Lambda calculus (λ-calculus), originally created by Alonzo Church, is the world’s smallest programming language. Despite not having numbers, strings, booleans, or any non-function datatype, lambda calculus can be used to represent any Turing Machine!

Lambda calculus is composed of 3 elements: **variables**, **functions**, and
**applications**.

Name | Syntax | Example | Explanation |
---|---|---|---|

Variable | `<name>` |
`x` |
a variable named “x” |

Function | `λ<parameters>.<body>` |
`λx.x` |
a function with parameter “x” and body “x” |

Application | `<function><variable or function>` |
`(λx.x)a` |
calling the function “λx.x” with argument “a” |

The most basic function is the identity function: `λx.x`

which is equivalent to
`f(x) = x`

. The first “x” is the function’s argument, and the second is the
body of the function.

- In the function
`λx.x`

, “x” is called a bound variable because it is both in the body of the function and a parameter. - In
`λx.y`

, “y” is called a free variable because it is never declared before hand.

Evaluation is done via β-Reduction, which is essentially lexically-scoped substitution.

When evaluating the
expression `(λx.x)a`

, we replace all occurrences of “x” in the function’s body
with “a”.

`(λx.x)a`

evaluates to:`a`

`(λx.y)a`

evaluates to:`y`

You can even create higher-order functions:

`(λx.(λy.x))a`

evaluates to:`λy.a`

Although lambda calculus traditionally supports only single parameter functions, we can create multi-parameter functions using a technique called currying.

`(λx.λy.λz.xyz)`

is equivalent to`f(x, y, z) = ((x y) z)`

Sometimes `λxy.<body>`

is used interchangeably with: `λx.λy.<body>`

It’s important to recognize that traditional **lambda calculus doesn’t have
numbers, characters, or any non-function datatype!**

There is no “True” or “False” in lambda calculus. There isn’t even a 1 or 0.

Instead:

`T`

is represented by: `λx.λy.x`

`F`

is represented by: `λx.λy.y`

First, we can define an “if” function `λbtf`

that
returns `t`

if `b`

is True and `f`

if `b`

is False

`IF`

is equivalent to: `λb.λt.λf.b t f`

Using `IF`

, we can define the basic boolean logic operators:

`a AND b`

is equivalent to: `λab.IF a b F`

`a OR b`

is equivalent to: `λab.IF a T b`

`NOT a`

is equivalent to: `λa.IF a F T`

*Note: IF a b c is essentially saying: IF((a b) c)*

Although there are no numbers in lambda calculus, we can encode numbers using Church numerals.

For any number n: `n = λf.f`

so:^{n}

`0 = λf.λx.x`

`1 = λf.λx.f x`

`2 = λf.λx.f(f x)`

`3 = λf.λx.f(f(f x))`

To increment a Church numeral,
we use the successor function `S(n) = n + 1`

which is:

`S = λn.λf.λx.f((n f) x)`

Using successor, we can define add:

`ADD = λab.(a S)b`

**Challenge:** try defining your own multiplication function!

Let S, K, I be the following functions:

`I x = x`

`K x y = x`

`S x y z = x z (y z)`

We can convert an expression in the lambda calculus to an expression in the SKI combinator calculus:

`λx.x = I`

`λx.c = Kc`

provided that`x`

does not occur free in`c`

`λx.(y z) = S (λx.y) (λx.z)`

Take the church number 2 for example:

`2 = λf.λx.f(f x)`

For the inner part `λx.f(f x)`

:

```
λx.f(f x)
= S (λx.f) (λx.(f x)) (case 3)
= S (K f) (S (λx.f) (λx.x)) (case 2, 3)
= S (K f) (S (K f) I) (case 2, 1)
```

So:

```
2
= λf.λx.f(f x)
= λf.(S (K f) (S (K f) I))
= λf.((S (K f)) (S (K f) I))
= S (λf.(S (K f))) (λf.(S (K f) I)) (case 3)
```

For the first argument `λf.(S (K f))`

:

```
λf.(S (K f))
= S (λf.S) (λf.(K f)) (case 3)
= S (K S) (S (λf.K) (λf.f)) (case 2, 3)
= S (K S) (S (K K) I) (case 2, 3)
```

For the second argument `λf.(S (K f) I)`

:

```
λf.(S (K f) I)
= λf.((S (K f)) I)
= S (λf.(S (K f))) (λf.I) (case 3)
= S (S (λf.S) (λf.(K f))) (K I) (case 2, 3)
= S (S (K S) (S (λf.K) (λf.f))) (K I) (case 1, 3)
= S (S (K S) (S (K K) I)) (K I) (case 1, 2)
```

Merging them up:

```
2
= S (λf.(S (K f))) (λf.(S (K f) I))
= S (S (K S) (S (K K) I)) (S (S (K S) (S (K K) I)) (K I))
```

Expanding this, we would end up with the same expression for the church number 2 again.

The SKI combinator calculus can still be reduced further. We can
remove the I combinator by noting that `I = SKK`

. We can substitute
all `I`

‘s with `SKK`

.

The SK combinator calculus is still not minimal. Defining:

```
ι = λf.((f S) K)
```

We have:

```
I = ιι
K = ι(ιI) = ι(ι(ιι))
S = ι(K) = ι(ι(ι(ιι)))
```

- A Tutorial Introduction to the Lambda Calculus
- Cornell CS 312 Recitation 26: The Lambda Calculus
- Wikipedia - Lambda Calculus
- Wikipedia - SKI combinator calculus
- Wikipedia - Iota and Jot

Got a suggestion? A correction, perhaps? Open an Issue on the GitHub Repo, or make a pull request yourself!

Originally contributed by Max Sun, and updated by 8 contributors.