Programming with Birds - There is a Bluebird in My Talk That Wants to Get Out (LambdAle 2018)

These are the slides for my talk at LambdAle 2018 in London, UK.

The whole idea here is to show people how Lambda Calculus works, how to encode data using functions and demonstrate its Turing-Completeness. I then proceed to explain combinators and replace the functions with combinators and then replace the combinators themselves with only S and K, the smallest combinator base possible for completeness.

This talk is inspired in the excellent "Programming With Nothing" by Tom Stuart, the main difference is that in this one I use JavaScript and I also added combinators and SK calculus. My take is a bit different, but Tom's talk is definitely worth watching (one of the best I've ever seen).

Transcript

1. 100
   1
   There is a Bluebird in my
   Talk that Wants to Get out
   Programming with birds
   THEWIZARDLUCAS
   

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2. 100
   2
   PROGRAMMING
   IS A GREAT
   EXPERIENCE
   

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3. 100
   3
   I want to make it awful.
   And I'm not the first one.
   Programming
   with nothing
   By Tom Stuart
   

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4. 100
   4
   I want to make it awful.
   But I want to take it one step further.
   z
   Programming
   with nothing
   By Tom Stuart
   

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5. 100
   5
   Ruin
   JavaScript
   Part 1
   

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6. 100
   6
   Ruin
   JavaScript
   Ruin
   JavaScript
   with birds
   Part 1 Part 2
   

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7. 100
   7
   Ruin
   JavaScript
   Ruin
   JavaScript
   with birds
   Ruin
   JavaScript
   and The
   Birds
   Part 1 Part 2 Part 3
   

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8. 100
   8
   Ruin
   JavaScript
   Ruin
   JavaScript
   with birds
   Ruin
   JavaScript
   and The
   Birds
   Part 1 Part 2 Part 3
   Apologise
   Part 4
   

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9. 100
   9
   Ruining JavaScript
   Part 1
   

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10. 100
    10
    We are spoiled
    We program with way too much:
    • Objects
    • Classes/Inheritance
    • Booleans
    • Numbers
    • Builtin functions
    • Builtin data structures
    

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11. 100
    11
    This is what we're
    going to rewrite
    T H E W R E C K
    

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12. 100
    12
    We are only allowed to use four
    kinds of tokens
    THE RULES
    1 2 3 4
    Identifier
    ID λID. E E E (E)
    Abstraction Application Grouping
    

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    13
    We are only allowed to use four
    kinds of tokens
    THE RULES
    1 2 3 4
    Identifier
    ID λID. E E E (E)
    Abstraction Application Grouping
    Yes, all functions only take
    one argument
    I will use assignments to
    make this more convenient In JavaScript this is E(E)
    

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14. 100
    14
    Lambda
    Calculus
    This is
    λ
    

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15. 100
    15
    Our first step
    T H E W R E C K
    

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16. 100
    16
    Replacing
    Booleans
    P a r t 1
    

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17. 100
    17
    Why do we need booleans anyway?
    Replacing Numbers
    if { this } else { that }
    const val = boolean ? "Yes": "No"
    

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18. 100
    18
    Why do we need booleans anyway?
    Replacing Numbers
    if { this } else { that }
    const val = boolean ? "Yes": "No"
    const TRUE = a => b => a
    Always selects the first value.
    

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19. 100
    19
    Why do we need booleans anyway?
    Replacing Numbers
    if { this } else { that }
    const val = boolean ? "Yes": "No"
    const FALSE = a => b => b
    Always selects the second value.
    

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20. 100
    20
    Booleans
    Replacing Numbers
    const FALSE = a => b => b
    const TRUE = a => b => a
    

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    21
    Booleans
    Replacing Numbers
    const SECOND = a => b => b
    const FIRST = a => b => a
    

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    22
    Booleans
    Replacing Numbers
    

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23. 100
    23
    Booleans
    Replacing Numbers
    What about the if keyword?
    

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24. 100
    24
    Booleans
    Replacing Numbers
    What about the if keyword?
    if { this } else { that }
    

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25. 100
    25
    Booleans
    Replacing Numbers
    What about the if keyword?
    BOOLEAN { this } else { that }
    

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    26
    Booleans
    Replacing Numbers
    What about the if keyword?
    BOOLEAN { this } { that }
    

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    27
    Booleans
    Replacing Numbers
    What about the if keyword?
    BOOLEAN { this } { that }
    The boolean itself is our this.
    

