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Lambda Calc Talk (Smartly.io Version)

Lambda Calc Talk (Smartly.io Version)

Slightly modified / updated version of my LC talk, delivered for the Smartly.io DevTalks in Helsinki, Finland. For the longer original version, see https://github.com/glebec/lambda-talk. For a follow-up code examples repo including the Z-combinator, see https://github.com/glebec/lambda-talk-practical.

Gabriel Lebec

October 15, 2019
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  1. as.js
    A FL O C K of FU N C T I O N S
    COMBINATORS, LAMBDA CALCULUS,
    & CHURCH ENCODINGS in JAVASCRIPT
    Smartly.io Edition

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  2. glebec
    glebec
    glebec
    glebec
    g_lebec

    Gabriel Lebec
    github.com/glebec/lambda-talk
    formerly @

    currently @
    presenting @
    Views and opinions in this presentation

    are my own and do not necessarily

    represent those of my employer.

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  3. a.a
    IDENTITY

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  4. λ
    JS I = a => a
    I := a.a

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  5. λ
    JS I(x) === ?
    I x = ?
    I := a.a
    I = a => a

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  6. λ
    JS I(x) === x
    I x = x
    I := a.a
    I = a => a
    (a.a)x = x
    (a => a)(x) === x

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  7. λ
    JS I(I) === ?
    I I = ?
    I := a.a
    I = a => a

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  8. λ
    JS I(I) === I
    I I = I
    I := a.a
    I = a => a
    (a.a)a.a = a.a

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  9. id 5 == 5

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  10. ?

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  11. a.a
    FUNCTION
    SIGNIFIER

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  12. a.a
    FUNCTION
    SIGNIFIER
    PARAMETER VARIABLE

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  13. a.a
    FUNCTION
    SIGNIFIER
    PARAMETER VARIABLE
    RETURN
    EXPRESSION

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  14. a.a
    FUNCTION
    SIGNIFIER
    PARAMETER VARIABLE
    RETURN
    EXPRESSION
    LAMBDA ABSTRACTION

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  15. View Slide

  16. -CALCULUS SYNTAX
    expression ::= variable identifier
    | expression expression application
    | variable . expression abstraction
    | ( expression ) grouping

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  17. λ JS

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  18. VARIABLES
    x x
    (a) (a)

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  19. f a f(a)
    f a b f(a)(b)
    (f a) b (f(a))(b)
    f (a b) f(a(b))
    APPLICATIONS

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  20. a.b a => b
    a.b x a => b(x)
    a.(b x) a => (b(x))
    (a.b) x (a => b)(x)
    a.b.a a => b => a
    a.(b.a) a => (b => a)
    ABSTRACTIONS

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  21. ((a.a)b.c.b)(x)e.f
    β-REDUCTION

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  22. ((a.a)b.c.b)(x)e.f
    β-REDUCTION

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  23. ((a.a)b.c.b)(x)e.f
    β-REDUCTION

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  24. ((a.a)b.c.b)(x)e.f
    β-REDUCTION

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  25. ((a.a)b.c.b)(x)e.f
    β-REDUCTION

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  26. ((a.a)b.c.b)(x)e.f
    β-REDUCTION

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  27. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    β-REDUCTION

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  28. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    β-REDUCTION

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  29. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    β-REDUCTION

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  30. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    β-REDUCTION

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  31. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    β-REDUCTION

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  32. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    β-REDUCTION

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  33. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    = (c.x) e.f
    β-REDUCTION

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  34. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    = (c.x) e.f
    β-REDUCTION

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  35. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    = (c.x) e.f
    β-REDUCTION

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  36. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    = (c.x) e.f
    β-REDUCTION

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  37. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    = (c.x) e.f
    β-REDUCTION

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  38. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    = (c.x) e.f
    β-REDUCTION

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  39. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    = (c.x) e.f
    = x
    β-REDUCTION

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  40. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    = (c.x) e.f
    = x
    β-REDUCTION

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  41. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    = (c.x) e.f
    = x
    β-REDUCTION
    β-NORMAL FORM

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  42. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    = (c.x) e.f
    = x
    β-REDUCTION*
    β-NORMAL FORM
    *not covered: evaluation order, variable collision avoidance

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  43. f.ff
    MOCKINGBIRD

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  44. λ
    JS M = f => f(f)
    M := f.ff

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  45. λ
    JS M(I) === ?
    M I = ?
    M := f.ff

