Lambda as JS, or A Flock of Functions: Combinators, Lambda Calculus, and Church Encodings in JavaScript

🔗 Repo: https://github.com/glebec/lambda-talk
🔗 Video Part I: https://youtu.be/3VQ382QG-y4
🔗 Video Part II: https://youtu.be/pAnLQ9jwN-E

A presentation given at Fullstack Academy, by instructor Gabriel Lebec. The Lambda Calculus is a symbol manipulation system which suffices to calculate anything calculable. Developed in the 1930s by Alonzo Church, this branch of pure math forms the basis of many functional programming languages (such as Scheme and Haskell), and is explained here through the familiar lens of JavaScript. Along the way, related concepts like combinatory logic (investigated by Haskell B. Curry) and encodings are examined, building up boolean logic and arithmetic from scratch.

(Errata: 307–311 & 313, the JS function `eq` should be written as `gt`.)

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
   

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2. glebec
   glebec
   glebec
   glebec
   g_lebec
   
   Gabriel Lebec
   github.com/glebec/lambda-talk
   

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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 = a => a
   I := a.a
   

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6. λ
   JS I = a => a
   I := a.a
   

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7. λ
   JS I = a => a
   I := a.a
   

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8. λ
   JS I = a => a
   I := a.a
   

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

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

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

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

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13. id 5 == 5
    

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14. ?
    

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15. a.a
    FUNCTION
    SIGNIFIER
    

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16. a.a
    FUNCTION
    SIGNIFIER
    PARAMETER VARIABLE
    

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17. a.a
    FUNCTION
    SIGNIFIER
    PARAMETER VARIABLE
    RETURN
    EXPRESSION
    

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

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

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20. λ JS
    →
    

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21. VARIABLES
    x x
    (a) (a)
    

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22. 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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23. 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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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
    = (c.x) e.f
    β-REDUCTION
    

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

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

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35. f.ff
    MOCKINGBIRD
    

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

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

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

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

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

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

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

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

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

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45. 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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46. λ
    JS
    ω ω = ω ω = ω ω = 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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47. a.b.c.b a => b => c => b
    abc.b a => b => c => b
    (a, b, c) => b
    =
    ABSTRACTIONS, again
    

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

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

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

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

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

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53. ((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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54. ab.a
    KESTREL
    

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

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

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

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

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

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

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61. 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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62. 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
    

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63. const 7 2 == 7
    

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

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65. 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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66. 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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67. 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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68. 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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69. ab.b
    KITE
    

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

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

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

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73. 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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74. 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
    

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75. ?
    

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

    K

    S

    C

    B
    Idiot

    Kestrel

    Starling

    Cardinal

    Bluebird
    Ibis?
    

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78. ?
    
    

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79. 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
    

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80. 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
    

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81. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    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
    

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82. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    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
    

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83. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    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
    

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84. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    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 )
    

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85. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    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
    

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86. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    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
    

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87. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    IN C O M P L E T E N E S S TH E O R E M S
    1931
    EN D I N G T H E SE A RC H FO R SU F F I C I E N T AX I O M S
    

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88. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    -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
    

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89. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    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
    

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90. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    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
    

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91. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    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
    

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92. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    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
    

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93. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    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
    

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94. 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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95. 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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96. CARDINAL
    fab.fba
    

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

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

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99. 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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100. 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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101. λ
     JS KI(I)(M) === M
     KI I M = M
     C := fab.fba
     C = f => a => b => f(b)(a)
     

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102. 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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103. 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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104. flip const 1 8 == 8
     

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105. -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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106. TM
     

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107. EVERYTHING
     CAN BE
     FUNCTIONS
     

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108. …
     TRUE FALSE NOT AND OR BEQ
     

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

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

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

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

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

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

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

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

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

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

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

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

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121. λ
     JS !p
     ! p
     

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122. λ
     JS not(p)
     NOT p
     

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

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124. λ
     JS not(T)
     NOT T
     F
     T
     F
     T
     

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125. λ
     JS not(F)
     NOT F
     F
     T
     F
     T
     