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    28
    Booleans
    Replacing Numbers
    Let's have some sugar
    

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    29
    What can we replace?
    Replacing arithmetics
    

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    30
    Replacing
    Numbers
    P a r t 1
    

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    31
    What is a number, anyway?
    Replacing Numbers
    2
    

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    32
    What is a number, anyway?
    Replacing Numbers
    2
    

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    33
    What is a number, anyway?
    Replacing Numbers
    2
    

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    34
    What is a number, anyway?
    Replacing Numbers
    2
    Representation Meaning Meaning
    

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    35
    How can we represent quantities with functions?
    Replacing Numbers
    

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    36
    How can we represent quantities with functions?
    Replacing Numbers
    Function applications!
    

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    37
    How can we represent quantities with functions?
    Replacing Numbers
    Function applications!
    

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    38
    How can we represent quantities with functions?
    Replacing Numbers
    Function applications!
    

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39. 100
    39
    How can we represent quantities with functions?
    Replacing Numbers
    Function applications!
    

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    40
    From meaning to representation
    Replacing Numbers
    

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    41
    From meaning to representation
    Replacing Numbers
    

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    42
    From meaning to representation
    Replacing Numbers
    

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    43
    From meaning to representation
    Replacing Numbers
    

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    44
    From meaning to representation
    Replacing Numbers
    

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    45
    What can we replace?
    Replacing arithmetics
    

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    46
    Replacing
    Arithmetics
    P a r t 1
    

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    47
    The successor function
    Replacing arithmetics
    const ONE = f => x => f(x)
    const TWO = f => x => f(f(x))
    

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    48
    The successor function
    Replacing arithmetics
    const ONE = f => x => f(x)
    const SUCCESSOR = n => f => x => f(n(f)(x))
    const TWO = f => x => f(f(x))
    

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    49
    The successor function
    Replacing arithmetics
    const ONE = f => x => f(x)
    SUCCESSOR(ONE)
    const TWO = f => x => f(f(x))
    const SUCCESSOR = n => f => x => f(n(f)(x))
    

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    50
    The successor function
    Replacing arithmetics
    const ONE = f => x => f(x)
    SUCCESSOR(ONE) f => x => f((ONE(f))(x))
    const TWO = f => x => f(f(x))
    const SUCCESSOR = n => f => x => f(n(f)(x))
    

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51. 100
    51
    The successor function
    Replacing arithmetics
    const ONE = f => x => f(x)
    SUCCESSOR(ONE) f => x => f((ONE(f))(x))
    f => x => f((x => f(x))(x))
    const TWO = f => x => f(f(x))
    const SUCCESSOR = n => f => x => f(n(f)(x))
    

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52. 100
    52
    The successor function
    Replacing arithmetics
    const ONE = f => x => f(x)
    f => x => f((ONE(f))(x))
    f => x => f((x => f(x))(x))
    f => x => f(f(x))
    const TWO = f => x => f(f(x))
    const SUCCESSOR = n => f => x => f(n(f)(x))
    SUCCESSOR(ONE)
    

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53. 100
    53
    The addition function
    Replacing arithmetics
    const TWO = f => x => f(f(x))
    const ONE = f => x => f(x)
    

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54. 100
    54
    The addition function
    Replacing arithmetics
    const TWICE = f => x => f(f(x))
    const ONE = f => x => f(x)
    

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55. 100
    55
    The addition function
    Replacing arithmetics
    twice(SUCCESSOR(ONE))
    ADDITION(TWO)(ONE)
    

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    56
    The multiplication function
    Replacing arithmetics
    

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    57
    The multiplication function
    Replacing arithmetics
    multiplication(THREE)(TWO)
    

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    58
    The multiplication function
    Replacing arithmetics
    THRICE(TWO)
    multiplication(THREE)(TWO)
    

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    59
    What can we replace?
    Replacing arithmetics
    

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    60
    What can we replace?
    Replacing arithmetics
    

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    61
    What is next?
    Replacing arithmetics
    Subtraction
    

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    62
    The predecessor function
    Replacing arithmetics
    Predecessor
    TWO ONE
    

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63. 100
    63
    The predecessor function
    Replacing arithmetics
    TWO ONE
    ?
    

