## My hobby: proof engineering

November 25, 2017

I’m a software engineer at a place. I like the work and the people, and I learn a lot from my teammates. Many of them work very hard, so much that they don’t enjoy programming for fun anymore. I still love recreational programming, but in a peculiar sense.

When I come home from work, I try to prove theorems in a proof assistant. Usually the theorems are related to functional programming or type systems.

It’s a masochistic hobby. Convincing a computer that a theorem is true can be quite difficult compared to convincing a human. But if I can get a computer to accept my proof, then a) I must have a pretty good understanding of it, and b) it really must be right!

I use the Coq theorem prover for formalizing proofs. Coq is a beautiful and simple language [1]—far simpler than the languages I use at work! Its beauty is hidden beneath a rather unsightly IDE [2]:

The proof is on the left. The green highlighted part has been verified by Coq. As I am working on a proof, I can ask Coq to step forward through it with

`Ctrl+Down`

, and I can use `Ctrl+Up`

to step backward. At any point if my proof is wrong, Coq will highlight the error in red. It’s similar to how a type checker works—actually, it’s exactly the same in a very deep sense. I’d love to explain what I mean by that, but I’ll have to save that for separate article.In the screenshot, I’m currently in the middle of a proof. The upper right pane shows what I need to prove (in this case,

`f x = x`

) and also what hypotheses I can use to prove it (shown above the horizontal line). The lower right pane shows helpful information if I request it, such as the type of something or a list of definitions/lemmas/theorems related to it.Coq is not a toy, although it can be fun like one. Many serious results have been proven using it, including the four color theorem and the correctness of a certain C compiler.

Recently I set up a small GitHub repository with a continuous integration system that checks my proofs when I push to a branch, preventing me from merging it into master if there is a mistake. It’s impossible to commit incorrect math to this repository [3]!

Proving theorems in a machine-checked language is invigorating once you get the hang of it. It feels like doing a puzzle. And when you finish a long proof of a nontrivial result, it’s a remarkable sensation unlike anything else.

### Case study: domain theory

When you write down a recursive function in any language, it’s actually not obvious that such a definition is mathematically well-defined. This week I was reading about the theoretical underpinning of recursive definitions: fixed points. Whenever you define something recursively, you are technically taking a fixed point of a continuous function (yes, you read that correctly!). There is some really cool math behind this called domain theory.

Before this week, I had a vague understanding of all this. I do a fair amount of functional programming, but I hardly knew anything about domain theory. Today, I finished proving the main result: the Kleene fixed-point theorem. This was a difficult proof for me, but now I consider myself pretty well-versed in the basic ideas of domain theory.

The proof is not very remarkable on paper, but convincing a computer to accept it is another matter entirely. I started by writing down the definitions below. The first step to understanding a theory is being able to formally define the main ideas! I don’t expect the casual reader to dissect all these definitions, but I include them here to give you a sense of what it looks like to write them in Coq:

(* Assumption: Let (T, leq) be a partially ordered set, or poset. A poset is a set with a binary relation which is reflexive, transitive, and antisymmetric. *) Parameter T : Set. Parameter leq : T -> T -> Prop. Axiom refl : forall x, leq x x. Axiom trans : forall x y z, leq x y -> leq y z -> leq x z. Axiom antisym : forall x y, leq x y -> leq y x -> x = y. (* A supremum of a subset of T is a least element of T which is greater than or equal to every element in the subset. This is also called a join or least upper bound. *) Definition supremum P x1 := (forall x2, P x2 -> leq x2 x1) /\ forall x3, (forall x2, P x2 -> leq x2 x3) -> leq x1 x3. (* A directed subset of T is a non-empty subset of T such that any two elements in the subset have an upper bound in the subset. *) Definition directed P := (exists x1, P x1) /\ forall x1 x2, P x1 -> P x2 -> exists x3, leq x1 x3 /\ leq x2 x3 /\ P x3. (* Assumption: Let the partial order be directed-complete. That means every directed subset has a supremum. *) Axiom directedComplete : forall P, directed P -> exists x, supremum P x. (* Assumption: Let T have a least element called bottom. This makes our partial order a pointed directed-complete partial order. *) Parameter bottom : T. Axiom bottomLeast : forall x, leq bottom x. (* A monotone function is one which preserves order. We only need to consider functions for which the domain and codomain are identical and have the same order relation, but this need not be the case for monotone functions in general. *) Definition monotone f := forall x1 x2, leq x1 x2 -> leq (f x1) (f x2). (* A function is Scott-continuous if it preserves suprema of directed subsets. We only need to consider functions for which the domain and codomain are identical and have the same order relation, but this need not be the case for continuous functions in general. *) Definition continuous (f : T -> T) := forall P x1, directed P -> supremum P x1 -> supremum (fun x2 => exists x3, P x3 /\ x2 = f x3) (f x1). (* This function performs iterated application of a function to bottom. *) Fixpoint approx f n := match n with | 0 => bottom | S n => f (approx f n) end.

