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---
category: Algorithms & Data Structures
name: Lambda Calculus
contributors:
- ["Max Sun", "http://github.com/maxsun"]
- ["Yan Hui Hang", "http://github.com/yanhh0"]
---
# Lambda Calculus
Lambda calculus (λ-calculus), originally created by
[Alonzo Church](https://en.wikipedia.org/wiki/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.
## Free vs. Bound Variables:
- 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:
Evaluation is done via
[β-Reduction](https://en.wikipedia.org/wiki/Lambda_calculus#Beta_reduction),
which is essentially lexically-scoped substitution.
When evaluating the
expression `(λx.x)a`, we replace all occurences 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](https://en.wikipedia.org/wiki/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!**
## Boolean Logic:
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`
`a NOT b` is equivalent to: `λa.IF a F T`
*Note: `IF a b c` is essentially saying: `IF((a b) c)`*
## Numbers:
Although there are no numbers in lambda calculus, we can encode numbers using
[Church numerals](https://en.wikipedia.org/wiki/Church_encoding).
For any number n: <code>n = λf.f<sup>n</sup></code> so:
`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!
## Get even smaller: SKI, SK and Iota
### SKI Combinator Calculus
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:
1. `λx.x = I`
2. `λx.c = Kc`
3. `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.
### SK Combinator Calculus
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`.
### Iota Combinator
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)))
```
## For more advanced reading:
1. [A Tutorial Introduction to the Lambda Calculus](http://www.inf.fu-berlin.de/lehre/WS03/alpi/lambda.pdf)
2. [Cornell CS 312 Recitation 26: The Lambda Calculus](http://www.cs.cornell.edu/courses/cs3110/2008fa/recitations/rec26.html)
3. [Wikipedia - Lambda Calculus](https://en.wikipedia.org/wiki/Lambda_calculus)
4. [Wikipedia - SKI combinator calculus](https://en.wikipedia.org/wiki/SKI_combinator_calculus)
5. [Wikipedia - Iota and Jot](https://en.wikipedia.org/wiki/Iota_and_Jot)
|