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    64
    Pairs
    Replacing arithmetics
    

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    65
    Pairs
    Replacing arithmetics
    First we store two values.
    

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66. 100
    66
    Pairs
    Replacing arithmetics
    Then we tell which one we want.
    

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    67
    Pairs
    Replacing arithmetics
    

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    68
    Pairs
    Replacing arithmetics
    

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    69
    Pairs
    Replacing arithmetics
    

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    70
    Pairs
    Replacing arithmetics
    

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    71
    Pairs
    Replacing arithmetics
    

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72. 100
    72
    Replacing arithmetics
    Incrementing Pairs
    { 0, 1 }
    { 0, 0 }
    

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    73
    Replacing arithmetics
    { 1, 2 }
    Incrementing Pairs
    { 0, 1 }
    { 0, 0 }
    

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74. 100
    74
    Replacing arithmetics
    { 1, 2 } { 2, 3 }
    Incrementing Pairs
    { 0, 1 }
    { 0, 0 }
    

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75. 100
    75
    Replacing arithmetics
    { 1, 2 } { 2, 3 }
    { 4, 5 }
    Incrementing Pairs
    { 0, 1 }
    { 0, 0 }
    

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76. 100
    76
    Replacing arithmetics
    { 1, 2 } { 2, 3 }
    { 4, 5 } { 5, 6 }
    Incrementing Pairs
    { 0, 1 }
    { 0, 0 }
    

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77. 100
    77
    Replacing arithmetics
    { 1, 2 } { 2, 3 }
    { 4, 5 } { 5, 6 }
    { 0, 0 }
    Thrice
    Incrementing Pairs
    { 0, 1 }
    { 0, 0 }
    

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78. 100
    78
    Replacing arithmetics
    { 1, 2 } { 2, 3 }
    { 4, 5 } { 5, 6 }
    { 0, 0 }
    Thrice { 0, 1 }
    Incrementing Pairs
    { 0, 1 }
    { 0, 0 }
    x1
    

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79. 100
    79
    Replacing arithmetics
    { 1, 2 } { 2, 3 }
    { 4, 5 } { 5, 6 }
    { 0, 0 }
    Thrice { 1, 2 }
    Incrementing Pairs
    { 0, 1 }
    { 0, 0 }
    x2
    

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80. 100
    80
    Replacing arithmetics
    { 1, 2 } { 2, 3 }
    { 4, 5 } { 5, 6 }
    { 0, 0 }
    Thrice { 2, 3 }
    Incrementing Pairs
    { 0, 1 }
    { 0, 0 }
    x3
    

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81. 100
    81
    Replacing arithmetics
    { 1, 2 } { 2, 3 }
    { 4, 5 } { 5, 6 }
    { 0, 0 }
    Thrice { 2, 3 }
    Incrementing Pairs
    { 0, 1 }
    { 0, 0 }
    x3
    

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82. 100
    82
    Replacing arithmetics
    { 1, 2 } { 2, 3 }
    { 4, 5 } { 5, 6 }
    { 0, 0 }
    Three { 2, 3 }
    Incrementing Pairs
    { 0, 1 }
    { 0, 0 }
    x3
    Predecessor
    

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83. 100
    83
    Replacing arithmetics
    The Predecessor Function
    N x INCREMENT_PAIR(0, 0)
    First of
    

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84. 100
    84
    Replacing arithmetics
    The Subtraction Function
    Predecessor of N K times
    

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85. 100
    85
    What can we replace?
    Replacing arithmetics
    

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    86
    What can we replace?
    Replacing arithmetics
    

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87. 100
    87
    Replacing
    Boolean
    Operators
    P a r t 1
    

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88. 100
    88
    AND { true } { true } is true
    AND
    Replacing Boolean Operators
    Any other combinations are false
    

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    89
    AND { true } { true } is true
    AND
    Replacing Boolean Operators
    Any other combinations are false
    

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90. 100
    90
    AND
    Replacing Boolean Operators
    If a is true b must be true
    If a is false it returns itself
    