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126. λ
     JS p F T
     F T
     p
     ( )
     ( )
     

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127. λ
     JS ( ) ( )
     T F T
     F T
     T
     K
     K
     KI
     KI
     

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128. λ
     JS ( ) ( )
     F F T
     F T
     F
     KI
     KI K
     K
     

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129. λ
     JS const not = p => p(F)(T)
     NOT := p.pFT
     

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130. 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 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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131. λ
     JS not(T) === F
     not(F) === T
     NOT T = F
     NOT F = T
     

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132. λ
     JS not(K) === KI
     not(KI) === K
     NOT K = KI
     NOT (KI) = K
     ab.a ba.a
     ba.a ab.a
     

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

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

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135. 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 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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136. 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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137. λ
     JS const and = ?
     AND := ?
     

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

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

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

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

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

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

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144. pq.p F
     q
     

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145. pq.p
     F
     q
     

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146. pq.p
     p
     q
     

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147. pq.p p
     q
     

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148. 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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149. λ
     JS const or = ?
     OR := ?
     

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

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

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152. λ
     JS const or = p => q => p(T)(¿)
     OR := pq.pT¿
     

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153. λ
     JS const or = p => q => p(T)(q)
     OR := pq.pTq
     

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

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155. 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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156. pq.ppq
     

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157. ( pq.ppq ) xy = ?
     

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158. ( pq.ppq ) xy = xxy
     

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159. ( pq.ppq ) xy = xxy
     ( ? ) xy = xxy
     

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160. ( pq.ppq ) xy = xxy
     ( ? ) xy = xxy
     

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161. ( pq.ppq ) xy = xxy
     M xy = xxy
     

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162. ( pq.ppq ) xy = xxy
     M xy = xxy
     

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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. 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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165. ( )
     pq.p( ) T
     T F F
     q
     q
     p => q => p(q(T)(F))(q(F)(T))
     

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166. ( )
     pq.p
     ( )
     T
     T F
     F
     q
     q
     

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167. ( )
     pq.p
     ( )
     T
     T
     F
     F
     q
     q
     

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

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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
     BOOLEAN EQUALITY
     

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173. pq.p
     ( )
     T
     F
     q
     q
     

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174. ( )
     pq.p
     q
     NOT
     q
     

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175. ( )
     pq.p q
     q NOT
     p => q => p(q)(not(q))
     

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176. 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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177. (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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178. 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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179. 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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180. 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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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. …
     ZERO ONE TWO THREE SUCC
     

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186. …
     ZERO ONCE TWICE THRICE SUCC
     

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187. λ
     JS n1 = f => a => f(a)
     n2 = f => a => f(f(a))
     n3 = f => a => f(f(f(a)))
     N1 := fa.fa
     N2 := fa.f(fa)
     N3 := fa.f(f(fa))
     

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188. λ
     JS n1(not)(T) = not(T) = ?
     N1 NOT T = NOT T = ?
     N1 := fa.fa
     N1 = f => a => f(a)
     

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189. λ
     JS n1(not)(T) = not(T) = F
     N1 NOT T = NOT T = F
     N1 := fa.fa
     N1 = f => a => f(a)
     

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190. λ
     JS n2(not)(T) = not(not(T)) = ?
     N2 NOT T = NOT (NOT T) = ?
     N2 := fa.f(fa)
     N2 = f => a => f(f(a))
     

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191. λ
     JS n2(not)(T) = not(not(T)) = T
     N2 NOT T = NOT (NOT T) = T
     N2 := fa.f(fa)
     N2 = f => a => f(f(a))
     

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192. λ
     JS n3(not)(T) = not(not(not(T))) = F
     N3 NOT T =
     NOT (NOT (NOT T))) = F
     N3 := fa.f(f(fa))
     N3 = f => a => f(f(f(a)))
     

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193. λ
     JS n0 = f => a => a
     n1 = f => a => f(a)
     n2 = f => a => f(f(a))
     n3 = f => a => f(f(f(a)))
     N0 := fa.a
     N1 := fa.fa
     N2 := fa.f(fa)
     N3 := fa.f(f(fa))
     

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194. λ
     JS n0(not)(T) = ?
     N0 NOT T = ?
     N0 := fa.a
     N3 = f => a => a
     

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195. λ
     JS n0(not)(T) = T
     N0 NOT T = T
     N0 := fa.a
     N3 = f => a => a
     

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196. 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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197. + 1 ?
     

