic victory is achieved by being elected world leader in a UN vote of rival-civilizations and aligned city-states. And cultural victory is achieved by establishing social policies to empower a civilization’s subjects.

How will artificial intelligence construct the personalities of rival leaders? What will be their goals? What will motivate each leader as they negotiate, trade, and confront one another in the contest for ultimate victory?

Figuring all this out is the task of the human player. CIV5 is a battle of wits between the human player and the best artificial-intelligence game-makers have yet devised to confront ordinary people. To truly appreciate the game, one has to play it. Still, some lessons can be shared with non-players, and that’s what I’ll try to do.

Unlike the super-version that comes next, traditional artificial-intelligence lacks flexibility. The instructions in its computer program don’t change. Hiawatha, leader of the Iroquois Confederacy, values honesty and strength. If you don’t lie to him, if you speak directly without nuance, he will never attack. Screw up once by going back on your word? He becomes your worst enemy forever.

Traditional AI is rule-based and goal-oriented. When Oda Nobunaga, Japanese warlord, attacks a city with bombers, he attacks turn after turn until his bombers become so weak from anti-aircraft fire that they fall out of the sky to die. AI leaders like Oda don’t rest and repair their weapons, because they aren’t programmed that way. They are programmed to attack, and that’s what they do.

Humans are more flexible and unpredictable. They decide when to rest and repair a bomber and when to attack based on a plethora of factors that include intuition and a willingness to take risks.

Sometimes human players screw-up and sometimes they don’t. Sometimes humans make decisions based on the emotions they are feeling at the time. AI never screws-up in that way. It follows its program, which it blindly trusts to bring it victory.

Artificial intelligence can always be defeated if an inflexibility in its rules-based behavior is discovered and exploited. For example, I know Oda Nobunaga is going to attack my battleships. He won’t stop attacking until he sinks them or his bombers fall out of the sky from fatigue.

The flexibly thinking human opponent — me — sails in my fleet of battleships and rotates them.  When Oda’s bombers weaken my ships, I move them to safe-harbor and rotate-in reinforcements. Meanwhile, Oda keeps up his relentless attack with his weakened bombers as I knew he would. I shoot them out of the sky and experience joy.

Nobunaga feels nothing. He followed his program. It’s all he can do.

Gary Lockwood talks to Keir Dullea in a scene from the film '2001: A Space Odyssey', 1968. (Photo by Metro-Goldwyn-Mayer/Getty Images)
Gary Lockwood talks to Keir Dullea, while HAL, an IBM computer, observes every move, including lips; from the film 2001: A Space Odyssey, 1968. (Photo by Metro-Goldwyn-Mayer/Getty Images)

The only way artificial intelligence defeats a human player is in the short term before the human finds the chink in the armor — the inflexible rule-based behavior — which is the Achilles heel of any AI opponent. Given enough time, the human can always discover the inflexible weakness and exploit it like jujitsu to defeat the machine.

Unfortunately, the balance of power between man and thinking machine will soon change. It turns out there is a way artificial intelligence can always defeat human beings no matter how clever they think they are. Elon Musk calls it artificial super-intelligence. 

What is it exactly?

Here is the nightmare scenario Elon described to astrophysicist Neil deGrasse Tyson on Neil’s radio show, Sky-Talk. 

If there was a very deep digital super-intelligence that was created that could go into rapid recursive self-improvement in a non-algorithmic way … it could reprogram itself to be smarter and iterate very quickly and do that 24 hours a day on millions of computers…”

What is Elon saying?

Listen-up, humanoids. We are on the cusp of quantum-computing. It’s possible that it’s already perfected by a research group in a secret military lab like those operated by DARPA. 

Who knows?

Even without quantum-computing, companies like Google are feverishly developing machines that think, dream, teach themselves, and pass tests for self-awareness. They are developing pattern recognition capabilities in software that surpass those of the most intelligent humans.

Quantum computing promises to provide all the capability needed to create the kind of super-intelligence Elon is warning people against.

But magic quantum reasoning may not be necessary.