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91. 100
    91
    It is true if any values are { true }
    OR
    Replacing Boolean Operators
    

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92. 100
    92
    It is true if any values are { true }
    OR
    Replacing Boolean Operators
    

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    93
    It is true if any values are { true }
    OR
    Replacing Boolean Operators
    If a is true it returns itself, otherwise tries b
    

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94. 100
    94
    NOT
    Replacing Boolean Operators
    Inverts any { true } and { false }
    

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    95
    NOT
    Replacing Boolean Operators
    Inverts any { true } and { false }
    

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    96
    NOT
    Replacing Boolean Operators
    Inverts any { true } and { false }
    

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    97
    NOT
    Replacing Boolean Operators
    Inverts any { true } and { false }
    Takes a boolean f and applies it to the arguments in reverse order
    

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    98
    NOT
    Replacing Boolean Operators
    Inverts the selector
    Takes a boolean f and applies it to the arguments in reverse order
    

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99. 100
    99
    NOT
    Replacing Boolean Operators
    Inverts the selector
    Takes a boolean f and applies it to the arguments in reverse order
    

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100. 100
     100
     IS ZERO
     Replacing Boolean Operators
     Checks if a is zero
     If n is zero, it will return the last TRUE
     

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101. 100
     101
     EQUALS
     Replacing Boolean Operators
     Checks if a equals b
     Checks if both n <= k and k <= n
     

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102. 100
     102
     GREATER THAN
     Replacing Boolean Operators
     The opposite of lesser or equal
     

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103. 100
     103
     What can we replace?
     Replacing Boolean Operators
     

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104. 100
     104
     What can we replace?
     Replacing Boolean Operators
     

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105. 100
     105
     Replacing everything with functions
     Ruining Javascript
     

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106. 100
     106
     Ruining Javascript
     Replacing everything with functions
     

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107. 100
     107
     Ruining Javascript
     Replacing everything with functions
     

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108. 100
     108
     Ruining Javascript
     Replacing everything with functions
     

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109. 100
     109
     Ruining Javascript
     Replacing everything with functions
     

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110. 100
     110
     Ruining Javascript
     Replacing everything with functions
     

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111. 100
     111
     Ruining Javascript
     We could replace the
     names of these functions
     by their definitions and
     remove all variable names
     Replacing everything with functions
     

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112. 100
     112
     Unleash Hell
     Ruining Javascript
     But I will save you from
     this pain for now
     

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113. 100
     113
     Ruining JavaScript With Birds
     Part 2
     

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114. 100
     114
     Combinators
     Our Birds
     Functions that don't
     have free variables.
     const combinator = a => b => a
     const withContext = a => b(a)
     a is bound and b is free
     both a and b are bound
     

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115. 100
     115
     To Mock a
     Mockingbird
     C o m b i n a t o r s a n d B i r d s
     

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116. 100
     116
     Combinators
     The Idiot Bird
     

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     117
     Combinators
     The Kestrel
     

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     118
     Combinators
     The Kestrel
     

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     119
     Combinators
     The Cardinal
     

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     120
     Combinators
     The Vireo
     

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     121
     Combinators
     The Bluebird
     

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     122
     Combinators
     The Thrush
     

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     123
     Combinators
     The Starling
     

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     124
     Combinators
     Replacing Functions (Successor)
     const SUCCESSOR = n => f => x => f(n(f)(x))
     

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     125
     Combinators
     const SUCCESSOR = n => f => B(f)(n(f))
     Replacing Functions (Successor)
     

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     126
     Combinators
     const SUCCESSOR = n => f => B(f)(n(f))
     Replacing Functions (Successor)
     

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     127
     Combinators
     const SUCCESSOR = n => f => B(f)(n(f))
     const SUCCESSOR = S(B)
     Replacing Functions (Successor)
     

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     Combinators
     Replacing Functions (Addition)
     const ADDITION = n => k => n(SUCESSOR)(k)
     

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     129
     Combinators
     Replacing Functions (Addition)
     const ADDITION = n => k => n(S(B))(k)
     

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     130
     Combinators
     Replacing Functions (Addition)
     const ADDITION = k => Th(S(B))(k)
     

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     131
     Combinators
     Replacing Functions (Addition)
     const ADDITION = C(Th)(Th(S(B)))
     

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     132
     Nice birds m8
     Ruining Javascript
     

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     133
     Ruining Javascript
     B Th
     C
     K
     KI
     V
     S
     Nice birds m8
     

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134. 100
     134
     Ruining Javascript
     Haskell's Data.Aviary.Birds
     This module is intended
     for illustration (the type
     signatures!) rather than
     utility.
     

