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.

Transcript

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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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
    

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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
    

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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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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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