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  46. λ
    JS M(I) === I(I)
    M I = I I
    M := f.ff
    (f.ff)a.a = (a.a)a.a

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  47. λ
    JS M(I) === I(I)
    && I(I) === ?
    M I = I I = ?
    M := f.ff

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  48. λ
    JS M(I) === I(I)
    && I(I) === I
    M I = I I = I
    M := f.ff
    (f.ff)a.a = (a.a)x.x = x.x

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  49. λ
    JS M(M) === ?
    M M = ?
    M := f.ff

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  50. λ
    JS M(M) === M(M)
    M M = M M
    M := f.ff
    (f.ff)g.gg = (g.gg)g.gg

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  51. λ
    JS M(M) === M(M) === ?
    M M = M M = ?
    M := f.ff

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  52. λ
    JS M(M) === M(M) ===
    M M = M M = M M = …
    // stack overflow
    M := f.ff
    M(M) === M(M) === M(M) === M(M) === M
    M(M) === M(M) === M(M) === M(M) === M
    M(M) === M(M) === M(M) === M(M) === M
    M(M) === M(M) === M(M) === M(M) === M
    M(M) === M(M) === M(M) === M(M) === M
    (f.ff)g.gg = (g.gg)g.gg = …

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  53. λ
    JS
    M M = M M = M M = Ω
    // stack overflow
    M := f.ff
    M(M) === M(M) === M(M) === M(M) === M
    M(M) === M(M) === M(M) === M(M) === M
    M(M) === M(M) === M(M) === M(M) === M
    M(M) === M(M) === M(M) === M(M) === M
    M(M) === M(M) === M(M) === M(M) === M

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  54. a.b.c.b a => b => c => b
    abc.b a => b => c => b
    (a, b, c) => b
    =
    ABSTRACTIONS, again

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  55. ((a.a)bc.b)(x)e.f
    β-REDUCTION, again

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  56. ((a.a)bc.b)(x)e.f
    β-REDUCTION, again

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  57. ((a.a)bc.b)(x)e.f
    β-REDUCTION, again

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  58. ((a.a)bc.b)(x)e.f
    β-REDUCTION, again

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  59. ((a.a)bc.b)(x)e.f
    β-REDUCTION, again

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  60. ((a.a)bc.b)(x)e.f
    β-REDUCTION, again

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  61. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    β-REDUCTION, again

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  62. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    β-REDUCTION, again

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  63. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    β-REDUCTION, again

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  64. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    β-REDUCTION, again

    View Slide

  65. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    β-REDUCTION, again

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  66. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    β-REDUCTION, again

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  67. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    = (c.x) e.f
    β-REDUCTION, again

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  68. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    = (c.x) e.f
    β-REDUCTION, again

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  69. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    = (c.x) e.f
    β-REDUCTION, again

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  70. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    = (c.x) e.f
    β-REDUCTION, again

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  71. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    = (c.x) e.f
    β-REDUCTION, again

    View Slide

  72. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    = (c.x) e.f
    β-REDUCTION, again

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  73. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    = (c.x) e.f
    = x
    β-REDUCTION, again

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  74. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    = (c.x) e.f
    = x
    β-REDUCTION, again

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  75. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    = (c.x) e.f
    = x
    β-REDUCTION, again
    β-NORMAL FORM

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  76. ab.a
    KESTREL

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  77. λ
    JS K = a => b => a
    K := ab.a
    = a.b.a

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  78. λ
    JS K(M)(I) === ?
    K M I = ?
    K := ab.a
    K = a => b => a

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  79. λ
    JS K(M)(I) === M
    K M I = M
    K := ab.a
    K = a => b => a
    (ab.a)(f.ff)x.x = f.ff

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  80. λ
    JS K(M)(I) === M
    K(I)(M) === ?
    K M I = M
    K I M = ?
    K := ab.a
    K = a => b => a

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  81. λ
    JS K(M)(I) === M
    K(I)(M) === I
    K M I = M
    K I M = I
    K := ab.a
    K = a => b => a
    (ab.a)(x.x)f.ff = x.x

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  82. const 7 2 == 7

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  83. λ
    JS K(I)(x) === ?
    K I x = ?
    K := ab.a
    K = a => b => a

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  84. λ
    JS K(I)(x) === I
    K I x = I
    K := ab.a
    K = a => b => a
    (ab.a)(i.i)x = (i.i)

