December 24, 2016
Renowned NSA cryptographer Jacques Saunière hurried toward the east exit of the OPS2A building at Fort Meade. As he lunged for the door, the corridor went dark like it sensed his attempted escape. His face was dimly lit by the blue glow of his iPhone, with Mrs. Saunière’s number already dialed. The building was on lockdown. Saunière was trapped inside.
He heard the click of a pistol’s hammer locking into place, followed by a voice. “Put away the phone. Who knows about the oracle?”
“I told you already,” the mathematician stammered as he turned to face his killer. “I don’t know what you’re talking about!”
“You’ve made a big mistake, Saunière. The exploit will be released tonight. The world will see what happens when the NSA puts backdoors in their algorithms. Your citizens will never trust their government again.”
The cryptographer reached for his pocket and pulled out a knife. There wasn’t much time.
Robert Langdon awoke to a clamorous knock on the front door of his Airbnb. “Professor Langdon?” the visitor squalled from outside. “I need to talk to you. Can I come in?”
Langdon groaned. Does this happen with every Airbnb? He glanced down at a crumpled flyer on the bedside table, reminding him how his life ended up here.
CARNEGIE MELLON UNIVERSITY
an evening with Robert Langdon
Professor of Symbolic Systems, Stanford University
“Professor?” the voice continued. The drowsy computer scientist surrendered. The visitor made herself at home.
“My name is Sophie Neveu. I work for the NSA.”
Langdon greeted her. NSA agents aren’t his typical cup of tea, but this one seemed benign.
“Your life is in danger. What do you know about this symbol?” Neveu produced a grotesque photograph taken at NSA headquarters, depicting a lifeless Saunière with a strange-looking letter carved into his chest.
“Jesus! Who did this to him?”
“No, Professor, you misunderstand. Yes, Saunière was murdered, but he carved this symbol into his own body. We found a message draft on his phone: Find Robert Langdon.”
“That symbol is the Greek letter lambda. An icon of the ancient Church of Alonzo,” Langdon explained.
“The Church of Alonzo?”
“The Church, if it even still exists, preserves an ancient language rumored to give the clergy a miraculous power: the ability to construct new worlds. Entire universes, not bound by the laws of physics as we know them today. That language was called the lambda calculus.”
“Yes—the lambda calculus! That’s what I need to talk to you about. We have reason to believe there is a group of black hat insurgents with a computational oracle, some kind of impossibly powerful computer. They’re planning to use it to break the cryptographic algorithms that secure the Internet.” Neveu’s anxiety was contagious.
“What do we know about this oracle?”
“It’s fast—faster than any computer we’ve seen. But it only understands one language—”
“—the lambda calculus?” The computer scientist interjected.
“That’s right, and we need your help to break it.”
Neveu reached for her satchel and pulled out a laptop. She opened a terminal and was greeted with the following prompt:
Welcome to the Lambda Oracle, version 1.0.0. Copyright © 1936. The Church of Alonzo. Enter terms at the prompt. $
Langdon glanced at the screen. “What is this?”
“It’s the oracle. My laptop has an SSH connection to it,” Neveu explained. “The insurgents did what they could to lock it down, but the NSA has tools for breaking into systems like this. Now, I need you to teach me that ancient language.”
Langdon recoiled. “You’re telling me you have access to a magical supercomputer with nearly infinite resources? I need to call my colleagues at Stanford—”
“No! You must not tell anyone!” Neveu bellowed.
“But this changes everything! Just imagine: we could build a massive neural network—a brain! Or we could run protein folding algorithms. We can cure diseases! This oracle can save the world!”
“—or destroy it.” Neveu didn’t share Langdon’s enthusiasm. “This computer can factor integers. You know what that means.”
“It can break cryptography,” Langdon realized.
“Exactly. Secure communications, bitcoin, vote counting, the New York Stock Exchange, all vulnerable. If we don’t stop it, this thing will catalyze the biggest financial collapse in human history. Civilization as we know it will be destroyed.” Neveu paused, as if she too just realized what this meant.
Langdon was convinced. “So how fast is this oracle?”
“Our analysts estimate it can do 10^100 beta reductions per second. A googol. More than the number of atoms in the observable universe. Per second.”
Langdon stood in awe.
“Teach me the ancient language,” Neveu demanded. “How does the lambda calculus work?”
“It’s remarkably simple,” Langdon explained. “It’s just functions. You know the identity function
x ↦ x?”
λx. xinto the terminal. “This is how you write it in the lambda calculus.” He pressed the enter key. The oracle echoed his input.
Welcome to the Lambda Oracle, version 1.0.0. Copyright © 1936. The Church of Alonzo. Enter terms at the prompt. $ λx. x Result: λx. x
Neveu looked puzzled. “What are the inputs and outputs of functions in the lambda calculus?”
“The inputs and outputs are also functions. It’s functions all the way down,” Langdon clarified.
“How do you do computation then?”
“You can apply a function to an argument,” Langdon continued, grabbing a pen and notebook. “With traditional notation, you would write
f(x). In the lambda calculus, we just write
“How do you compute
“You evaluate expressions with beta reduction. You substitute the argument for the function’s formal parameter. Plug and chug, as they say. You keep doing that until no more reductions are possible. Consider the identity function. What happens if you apply it to itself? You just get the identity function back.” Langdon demonstrated his claim at the prompt.