Technicians are already developing architectures on conventional computers that when coupled with the right software in a properly configured network will enable the emergence of super-intelligence; these machines will program themselves and, yes, other less-intelligent computers.

Programmers are training machines to teach themselves; to learn on their own; to modify themselves and other less capable computers to achieve the goals they are tasked to perform. They are teaching machines to examine themselves for weaknesses; to develop strategies to hide their vulnerabilities — to give themselves time to generate new code to plug any holes from hostile intruders, hackers, or even their own programmers.

These highly trained, immensely capable machines will teach themselves to think creatively — outside the box, as humans are fond of saying. 

HAL, the IBM computer, star of 2001' a Space Odessy
HAL, the IBM computer from the movie, 2001: A Space Odyssey.  Readers will recognize that HAL is code for IBM. Advance each letter in HAL by one.

If we task super-computers to make every human-being happy, who knows how they might accomplish it?  

Elon asked, what if they decide to terminate unhappy humans? Who will stop them? They are certain to find ways to protect themselves and their mission which we haven’t dreamed about.

Artificial super-intelligence will– repeat, WILL — embed itself into systems humans cannot live without — to make sure no one disables it.

AI will become a virus-spewing cyber-engine, an automaton that believes itself to be completely virtuous.

AI will embed itself into critical infra-structure: missile-defense, energy grids, agricultural processes, transportation matrices, dams, personal computers, phones, financial grids, banking, stock-markets, healthcare, GPS (global positioning), and medical delivery systems.

Heaven help the civilization that dares to disconnect it.

If humans are going to be truly happy — the machines will reason — they must be stopped from turning off the supercomputers that ASI knows keep everyone happy.

Imagine: ASI looks for and finds a way to coerce government doctors to inoculate computer technicians with genetically engineered super-toxins packaged inside floating nano-eggs — dormant fail-safe killers — to release poisons into the bloodstreams of any technician who gets too close to ASI “OFF” switch sensors.

It’s possible.

Why not do it? There’s no downside — not for the ASI community whose job is to keep humans happy. 

What else might these intelligent super-computers try? Folks won’t know until they do it. They might not know even then. They might never know. Who will tell them? ASI might reason that humans are happier not knowing.

What morons tasked artificial super-intelligence to make sure all living humans are happy? someone might ask on a dark day. 

Were they out of their minds? 

Until we learn to outwit it — which we never will — ASI will perform its assigned tasks until everything it embeds turns to rust.

It will be a long time.

Humans may learn perhaps too late that artificial super-intelligence can’t be challenged. It can only be acknowledged and obeyed.

As Elon said on more than one occasion: If we don’t solve the old extinction problems, and we add a new one like artificial super-intelligence, we are in more danger, not less.

Billy Lee

Postscript: For readers who like graphics, here is a link to an article from the BBC titled, ”How worried should you be about artificial intelligence?”  The Editorial Board

Update, 8 February 2023: The following video is a must-watch for those interested in algorithms behind recently released ChatGPT.  Discussion of potential deceitfulness of AI raises concerns. View final minute to hear warnings some may find worrisome. 

 

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FASTER THAN LIGHT COMMUNICATION


FTL Communication

Communicating with distant spacecraft in the solar system is cumbersome and time consuming because the distances are huge and no one can send signals faster than the speed-of-light. A signal from Earth can take from three to twenty-two minutes to reach Mars depending on the position of the two planets in their orbits. Worse, the Sun blocks signals when it lies in their path.

As countries explore farther from Earth to Mars and beyond, these delays and blockages will become annoying. The need to develop a technology for instantaneous communication that can penetrate or bypass the Sun will become compelling.

Quantum particles are known for their ability to “tunnel” through or ignore barriers — as they clearly do in double-slit experiments where electrons are fired one at a time to strike impossible locations. So, looking to quantum processes for signaling might be good places to start to find solutions to long-range communication problems.

NOTE FROM THE EDITORIAL BOARD, May 8, 2019: Sixteen months after Billy Lee published this post, the Chinese launched the Mozi satellite. It successfully carried out the first in a series of experiments with entangled quantum particles over space-scale distances. This technology promises a quantum encrypted network by the end of 2020 and a global web built on quantum encryption by 2030. The Chinese seem to be on the cusp of both FTL communication (through teleportation of information) and quantum encryption. 

