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
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.
3. a.a IDENTITY
4. λ JS I = a => a I := a.a
5. λ JS I(x) === ? I x = ? I := a.a I = a => a
6. λ JS I(x) === x I x = x I := a.a I = a => a (a.a)x = x (a => a)(x) === x
7. λ JS I(I) === ? I I = ? I := a.a I = a => a
8. λ JS I(I) === I I I = I I := a.a I = a => a (a.a)a.a = a.a
9. id 5 == 5
10. ?
11. a.a FUNCTION SIGNIFIER
12. a.a FUNCTION SIGNIFIER PARAMETER VARIABLE
13. a.a FUNCTION SIGNIFIER PARAMETER VARIABLE RETURN EXPRESSION
14. a.a FUNCTION SIGNIFIER PARAMETER VARIABLE RETURN EXPRESSION LAMBDA ABSTRACTION
16. -CALCULUS SYNTAX expression ::= variable identifier | expression expression application | variable . expression abstraction | ( expression ) grouping
17. λ JS →
18. VARIABLES x x (a) (a)
19. f a f(a) f a b f(a)(b) (f a) b (f(a))(b) f (a b) f(a(b)) APPLICATIONS
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
21. ((a.a)b.c.b)(x)e.f β-REDUCTION
22. ((a.a)b.c.b)(x)e.f β-REDUCTION
23. ((a.a)b.c.b)(x)e.f β-REDUCTION
24. ((a.a)b.c.b)(x)e.f β-REDUCTION
25. ((a.a)b.c.b)(x)e.f β-REDUCTION
26. ((a.a)b.c.b)(x)e.f β-REDUCTION
27. ((a.a)b.c.b)(x)e.f = (b.c.b) (x)e.f β-REDUCTION
28. ((a.a)b.c.b)(x)e.f = (b.c.b) (x)e.f β-REDUCTION
29. ((a.a)b.c.b)(x)e.f = (b.c.b) (x)e.f β-REDUCTION
30. ((a.a)b.c.b)(x)e.f = (b.c.b) (x)e.f β-REDUCTION
31. ((a.a)b.c.b)(x)e.f = (b.c.b) (x)e.f β-REDUCTION
32. ((a.a)b.c.b)(x)e.f = (b.c.b) (x)e.f β-REDUCTION
33. ((a.a)b.c.b)(x)e.f = (b.c.b) (x)e.f = (c.x) e.f β-REDUCTION
34. ((a.a)b.c.b)(x)e.f = (b.c.b) (x)e.f = (c.x) e.f β-REDUCTION
35. ((a.a)b.c.b)(x)e.f = (b.c.b) (x)e.f = (c.x) e.f β-REDUCTION
36. ((a.a)b.c.b)(x)e.f = (b.c.b) (x)e.f = (c.x) e.f β-REDUCTION
37. ((a.a)b.c.b)(x)e.f = (b.c.b) (x)e.f = (c.x) e.f β-REDUCTION
38. ((a.a)b.c.b)(x)e.f = (b.c.b) (x)e.f = (c.x) e.f β-REDUCTION
39. ((a.a)b.c.b)(x)e.f = (b.c.b) (x)e.f = (c.x) e.f = x β-REDUCTION
40. ((a.a)b.c.b)(x)e.f = (b.c.b) (x)e.f = (c.x) e.f = x β-REDUCTION
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
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
43. f.ff MOCKINGBIRD
44. λ JS M = f => f(f) M := f.ff
45. λ JS M(I) === ? M I = ? M := f.ff
46. λ JS M(I) === I(I) M I = I I M := f.ff (f.ff)a.a = (a.a)a.a
47. λ JS M(I) === I(I) && I(I) === ? M I = I I = ? M := f.ff
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
49. λ JS M(M) === ? M M = ? M := f.ff
50. λ JS M(M) === M(M) M M = M M M := f.ff (f.ff)g.gg = (g.gg)g.gg
51. λ JS M(M) === M(M) === ? M M = M M = ? M := f.ff
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 = …
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
54. a.b.c.b a => b => c => b abc.b a => b => c => b (a, b, c) => b = ABSTRACTIONS, again
55. ((a.a)bc.b)(x)e.f β-REDUCTION, again
56. ((a.a)bc.b)(x)e.f β-REDUCTION, again
57. ((a.a)bc.b)(x)e.f β-REDUCTION, again
58. ((a.a)bc.b)(x)e.f β-REDUCTION, again
59. ((a.a)bc.b)(x)e.f β-REDUCTION, again
60. ((a.a)bc.b)(x)e.f β-REDUCTION, again
61. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f β-REDUCTION, again
62. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f β-REDUCTION, again
63. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f β-REDUCTION, again
64. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f β-REDUCTION, again
65. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f β-REDUCTION, again
66. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f β-REDUCTION, again
67. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f = (c.x) e.f β-REDUCTION, again
68. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f = (c.x) e.f β-REDUCTION, again
69. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f = (c.x) e.f β-REDUCTION, again
70. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f = (c.x) e.f β-REDUCTION, again
71. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f = (c.x) e.f β-REDUCTION, again
72. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f = (c.x) e.f β-REDUCTION, again
73. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f = (c.x) e.f = x β-REDUCTION, again
74. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f = (c.x) e.f = x β-REDUCTION, again
75. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f = (c.x) e.f = x β-REDUCTION, again β-NORMAL FORM
76. ab.a KESTREL
77. λ JS K = a => b => a K := ab.a = a.b.a
78. λ JS K(M)(I) === ? K M I = ? K := ab.a K = a => b => a
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
80. λ JS K(M)(I) === M K(I)(M) === ? K M I = M K I M = ? K := ab.a K = a => b => a
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
82. const 7 2 == 7
83. λ JS K(I)(x) === ? K I x = ? K := ab.a K = a => b => a
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)
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
86. λ JS K(I)(x)(y) === I(y) && I(y) === ? K I x y = I y = ? K := ab.a K = a => b => a
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
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
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
90. λ JS K I x y = y K := ab.a K = a => b => a K(I)(x)(y) === y
91. ab.b KITE
92. λ JS KI = a => b => b KI = K(I) KI := ab.b = K I
93. λ JS KI(M)(K) === ? KI M K = ? KI := ab.b KI = a => b => b
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
95. λ JS KI(M)(K) === K KI(K)(M) === ? KI M K = K KI K M = ? KI := ab.b KI = a => b => b
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
97. ?
98. SCHÖNFINKEL CURRY SMULLYAN Identitätsfunktion Konstante Funktion verSchmelzungsfunktion verTauschungsfunktion Zusammensetzungsf. I
 K
 S
 C
 B Idiot
 Kestrel
 Starling
 Cardinal
 Bluebird Ibis?
100. ?
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
119. CARDINAL fab.fba
120. λ JS C = f => a => b => f(b)(a) C := fab.fba
121. λ JS C(K)(I)(M) === ? C K I M = ? C := fab.fba C = f => a => b => f(b)(a)
122. λ JS C(K)(I)(M) === M C K I M = M C := fab.fba C = f => a => b => f(b)(a)
123. λ JS C(K)(I)(M) === M C K I M = M C := fab.fba C = f => a => b => f(b)(a)
124. λ JS KI(I)(M) === M KI I M = M C := fab.fba C = f => a => b => f(b)(a)
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
126. flip const 1 8 == 8
127. so?
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
129. TM
130. EVERYTHING CAN BE FUNCTIONS *though not everything IS or
 SHOULD BE functions **but maybe more than you expect
131. !x == y || (a && z)
132. !x == y || (a && z)
133. how‽
134. λ JS const result = bool ? exp1 : exp2
135. λ JS const result = bool ? exp1 : exp2 // true
136. λ JS const result = bool ? exp1 : exp2 // false
137. λ JS const result = bool ? exp1 : exp2 result := ?