Next, I proved a few lemmas. To keep the article short, I will only state them; the full proofs are here.

(* We will need this simple lemma about pairs of natural numbers. *) Lemma natDiff : forall n1 n2, exists n3, n1 = add n2 n3 \/ n2 = add n1 n3. (* The supremum of a subset of T, if it exists, is unique. *) Lemma supremumUniqueness : forall P x1 x2, supremum P x1 -> supremum P x2 -> x1 = x2. (* Scott-continuity implies monotonicity. *) Lemma continuousImpliesMonotone : forall f, continuous f -> monotone f. (* Iterated applications of a monotone function f to bottom form an ω-chain, which means they are a totally ordered subset of T. This ω-chain is called the ascending Kleene chain of f. *) Lemma omegaChain : forall f n m, monotone f -> leq (approx f n) (approx f m) \/ leq (approx f m) (approx f n). (* The ascending Kleene chain of f is directed. *) Lemma kleeneChainDirected : forall f, monotone f -> directed (fun x2 => exists n, x2 = approx f n).

To give you an idea of what an actual proof looks like in Coq, below is a proof of one of the easier lemmas above. The proof is virtually impossible to read without stepping through it interactively, so don’t worry if it doesn’t make any sense to you here.

Lemma supremumUniqueness : forall P x1 x2, supremum P x1 -> supremum P x2 -> x1 = x2. Proof. intros. unfold supremum in H; destruct H. unfold supremum in H0; destruct H0. apply H1 in H0. apply H2 in H. apply antisym; auto. Qed.

And finally, this is the main result I proved (the proof is here):

(* The Kleene fixed-point theorem states that the least fixed-point of a Scott- continuous function over a pointed directed-complete partial order exists and is the supremum of the ascending Kleene chain. *) Theorem kleene : forall f, continuous f -> exists x1, supremum (fun x2 => exists n, x2 = approx f n) x1 /\ f x1 = x1 /\ (forall x2, f x2 = x2 -> leq x1 x2).

There’s nothing better than typing

`Qed`

at the end of a long proof and watching it turn green as Coq verifies it. What’s really amazing to me is that Stephen Kleene probably proved this without the help of a computer, but I was able to verify it with the highest possible scrutiny. It’s as if he wrote an impressive program without ever having run it, and it turned out not to have any bugs! This is more than just an analogy. When you work with Coq, you become intimately familiar with the interpretation of propositions as types and proofs as programs.### Case study: soundness of a type system

A successful formula for writing papers about programming languages theory is the following:

- Define the syntax of a small but novel language (usually based on the lambda calculus).
- Define a type system for the language (a set of rules that specify which programs are legal).
- Define a semantics for the language (i.e., describe what happens when you run a program).
- Prove that the type system prevents programs from “getting stuck” according to the semantics.

The theorem that is proven in that last step is called

*soundness*. A few months ago, I wanted to try doing a formally-verified soundness proof.The simply-typed lambda calculus is essentially the smallest programming language with higher-order functions and a meaningful type system [4]. I started by defining the abstract syntax of the language in Coq:

Inductive term : Set := | eUnit : term | eVar : id -> term | eAbs : id -> type -> term -> term | eApp : term -> term -> term with type : Set := | tUnit : type | tArrow : type -> type -> type.

To keep this article short, I won’t describe how the language works. There are many good introductions. My favorite is Chapter 9 of Benjamin C. Pierce’s wonderful book called

*Types and Programming Languages*.Below are the typing rules. You can add whatever base types you like (e.g., Booleans, integers, etc.) to the simply-typed lambda calculus, but to keep things simple I only have a rather useless “unit” type which has only a single value (akin to an empty tuple).