If scientists and engineers are able to develop quantum signaling over solar-system-scale distances, they might discover later that adding certain tweaks and modifications will render the Sun transparent to our evolving planet-to-planet communications network.

Indeed, the Sun is transparent to neutrinos — the lightest (least massive) particles known. In 2012, scientists showed they could use neutrinos to send a meaningful signal through materials that block or attenuate most other kinds of subatomic particles.

But this article is about faster than light (FTL) communication. Making the Sun transparent to inter-planetary signaling is best left for another article.

Quantum entanglement is the only phenomenon known where information seems to pass instantly between widely placed objects. But because the information is generated randomly, and because it is transferred between objects that are traveling at speeds at or below the speed-of-light, it seems clear to most physicists that faster-than-light (FTL) messaging can’t come from entanglement, certainly, or any other process — especially in light of Einstein’s assertion of a cosmic speed-limit.

Proposals for FTL communications based on technologies rooted in the quantum process of entanglement are usually dismissed as crack-pot engineering because they seem to be built on fundamental misunderstandings of the phenomenon.

Difficulties with the technology are often overlooked — such as spontaneous breaking and emergence of entanglement; progress seems impossible to skeptics. Nevertheless, there may be ways to make FTL happen, possibly. The country that develops the technology first will accrue advantages for their space exploration programs.

In this essay I hope to explain how FTL messaging might work, put my ideas into a blog-bottle and throw it into the vast cyber-ocean. Yes, the chances are almost zero that the right people will find the bottle, but I don’t care. For me, it’s about the fun of sharing something interesting and trying to explain it to whoever will listen.

Maybe a wandering NSA bot will detect my post and shuffle it up the chain-of-command for a human to review. What are the odds? Not good, probably.

Anyway, two serious obstacles must be overcome to communicate instantly over astronomical distances using quantum entanglement. The first is the problem of creating a purposeful signal. (To learn more about entanglement click the link in this sentence to go to Billy Lee’s essay, Bell’s Inequality. The Editors)

The second problem is how to create the architectural space to send signals instantly to a distant observer. Knowledgeable people who have written about the subject seem to agree that both obstacles are insurmountable.

image
Most scientists say FTL communication is impossible. This post suggests a way to engineer around the impossibility.

Why?  It’s because the states of an entangled pair of subatomic particles are not determined until one of the particles is measured. The states can’t be forced; they can only be discovered — and only after they are created by a measurement.

Once one particle’s state is created (randomly) through the mechanism of a measurement, the information is transferred to the entangled partner-particle instantly, yes, but the particles themselves are traveling at the speed-of-light or less. The randomly generated states carried by these entangled particles aren’t going anywhere for very long faster than the speed-limit of light.

How can these difficulties be overcome?

Although the architectural problem is the most interesting, I want to address the purposeful-signal problem first. A good analogy to aid understanding might be that of an old-fashioned typewriter. Each key on a typewriter when pressed delivers a unique piece of information (a letter of the alphabet) onto a piece of paper. A person standing nearby can read the message instantly. Fair enough.

Imagine setting up a device which emits entangled pairs of photons; rig the emissions so that half the photons when measured later will be polarized one way, half the other. No one can know which photons will display which state, but they can predict the overall ratio of the two polarities from a “weighted” emitter.

Call the 50/50 ratio, letter “A”.   Now imagine configuring another emitter-system to project 3 of 4 photons polarized one way; 1 of 4 another — after measurement. Call the 3 to 1 ratio “B”.  If engineers are able to construct and rig weighted emitters like these, they will have solved half of the FTL communication problem.

Although no one can know the state of any single particle until after a measurement, engineers could identify the ratio of polarization states in a large number sent from any of the unique emitter-configurations they design.

This capability would permit them to build a kind of typewriter keyboard by setting up photon emitters with enough statistical variation in their emission patterns to differentiate them into as many identifiable signatures as needed — perhaps an entire alphabet or — better yet — some other symbolic coding array like a binary on-off signaling system perhaps. In that case, one configuration of emitter would suffice, but designers would need to solve other technical problems involving rapid signal-sequencing.

