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
2. glebec glebec glebec glebec g_lebec Gabriel Lebec github.com/glebec/lambda-talk
3. a.a IDENTITY
4. λ JS I = a => a I := a.a
5. λ JS I = a => a I := a.a
6. λ JS I = a => a I := a.a
7. λ JS I = a => a I := a.a
8. λ JS I = a => a I := a.a
9. λ JS I(x) === ? I x = ? I := a.a I = a => a
10. λ JS I(x) === x I x = x I := a.a I = a => a
11. λ JS I(I) === ? I I = ? I := a.a I = a => a
12. λ JS I(I) === I I I = I I := a.a I = a => a
13. id 5 == 5
14. ?
15. a.a FUNCTION SIGNIFIER
16. a.a FUNCTION SIGNIFIER PARAMETER VARIABLE
17. a.a FUNCTION SIGNIFIER PARAMETER VARIABLE RETURN EXPRESSION
18. a.a FUNCTION SIGNIFIER PARAMETER VARIABLE RETURN EXPRESSION LAMBDA ABSTRACTION
19. -CALCULUS SYNTAX expression ::= variable identifier | expression expression application | variable . expression abstraction | ( expression ) grouping
20. λ JS →
21. VARIABLES x x (a) (a)
22. f a f(a) f a b f(a)(b) (f a) b (f(a))(b) f (a b) f(a(b)) APPLICATIONS
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
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 = (c.x) e.f β-REDUCTION
31. ((a.a)b.c.b)(x)e.f = (b.c.b) (x)e.f = (c.x) e.f β-REDUCTION
32. ((a.a)b.c.b)(x)e.f = (b.c.b) (x)e.f = (c.x) e.f β-REDUCTION
33. ((a.a)b.c.b)(x)e.f = (b.c.b) (x)e.f = (c.x) e.f = x β-REDUCTION
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
35. f.ff MOCKINGBIRD
36. λ JS M = f => f(f) M := f.ff
37. λ JS M = f => f(f) M := f.ff
38. λ JS M = f => f(f) M := f.ff
39. λ JS M = f => f(f) M := f.ff
40. λ JS M(I) === ? M I = ? M := f.ff
41. λ JS M(I) === I(I) && I(I) === ? M I = I I = ? M := f.ff
42. λ JS M(I) === I(I) && I(I) === I M I = I I = I M := f.ff
43. λ JS M(M) === ? M M = ? M := f.ff
44. λ JS M(M) === M(M) === ? M M = M M = ? M := f.ff
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
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
47. a.b.c.b a => b => c => b abc.b a => b => c => b (a, b, c) => b = ABSTRACTIONS, again
48. ((a.a)bc.b)(x)e.f β-REDUCTION, again
49. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f β-REDUCTION, again
50. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f β-REDUCTION, again
51. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f = (c.x) e.f β-REDUCTION, again
52. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f = (c.x) e.f β-REDUCTION, again
53. ((a.a)bc.b)(x)e.f = (bc.b) (x)e.f = (c.x) e.f = x β-REDUCTION, again β-NORMAL FORM
54. ab.a KESTREL
55. λ JS K = a => b => a K := ab.a = a.b.a
56. λ JS K = a => b => a K := ab.a = a.b.a
57. λ JS K = a => b => a K := ab.a = a.b.a
58. λ JS K = a => b => a K := ab.a = a.b.a
59. λ JS K(M)(I) === ? K M I = ? K := ab.a K = a => b => a
60. λ JS K(M)(I) === M K M I = M K := ab.a K = a => b => a
61. λ JS K(M)(I) === M K(I)(M) === ? K M I = M K I M = ? K := ab.a K = a => b => a
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
63. const 7 2 == 7
64. λ JS K(I)(x) === I K I x = I K := ab.a K = a => b => a
65. λ JS K(I)(x)(y) === I(y) && I(y) === ? K I x y = I y = ? K := ab.a K = a => b => a
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
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
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
69. ab.b KITE
70. λ JS KI = a => b => b KI = K(I) KI := ab.b = K I
71. λ JS KI(M)(K) === ? KI M K = ? KI := ab.b KI = a => b => b
72. λ JS KI(M)(K) === K KI M K = K KI := ab.b KI = a => b => b
73. λ JS KI(M)(K) === K KI(K)(M) === ? KI M K = K KI K M = ? KI := ab.b KI = a => b => b
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
75. ?
76. SCHÖNFINKEL CURRY SMULLYAN Identitätsfunktion Konstante Funktion verSchmelzungsfunktion verTauschungsfunktion Zusammensetzungsf. I
 K
 S
 C
 B Idiot
 Kestrel
 Starling
 Cardinal
 Bluebird Ibis?
78. ?