Inductive hasType : context -> term -> type -> Prop := | htUnit : forall c, hasType c eUnit tUnit | htVar : forall x t c, lookupVar c x = Some t -> hasType c (eVar x) t | htAbs : forall e x t1 t2 c, hasType (cExtend c x t1) e t2 -> hasType c (eAbs x t1 e) (tArrow t1 t2) | htApp : forall e1 e2 t1 t2 c, hasType c e1 t1 -> hasType c e2 (tArrow t1 t2) -> hasType c (eApp e2 e1) t2.

To specify the semantics, I defined a

*small-step operational semantics*. The main idea is that you define a relation which says what steps a program takes when it is run. There are many ways to define the semantics of a programming language, and this is just one of them.Inductive step : term -> term -> Prop := | sBeta : forall e1 e2 x t, value e2 -> step (eApp (eAbs x t e1) e2) (sub e1 x e2) | sApp1 : forall e1 e2 e3, step e1 e2 -> step (eApp e1 e3) (eApp e2 e3) | sApp2 : forall e1 e2 e3, value e1 -> step e2 e3 -> step (eApp e1 e2) (eApp e1 e3).

How do we know when a program has finished running? We just have to define what kinds of programs are “fully evaluated.”

Inductive value : term -> Prop := | vUnit : value eUnit | vAbs : forall e x t, value (eAbs x t e).

This states that a

*value*(i.e., an expression which has been fully evaluated) is either the unit constant or a function. So function applications are not values, since they can be reduced.An expression is in

*normal form*if there are no more steps it can take:Definition normalForm e1 := ~ exists e2, step e1 e2.

A

*stuck*program is one which is in normal form but is not a value:Definition stuck e := normalForm e /\ ~ value e.

Soundness means a program that is well-typed will never get stuck (

`stepStar`

is the transitive reflexive closure of the `step`

relation defined above):Theorem soundness : forall e1 e2 t, hasType cEmpty e1 t -> stepStar e1 e2 -> ~ stuck e2.

The proof, including definitions and lemmas, is about 500 lines long. For such a simple language, this was quite a difficult undertaking! I recall working on this for an entire weekend, probably 8 hours both days. Now I really appreciate when research papers include a formally verified soundness proof.

### Conclusion

I think of Coq as an extension of my own ability to reason logically. When I’m in doubt about something, I open the Coq IDE and try to prove it. I think the reason this is so valuable to me is that I often mull over functional programming, types, logic, algorithms, etc. These kinds of things are well-suited to formalization.

Unfortunately, machine-checked theorem proving is not very easy to learn, which is probably why it’s so esoteric. It’s difficult to prove nontrivial results without a graduate-level understanding of type theory (even then it’s still difficult). The most accessible introduction is Benjamin C. Pierce’s Software Foundations, which is freely available online. After that, read Adam Chlipala’s Certified Programming with Dependent Types, which is also freely available online.

I am not an expert in Coq by any means, and my proofs are probably longer and clunkier than they need to be. I only know basic techniques, but a nice thing about logic is that the underlying rules are very simple. I have reached a point where I can prove anything in Coq that I can rigorously prove on paper (given enough time). The two proofs I described in this article took me a few days each, and I’m steadily getting better.

### Footnotes

- Strictly speaking, the language itself is called
*Gallina*, and “Coq” refers to the system as a whole. - There is also an Emacs-based interface, but I’m a Vim user. A friend helped me set it up with Spacemacs, but I still found navigating proofs in CoqIDE to be more a little more intuitive.
- I should know better than to say it’s
*impossible*. Coq allows you to add your own axioms, which is dangerous because your axioms might be inconsistent. - One could argue that the SK combinator calculus is smaller and still fits this description.

## What are covariance and contravariance?

July 21, 2017

Subtyping is a tricky topic in programming language theory. The trickiness comes from a pair of frequently misunderstood phenomena called

*covariance*and*contravariance*. This article will explain what these terms mean.The following notation will be used:

`A ≼ B`

means`A`

is a subtype of`B`

.`A → B`

is the type of functions for which the argument type is`A`

and the return type is`B`

.`x : A`

means`x`

has type`A`

.