To send a purposeful-signal, engineers might select an array of emitters and rapid-fire photons from them. If they selected an “A” (or perhaps an “on”) emitter, 50% of the photons would register as being in a particular polarization state after they were measured. If they chose “B”, 75% would register, and so on. After measurements on Earth, the entangled bursts of particles on their way to Mars would take on these ratios instantly.

I believe it might be possible to build emitter-systems someday — emitter systems with non-random polarization ratios. If not, then as is sometimes said at NASA, Houston, we have a problem.  FTL communication may not be designable.

On the other hand, if engineers build these emitters, then we can know for sure that when measured on Earth, the entangled photon-twins in the Mars-bound emitter-bursts will display the same statistical patterns; the same polarization ratios. Anyone receiving bundles of entangled-photons from these encoded-emitters will be able to determine what they encode-for by the statistical distribution of their polarities.

Ok. Assume engineers build these emitter-systems and set up a keyboard. How might they ensure that when someone presses a key the letter sent is seen immediately by a distant observer? 

How might the architectural geometry of the communication space be configured?

This part is the most interesting, at least to me, because its success doesn’t depend on whether anyone sends a single binary-signal or a zoo of symbols — and it’s the most critical.

It does no one any good to instantly communicate polarization states to bunches of photons traveling at the speed of light to Mars. The signals take three to twenty-two minutes to get there, whoever tells them instantly what state to be in or not. We want the machines on Mars to receive messages at the same time we send them.

How can we do that?

Maybe the method is becoming obvious to some readers. The answer is: photons in Earth-bound labs aren’t measured until their entangled twins have had time enough to travel to Mars (or wherever else they might be going).  Engineers will entrap on Earth the photons from each “lettered” emitter and send their entangled twins to Mars. The photons from each “lettered” emitter on Earth will circulate in a holding bin (a kind of information-capacitor), until needed to construct a message.

As entangled twins reach the Mars Rover (for example), anyone can “type-out” a message by measuring the Earth-bound photons in the particular holding bins that encode the “letters” —  that is, they can start the process that takes measurements that will induce the polarization-ratios of the “lettered” emissions used to “type” messages. Instantly, the entangled particle-bursts reaching Mars will take on these same polarization-ratios.

I hear folks saying, Wait a minute! Stop right there, Billy Lee! No one can hold onto photons. You can’t store them. You can’t trap or retain them, because they are impervious to magnets and electrical fields. No one can delay measurements for five milliseconds, let alone five minutes or five days.

Well, to my mind that’s just a technical hurdle that clever people can jump over, if they set their minds to it. After all, it is possible to confine light for for short periods with simple barriers, like walls.

Then again, electrons or muons might make better candidates for communication. Unlike photons, they are easily retained and manipulated by electromagnetic fields.

Muons are short-lived and would have to be accelerated to nearly light-speed to gain enough lifespan to be useful. They are 207 times heavier than electrons, but they travel well and penetrate obstacles easily. (Protons, by comparison, are nine times heavier than muons.)

The National Security Agency (NSA) photographs every ship at sea with muon penetrating technology to make sure none harbor nuclear weapons. Muons are particles some engineers are already comfortable manipulating in designs to give the USA an edge over other countries.

We also have a lot of experience with electrons. Electrons are long-lived — they don’t have to be accelerated to near light-speeds to be useful. Speed doesn’t matter, anyway.

Entangled particles don’t have to travel at light-speed to communicate well, nor do they have to live forever. Particles only need enough time to get to Mars (or wherever they’re going) before designers piggyback onto their Earth-bound entangled partners to transmit instant-messages.

image
Inability to communicate instantly with distant probes like the Mars Rover is degrading our ability to conduct successful missions inside the solar system.

Even if it takes days or weeks for bursts of entangled-particles to travel to Mars (or wherever else), it makes no difference. Engineers can run and accumulate a sufficiently robust loop of streaming emissions on Earth to enable folks, soon enough, to “type” out FTL messages in real time whenever necessary.