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198. PEANO NUMBERS
     SUCC N1 = N2
     SUCC N2 = N3
     SUCC (SUCC N1) = N3
     

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199. λ
     JS succ = n => ?
     SUCC := n.?
     SUCC N1 = N2
     

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200. λ
     JS succ = n => ?
     SUCC := n.?
     SUCC fa.fa = fa.f(fa)
     

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201. λ
     JS succ = n => f => a => f(n(f)(a))
     SUCC := nfa.f(nfa)
     SUCC fa.fa = fa.f(fa)
     

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202. λ
     JS succ = n => f => a => f(n(f)(a))
     SUCC := nfa.f(nfa)
     SUCC fa.fa = fa.f(fa)
     

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203. λ
     JS succ = n => f => a => f(n(f)(a))
     SUCC := nfa.f(nfa)
     SUCC fa.fa = fa.f(fa)
     

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204. SUCC N2 = (nfa.f(nfa)) N2
     = fa.f(N2 f a)
     = fa.f(f(f a)
     = N3
     

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205. CHURCH ARITHMETIC
     Name -Calculus Use
     SUCC 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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206. BLUEBIRD
     fga.f(ga)
     

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207. λ
     JS B = f => g => a => f(g(a))
     B := fga.f(ga)
     

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208. λ
     JS B(not)(not)(T) === ?
     B NOT NOT T = ?
     B := fga.f(ga)
     B = f => g => a => f(g(a))
     

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209. λ
     JS B(not)(not)(T) === T
     B NOT NOT T = T
     B := fga.f(ga)
     B = f => g => a => f(g(a))
     

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210. odd = not . even
     

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211. 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 (.)
     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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212. λ
     JS succ = n => f => a => f(n(f)(a))
     SUCC := nfa.f(nfa)
     

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213. λ
     JS succ = n => f => B(f)(n(f))
     SUCC := nf.Bf(nf)
     

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214. 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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215. …
     ADD MULT POW
     

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216. λ
     JS add = n => k => ?
     ADD := nk.?
     ADD N3 N5 = SUCC (SUCC (SUCC N5))
     

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217. λ
     JS add = n => k => ?
     ADD := nk.?
     ADD N3 N5 = (SUCC ∘ SUCC ∘ SUCC) N5
     

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218. λ
     JS add = n => k => ?
     ADD := nk.?
     ADD N3 N5 = N3 SUCC N5
     

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219. λ
     JS add = n => k => n(succ)(k)
     ADD := nk.n SUCC k
     ADD N3 N5 = N3 SUCC N5
     

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220. ADD N3 N5 = N3 SUCC N5
     = THRICE SUCC FIVEFOLD
     = SUCC (SUCC (SUCC FIVEFOLD)))
     = EIGHTFOLD
     

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221. 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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222. λ
     JS mult = n => k => ?
     MULT := nk.?
     MULT N2 N3 = N6
     

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223. λ
     JS mult = n => k => ?
     MULT := nk.?
     MULT N2 N3 f a = N6 f a
     

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224. λ
     JS mult = n => k => ?
     MULT := nk.?
     MULT N2 N3 f a = (f ∘ f ∘ f ∘ f ∘ f ∘ f) a
     

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225. λ
     JS mult = n => k => ?
     MULT := nk.?
     MULT N2 N3 f a = ((f ∘ f ∘ f) ∘ (f ∘ f ∘ f)) a
     

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226. λ
     JS mult = n => k => ?
     MULT := nk.?
     MULT N2 N3 f a = ((N3 f) ∘ (N3 f)) a
     

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227. λ
     JS mult = n => k => ?
     MULT := nk.?
     MULT N2 N3 f a = N2 (N3 f) a
     

