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Last active Jul 30, 2019
 .ipynb_checkpoints/ node_modules/

# The Lambda Calculus!

Lambda calculus can be used to encode any program. Fundamentally it's the idea that everything can be build from a function of one argument. By using functions, data can be encoded.

## Booleans

Start with a little bit of binary logic. Booleans are a common abstraction in programming languages which are easily expressed in lambda calculus. A boolean can be thought of as a binary construct that chooses between two values. When true, it chooses one value, and when false it chooses another.

```troo = x => y => x
falz = x => y => y```

Add a little helper function to display the boolean as a value.

```to_bool = b => b('TRUE')('FALSE')

to_bool(troo)
// TRUE
to_bool(falz)
// FALSE```

## Boolean Operators

With such an encoding for booleans, you can start to develop some expressions of boolean logic. How about unary `not` (`!`), the boolean flipping operator?

```not = b => b(falz)(troo)

to_bool(not(troo))
// FALSE
to_bool(not(falz))
// TRUE```

Neat! Now, take a shot at a couple binary operators, `and` and `or`. Here are the truth tables for reference.

### AND

b1 b2 result
true true true
true false false
false true false
false false False

### OR

b1 b2 result
true true true
true false true
false true true
false false False

With that, encode their meaning.

```and = b1 => b2 => b1(b2)(b1)
or = b1 => b2 => b1(b1)(b2)

to_bool(and(troo)(falz))
// FALSE
to_bool(or(troo)(falz))
// TRUE```

It works! Now that we have some boolean logic, it shouldn't be too hard to implement everyone's favorite conditional operation: if/then/else.

```// if(condition)(truecase)(falsecase)
iff = c => thn => els => c(thn)(els)

// cheating a bit with strings here...
iff(and(troo)(falz))('ya')('no')
// no
iff(or(troo)(falz))('ya')('no')
// ya```

Not bad for a bunch of one argument functions. What else can we do?

## Numbers

Another fun topic to explore is numbers and arithmetic. Like booleans (and everything for that matter), there must be an encoding for numbers as functions. This so-called "Church encoding" is based on counting function applications.

Consider zero.

`zero = f => x => x`

Zero applies a function, `f`, zero times to the value. Similarly one applies the function once, two twice, and so on.

```one = f => x => f(x)
two = f => x => f(f(x))
//...```

It's convenient to define a helper function that counts the function applications for visualization.

```to_int = n => n(i => i+1)(0)

to_int(zero)
// 0
to_int(one)
// 1
to_int(two)
// 2```

Also, if you look closely, you'll recognize that each successive number if based on its predecessor. It's just one more application of `f` on the result of the preceding number.

```zero = f => x => x
one = f => x => f(zero(f)(x))
two = f => x => f(one(f)(x))
// ...```

In fact, this routine of determining the successor of a number can be generalized. Do you see the pattern? The predecessor is hard-coded in the encoding of each number. Extract that value as an argument to another function that produces the successor.

```succ = n => f => x => f(n(f)(x))
three = succ(two)
four = succ(three)

to_int(three)
// 3
to_int(four)
// 4```

### Basic Arithmetic

Alright, numbers are a matter of counting function applications. So it's pretty easy to imagine some basic arithmetic. Addition, for example, is `m` applications followed by `n` more applications. Multiplication would be `m` applications repeated `n` times.

```add = m => n => f => x => n(f)(m(f)(x))

res = add(two)(three)
to_int(res)
// 5; whoa! that hadn't event be defined yet as a known number :mindblown:

mul = m => n => f => x => n(m(f))(x)
// Interestingly, since `x` is the last value applied, it can be dropped and the expression
// reduced to:
//   mul = m => => f => x => n(m(f))