### A motivating question

Suppose I have these three types:

Greyhound ≼ Dog ≼ Animal

So

`Greyhound`

is a subtype of `Dog`

, and `Dog`

is a subtype of `Animal`

. Subtyping is usually transitive, so we’ll say `Greyhound`

is also a subtype of `Animal`

.**Question:**Which of the following types could be subtypes of

`Dog → Dog`

?`Greyhound → Greyhound`

`Greyhound → Animal`

`Animal → Animal`

`Animal → Greyhound`

How do we answer this question? Let

`f`

be a function which takes a `Dog → Dog`

function as its argument. We don’t care about the return type. For concreteness, we can say `f : (Dog → Dog) → String`

.Now I want to call

`f`

with some function `g`

. Let’s see what happens when `g`

has each of the four types above.**1. Suppose**

`g : Greyhound → Greyhound`

. Is `f(g)`

type safe?No, because

`f`

might try to call its argument (`g`

) with a different subtype of `Dog`

, like a `GermanShepherd`

.**2. Suppose**

`g : Greyhound → Animal`

. Is `f(g)`

type safe?No, for the same reason as (1).

**3. Suppose**

`g : Animal → Animal`

. Is `f(g)`

type safe?No, because

`f`

might call its argument (`g`

) and then try to make the return value bark. Not every `Animal`

can bark.**4. Suppose**

`g : Animal → Greyhound`

. Is `f(g)`

type safe?Yes—this one is safe.

`f`

might call its argument (`g`

) with any kind of `Dog`

, and all `Dog`

s are `Animal`

s. Likewise, it may assume the result is a `Dog`

, and all `Greyhound`

s are `Dog`

s.### What’s going on?

So this is safe:

(Animal → Greyhound) ≼ (Dog → Dog)

The return types are straightforward:

`Greyhound`

is a subtype of `Dog`

. But the argument types are flipped around: `Animal`

is a *supertype*of`Dog`

!To state this strange behavior in the proper jargon, we allow function types to be

*covariant*in their return type and*contravariant*in their argument type. Covariance in the return type means`A ≼ B`

implies `(T → A) ≼ (T → B)`

(`A`

stays on the left of the `≼`

, and `B`

stays on the right). Contravariance in the argument type means `A ≼ B`

implies `(B → T) ≼ (A → T)`

(`A`

and `B`

flipped sides).**Fun fact:**In TypeScript, argument types are

*bivariant*(both covariant and contravariant), which is unsound (although now in TypeScript 2.6 you can fix this with

`--strictFunctionTypes`

or `--strict`

). Eiffel also got this wrong, making argument types covariant instead of contravariant.### What about other types?

**Question:**Could

`List<Dog>`

be a subtype of `List<Animal>`

?The answer is a little nuanced. If lists are immutable, then it’s safe to say yes. But if lists are mutable, then definitely not!

Why? Suppose I need a

`List<Animal>`

and you pass me a `List<Dog>`

. Since I think I have a `List<Animal>`

, I might try to insert a `Cat`

into it. Now your `List<Dog>`

has a `Cat`

in it! The type system should not allow this.Formally: we can allow the type of immutable lists to be covariant in its type parameter, but the type of mutable lists must be

*invariant*(neither covariant nor contravariant) in its type parameter.**Fun fact:**In Java, arrays are both mutable and covariant. This is, of course, unsound.

## Type safe dimensional analysis in Haskell

July 16, 2017

Years ago my colleague Gustavo asked how I would represent physical units like

`m/s`

or `kg*m/s^2`

as types so the compiler can check that they match up and cancel correctly. F# supports this natively, but it felt weird to have it baked into the type system. It seemed too ad hoc, though I didn’t know of anything better.Today I was thinking about this again, and I found a way to do it in Haskell. The main idea is to represent units of measure as function spaces. For example, the unit

`m/s`

can be encoded as the type `Seconds -> Meters`

. The numerator becomes the return type, and the denominator the argument type. Function types can be composed to form more interesting units, such as `Seconds -> Seconds -> Meters`

for acceleration. Products can be represented by higher-order functions. A special type class will enable us to easily convert a `Double`

into a quantity with any units of measure and vice versa.There are a handful of packages on Hackage for doing dimensional analysis. This article will demonstrate a simple, portable way to do it without relying on any language extensions. As we will see in

