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121 lines
3.8 KiB
Markdown
121 lines
3.8 KiB
Markdown
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---
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category: Algorithms & Data Structures
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name: Lambda Calculus
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contributors:
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- ["Max Sun", "http://github.com/maxsun"]
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---
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# Lambda Calculus
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Lambda calculus (λ-calculus), originally created by
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[Alonzo Church](https://en.wikipedia.org/wiki/Alonzo_Church),
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is the world's smallest programming language.
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Despite not having numbers, strings, booleans, or any non-function datatype,
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lambda calculus can be used to represent any Turing Machine!
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Lambda calculus is composed of 3 elements: **variables**, **functions**, and
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**applications**.
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| Name | Syntax | Example | Explanation |
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|-------------|------------------------------------|-----------|-----------------------------------------------|
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| Variable | `<name>` | `x` | a variable named "x" |
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| Function | `λ<parameters>.<body>` | `λx.x` | a function with parameter "x" and body "x" |
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| Application | `<function><variable or function>` | `(λx.x)a` | calling the function "λx.x" with argument "a" |
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The most basic function is the identity function: `λx.x` which is equivalent to
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`f(x) = x`. The first "x" is the function's argument, and the second is the
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body of the function.
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## Free vs. Bound Variables:
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- In the function `λx.x`, "x" is called a bound variable because it is both in
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the body of the function and a parameter.
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- In `λx.y`, "y" is called a free variable because it is never declared before hand.
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## Evaluation:
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Evaluation is done via
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[β-Reduction](https://en.wikipedia.org/wiki/Lambda_calculus#Beta_reduction),
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which is essentially lexically-scoped substitution.
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When evaluating the
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expression `(λx.x)a`, we replace all occurences of "x" in the function's body
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with "a".
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- `(λx.x)a` evaluates to: `a`
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- `(λx.y)a` evaluates to: `y`
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You can even create higher-order functions:
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- `(λx.(λy.x))a` evaluates to: `λy.a`
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Although lambda calculus traditionally supports only single parameter
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functions, we can create multi-parameter functions using a technique called
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[currying](https://en.wikipedia.org/wiki/Currying).
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- `(λx.λy.λz.xyz)` is equivalent to `f(x, y, z) = x(y(z))`
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Sometimes `λxy.<body>` is used interchangeably with: `λx.λy.<body>`
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----
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It's important to recognize that traditional **lambda calculus doesn't have
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numbers, characters, or any non-function datatype!**
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## Boolean Logic:
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There is no "True" or "False" in lambda calculus. There isn't even a 1 or 0.
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Instead:
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`T` is represented by: `λx.λy.x`
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`F` is represented by: `λx.λy.y`
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First, we can define an "if" function `λbtf` that
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returns `t` if `b` is True and `f` if `b` is False
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`IF` is equivalent to: `λb.λt.λf.b t f`
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Using `IF`, we can define the basic boolean logic operators:
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`a AND b` is equivalent to: `λab.IF a b F`
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`a OR b` is equivalent to: `λab.IF a T b`
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`a NOT b` is equivalent to: `λa.IF a F T`
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*Note: `IF a b c` is essentially saying: `IF(a(b(c)))`*
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## Numbers:
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Although there are no numbers in lambda calculus, we can encode numbers using
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[Church numerals](https://en.wikipedia.org/wiki/Church_encoding).
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For any number n: <code>n = λf.f<sup>n</sup></code> so:
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`0 = λf.λx.x`
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`1 = λf.λx.f x`
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`2 = λf.λx.f(f x)`
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`3 = λf.λx.f(f(f x))`
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To increment a Church numeral,
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we use the successor function `S(n) = n + 1` which is:
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`S = λn.λf.λx.f((n f) x)`
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Using successor, we can define add:
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`ADD = λab.(a S)n`
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**Challenge:** try defining your own multiplication function!
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## For more advanced reading:
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1. [A Tutorial Introduction to the Lambda Calculus](http://www.inf.fu-berlin.de/lehre/WS03/alpi/lambda.pdf)
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2. [Cornell CS 312 Recitation 26: The Lambda Calculus](http://www.cs.cornell.edu/courses/cs3110/2008fa/recitations/rec26.html)
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3. [Wikipedia - Lambda Calculus](https://en.wikipedia.org/wiki/Lambda_calculus)
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