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228. λ
     JS mult = n => k => ?
     MULT := nk.?
     MULT N2 N3 f a = N2 (N3 f) a
     

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229. λ
     JS mult = n => k => ?
     MULT := nk.?
     MULT N2 N3 f = N2 (N3 f)
     

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230. λ
     JS mult = n => k => n(k(f))
     MULT := nkf.n(kf)
     MULT N2 N3 f = N2 (N3 f)
     

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231. MULT N2 N3 f = N2 (N3 f)
     = TWICE (THRICE f)
     = (f ∘ f ∘ f) ∘ (f ∘ f ∘ f)
     = SIXFOLD f
     

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232. 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) 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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233. λ
     JS
     MULT := nkf.n(kf)
     MULT N2 N3 f = N2 (N3 f)
     mult = n => k => n(k(f))
     

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234. λ
     JS
     MULT := nkf.n(kf)
     MULT N2 N3 f = (N2 ∘ N3) f
     mult = n => k => n(k(f))
     

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235. λ
     JS
     MULT := nkf.n(kf)
     MULT N2 N3 f = (N2 ∘ N3) f
     mult = n => k => n(k(f))
     

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236. λ
     JS
     MULT := nkf.n(kf)
     MULT N2 N3 = N2 ∘ N3
     mult = n => k => n(k(f))
     

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237. λ
     JS
     MULT := nkf.n(kf)
     MULT N2 N3 = B N2 N3
     mult = n => k => n(k(f))
     

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238. λ
     JS
     MULT := nkf.n(kf)
     MULT N2 N3 = B N2 N3
     mult = n => k => n(k(f))
     

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239. λ
     JS
     MULT := nkf.n(kf)
     MULT = B
     mult = n => k => n(k(f))
     

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240. MULT := B
     

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241. Mult := n k f . n ( k f)
     = B = f g a . f ( g a)
     -EQUIVALENCE
     

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242. 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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243. λ
     JS pow = n => k => ?
     POW := nk.?
     POW N2 N3 = N8
     

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244. λ
     JS pow = n => k => ?
     POW := nk.?
     POW N2 N3 = N2 × N2 × N2
     

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245. λ
     JS pow = n => k => ?
     POW := nk.?
     POW N2 N3 = N2 ∘ N2 ∘ N2
     

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246. λ
     JS pow = n => k => ?
     POW := nk.?
     POW N2 N3 = N3 N2
     

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247. λ
     JS pow = n => k => k(n)
     POW := nk.kn
     POW N2 N3 = N3 N2
     

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248. POW N2 N3 = N3 N2
     = THRICE TWICE
     = TWICE ∘ TWICE ∘ TWICE
     = EIGHTFOLD
     

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249. 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 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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250. THRUSH
     af.fa
     

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251. 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 hold an argument
     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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252. 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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253. 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 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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254. 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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255. …
     ISZERO
     

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256. IS0 N0 = T
     IS0 N1 = F
     IS0 N2 = F
     …
     

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257. λ
     JS is0 = n => ?
     IS0 := n.?
     

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258. λ
     JS is0 = n => n(func)(arg)
     IS0 := n.n func arg
     

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259. λ
     JS is0 = n => n(func)(arg)
     IS0 := n.n func arg
     N0
     

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260. λ
     JS is0 = n => n(func)(T)
     IS0 := n.n func T
     N0
     

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261. λ
     JS is0 = n => n(func)(T)
     IS0 := n.n func T
     N1
     F
     

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262. λ
     JS is0 = n => n(func)(T)
     IS0 := n.n func T
     N2
     F
     

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263. λ
     JS is0 = n => n(func)(T)
     IS0 := n.n func T
     N > 0
     F
     

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264. λ
     JS is0 = n => n(K(F))(T)
     IS0 := n.n (KF) T
     

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265. λ
     JS is0 = n => n(K(F))(T)
     IS0 := n.n (KF) T
     IS0 N3 = KF(KF(KF(T))) = F
     

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266. λ
     JS is0 = n => n(K(F))(T)
     IS0 := n.n (KF) T
     IS0 N3 = KF(KF(KF(T))) = F
     

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267. λ
     JS is0 = n => n(K(F))(T)
     IS0 := n.n (KF) T
     IS0 N3 = KF(KF(KF(T))) = F
     

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268. 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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269. + × ^
     

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270. – 1 ?
     

