2017-10-19 09:55:43 +03:00
|
|
|
|
---
|
|
|
|
|
|
category: Algorithms & Data Structures
|
|
|
|
|
|
name: Lambda Calculus
|
|
|
|
|
|
contributors:
|
|
|
|
|
|
- ["Max Sun", "http://github.com/maxsun"]
|
2018-07-15 10:33:01 +03:00
|
|
|
|
- ["Yan Hui Hang", "http://github.com/yanhh0"]
|
2017-10-19 09:55:43 +03:00
|
|
|
|
---
|
|
|
|
|
|
|
|
|
|
|
|
# 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
|
2022-12-10 18:05:34 +03:00
|
|
|
|
expression `(λx.x)a`, we replace all occurrences of "x" in the function's body
|
2017-10-19 09:55:43 +03:00
|
|
|
|
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).
|
|
|
|
|
|
|
2018-10-08 20:57:53 +03:00
|
|
|
|
- `(λx.λy.λz.xyz)` is equivalent to `f(x, y, z) = ((x y) z)`
|
2017-10-19 09:55:43 +03:00
|
|
|
|
|
|
|
|
|
|
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`
|
|
|
|
|
|
|
2019-03-17 20:36:27 +03:00
|
|
|
|
`NOT a` is equivalent to: `λa.IF a F T`
|
2017-10-19 09:55:43 +03:00
|
|
|
|
|
2018-10-09 14:08:41 +03:00
|
|
|
|
*Note: `IF a b c` is essentially saying: `IF((a b) c)`*
|
2017-10-19 09:55:43 +03:00
|
|
|
|
|
|
|
|
|
|
## 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:
|
|
|
|
|
|
|
2018-11-22 02:22:52 +03:00
|
|
|
|
`ADD = λab.(a S)b`
|
2017-10-19 09:55:43 +03:00
|
|
|
|
|
|
|
|
|
|
**Challenge:** try defining your own multiplication function!
|
|
|
|
|
|
|
2018-07-15 10:33:01 +03:00
|
|
|
|
## 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`
|
2021-11-14 14:40:54 +03:00
|
|
|
|
2. `λx.c = Kc` provided that `x` does not occur free in `c`
|
2018-07-15 10:33:01 +03:00
|
|
|
|
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)`:
|
2019-02-17 07:10:49 +03:00
|
|
|
|
|
2018-07-15 10:33:01 +03:00
|
|
|
|
```
|
|
|
|
|
|
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:
|
2019-02-17 07:10:49 +03:00
|
|
|
|
|
2018-07-15 10:33:01 +03:00
|
|
|
|
```
|
|
|
|
|
|
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))`:
|
2019-02-17 07:10:49 +03:00
|
|
|
|
|
2018-07-15 10:33:01 +03:00
|
|
|
|
```
|
|
|
|
|
|
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)`:
|
2019-02-17 07:10:49 +03:00
|
|
|
|
|
2018-07-15 10:33:01 +03:00
|
|
|
|
```
|
|
|
|
|
|
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:
|
2019-02-17 07:10:49 +03:00
|
|
|
|
|
2018-07-15 10:33:01 +03:00
|
|
|
|
```
|
|
|
|
|
|
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)))
|
|
|
|
|
|
```
|
|
|
|
|
|
|
2017-10-19 09:55:43 +03:00
|
|
|
|
## 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)
|
2018-07-15 10:33:01 +03:00
|
|
|
|
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)
|