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  85. λ
    JS K(I)(x)(y) === I(y)
    K I x y = I y
    K := ab.a
    K = a => b => a
    (ab.a)(i.i)xy = (i.i)y

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  86. λ
    JS K(I)(x)(y) === I(y)
    && I(y) === ?
    K I x y = I y = ?
    K := ab.a
    K = a => b => a

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  87. λ
    JS
    K I x y = I y = y
    K := ab.a
    K = a => b => a
    K(I)(x)(y) === I(y)
    && I(y) === y
    (ab.a)(i.i)xy = (i.i)y = y

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  88. λ
    JS
    K I x y = I y = y
    K := ab.a
    K = a => b => a
    K(I)(x)(y) === I(y)
    && I(y) === y

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  89. λ
    JS
    K I x y = I y = y
    K := ab.a
    K = a => b => a
    K(I)(x)(y) === I(y)
    && I(y) === y

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  90. λ
    JS
    K I x y = y
    K := ab.a
    K = a => b => a
    K(I)(x)(y) === y

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  91. ab.b
    KITE

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  92. λ
    JS KI = a => b => b
    KI = K(I)
    KI := ab.b
    = K I

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  93. λ
    JS KI(M)(K) === ?
    KI M K = ?
    KI := ab.b
    KI = a => b => b

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  94. λ
    JS KI(M)(K) === K
    KI M K = K
    KI := ab.b
    KI = a => b => b
    (ab.b)(f.ff)ab.a = ab.a

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  95. λ
    JS KI(M)(K) === K
    KI(K)(M) === ?
    KI M K = K
    KI K M = ?
    KI := ab.b
    KI = a => b => b

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  96. λ
    JS KI(M)(K) === K
    KI(K)(M) === M
    KI M K = K
    KI K M = M
    KI := ab.b
    KI = a => b => b
    (ab.b)(f.ff)ab.a = f.ff

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  97. ?

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  98. SCHÖNFINKEL CURRY SMULLYAN
    Identitätsfunktion
    Konstante Funktion
    verSchmelzungsfunktion
    verTauschungsfunktion
    Zusammensetzungsf.
    I

    K

    S

    C

    B
    Idiot

    Kestrel

    Starling

    Cardinal

    Bluebird
    Ibis?

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  99. View Slide

  100. ?

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  101. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    TH E FO R M A L I Z AT I O N O F
    MAT H E M AT I C A L LO G I C
    PÉTER

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  102. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    FO R M A L NO TAT I O N FO R FU N C T I O N S
    1889
    PE A N O AR I T H M E T I C
    PÉTER

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  103. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL CURRY CHURCH
    GÖDEL
    TURING
    KLEENE
    AX I O M AT I C LO G I C · FN NO TAT I O N
    FU N C T I O N S A S GR A P H S · CU R RY I N G
    1891
    VON NEUMANN ROSSER
    PÉTER

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  104. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL CURRY CHURCH
    GÖDEL
    TURING
    KLEENE
    PR I N C I P I A MAT H E M AT I C A
    1910
    RU S S E L L ’S PA R A D OX · FN NO TAT I O N
    VON NEUMANN ROSSER
    PÉTER

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  105. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL CURRY CHURCH
    GÖDEL
    TURING
    KLEENE
    CO M B I N AT O RY LO G I C
    CO M B I N AT O R S · CU R RY I N G
    1920
    VON NEUMANN ROSSER
    PÉTER

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  106. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL CURRY CHURCH
    GÖDEL
    TURING
    KLEENE
    FU N C T I O N A L SY S T E M O F SE T TH E O RY
    1925
    (OV E R L A P P E D W I T H CO M B I N AT O RY LO G I C )
    VON NEUMANN ROSSER
    PÉTER

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  107. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL CURRY CHURCH
    GÖDEL
    TURING
    KLEENE
    CO M B I N AT O RY LO G I C (AG A I N )
    CO M B I N AT O R S · M A N Y C O N T R I B U T I O N S
    1926
    VON NEUMANN ROSSER
    PÉTER

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  108. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL CURRY CHURCH
    GÖDEL
    TURING
    KLEENE
    D I S C OV E R S SC H Ö N F I N K E L
    “This paper anticipates much of what I have done.”
    1927
    VON NEUMANN ROSSER
    PÉTER

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  109. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL CURRY CHURCH
    GÖDEL
    TURING
    KLEENE
    IN C O M P L E T E N E S S TH E O R E M S
    1931
    GE N E R A L RE C U R S I O N TH E O RY
    VON NEUMANN ROSSER
    PÉTER

