Merge pull request #2921 from maxsun/lambda-calculus

[lambda-calculus/en] Added lambda calculus file
2024-11-23 22:27:35 +03:00 · 2018-02-28 14:29:38 +05:45 · 2018-02-28 14:29:38 +05:45 · 8c10a04b5e
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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)