*Example 3*, this implementation sometimes requires the programmer to provide proofs of unit equivalence (which can be tedious at times). This is less convenient than other libraries, but it’s an interesting exhibition of the power of vanilla Haskell.### Units as function spaces

First, we define some types for base units. We will never instantiate these types, so we don’t specify any constructors. They are only used for type checking.

data Meter data Kilogram data Second

Next, we need types which actually represent quantities with units. For the base units above, we define

`BaseQuantity`

as follows:newtype BaseQuantity a = BaseQuantity Double

It’s just a wrapper for

`Double`

, but written as a phantom type. The type parameter `a`

keeps track of the base unit. For example, `BaseQuantity Meter`

is a type which represents a length.A quotient

`a / b`

of two units `a`

and `b`

will be represented by the function space `b -> a`

. For example, `m / s`

becomes `BaseQuantity Second -> BaseQuantity Meter`

. To make intentions clear, we formalize this idea as a type synonym:type Quotient a b = b -> a

We also need a

`BaseQuantity`

for dimensionless quantities like π. We could define a new base type for this, but `()`

does the job nicely:type Dimensionless = BaseQuantity ()

We can also define multiplicative inverse

`a^-1`

as the quotient `1 / a`

.type Inverse a = Quotient Dimensionless a

A product

`a * b`

can be represented as `a / b^-1`

:type Product a b = Quotient a (Inverse b)

A helpful synonym to make square units like

`m^2`

easier to read:type Square a = Product a a

All quantities have some numeric value. We formalize this in Haskell using a type class:

class Quantity a where construct :: Double -> a destruct :: a -> Double

The instance for base quantities is trivial, since

`BaseQuantity`

just wraps a `Double`

:instance Quantity (BaseQuantity a) where construct = BaseQuantity destruct (BaseQuantity x) = x

Quotients of quantities are quantities as well. To construct a

`Quotient`

from a `Double`

, we define a function which destructs its argument, multiplies the result by the given `Double`

, and constructs a quantity of the return type (the numerator unit). To destruct a `Quotient`

, we first construct `1`

in the denominator unit (the argument type), use the quotient to convert it into the numerator unit, and destruct the result.instance (Quantity q, Quantity r) => Quantity (q -> r) where construct x = \y -> construct (x * (destruct y)) destruct x = destruct (x (construct 1))

We need to define an axiom that allows us to rearrange quotients:

-- a / (b / c) = c / (b / a) quotientAxiom :: (Quantity a, Quantity b, Quantity c) => Quotient a (Quotient b c) -> Quotient c (Quotient b a) quotientAxiom = construct . destruct

*Exercise for the reader:*Do we need this axiom, or is there a way to derive it without destructing any quantities? Alternatively, are there any other axioms we need?

And finally, we define the familiar arithmetic operations:

-- We can add two quantities of the same unit. infixl 6 .+. (.+.) :: Quantity a => a -> a -> a (.+.) x y = construct $ (destruct x) + (destruct y) -- We can subtract two quantities of the same unit. infixl 6 .-. (.-.) :: Quantity a => a -> a -> a (.-.) x y = construct $ (destruct x) - (destruct y) -- We can multiply any two quantities. infixl 7 .*. (.*.) :: (Quantity a, Quantity b) => a -> b -> Product a b (.*.) x y = \z -> construct $ destruct (z y) * destruct x -- We can divide any two quantities. infixl 7 ./. (./.) :: (Quantity a, Quantity b) => a -> b -> Quotient a b (./.) x y = \z -> construct $ (destruct z) * (destruct x) / (destruct y)

Other primitive operations, such as comparison, can be defined similarly.

