Lambda Calc Talk (ConsenSys Condensed Version)

Truncated version of my LC talk, delivered for a remote talk. For the longer original version, see https://github.com/glebec/lambda-talk. For the 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
   ConsenSys Crash Course

   Edition™
   

   View Slide

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

   View Slide

3. a.a
   IDENTITY
   

   View Slide

4. λ
   JS I = a => a
   I := a.a
   

   View Slide

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

   View Slide

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

   View Slide

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

   View Slide

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

   View Slide

9. id 5 == 5
   

   View Slide

10. ?
    

    View Slide

11. a.a
    FUNCTION
    SIGNIFIER
    

    View Slide

12. a.a
    FUNCTION
    SIGNIFIER
    PARAMETER VARIABLE
    

    View Slide

13. a.a
    FUNCTION
    SIGNIFIER
    PARAMETER VARIABLE
    RETURN
    EXPRESSION
    

    View Slide

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

    View Slide

15. ConsenSys Crash Course

    Edition™
    

    View Slide

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

    View Slide

17. λ JS
    →
    

    View Slide

18. VARIABLES
    x x
    (a) (a)
    

    View Slide

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

    View Slide

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
    

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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
    

    View Slide

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
    

    View Slide

43. f.ff
    MOCKINGBIRD
    

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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
    

    View Slide

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
    

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

76. ab.a
    KESTREL
    

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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
    

    View Slide

82. const 7 2 == 7
    

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

90. ab.b
    KITE
    

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

95. 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
    

    View Slide

96. ?
    

    View Slide

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

    K

    S

    C

    B
    Idiot

    Kestrel

    Starling

    Cardinal

    Bluebird
    Ibis?
    

    View Slide

98. View Slide

99. ?
    
    

    View Slide

100. 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
     

     View Slide

101. 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
     

     View Slide

102. 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
     

     View Slide

103. 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
     

     View Slide

104. 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
     GE N E R A L RE C U R S I O N TH E O RY
     

     View Slide

105. RÓZSA PÉTER (POLITZER)
     RE C U R S I V E FU N C T I O N TH E O RY
     1932
     RE K U R S I V E FU N K T I O N E N
     ConsenSys Crash Course

     Edition™
     

     View Slide

106. 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
     

     View Slide

107. 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
     

     View Slide

108. 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
     

     View Slide

109. 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
     

     View Slide

110. 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
     

     View Slide

111. 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
     

     View Slide

112. 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
     

     View Slide

113. 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
     

     View Slide

114. CARDINAL
     fab.fba
     

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

120. 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
     

     View Slide

121. flip const 1 8 == 8
     

     View Slide

122. so?
     

     View Slide

123. -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
     

     View Slide

124. TM
     

     View Slide

125. EVERYTHING
     CAN BE
     FUNCTIONS
     ConsenSys Crash Course

     Edition™: "but note,

     not everything IS or

     SHOULD be functions"
     

     View Slide

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

     View Slide

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

     View Slide

128. how‽
     

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

142. λ
     JS p
     

     View Slide

143. λ
     JS !p
     

     View Slide

144. λ
     JS !p
     ! p
     

     View Slide

145. λ
     JS not(p)
     NOT p
     

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

149. 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
     

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

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

     View Slide

157. 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
     

     View Slide

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

     View Slide

159. 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))
     

     View Slide

160. 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))
     

     View Slide

161. 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))
     

     View Slide

162. 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))
     

     View Slide

163. 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))
     

     View Slide

164. 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))
     

     View Slide

165. 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))
     

     View Slide

166. 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))
     

     View Slide

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

     View Slide

168. QUESTION
     how many combinators

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

     View Slide

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

     View Slide

170. THE
     SK
     COMBINATOR
     CALCULUS
     

     View Slide

171. I = ?
     

     View Slide

172. I = S K K
     

     View Slide

173. I = S K K
     V = ?
     

     View Slide

174. I = S K K
     V = (S(K((S((S(K((

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

     View Slide

175. IOTA
     ι := f.(f abc.ac(bc))ab.a
     I := ιι
     K := ι(ι(ιι))
     S := ι(ι(ι(ιι)))
     

     View Slide

176. seriously though,
     why?
     

     View Slide

177. View Slide

178. View Slide

179. View Slide

180. View Slide

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

     View Slide

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

     View Slide

183. pseudo-recursive
     functorial to invoke
     THE Z FIXED-POINT COMBINATOR
     Z = f => M(x => f(v => x(x)(v)))
     “engine” which
     invents recursion
     ConsenSys Crash Course

     Edition™
     

     View Slide

184. “functorial?”
     “pseudorecursive?”
     

     View Slide

185. RECURSIVE
     fact =
     

     n =>
     

     (n > 0)
     ? n * (n - 1)
     : 1
     fact
     

     View Slide

186. 

     (n > 0)
     ? n * (n - 1)
     : 1
     “PSEUDO-RECURSIVE”
     pseudoFact =
     

     n =>
     f
     f =>
     replaced recursive
     call with parameter
     

     View Slide

187. RECURSIVE VS.PSUEDO-RECURSIVE
     rec = a => ___ rec(x)

     

     pseudoRec = step => a => ___ step(x)
     rec = Z(pseudoRec)
     rec(input) -> result
     

     View Slide

188. so, what is the
     use case?
     

     View Slide

189. RECURSION
     DECORATORS
     LOGGING

     LIMITING
     DEBUGGING
     MEMOIZING
     

     View Slide

190. DEMO
     

     View Slide

191. …
     ADDENDUM
     

     View Slide

192. 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 (.) . (.)
     

     View Slide

193. 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
     

     View Slide

194. 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
     

     View Slide

195. 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
     

     View Slide

196. 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
     

     View Slide

197. 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
     

     View Slide

198. 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
     

     View Slide