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     135
     Ruining Javascript
     Nice birds m8
     

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     136
     Ruining JavaScript and the Birds
     Part 3
     

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     137
     WHAT IF I TOLD YOU
     YOU ONLY NEED
     TWO COMBINATORS?
     Yes, that's right. Two.
     

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138. SK
     SK Calculus
     S K
     

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139. 100
     139
     SK Calculus
     Replacing Numbers
     People also call it SKI Calculus
     

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140. 100
     140
     SK Calculus
     Replacing Numbers
     People also call it SKI Calculus
     Because it's more convenient to have I
     

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     141
     SK Calculus
     Replacing Numbers
     People also call it SKI Calculus
     const I = S(K)(K)
     Because it's more convenient to have I
     

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     142
     SK Calculus
     Replacing Numbers
     const KI = K(S(K)(K))
     

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     143
     SK Calculus
     Replacing Numbers
     const KI = K(I)
     

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     144
     SK Calculus
     Replacing Numbers
     const KI = K(I)
     const B = S(K(S))(K)
     

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     145
     SK Calculus
     Replacing Numbers
     const KI = K(I)
     const B = S(K(S))(K)
     const V = PAIRING
     

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     146
     SK Calculus
     Replacing Numbers
     const V =
     ((S(K((S((S(K((S(KS)
     )K)))S))(KK))))
     ((S(K(S((SK)K))))K))
     

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     147
     SK Calculus
     Replacing Numbers
     const V =
     ((S(K((S((S(K((S(KS)
     )K)))S))(KK))))
     ((S(K(S((SK)K))))K))
     Easy.
     

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     148
     What does this mean?
     

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149. 100
     149
     What does this mean?
     If we can replace all code by functions
     

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150. 100
     150
     What does this mean?
     If we can replace all code by functions
     Replace all functions by combinators
     

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151. 100
     151
     What does this mean?
     If we can replace all code by functions
     Replace all functions by combinators
     And replace all combinators by S and K
     

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152. 100
     152
     What does this mean?
     If we can replace all code by functions
     Replace all functions by combinators
     And replace all combinators by S and K
     Then we can replace all code by S and K
     

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153. 100
     153
     What does this mean?
     https://crypto.stanford.edu/~blynn/lambda/sk.html
     http://xn--wxak1a.com/blog/Combinators.html
     

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154. 100
     154
     Apologising
     Part 4
     

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155. 100
     155
     Apologising
     Part 4
     Functional Programming
     Combinatory Logic
     Computability Theory History of Math
     Compiler Theory
     Language Design
     Recursion
     Wanting to frame

     Gödel pictures to
     hang in your room
     

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156. 100
     156
     Thank you!
     Shhh, no tears. Only lambdas now.
     @THEWIZARDLUCAS (TWITTER)
     @LUCASFCOSTA (GITHUB)
     LUCASFCOSTA.COM
     

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     157
     References
     http://codon.com/programming-with-nothing
     http://www.angelfire.com/tx4/cus/combinator/birds.html
     https://bit.ly/2xpcPKn A Flock of Functions - Gabriel Lebec
     List of Notorious Combinators
     The blog post for the talk I mentioned in the beginning
     https://speakerdeck.com/tomstuart/programming-with-nothing Slides for the talk "Programming with Nothing"
     https://amzn.to/2BVVsa1 To Mock a Mockingbird - Raymond Smullyan
     http://computationbook.com Understanding Computation - Tom Stuart
     https://www.youtube.com/watch?v=_kYGDJSm0gE - ASU Lectures - Lambda Calculus by Adam Doupé
     

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     Thank you!
     Shhh, no tears. Only lambdas now.
     @THEWIZARDLUCAS (TWITTER)
     @LUCASFCOSTA (GITHUB)
     LUCASFCOSTA.COM
     

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