### Examples

The examples below will use these helpful type synonyms:

type Length = BaseQuantity Meter type Mass = BaseQuantity Kilogram type Time = BaseQuantity Second type Area = Square Length type Velocity = Quotient Length Time

#### Example 1: Tracking units in types

Let’s calculate the area of a table.

tableWidth :: Length tableWidth = construct 1.5 tableHeight :: Length tableHeight = construct 2.5 tableArea :: Area tableArea = tableWidth .*. tableHeight

Calculations with quantities are type safe. Suppose we defined the mass of the table:

tableMass :: Mass tableMass = construct 150

Then the following is a type error:

tableArea :: Area tableArea = tableWidth .*. tableMass -- Error: Couldn't match type ‘Kilogram’ with ‘Meter’ -- Expected type: Area -- Actual type: Product Length Mass

#### Example 2: Quantities as functions

Suppose we want to calculate how far a train will travel, given its velocity and the duration of the trip.

trainVelocity :: Velocity trainVelocity = construct 30 tripDuration :: Time tripDuration = construct 5000

Here we demonstrate the correspondence between quantities and functions.

`Velocity`

is a synonym for `Length / Time`

, but it’s also a function from `Time`

to `Length`

. Given a `Time`

, we can simply “apply” a `Velocity`

to it to get a `Length`

:tripDistance :: Length tripDistance = trainVelocity tripDuration

So multiplication of

`a / b`

by `b`

is just function application.#### Example 3: Manual proofs of unit equality

Let’s define a function that takes a

`Length`

and a `Velocity`

and returns a `Time`

. First try:calculateDuration :: Length -> Velocity -> Time calculateDuration distance velocity = distance ./. velocity -- Error: Couldn't match type ‘Velocity -> Length’ with ‘BaseQuantity Second’ -- Expected type: Time -- Actual type: Quotient Length Velocity

Haskell doesn’t know that

`Length / Velocity = Time`

. If this is indeed true, there will be a way to manipulate the program (without destructing the quantity) to make it type check by swapping arguments, defining new functions, using the `quotientAxiom`

, etc. This is similar to what one would do in a proof assistant like Agda or Coq.So we have:

distance ./. velocity :: Length / Velocity

`Velocity`

is a type synonym for `Length / Time`

, so we actually have:distance ./. velocity :: Length / (Length / Time)

We can apply the

`quotientAxiom`

to get:quotientAxiom (distance ./. velocity) :: Time / (Length / Length)

Under the interpretation of units as function spaces, we have:

quotientAxiom (distance ./. velocity) :: (Length -> Length) -> Time

We can apply

`id`

to cancel the `Length`

s and get a `Time`

. Putting it all together:calculateDuration :: Length -> Velocity -> Time calculateDuration distance velocity = quotientAxiom (distance ./. velocity) id

Now we can calculate how long the trip from

*Example 2*would take if the train traveled at 40 m/s instead of 30 m/s.fasterVelocity :: Velocity fasterVelocity = construct 40 shorterDuration :: Time shorterDuration = calculateDuration tripDistance fasterVelocity

#### Example 4: Destructing the results

When we are done calculating, we can convert the results back to

`Double`

s with the `destruct`

function. Let’s print the results of the first three examples:main = do putStrLn $ "tableArea: " ++ (show $ destruct $ tableArea) ++ " m^2" putStrLn $ "tripDistance: " ++ (show $ destruct $ tripDistance) ++ " m" putStrLn $ "shorterDuration: " ++ (show $ destruct $ shorterDuration) ++ " s"

The output is:

tableArea: 3.75 m^2 tripDistance: 150000.0 m shorterDuration: 3750.0 s

### Conclusion

We can do type safe dimensional analysis by encoding units as function spaces. The basic pattern is:

- Construct whatever quantities you want using the
`construct`

function. A type annotation can be used to specify the units, or you can let type inference figure out the units automatically. - Do type safe operations on these quantities using the operations
`.+.`

,`.*.`

, etc. You may need to manually rearrange units or cancel them to satisfy the type checker, but this will always be possible if your units are actually correct (exercise for the reader: prove it!). - When you are done with calculation and want to use the resulting quantities, the
`destruct`

function will convert them into`Double`

s.

It’s inconvenient that we sometimes have to provide manual proofs of unit equivalence. It would be nice if rearranging and canceling units was completely automatic. But at least we have type safety!