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  110. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL CURRY CHURCH TURING
    KLEENE
    VON NEUMANN ROSSER
    RE C U R S I V E FU N C T I O N TH E O RY
    RE K U R S I V E FU N K T I O N E N
    1932
    GÖDEL PÉTER

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  111. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL CURRY CHURCH
    GÖDEL
    TURING
    KLEENE
    -CA L C U L U S
    AN EF F E C T I V E MO D E L O F CO M P U TAT I O N
    1932
    VON NEUMANN ROSSER
    PÉTER

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  112. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    I N C O N S I S T E N C Y O F S P E C I A L I Z E D
    1931–1936
    C O N S I S T E N C Y O F P U R E
    VON NEUMANN PÉTER

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  113. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL CURRY CHURCH
    GÖDEL
    TURING
    KLEENE
    SO LV E S T H E DE C I S I O N PRO B L E M
    V I A T H E -CA L C U L U S
    1936
    VON NEUMANN ROSSER
    PÉTER

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  114. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL CURRY CHURCH
    GÖDEL
    TURING
    KLEENE
    SO LV E S T H E DE C I S I O N PRO B L E M
    1936
    V I A T H E TU R I N G MAC H I N E
    VON NEUMANN ROSSER
    PÉTER

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  115. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL CURRY CHURCH
    GÖDEL
    TURING
    KLEENE
    ES TA B L I S H E S T H E CH U RC H -TU R I N G TH E S I S
    1936
    -CA L C U L U S 㱻 TU R I N G MAC H I N E
    VON NEUMANN ROSSER
    PÉTER

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  116. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL CURRY CHURCH
    GÖDEL
    TURING
    KLEENE
    O B TA I N S PH D U N D E R CH U RC H
    1936–1938
    PU B L I S H E S 1S T FI X E D -PO I N T CO M B I N AT O R
    VON NEUMANN ROSSER
    PÉTER

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  117. COMBINATORS
    functions with no free variables
    b.b combinator
    b.a not a combinator
    ab.a combinator
    a.ab not a combinator
    abc.c(e.b) combinator

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  118. COMBINATORS
    Sym. Bird -Calculus Use Haskell
    I Idiot a.a identity id
    M Mockingbird f.ff self-application (cannot define)
    K Kestrel ab.a first, const const
    KI Kite ab.b = KI second const id
    C Cardinal fab.fba reverse arguments flip
    B Bluebird fga.f(ga) 1°-1° composition (.)
    B1
    Blackbird fgab.f(gab) = BBB 1°-2° composition (.) . (.)
    Th
    Thrush af.fa = CI hold an argument flip id
    V Vireo abf.fab = BCT hold a pair of args flip . flip id

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  119. CARDINAL
    fab.fba

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  120. λ
    JS C = f => a => b => f(b)(a)
    C := fab.fba

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  121. λ
    JS C(K)(I)(M) === ?
    C K I M = ?
    C := fab.fba
    C = f => a => b => f(b)(a)

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  122. λ
    JS C(K)(I)(M) === M
    C K I M = M
    C := fab.fba
    C = f => a => b => f(b)(a)

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  123. λ
    JS C(K)(I)(M) === M
    C K I M = M
    C := fab.fba
    C = f => a => b => f(b)(a)

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  124. λ
    JS KI(I)(M) === M
    KI I M = M
    C := fab.fba
    C = f => a => b => f(b)(a)

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  125. COMBINATORS
    Sym. Bird -Calculus Use Haskell
    I Idiot a.a identity id
    M Mockingbird f.ff self-application (cannot define)
    K Kestrel ab.a first, const const
    KI Kite ab.b = KI = CK second const id
    C Cardinal fab.fba reverse arguments flip
    B Bluebird fga.f(ga) 1°-1° composition (.)
    B1
    Blackbird fgab.f(gab) = BBB 1°-2° composition (.) . (.)
    Th
    Thrush af.fa = CI hold an argument flip id
    V Vireo abf.fab = BCT hold a pair of args flip . flip id

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  126. flip const 1 8 == 8

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  127. so?

