From 8676459adb14d9830259c83edbfcbc0988277d49 Mon Sep 17 00:00:00 2001
From: Max Sun <maxsun@airbears2-10-142-68-88.airbears2.1918.berkeley.edu>
Date: Wed, 18 Oct 2017 23:55:43 -0700
Subject: Added lambda calculus
---
lambda-calculus.html.markdown | 121 ++++++++++++++++++++++++++++++++++++++++++
1 file changed, 121 insertions(+)
create mode 100644 lambda-calculus.html.markdown
(limited to 'lambda-calculus.html.markdown')
diff --git a/lambda-calculus.html.markdown b/lambda-calculus.html.markdown
new file mode 100644
index 00000000..6103c015
--- /dev/null
+++ b/lambda-calculus.html.markdown
@@ -0,0 +1,121 @@
+---
+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)
\ No newline at end of file
--
cgit v1.2.3
From 3ab2e88b4af9d072fbd2e0e2b9c0fe849d46892b Mon Sep 17 00:00:00 2001
From: YAN HUI HANG <yanhh0@163.com>
Date: Sun, 15 Jul 2018 15:33:01 +0800
Subject: SKI, SK and Iota
---
lambda-calculus.html.markdown | 95 ++++++++++++++++++++++++++++++++++++++++++-
1 file changed, 94 insertions(+), 1 deletion(-)
(limited to 'lambda-calculus.html.markdown')
diff --git a/lambda-calculus.html.markdown b/lambda-calculus.html.markdown
index 6103c015..72ed78ba 100644
--- a/lambda-calculus.html.markdown
+++ b/lambda-calculus.html.markdown
@@ -3,6 +3,7 @@ category: Algorithms & Data Structures
name: Lambda Calculus
contributors:
- ["Max Sun", "http://github.com/maxsun"]
+ - ["Yan Hui Hang", "http://github.com/yanhh0"]
---
# Lambda Calculus
@@ -114,8 +115,100 @@ Using successor, we can define add:
**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)
\ No newline at end of file
+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)
--
cgit v1.2.3
From 3a48077c16b2a7f083cd096cf146e16b9bcebdb1 Mon Sep 17 00:00:00 2001
From: Chris Zimmerman <chris@chriszimmerman.net>
Date: Mon, 8 Oct 2018 13:57:53 -0400
Subject: Fixes lambda calculus evaluation formula by fixing associativity
---
lambda-calculus.html.markdown | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
(limited to 'lambda-calculus.html.markdown')
diff --git a/lambda-calculus.html.markdown b/lambda-calculus.html.markdown
index 72ed78ba..79dc20e7 100644
--- a/lambda-calculus.html.markdown
+++ b/lambda-calculus.html.markdown
@@ -55,7 +55,7 @@ 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))`
+- `(λx.λy.λz.xyz)` is equivalent to `f(x, y, z) = ((x y) z)`
Sometimes `λxy.<body>` is used interchangeably with: `λx.λy.<body>`
--
cgit v1.2.3
From 3f685318635b69d1492238378cd90c5908264b6c Mon Sep 17 00:00:00 2001
From: Chris Zimmerman <chris@chriszimmerman.net>
Date: Tue, 9 Oct 2018 07:08:41 -0400
Subject: Fixes associativity issue in Lambda Calculus doc.
---
lambda-calculus.html.markdown | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
(limited to 'lambda-calculus.html.markdown')
diff --git a/lambda-calculus.html.markdown b/lambda-calculus.html.markdown
index 79dc20e7..3d080de7 100644
--- a/lambda-calculus.html.markdown
+++ b/lambda-calculus.html.markdown
@@ -87,7 +87,7 @@ Using `IF`, we can define the basic boolean logic operators:
`a NOT b` is equivalent to: `λa.IF a F T`
-*Note: `IF a b c` is essentially saying: `IF(a(b(c)))`*
+*Note: `IF a b c` is essentially saying: `IF((a b) c)`*
## Numbers:
--
cgit v1.2.3
From 34b2ab3a8b8ab345193585fd686e2ddbf6d36169 Mon Sep 17 00:00:00 2001
From: Nick Henderson <nick.henderson@gmail.com>
Date: Wed, 21 Nov 2018 15:22:52 -0800
Subject: propose correction for ADD
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit
I think it should be: `ADD = λab.(a S)b`
---
lambda-calculus.html.markdown | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
(limited to 'lambda-calculus.html.markdown')
diff --git a/lambda-calculus.html.markdown b/lambda-calculus.html.markdown
index 3d080de7..8343d62a 100644
--- a/lambda-calculus.html.markdown
+++ b/lambda-calculus.html.markdown
@@ -111,7 +111,7 @@ we use the successor function `S(n) = n + 1` which is:
Using successor, we can define add:
-`ADD = λab.(a S)n`
+`ADD = λab.(a S)b`
**Challenge:** try defining your own multiplication function!
--
cgit v1.2.3
From ef11bde030bc530f713d35f1ff03a248ea658b56 Mon Sep 17 00:00:00 2001
From: nichijou <angusbike@gmail.com>
Date: Sun, 17 Feb 2019 12:10:49 +0800
Subject: markdown syntax issue
---
lambda-calculus.html.markdown | 5 +++++
1 file changed, 5 insertions(+)
(limited to 'lambda-calculus.html.markdown')
diff --git a/lambda-calculus.html.markdown b/lambda-calculus.html.markdown
index 8343d62a..a5535d6c 100644
--- a/lambda-calculus.html.markdown
+++ b/lambda-calculus.html.markdown
@@ -139,6 +139,7 @@ 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)
@@ -147,6 +148,7 @@ For the inner part `λx.f(f x)`:
```
So:
+
```
2
= λf.λx.f(f x)
@@ -156,6 +158,7 @@ So:
```
For the first argument `λf.(S (K f))`:
+
```
λf.(S (K f))
= S (λf.S) (λf.(K f)) (case 3)
@@ -164,6 +167,7 @@ For the first argument `λf.(S (K f))`:
```
For the second argument `λf.(S (K f) I)`:
+
```
λf.(S (K f) I)
= λf.((S (K f)) I)
@@ -174,6 +178,7 @@ For the second argument `λf.(S (K f) I)`:
```
Merging them up:
+
```
2
= S (λf.(S (K f))) (λf.(S (K f) I))
--
cgit v1.2.3
From 753325355a8e22b050fb32e3e48d74cb20d92403 Mon Sep 17 00:00:00 2001
From: Brian Berns <brianberns@gmail.com>
Date: Sun, 17 Mar 2019 13:36:27 -0400
Subject: Update lambda-calculus.html.markdown
---
lambda-calculus.html.markdown | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
(limited to 'lambda-calculus.html.markdown')
diff --git a/lambda-calculus.html.markdown b/lambda-calculus.html.markdown
index a5535d6c..53a7a7cd 100644
--- a/lambda-calculus.html.markdown
+++ b/lambda-calculus.html.markdown
@@ -85,7 +85,7 @@ Using `IF`, we can define the basic boolean logic operators:
`a OR b` is equivalent to: `λab.IF a T b`
-`a NOT b` is equivalent to: `λa.IF a F T`
+`NOT a` is equivalent to: `λa.IF a F T`
*Note: `IF a b c` is essentially saying: `IF((a b) c)`*
--
cgit v1.2.3