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---
category: Algorithms & Data Structures
name: Lambda Calculus
contributors:
- ["Max Sun", "http://github.com/maxsun"]
---
# 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)n`
**Challenge:** try defining your own multiplication function!
## 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)