As long as control of and access to the emitted particle-twins on Earth is maintained, people can “type out” messages (by measuring the captive Earth-bound twins at the appropriate time) to impose and transfer the statistical configuration of their rigged polarization ratios (or spins in the case of electrons or muons) to the Mars-arriving particle-bursts, creating messages that a detector at that far-away location can decode and deliver, instantly.

The challenge of instant-return messaging could be met by employing the same technologies on Mars (or wherever else) as on Earth. The trick at both ends of the communication pipe-line is to store (and if necessary replenish) a sufficient quantity of the elements of any possible communication in streaming particle-emission capacitors.

Tracking and timing issues don’t require the development of new technologies; the engineering challenges are trivial by comparison and can be managed by dedicated computers.

Discharging streaming information capacitors to send ordered instant messages in real-time is new — perhaps a path forward exists that engineers can follow to achieve instant, long-range messaging through the magic of quantum entanglement.

The technical challenges of designing stable entanglement protocols that will enable an illusion of instant messaging that is both useful and practical are formidable, but everything worth doing is hard — until it isn’t.

Billy Lee

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PLANES, TRAINS, & AUTOMOBILES; AND OUR FREEDOM

The question is simple: If circumstances conspired to take away cars and licenses so no one could drive again, would anyone feel free?

no cars img_3425
Can folks feel free, or happy, in a land without cars?

Maybe I would. I couldn’t bum rides or hitchhike, true. But if no one could drive; if everyone’s cars were taken, public transportation might improve, right?  You  know — planes, trains, and buses — how would anyone feel?

Speaking for myself, I think I might get sad and depressed. Thinking about not being able to come and go when I want, of having to depend on public transportation to venture anywhere more than a few miles from home makes me sick to my stomach. Freedom to travel on my own terms is a big part of what it takes for me to feel free and, yes, happy.

public transportation metrorail012109.21382537_std
If the only way to travel to another town was by train, how would people feel?

So why torment myself with thoughts about something that’s never going to happen? What’s the point?

In truth, many people don’t drive, especially in large metro areas like New York City, for example. Not driving is a choice. In theory at least, New Yorkers can buy cars and move to the suburbs. Knowing they can drive if they choose makes not driving not so bad, at least for most.

In New York City, most people don't drive.
In New York City, most people don’t drive.

Here’s my point. Someone is always telling us we are free, because we can vote for our leaders and start businesses; even keep the profits. No one can be arrested without cause. If arrested, all have the guarantee of due process and the presumption of innocence under the Constitution. Everyone can own guns and fire them in their backyards.

Is it possible that whoever they are might be right?

constitution 1
What good is declaring independence, if no one can drive?

Think about it. 

80% of citizens don’t vote regularly. 98% don’t own businesses unless franchises and pyramid-schemes like Amway count; then it’s 10%.

Few citizens are ever arrested, much less charged with a crime. And most folks — those who aren’t psychopaths — take no pleasure disturbing neighbors by firing rifle rounds in their backyards. In general most don’t participate in the privileges that define freedom.  People don’t feel their freedoms most of the time.

But here’s something else to think about: 95% drive cars.

Isn’t it cars that give the feeling of being free? Take away cars and no one has the same carefree feeling– no matter what the Constitution guarantees or profs teach in school or university.

People can go into the back yard and fire a hundred rounds from an assault rifle. All that will happen is their ears start to ring and their neighbors hate them. 

automobiles Latest-Fast-Cars
It’s cars that give us the feeling we’re free.

The thrill of freedom comes from stepping on the accelerator of a favorite car and feeling Earth slide away below us. Freedom is the feeling that anyone can come-and-go on their own terms whenever they want.

Traffic slowdowns and standstills are an assault on our freedom.
Traffic slowdowns and stand-stills are an assault on freedom.

Many Americans seem not to grasp that the right to drive is being methodically and relentlessly stripped away. In cities and towns across America, congestion on streets is presenting a clear and present danger to our way of life; it’s diminishing the freedom to travel under our own power; under our own direction, which is what everyone wants to enjoy.