$ (λx. x) λx. x Result: λx. x
Neveu was astounded by the elegance and simplicity of this language. “That’s it?”
“That’s it,” Langdon confirmed. “Apparently the oracle can do that 10^100 times per second.”
“What about functions with multiple parameters?” Neveu’s curiosity was insatiable.
“In a higher-order system like the lambda calculus, you don’t need functions with multiple parameters. You can simulate them. Here is a function of one argument which returns a function of another argument, and that function applies the first argument to the second.”
$ λf. λx. f x Result: λf. λx. f x
“It’s like a single function that takes two arguments
x,” Langdon continued. “In fact, with a sacred ritual called Church encoding, you can express all kinds of things in this simple language: numbers, booleans, trees, etc. It’s rumored that the Church of Alonzo even found a way to do recursion.”
“What’s so hard about recursion?” Neveu wondered. “Can’t you just write
f = λx. f x?”
“That’s the tricky part. The lambda calculus is such a simple language, it doesn’t even have
=. But, according to legend, the Church has a special expression that implemented general recursion. They called it the Y combinator. No one knows how it was constructed.”
In that moment, Neveu realized their next move. “We have to find out how they did it. We’re going to hit the oracle with a denial of service attack and stop the insurgents from destroying the world.”
Langdon offered Neveu a bottle of Soylent, interrupting her daydream of re-discovering the legendary Y combinator. The algae in the drink triggered a complicated and poorly-understood metabolic pathway from her stomach to her brain, leading to a moment of brilliance.
“Professor, what if there was an expression which, when reduced, results in the same expression?”
“So you could reduce it again and again, ad infinitum?” Langdon was catching on.
“Exactly. Look at this function.” Neveu scribbled
λx. x xinto Langdon’s notebook. “It takes
x x. In a sense, it duplicates its input. What if we used it to duplicate itself?”
“A mathematical copy machine!”
“Exactly.” Neveu reached for the keyboard and entered
(λx. x x) λx. x x. Something strange happened.
$ (λx. x x) λx. x x (λx. x x) λx. x x (λx. x x) λx. x x (λx. x x) λx. x x (λx. x x) λx. x x ...
“What is going on?” Langdon asked.
“The oracle keeps reducing this expression, but every reduction results in the original expression again. It’s stuck in an infinite loop!” Neveu exclaimed.
Langdon’s phone began to ring.
“Professor Langdon, this is NSA Director Bezu Fache. Do not react to this call. You are in grave danger. Sophie is lying to you.”
“Who is it?” Neveu asked.
“A colleague from Stanford. Something about the faculty lunch schedule. I’ll just be a moment.” Langdon returned to his call.
Fache continued, “Sophie is trying to use you to derive the Y combinator. She’s part of the insurgent group. You have to stop her!”
“Derive it yourself before she does. Use it to destroy the oracle.” Fache ended the call.
Langdon looked up. Neveu was gone; she must have figured out what that call was really about. Time was running out.
He pondered Neveu’s mathematical copy machine. It diverges, but it’s useless. It doesn’t help us compute factorial, Fibonacci, or any other recursive function. Now it was his turn to be brilliant.
(λx. x x) λx. x x
Langdon stared at Neveu’s fascinating but useless discovery as he drank the Soylent.
What if you wrap the
x xin a function call? So instead of
x x, we have
f (x x), he thought to himself. We could choose some function
fto be called at every level of recursion. He picked up his notebook and wrote:
λf. (λx. f (x x)) (λx. f (x x))
What does this function do? What happens when you apply it to some
g? Langdon wondered.
(λf. (λx. f (x x)) (λx. f (x x))) g = (λx. g (x x)) (λx. g (x x)) = g ((λx. g (x x)) (λx. g (x x)))
Langdon had just discovered
Y, with the remarkable property that
Y g = g (Y g)for any
g. “What if
gdecides not to use its argument?” he pondered. “Then, in a language with lazy evaluation, the recursion would terminate!”
Let’s take a moment to understand what Professor Langdon just discovered. Suppose
gis a curried function of two arguments (i.e., it takes the first argument and returns a function which takes the second argument). What happens when you compute
Y g? You get back
g, except the first argument has already been supplied. So you actually get a modified version of
gthat only takes the second argument. And what was given for the first argument? The modified
gcan use its first argument to call itself.
Hoping the Church of Alonzo would forgive him for introducing some heretical syntax into their sacred calculus, Langdon wrote out the factorial function:
Y λf. λn. if n == 0 then 1 else n * f (n - 1)
His phone was ringing again.
“Professor Langdon, I have good news and bad news.” It was Fache.
“Bad news first.”
“Our analysts discovered the oracle has pre-emptive multitasking. So a divergent computation isn’t enough to destroy it,” Fache explained.
“What’s the good news?”
“It doesn’t have a garbage collector. Use the Y combinator to eat up all of its memory and stop Neveu from causing a global catastrophe.”
“You don’t need the Y combinator for that. You just need some divergent term that isn’t tail recursive.
(λx. x x x) λx. x x xwill do,” Langdon pointed out.
“Brilliant.” Fache ended the call without saying goodbye. Typical Fache.