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271. PRED := n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0
     

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272. N0: N0 (g.IS0 (g N1) ) (K N0) N0
     I
     (B SUCC g)
     N0
     N1: N1 (g.IS0 (g N1) ) (K N0) N0
     I
     (B SUCC g)
     N0
     N2: N2 (g.IS0 (g N1) ) (K N0) N0
     I
     (B SUCC g)
     N1
     N3: N3 (g.IS0 (g N1) ) (K N0) N0
     I
     (B SUCC g)
     N2
     (K N0)
     I
     (SUCC ∘ I)
     PRED := n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0
     

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273. …
     PAIR FST SND PHI
     

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274. VIREO
     abf.fab
     

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275. λ
     JS V = a => b => f => f(a)(b)
     V := abf.fab
     

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276. λ
     JS V(I)(M)
     // f => f(I)(M)
     V I M
     = (f.f I M)
     V := abf.fab
     V = a => b => f => f(a)(b)
     

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277. λ
     JS V(I)(M)(K)
     // (f => f(I)(M))(K) === ?
     V I M K
     = (f.f I M) K = ?
     V := abf.fab
     V = a => b => f => f(a)(b)
     

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278. λ
     JS V(I)(M)(K)
     // (f => f(I)(M))(K) === I
     V I M K
     = (f.f I M) K = I
     V := abf.fab
     V = a => b => f => f(a)(b)
     

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279. λ
     JS
     V I M KI
     = (f.f I M) KI = M
     V := abf.fab
     V = a => b => f => f(a)(b)
     V(I)(M)(KI)
     // (f => f(I)(M))(KI) === M
     

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280. View Slide

281. 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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282. 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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283. FST := p.pK
     SND := p.p(KI)
     

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284. 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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285. λ
     JS phi = p => pair (snd(p)) (succ(snd(p)))
     PHI := p.V (SND p) (SUCC (SND p))
     

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286. Φ := p.PAIR (SND p) (SUCC (SND p))
     Φ (M, N7) = ?
     

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287. Φ := p.PAIR (SND p) (SUCC (SND p))
     Φ (M, N7) = ( , )
     

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288. Φ := p.PAIR (SND p) (SUCC (SND p))
     Φ (M, N7) = (N7, )
     

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289. Φ := p.PAIR (SND p) (SUCC (SND p))
     Φ (M, N7) = (N7, N8)
     

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290. Φ := p.PAIR (SND p) (SUCC (SND p))
     Φ (M, N7) = (N7, N8)
     Φ (N9, N2) = (N2, N3)
     

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291. 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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292. Φ := p.PAIR (SND p) (SUCC (SND p))
     Φ (M, N7) = (N7, N8)
     Φ (N9, N2) = (N2, N3)
     Φ (N0, N0) = (N0, N1)
     

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293. Φ := p.PAIR (SND p) (SUCC (SND p))
     Φ (M, N7) = (N7, N8)
     Φ (N9, N2) = (N2, N3)
     Φ (N0, N0) = (N0, N1)
     Φ (N0, N1) = (N1, N2)
     

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294. Φ := p.PAIR (SND p) (SUCC (SND p))
     Φ (M, N7) = (N7, N8)
     Φ (N9, N2) = (N2, N3)
     Φ (N0, N0) = (N0, N1)
     Φ (N0, N1) = (N1, N2)
     Φ (N1, N2) = (N2, N3)
     

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295. Φ := p.PAIR (SND p) (SUCC (SND p))
     Φ (M, N7) = (N7, N8)
     Φ (N9, N2) = (N2, N3)
     Φ (N0, N0) = (N0, N1)
     Φ (N0, N1) = (N1, N2)
     Φ (N1, N2) = (N2, N3)
     N8 Φ (N0, N0) = ?
     