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  128. -CALCULUS
    abstract symbol rewriting
    functional computation
    TURING MACHINE
    hypothetical device
    state-based computation
    (f.ff)a.a
    purely functional programming languages
    higher-level machine-centric languages
    assembly languages
    machine code
    higher-level abstract stateful languages
    real computers

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  129. TM

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  130. EVERYTHING
    CAN BE
    FUNCTIONS
    *though not everything IS or

    SHOULD BE functions
    **but maybe more than you expect

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  131. !x == y || (a && z)

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  132. !x == y || (a && z)

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  133. how‽

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  134. λ
    JS const result = bool ? exp1 : exp2

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  135. λ
    JS const result = bool ? exp1 : exp2
    // true

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  136. λ
    JS const result = bool ? exp1 : exp2
    // false

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  137. λ
    JS const result = bool ? exp1 : exp2
    result := ?

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  138. λ
    JS const result = bool ? exp1 : exp2
    result := bool ? exp1 : exp2

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  139. λ
    JS const result = bool ? exp1 : exp2
    result := bool ? exp1 : exp2

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  140. λ
    JS const result = bool ? exp1 : exp2
    result := bool exp1 exp2

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  141. λ
    JS const result = bool (exp1) (exp2)
    result := func exp1 exp2

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  142. λ
    JS
    result := func exp1 exp2
    const result = bool (exp1) (exp2)
    // true

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  143. λ
    JS
    result := func exp1 exp2
    const result = bool (exp1) (exp2)
    // false

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  144. λ
    JS const result = bool (exp1) (exp2)
    result := func exp1 exp2
    TRUE
    FALSE

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  145. λ
    JS const result = bool (exp1) (exp2)
    result := func exp1 exp2
    K
    KI

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  146. λ
    JS const T = K
    const F = KI
    TRUE := K
    FALSE := KI = C K
    CHURCH ENCODINGS: BOOLEANS

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  147. λ
    JS p

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  148. λ
    JS !p

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  149. λ
    JS !p
    ! p

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  150. λ
    JS not(p)
    NOT p

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  151. λ
    JS not(p)
    NOT p
    F
    T
    F
    T

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  152. λ
    JS C(K) (chirp)(tweet) === tweet
    C(KI)(chirp)(tweet) === chirp
    C K = KI
    C (KI) = K

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  153. λ
    JS C(T) (chirp)(tweet) === tweet
    C(F) (chirp)(tweet) === chirp
    C T = F
    C F = T

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  154. CHURCH ENCODINGS: BOOLEANS
    Sym. Name -Calculus Use
    T TRUE ab.a = K encoding for true
    F FALSE ab.b = KI = CK encoding for false
    NOT p.pFT or C negation
    AND pq.pqF or pq.pqp conjunction
    OR pq.pTq or pq.ppq disjunction
    BEQ pq.p q (NOT q) equality

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  155. λ
    JS const and = ? => ?
    AND := ?.?

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  156. λ
    JS const and = p => q => ?
    AND := pq.?

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  157. λ
    JS const and = p => q => p(?)(¿)
    AND := pq.p?¿

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  158. λ
    JS const and = p => q => p(?)(¿)
    AND := pq.p?¿
    F
    F

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  159. λ
    JS const and = p => q => p(?)(F)
    AND := pq.p?F

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  160. λ
    JS const and = p => q => p(?)(F)
    AND := pq.p?F
    T
    T

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  161. λ
    JS const and = p => q => p(q)(F)
    AND := pq.pqF

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  162. λ
    JS const and = p => q => p(q)(p)
    AND := pq.pqp

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  163. CHURCH ENCODINGS: BOOLEANS
    Sym. Name -Calculus Use
    T TRUE ab.a = K encoding for true
    F FALSE ab.b = KI = CK encoding for false
    NOT p.pFT or C negation
    AND pq.pqF or pq.pqp conjunction
    OR pq.pTq or pq.ppq disjunction
    BEQ pq.p q (NOT q) equality

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  164. λ
    JS const or = p => q => …
    OR := pq.…

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  165. λ
    JS const or = p => q => p(?)(¿)
    OR := pq.p?¿

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  166. λ
    JS const or = p => q => p(p)(q)
    OR := pq.ppq

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  167. CHURCH ENCODINGS: BOOLEANS
    Sym. Name -Calculus Use
    T TRUE ab.a = K encoding for true
    F FALSE ab.b = KI = CK encoding for false
    NOT p.pFT or C negation
    AND pq.pqF or pq.pqp conjunction
    OR pq.pTq or pq.ppq = M* disjunction
    BEQ pq.p q (NOT q) equality