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
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
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
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
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
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 )
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
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
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
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
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
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
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
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
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
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
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
96. CARDINAL fab.fba
97. λ JS C = f => a => b => f(b)(a) C := fab.fba
98. λ JS C(K)(I)(M) === ? C K I M = ? C := fab.fba C = f => a => b => f(b)(a)
99. λ JS C(K)(I)(M) === M C K I M = M C := fab.fba C = f => a => b => f(b)(a)
100. λ JS C(K)(I)(M) === M C K I M = M C := fab.fba C = f => a => b => f(b)(a)
101. λ JS KI(I)(M) === M KI I M = M C := fab.fba C = f => a => b => f(b)(a)
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
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
104. flip const 1 8 == 8
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
106. TM
107. EVERYTHING CAN BE FUNCTIONS
108. … TRUE FALSE NOT AND OR BEQ
109. λ JS const result = bool ? exp1 : exp2
110. λ JS const result = bool ? exp1 : exp2 // true
111. λ JS const result = bool ? exp1 : exp2 // false
112. λ JS const result = bool ? exp1 : exp2 result := ?
113. λ JS const result = bool ? exp1 : exp2 result := bool ? exp1 : exp2
114. λ JS const result = bool ? exp1 : exp2 result := bool exp1 exp2
115. λ JS const result = bool (exp1) (exp2) result := func exp1 exp2
116. λ JS result := func exp1 exp2 const result = bool (exp1) (exp2) // true
117. λ JS result := func exp1 exp2 const result = bool (exp1) (exp2) // false
118. λ JS const result = bool (exp1) (exp2) result := func exp1 exp2 TRUE FALSE
119. λ JS const result = bool (exp1) (exp2) result := func exp1 exp2 K KI
120. λ JS const T = K const F = KI TRUE := K FALSE := KI = C K CHURCH ENCODINGS: BOOLEANS
121. λ JS !p ! p
122. λ JS not(p) NOT p
123. λ JS not(p) NOT p F T F T
124. λ JS not(T) NOT T F T F T
125. λ JS not(F) NOT F F T F T
126. λ JS p F T F T p ( ) ( )
127. λ JS ( ) ( ) T F T F T T K K KI KI
128. λ JS ( ) ( ) F F T F T F KI KI K K
129. λ JS const not = p => p(F)(T) NOT := p.pFT
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
131. λ JS not(T) === F not(F) === T NOT T = F NOT F = T
132. λ JS not(K) === KI not(KI) === K NOT K = KI NOT (KI) = K ab.a ba.a ba.a ab.a
133. λ JS C(K) (chirp)(tweet) === tweet C(KI)(chirp)(tweet) === chirp C K = KI C (KI) = K
134. λ JS C(T) (chirp)(tweet) === tweet C(F) (chirp)(tweet) === chirp C T = F C F = T
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
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
137. λ JS const and = ? AND := ?
138. λ JS const and = p => q => ? AND := pq.?
139. λ JS const and = p => q => p(?)(¿) AND := pq.p?¿
140. λ JS const and = p => q => p(?)(¿) AND := pq.p?¿ F F
141. λ JS const and = p => q => p(?)(F) AND := pq.p?F
142. λ JS const and = p => q => p(?)(F) AND := pq.p?F T T
143. λ JS const and = p => q => p(q)(F) AND := pq.pqF
144. pq.p F q
145. pq.p F q
146. pq.p p q
147. pq.p p q
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
149. λ JS const or = ? OR := ?
150. λ JS const or = p => q => … OR := pq.…
151. λ JS const or = p => q => p(?)(¿) OR := pq.p?¿
152. λ JS const or = p => q => p(T)(¿) OR := pq.pT¿
153. λ JS const or = p => q => p(T)(q) OR := pq.pTq
154. λ JS const or = p => q => p(p)(q) OR := pq.ppq
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
156. pq.ppq
157. ( pq.ppq ) xy = ?
158. ( pq.ppq ) xy = xxy
159. ( pq.ppq ) xy = xxy ( ? ) xy = xxy
160. ( pq.ppq ) xy = xxy ( ? ) xy = xxy
161. ( pq.ppq ) xy = xxy M xy = xxy
162. ( pq.ppq ) xy = xxy M xy = xxy
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. 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
165. ( ) pq.p( ) T T F F q q p => q => p(q(T)(F))(q(F)(T))
166. ( ) pq.p ( ) T T F F q q
167. ( ) pq.p ( ) T T F F q q
168. ( ) pq.p ( ) T T F F q q
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 BOOLEAN EQUALITY
173. pq.p ( ) T F q q
174. ( ) pq.p q NOT q
175. ( ) pq.p q q NOT p => q => p(q)(not(q))
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
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)
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))
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))
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))
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. … ZERO ONE TWO THREE SUCC
186. … ZERO ONCE TWICE THRICE SUCC
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))
188. λ JS n1(not)(T) = not(T) = ? N1 NOT T = NOT T = ? N1 := fa.fa N1 = f => a => f(a)
189. λ JS n1(not)(T) = not(T) = F N1 NOT T = NOT T = F N1 := fa.fa N1 = f => a => f(a)
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))
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))
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)))
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))
194. λ JS n0(not)(T) = ? N0 NOT T = ? N0 := fa.a N3 = f => a => a
195. λ JS n0(not)(T) = T N0 NOT T = T N0 := fa.a N3 = f => a => a
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
197. + 1 ?