138. λ JS const result = bool ? exp1 : exp2 result := bool ? exp1 : exp2
139. λ JS const result = bool ? exp1 : exp2 result := bool ? exp1 : exp2
140. λ JS const result = bool ? exp1 : exp2 result := bool exp1 exp2
141. λ JS const result = bool (exp1) (exp2) result := func exp1 exp2
142. λ JS result := func exp1 exp2 const result = bool (exp1) (exp2) // true
143. λ JS result := func exp1 exp2 const result = bool (exp1) (exp2) // false
144. λ JS const result = bool (exp1) (exp2) result := func exp1 exp2 TRUE FALSE
145. λ JS const result = bool (exp1) (exp2) result := func exp1 exp2 K KI
146. λ JS const T = K const F = KI TRUE := K FALSE := KI = C K CHURCH ENCODINGS: BOOLEANS
147. λ JS p
148. λ JS !p
149. λ JS !p ! p
150. λ JS not(p) NOT p
151. λ JS not(p) NOT p F T F T
152. λ JS C(K) (chirp)(tweet) === tweet C(KI)(chirp)(tweet) === chirp C K = KI C (KI) = K
153. λ JS C(T) (chirp)(tweet) === tweet C(F) (chirp)(tweet) === chirp C T = F C F = T
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
155. λ JS const and = ? => ? AND := ?.?
156. λ JS const and = p => q => ? AND := pq.?
157. λ JS const and = p => q => p(?)(¿) AND := pq.p?¿
158. λ JS const and = p => q => p(?)(¿) AND := pq.p?¿ F F
159. λ JS const and = p => q => p(?)(F) AND := pq.p?F
160. λ JS const and = p => q => p(?)(F) AND := pq.p?F T T
161. λ JS const and = p => q => p(q)(F) AND := pq.pqF
162. λ JS const and = p => q => p(q)(p) AND := pq.pqp
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
164. λ JS const or = p => q => … OR := pq.…
165. λ JS const or = p => q => p(?)(¿) OR := pq.p?¿
166. λ JS const or = p => q => p(p)(q) OR := pq.ppq
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
168. ( ) pq.p( ) T T F F q q p => q => p(q(T)(F))(q(F)(T))
169. ( ) pq.p ( ) T T F F q q
170. ( ) pq.p ( ) T T F F q q
171. ( ) pq.p ( ) T T F F q q
172. ( ) pq.p ( ) T T F F q q
173. ( ) pq.p ( ) T T F F q q
174. ( ) pq.p ( ) T T F F q q
175. ( ) pq.p ( ) T T F F q q BOOLEAN EQUALITY
176. pq.p ( ) T F q q
177. ( ) pq.p q NOT q
178. ( ) pq.p q q NOT p => q => p(q)(not(q))
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
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)
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))
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))
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))
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))
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))
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))
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))
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))
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
190. QUESTION how many combinators are needed to form a basis?
191. QUESTION how many combinators
 are needed to form a basis? 20? 10? 5?
192. STARLING · KESTREL S := abc.ac(bc) K := ab.a
193. THE SK COMBINATOR CALCULUS
194. I = ?
195. I = S K K
196. I = S K K V = ?
197. I = S K K V = (S(K((S((S(K((
 S(KS))K)))S))(KK)))) ((S(K(S((SK)K))))K)
198. IOTA ι := f.(f abc.ac(bc))xy.x I := ιι K := ι(ι(ιι)) S := ι(ι(ι(ιι)))
199. seriously though, why?
204. f.(x.f(xx))(x.f(xx)) THE Y FIXED-POINT COMBINATOR
205. f.(x.f(v.xxv))(x.f(v.xxv)) THE Z FIXED-POINT COMBINATOR
206. … ADDENDUM
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 (.) . (.)
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
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
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
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
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
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