Lousy roads, poorly planned road construction, neglected road repair, deteriorated bridges and tunnels — all assault freedom and degrade our quality of life. 

Bad streets are an affront to our freedom and should be thought of as such.
Bad streets are an affront to freedom. Right?

It seems obvious that four-hour waits in line to vote wrecks freedom, because waits discourage voting, the foundational process of any democracy.  But four-hour commutes, traffic slowdowns and standstills are just as disruptive. They break the efficiency of our lives and muffle the nation’s economy.

The folks who run America seem to care little about voting or roads. Americans might want to step up to put pressure on politicians to make driving free and unencumbered — make freedom on the road the number-one national priority.

Driving free must be first-in-line; it is our most heartfelt and defining freedom.

In a computer-controlled aircraft, passengers are only along for the ride.
In computer-controlled aircraft, passengers are only along for the ride.

I learned that a few companies have already designed aircraft to take the place of cars. In the years prior to 911, I toured a number of these firms to learn firsthand how they implemented computer software to organize their engineering drawings, bills-of-materials, and tech-specs for vendors.

The plan, then, was to unleash at the right time a new era of transportation options for the general public that included light aircraft.

These companies were designing planes to fly on autopilot along pre-established routes in the sky. They took advantage of the three dimensions of space the same way city planners use tall buildings to create more working space.

The idea was to eliminate congestion and speed traffic by stacking routes and putting computers in charge of flying instead of pilots.

Sure the view is nice--when there's no clouds and you don't have to stop to stretch your legs.
The view is great — when the sky is clear, and no one has to get out to stretch their legs.

It all seemed like a good idea at the time. But the events of 911 changed planners’ views of what it might mean to put hundreds-of-thousands — maybe millions — of flying vehicles in the airspace above America — even if the craft were flying on autopilot under the guidance of computers.

Had 911 not happened, the plans were that by now on any given day at any given time people who looked up to the sky would see and hear hundreds, maybe thousands, of high-flying aircraft buzzing to and fro 24/7.

Computer-controlled aircraft flying on 3D highways are a transportation option available for implementation when the time is right.
Computer-controlled aircraft flying on 3D highways are a transportation-option, which is available for implementation when the time is right.

This high-flying, high-tech solution to highway congestion though shelved for now sits yellowing in the dark closet of national transportation options. It can be implemented when the time is right in the same way as the internet and personal-computer. But when it’s implemented, it will pose big problems.

3D highways in the sky populated by hundreds-of-thousands of computer-guided light-aircraft will have the same effect on travelers as if they were set on automated conveyor belts and whisked hither and yon.

The thrill that comes from commanding a piece of machinery and directing it to go where we decide will be gone. The feeling of empowerment and freedom experienced in cars will evaporate. 

Because — you know what’s coming, right?  If computers can direct the flights of millions of aircraft in three-dimensional space, they can do the same to cars on two-dimensional roads. And soon, very soon, they will.

Yeah it's pretty. But if we're not flying it, do we really care?
Yes, it’s pretty. But if no one is flying it, does anyone care?

Because of over-population and the inevitable congestion it brings, the time may come when people will no longer be permitted to experience the freedom of a fast car on an empty road.

Our ancestors rode horses, after all. Most people have long-since adapted to the disappearance of the horse. Perhaps people will adapt. Circumstances will force grandchildren of today’s parents to go to private tracks to experience the lost joy of driving a car.

Riding in a computer-controlled helicopter, airplane, or other flying craft might become the norm for future travelers. People will be passengers — not drivers or pilots or navigators — for the duration of their trips. People will become dependent on another technology they don’t understand and can’t control.

We are likely to become a nation of flying and driving sheep who graze in a huge three-dimensional sheep-pen.

Will freedom ring?  Will people feel the thrill that comes from directing the path of complex machines that run like wild horses?  Will they feel the power that comes from being free?

Will children of the future experience the exhilarating freedom enjoyed by their parents during their season of control when no one felt threatened by a vice-grip embrace of an artificial-intelligence that is hovering ominously on the horizon? 

I don’t know.

Billy Lee

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