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296. Φ := p.PAIR (SND p) (SUCC (SND p))
     Φ (M, N7) = (N7, N8)
     Φ (N9, N2) = (N2, N3)
     Φ (N0, N0) = (N0, N1)
     Φ (N0, N1) = (N1, N2)
     Φ (N1, N2) = (N2, N3)
     N8 Φ (N0, N0) = (N7, N8)
     

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297. FST ( )
     Φ := p.PAIR (SND p) (SUCC (SND p))
     Φ (M, N7) = (N7, N8)
     Φ (N9, N2) = (N2, N3)
     Φ (N0, N0) = (N0, N1)
     Φ (N0, N1) = (N1, N2)
     Φ (N1, N2) = (N2, N3)
     N8 Φ (N0, N0) = FST (N7, N8) = N7
     

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298. FST ( )
     Φ := p.PAIR (SND p) (SUCC (SND p))
     Φ (M, N7) = (N7, N8)
     Φ (N9, N2) = (N2, N3)
     Φ (N0, N0) = (N0, N1)
     Φ (N0, N1) = (N1, N2)
     Φ (N1, N2) = (N2, N3)
     N8 Φ (N0, N0) = FST (N7, N8) = N7
     

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299. λ
     JS pred = n => fst(n(phi)(pair(n0)(n0)))
     PRED := n.FST (n Φ (PAIR N0 N0))
     

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300. 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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301. …
     SUB LEQ EQ GT
     

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302. λ
     JS sub = n => k => k(pred)(n)
     SUB := nk.k PRED n
     

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303. 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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304. λ
     JS leq = n => k => is0(sub(n)(k))
     LEQ := nk.IS0 (SUB n k)
     

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305. λ
     JS eq = n => k => and(leq(n)(k))(leq(k)(n))
     EQ := nk.AND(LEQ n k)(LEQ k n)
     

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306. 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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307. λ
     JS eq = n => k => not(leq(n)(k))
     GT := nk.NOT (LEQ n k)
     

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308. λ
     JS eq = n => k => not(leq(n)(k))
     GT := nk.NOT (LEQ n k)
     

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309. λ
     JS eq = B(not)(leq) ???
     GT := B NOT LEQ ???
     

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310. λ
     JS eq = B(not)(leq)
     GT := B NOT LEQ
     

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311. λ
     JS eq = n => k => not(leq(n)(k))
     GT := nk.NOT (LEQ n k)
     

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312. BLACKBIRD
     fgab.f(gab)
     

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313. λ
     JS eq = B1(not)(leq)
     GT := B1
     NOT LEQ
     B1
     = fgab.f(gab)
     

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314. 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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315. 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) 1°←2° composition
     

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316. 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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317. B C K I
     KI = K I = C K
     B1
     = B B B
     Th
     = C I
     V = B C Th
     = B C (C I)
     

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318. QUESTION
     how many combinators

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

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

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320. THE
     SK
     COMBINATOR
     CALCULUS
     

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321. I = S K K
     

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322. 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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323. seriously though,
     why?
     

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324. View Slide

325. View Slide

326. View Slide

327. View Slide

328. …
     ADDENDUM
     

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329. 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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330. 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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331. 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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332. 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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333. 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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334. 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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335. f.M(x.f(Mx))
     THE Y FIXED-POINT COMBINATOR
     

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336. EVALUATION STRATEGIES
     (ab.b)((f.ff)f.ff) x
     = (b.b) x
     = x
     (ab.b)((f.ff)f.ff)x
     = (ab.b)((f.ff)f.ff)x
     = (ab.b)((f.ff)f.ff)x
     = (ab.b)((f.ff)f.ff)x
     = (ab.b)((f.ff)f.ff)x
     C A L L B Y N A M E
     (apply to args before reduction)
     C A L L B Y VA L U E
     (reduce args before application)
     (AKA normal order; lazy) (AKA applicative order; strict)
     

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337. f.M(x.f(v.Mxv))
     THE Z FIXED-POINT COMBINATOR
     

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338. 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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