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  168. ( )
    pq.p( ) T
    T F F
    q
    q
    p => q => p(q(T)(F))(q(F)(T))

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  169. ( )
    pq.p
    ( )
    T
    T F
    F
    q
    q

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  170. ( )
    pq.p
    ( )
    T
    T
    F
    F
    q
    q

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  171. ( )
    pq.p
    ( )
    T
    T
    F
    F
    q
    q

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  172. ( )
    pq.p
    ( )
    T
    T
    F
    F
    q
    q

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  173. ( )
    pq.p
    ( )
    T
    T
    F
    F
    q
    q

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  174. ( )
    pq.p
    ( )
    T
    T
    F
    F
    q
    q

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  175. ( )
    pq.p
    ( )
    T
    T
    F
    F
    q
    q
    BOOLEAN EQUALITY

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  176. pq.p
    ( )
    T
    F
    q
    q

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  177. ( )
    pq.p
    q
    NOT
    q

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  178. ( )
    pq.p q
    q NOT
    p => q => p(q)(not(q))

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  179. CHURCH ENCODINGS: BOOLEANS
    Sym. Name -Calculus Use
    T TRUE ab.a = K encoding for true
    F FALSE ab.b = KI = CK encoding for false
    NOT p.pFT or C negation
    AND pq.pqF or pq.pqp conjunction
    OR pq.pTq or pq.ppq = M* disjunction
    BEQ pq.p q (NOT q) equality

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  180. (ONE OF) DE MORGAN'S LAWS
    ¬(P ∧ Q) = (¬P) ∨ (¬Q)
    BEQ (NOT (AND p q)) (OR (NOT p) (NOT q))
    !(p && q) === (!p) || (!q)

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  181. BEQ (NOT (AND p q)) (OR (NOT p) (NOT q))
    (xy.x y ((fab.fba) y))

    ((fab.fba) ((xy.xyx) p q))

    ((f.ff) ((fab.fba) p) ((fab.fba) q))

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  182. BEQ (NOT (AND p q)) (OR (NOT p) (NOT q))
    (xy.x y ((fab.fba) y))

    ((fab.fba) ((xy.xyx) p q))

    ((f.ff) ((fab.fba) p) ((fab.fba) q))

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  183. BEQ (NOT (AND p q)) (OR (NOT p) (NOT q))
    (xy.x y ((fab.fba) y))

    ((fab.fba) ((xy.xyx) p q))

    ((f.ff) ((fab.fba) p) ((fab.fba) q))

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  184. BEQ (NOT (AND p q)) (OR (NOT p) (NOT q))
    (xy.x y ((fab.fba) y))

    ((fab.fba) ((xy.xyx) p q))

    ((f.ff) ((fab.fba) p) ((fab.fba) q))

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  185. BEQ (NOT (AND p q)) (OR (NOT p) (NOT q))
    (xy.x y ((fab.fba) y))

    ((fab.fba) ((xy.xyx) p q))

    ((f.ff) ((fab.fba) p) ((fab.fba) q))

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  186. BEQ (NOT (AND p q)) (OR (NOT p) (NOT q))
    (xy.x y ((fab.fba) y))

    ((fab.fba) ((xy.xyx) p q))

    ((f.ff) ((fab.fba) p) ((fab.fba) q))

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  187. BEQ (NOT (AND p q)) (OR (NOT p) (NOT q))
    (xy.x y ((fab.fba) y))

    ((fab.fba) ((xy.xyx) p q))

    ((f.ff) ((fab.fba) p) ((fab.fba) q))

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  188. BEQ (NOT (AND p q)) (OR (NOT p) (NOT q))
    (xy.x y ((fab.fba) y))

    ((fab.fba) ((xy.xyx) p q))

    ((f.ff) ((fab.fba) p) ((fab.fba) q))

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  189. WHAT ELSE CAN WE INVENT?
    numbers
    arithmetic
    data structures
    type systems
    recursion
    Sorry… can't fit them all in today!
    See part II online

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  190. QUESTION
    how many combinators
    are needed to form a basis?

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  191. QUESTION
    how many combinators

    are needed to form a basis?
    20? 10? 5?

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  192. STARLING · KESTREL
    S := abc.ac(bc)
    K := ab.a

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  193. THE
    SK
    COMBINATOR
    CALCULUS

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  194. I = ?

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  195. I = S K K

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  196. I = S K K
    V = ?