## Decomposing a string into its elements

December 29, 2016

My colleague Esther proposed the following challenge: given a string, decompose it into elemental symbols from the periodic table (if possible). For example,

`Hi Esther`

becomes `H I Es Th Er`

. In general there might be no solutions, one solution, or several.I implemented it in Haskell with dynamic programming. The

`elementize`

function does all the work, using the list monad to compute all possible solutions.import Data.Char (isLetter, toLower) import Data.Function.Memoize (memoFix) import Data.List (intercalate, isPrefixOf) import System.Environment (getArgs) elements = [ "H" , "He" , "Li" , "Be" , "B" , "C" , "N" , "O" , "F" , "Ne" , "Na" , "Mg" , "Al" , "Si" , "P" , "S" , "Cl" , "Ar" , "K" , "Ca" , "Sc" , "Ti" , "V" , "Cr" , "Mn" , "Fe" , "Co" , "Ni" , "Cu" , "Zn" , "Ga" , "Ge" , "As" , "Se" , "Br" , "Kr" , "Rb" , "Sr" , "Y" , "Zr" , "Nb" , "Mo" , "Tc" , "Ru" , "Rh" , "Pd" , "Ag" , "Cd" , "In" , "Sn" , "Sb" , "Te" , "I" , "Xe" , "Cs" , "Ba" , "La" , "Ce" , "Pr" , "Nd" , "Pm" , "Sm" , "Eu" , "Gd" , "Tb" , "Dy" , "Ho" , "Er" , "Tm" , "Yb" , "Lu" , "Hf" , "Ta" , "W" , "Re" , "Os" , "Ir" , "Pt" , "Au" , "Hg" , "Tl" , "Pb" , "Bi" , "Po" , "At" , "Rn" , "Fr" , "Ra" , "Ac" , "Th" , "Pa" , "U" , "Np" , "Pu" , "Am" , "Cm" , "Bk" , "Cf" , "Es" , "Fm" , "Md" , "No" , "Lr" , "Rf" , "Db" , "Sg" , "Bh" , "Hs" , "Mt" , "Ds" , "Rg" , "Cn" , "Uut", "Uuq", "Uup", "Uuh", "Uus" , "Uuo" ] elementize = memoFix (\fn str -> if null str then return [] else do element <- filterPrefixes elements str soln <- fn $ drop (length element) str return $ element : soln) where toLowerStr = map toLower filterPrefixes prefixes str = filter (\x -> isPrefixOf (toLowerStr x) (toLowerStr str)) prefixes main = do args <- getArgs putStrLn $ intercalate"\n" $ map (intercalate " ") (elementize $ filter isLetter $ concat args)

To compile the program, you need the memoize package. Once compiled, it can be run like this:

$ ./elementizer hi esther H I Es Th Er

**Fun fact:**the longest word (in the macOS words file) that can be decomposed into elements is

*nonrepresentationalism*, and it has 4 decompositions:

`No N Re P Re Se N Ta Ti O Na Li Sm`

`N O N Re P Re Se N Ta Ti O Na Li Sm`

`No N Re P Re Se N Ta Ti O N Al I Sm`

`N O N Re P Re Se N Ta Ti O N Al I Sm`

## The Di Vergent Code

December 24, 2016

### Prologue

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.

### Chapter 1

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

proudly presents

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.”

### Chapter 2

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.

### Chapter 3

“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`

?”“Sure.”

Langdon entered

`λx. x`

into 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. “In classical mathematics, you would write

`f(x)`

. In the lambda calculus, we just write `f x`

.”“How do you compute

`f x`

?”“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

`f`

and `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.”

### Chapter 4

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 x`

into Langdon’s notebook. “It takes `x`

and produces `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.

### Chapter 5

“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!”

“How?”

“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.### Chapter 6

(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 x`

in a function call? So instead of `x x`

, we have `f (x x)`

,” he spoke to himself—aloud—because it’s not weird to talk to yourself if it helps you think. “We could choose some function `f`

to 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*Langdon wondered.

`g`

?(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 `g`

decides 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

`g`

is 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 `g`

that only takes the second argument. And what was given for the first argument? The modified `g`

! So `g`

can use its first argument to call itself.Hoping the Church of Alonzo would forgive him for mixing a little Python into their sacred calculus, Langdon wrote out the factorial function:

Y λf. λn. 1 if n == 0 else n * f (n - 1) end

His phone was ringing again.

### Chapter 7

“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 x`

will do,” Langdon pointed out.“Brilliant.” Fache ended the call without saying goodbye. Typical Fache.