res = mul(two)(three)
to_int(res)
// 6```
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 succ = n => f => x => f(n(f)(x)) zero = f => x => x one = succ(zero) two = succ(one) three = succ(two) four = succ(three) add = m => n => f => x => n(f)(m(f)(x)) mul = m => n => f => n(m(f)) exp = b => e => e(b) to_int = n => n(i => i+1)(0) troo = x => _ => x falz = _ => y => y not = b => b(falz)(troo) and = b1 => b2 => b1(b2)(b1) or = b1 => b2 => b1(b1)(b2) iff = c => thn => els => c(thn)(els) to_bool = b => b('TRUE')('FALSE') is_zero = n => n(_ => falz)(troo) is_even = n => n(not)(troo) pair = l => r => f => f(l)(r) left = p => p(troo) right = p => p(falz) nil = pair(troo)(troo) is_empty = left cons = i => l => pair(falz)(pair(i)(l)) head = l => left(right(l)) tail = l => right(right(l)) double = n => add(n)(n) one_two = cons(one)(cons(two)(nil)) // [1,2] foldl = f => v => l => iff(is_empty(l))(()=> v )(()=> foldl(f)(f(v)(head(l)))(tail(l)) )() foldr = f => v => l => iff(is_empty(l))(()=> v )(()=> f(head(l))(foldr(f)(v)(tail(l))) )() each = f => foldl(_ => i => f(i))(nil) eachr= f => foldr(i => _ => f(i))(nil) count = foldl(c => _ => add(c)(one))(zero) sum = foldl(add)(zero) map = f => foldr(i => r => cons(f(i))(r))(nil) print_n = n => console.log(to_int(n)) print_l = each(print_n)
 { "dependencies": { "global": "^4.4.0", "ijavascript": "^5.1.0" } }
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"https://registry.yarnpkg.com/simple-concat/-/simple-concat-1.0.0.tgz#7344cbb8b6e26fb27d66b2fc86f9f6d5997521c6" simple-get@^2.7.0: version "2.8.1" resolved "https://registry.yarnpkg.com/simple-get/-/simple-get-2.8.1.tgz#0e22e91d4575d87620620bc91308d57a77f44b5d" dependencies: decompress-response "^3.3.0" once "^1.3.1" simple-concat "^1.0.0" string-width@^1.0.1: version "1.0.2" resolved "https://registry.yarnpkg.com/string-width/-/string-width-1.0.2.tgz#118bdf5b8cdc51a2a7e70d211e07e2b0b9b107d3" dependencies: code-point-at "^1.0.0" is-fullwidth-code-point "^1.0.0" strip-ansi "^3.0.0" "string-width@^1.0.2 || 2": version "2.1.1" resolved "https://registry.yarnpkg.com/string-width/-/string-width-2.1.1.tgz#ab93f27a8dc13d28cac815c462143a6d9012ae9e" dependencies: is-fullwidth-code-point "^2.0.0" strip-ansi "^4.0.0" string_decoder@~1.1.1: version "1.1.1" resolved "https://registry.yarnpkg.com/string_decoder/-/string_decoder-1.1.1.tgz#9cf1611ba62685d7030ae9e4ba34149c3af03fc8" dependencies: safe-buffer "~5.1.0" strip-ansi@^3.0.0, strip-ansi@^3.0.1: version "3.0.1" resolved "https://registry.yarnpkg.com/strip-ansi/-/strip-ansi-3.0.1.tgz#6a385fb8853d952d5ff05d0e8aaf94278dc63dcf" dependencies: ansi-regex "^2.0.0" strip-ansi@^4.0.0: version "4.0.0" resolved "https://registry.yarnpkg.com/strip-ansi/-/strip-ansi-4.0.0.tgz#a8479022eb1ac368a871389b635262c505ee368f" dependencies: ansi-regex "^3.0.0" strip-json-comments@~2.0.1: version "2.0.1" resolved "https://registry.yarnpkg.com/strip-json-comments/-/strip-json-comments-2.0.1.tgz#3c531942e908c2697c0ec344858c286c7ca0a60a" tar-fs@^1.13.0: version "1.16.3" resolved "https://registry.yarnpkg.com/tar-fs/-/tar-fs-1.16.3.tgz#966a628841da2c4010406a82167cbd5e0c72d509" dependencies: chownr "^1.0.1" mkdirp "^0.5.1" pump "^1.0.0" tar-stream "^1.1.2" tar-stream@^1.1.2: version "1.6.2" resolved "https://registry.yarnpkg.com/tar-stream/-/tar-stream-1.6.2.tgz#8ea55dab37972253d9a9af90fdcd559ae435c555" dependencies: bl "^1.0.0" buffer-alloc "^1.2.0" end-of-stream "^1.0.0" fs-constants "^1.0.0" readable-stream "^2.3.0" to-buffer "^1.1.1" xtend "^4.0.0" to-buffer@^1.1.1: version "1.1.1" resolved "https://registry.yarnpkg.com/to-buffer/-/to-buffer-1.1.1.tgz#493bd48f62d7c43fcded313a03dcadb2e1213a80" tunnel-agent@^0.6.0: version "0.6.0" resolved "https://registry.yarnpkg.com/tunnel-agent/-/tunnel-agent-0.6.0.tgz#27a5dea06b36b04a0a9966774b290868f0fc40fd" dependencies: safe-buffer "^5.0.1" util-deprecate@~1.0.1: version "1.0.2" resolved "https://registry.yarnpkg.com/util-deprecate/-/util-deprecate-1.0.2.tgz#450d4dc9fa70de732762fbd2d4a28981419a0ccf" uuid@3: version "3.3.2" resolved "https://registry.yarnpkg.com/uuid/-/uuid-3.3.2.tgz#1b4af4955eb3077c501c23872fc6513811587131" which-pm-runs@^1.0.0: version "1.0.0" resolved "https://registry.yarnpkg.com/which-pm-runs/-/which-pm-runs-1.0.0.tgz#670b3afbc552e0b55df6b7780ca74615f23ad1cb" wide-align@^1.1.0: version "1.1.3" resolved "https://registry.yarnpkg.com/wide-align/-/wide-align-1.1.3.tgz#ae074e6bdc0c14a431e804e624549c633b000457" dependencies: string-width "^1.0.2 || 2" wrappy@1: version "1.0.2" resolved "https://registry.yarnpkg.com/wrappy/-/wrappy-1.0.2.tgz#b5243d8f3ec1aa35f1364605bc0d1036e30ab69f" xtend@^4.0.0: version "4.0.2" resolved "https://registry.yarnpkg.com/xtend/-/xtend-4.0.2.tgz#bb72779f5fa465186b1f438f674fa347fdb5db54" zeromq@4: version "4.6.0" resolved "https://registry.yarnpkg.com/zeromq/-/zeromq-4.6.0.tgz#7b1ca19cf740f484abfb9358430728f1339f03db" dependencies: nan "^2.6.2" prebuild-install "^2.2.2"
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