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  197. I = S K K
    V = (S(K((S((S(K((

    S(KS))K)))S))(KK))))
    ((S(K(S((SK)K))))K)

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  198. IOTA
    ι := f.(f abc.ac(bc))xy.x
    I := ιι
    K := ι(ι(ιι))
    S := ι(ι(ι(ιι)))

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  199. seriously though,
    why?

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  200. View Slide

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  203. View Slide

  204. f.(x.f(xx))(x.f(xx))
    THE Y FIXED-POINT COMBINATOR

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  205. f.(x.f(v.xxv))(x.f(v.xxv))
    THE Z FIXED-POINT COMBINATOR

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  206. ADDENDUM

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  207. COMBINATORS
    Sym. Bird -Calculus Use Haskell
    I Idiot a.a identity id
    M Mockingbird f.ff self-application (cannot define)
    K Kestrel ab.a true, first, const const
    KI Kite ab.b = KI = CK false, second const id
    C Cardinal fab.fba reverse arguments flip
    B Bluebird fga.f(ga) 1°←1° composition (.)
    Th
    Thrush af.fa = CI hold an argument flip id
    V Vireo abf.fab = BCT hold a pair of args flip . flip id
    B1
    Blackbird fgab.f(gab) = BBB 1°←2° composition (.) . (.)

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  208. CHURCH ENCODINGS: BOOLEANS
    Sym. Name -Calculus Use
    T TRUE ab.a = K = C(KI) encoding for true
    F FALSE ab.b = KI = CK encoding for false
    NOT p.pFT or C negation
    AND pq.pqF or pq.pqp conjunction
    OR pq.pTq or pq.ppq = M* disjunction
    BEQ pq.p q (NOT q) equality

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  209. CHURCH ENCODINGS: NUMERALS
    Sym. Name -Calculus Use
    N0 ZERO fa.a = F apply f no times to a
    N1 ONCE fa.f a = I* apply f once to a
    N2 TWICE fa.f (f a) apply 2-fold f to a
    N3 THRICE fa.f (f (f a)) apply 3-fold f to a
    N4 FOURFOLD fa.f (f (f (f a))) apply 4-fold f to a
    N5 FIVEFOLD fa.f (f (f (f (f a))))) apply 5-fold f to a

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  210. CHURCH ARITHMETIC
    Name -Calculus Use
    SUCC nf.B f (nf) = nfa.f(nfa) successor of n
    ADD nk.n SUCC k = nkf.B (n f) (k f) addition of n and k
    MULT nkf.n(kf) = B multiplication of n and k
    POW nk.kn = Th
    raise n to the power of k
    PRED n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0 predecessor of n
    PRED n.FST (n Φ (PAIR N0 N0)) predecessor of n (easier)
    SUB nk.k PRED n subtract k from n

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  211. CHURCH ARITHMETIC: BOOLEAN OPS
    Name -Calculus Use
    IS0 n.n (K F) T test if n = 0
    LEQ nk.IS0 (SUB n k) test if n <= k
    EQ nk.AND (LEQ n k) (LEQ k n) test if n = k
    GT nk.B1
    NOT LEQ test if n > k

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  212. CHURCH PAIRS
    Sym. Name -Calculus Use
    PAIR abf.fab = V pair two arguments
    FST p.pK extract first of pair
    SND p.p(KI) extract second of pair
    Φ PHI p.PAIR (SND p) (SUCC (SND p) copy 2nd to 1st, inc 2nd
    SET1ST cp.PAIR c (SND p) set first, immutably
    SET2ND cp.PAIR (FST p) c set second, immutably

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  213. ADDITIONAL RESOURCES
    Combinator Birds · Rathman · http://bit.ly/2iudab9
    To Mock a Mockingbird · Smullyan · http://amzn.to/2g9AlXl
    To Dissect a Mockingbird · Keenan · http://dkeenan.com/Lambda
    .:.

    A Tutorial Introduction to the Lambda Calculus · Rojas · http://bit.ly/1agRC97
    Lambda Calculus · Wikipedia · http://bit.ly/1TsPkGn
    The Lambda Calculus · Stanford · http://stanford.io/2vtg8hp
    .:.

    History of Lambda-calculus and Combinatory Logic
    Cardone, Hindley · http://bit.ly/2wCxv4k
    .:.

    An Introduction to Functional Programming

    through Lambda Calculus · Michaelson · http://amzn.to/2vtts56

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