198. PEANO NUMBERS SUCC N1 = N2 SUCC N2 = N3 SUCC (SUCC N1) = N3
199. λ JS succ = n => ? SUCC := n.? SUCC N1 = N2
200. λ JS succ = n => ? SUCC := n.? SUCC fa.fa = fa.f(fa)
201. λ JS succ = n => f => a => f(n(f)(a)) SUCC := nfa.f(nfa) SUCC fa.fa = fa.f(fa)
202. λ JS succ = n => f => a => f(n(f)(a)) SUCC := nfa.f(nfa) SUCC fa.fa = fa.f(fa)
203. λ JS succ = n => f => a => f(n(f)(a)) SUCC := nfa.f(nfa) SUCC fa.fa = fa.f(fa)
204. SUCC N2 = (nfa.f(nfa)) N2 = fa.f(N2 f a) = fa.f(f(f a) = N3
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
206. BLUEBIRD fga.f(ga)
207. λ JS B = f => g => a => f(g(a)) B := fga.f(ga)
208. λ JS B(not)(not)(T) === ? B NOT NOT T = ? B := fga.f(ga) B = f => g => a => f(g(a))
209. λ JS B(not)(not)(T) === T B NOT NOT T = T B := fga.f(ga) B = f => g => a => f(g(a))
210. odd = not . even
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
212. λ JS succ = n => f => a => f(n(f)(a)) SUCC := nfa.f(nfa)
213. λ JS succ = n => f => B(f)(n(f)) SUCC := nf.Bf(nf)
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
215. … ADD MULT POW
216. λ JS add = n => k => ? ADD := nk.? ADD N3 N5 = SUCC (SUCC (SUCC N5))
217. λ JS add = n => k => ? ADD := nk.? ADD N3 N5 = (SUCC ∘ SUCC ∘ SUCC) N5
218. λ JS add = n => k => ? ADD := nk.? ADD N3 N5 = N3 SUCC N5
219. λ JS add = n => k => n(succ)(k) ADD := nk.n SUCC k ADD N3 N5 = N3 SUCC N5
220. ADD N3 N5 = N3 SUCC N5 = THRICE SUCC FIVEFOLD = SUCC (SUCC (SUCC FIVEFOLD))) = EIGHTFOLD
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
222. λ JS mult = n => k => ? MULT := nk.? MULT N2 N3 = N6
223. λ JS mult = n => k => ? MULT := nk.? MULT N2 N3 f a = N6 f a
224. λ JS mult = n => k => ? MULT := nk.? MULT N2 N3 f a = (f ∘ f ∘ f ∘ f ∘ f ∘ f) a
225. λ JS mult = n => k => ? MULT := nk.? MULT N2 N3 f a = ((f ∘ f ∘ f) ∘ (f ∘ f ∘ f)) a
226. λ JS mult = n => k => ? MULT := nk.? MULT N2 N3 f a = ((N3 f) ∘ (N3 f)) a
227. λ JS mult = n => k => ? MULT := nk.? MULT N2 N3 f a = N2 (N3 f) a
228. λ JS mult = n => k => ? MULT := nk.? MULT N2 N3 f a = N2 (N3 f) a
229. λ JS mult = n => k => ? MULT := nk.? MULT N2 N3 f = N2 (N3 f)
230. λ JS mult = n => k => n(k(f)) MULT := nkf.n(kf) MULT N2 N3 f = N2 (N3 f)
231. MULT N2 N3 f = N2 (N3 f) = TWICE (THRICE f) = (f ∘ f ∘ f) ∘ (f ∘ f ∘ f) = SIXFOLD f
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
233. λ JS MULT := nkf.n(kf) MULT N2 N3 f = N2 (N3 f) mult = n => k => n(k(f))
234. λ JS MULT := nkf.n(kf) MULT N2 N3 f = (N2 ∘ N3) f mult = n => k => n(k(f))
235. λ JS MULT := nkf.n(kf) MULT N2 N3 f = (N2 ∘ N3) f mult = n => k => n(k(f))
236. λ JS MULT := nkf.n(kf) MULT N2 N3 = N2 ∘ N3 mult = n => k => n(k(f))
237. λ JS MULT := nkf.n(kf) MULT N2 N3 = B N2 N3 mult = n => k => n(k(f))
238. λ JS MULT := nkf.n(kf) MULT N2 N3 = B N2 N3 mult = n => k => n(k(f))
239. λ JS MULT := nkf.n(kf) MULT = B mult = n => k => n(k(f))
240. MULT := B
241. Mult := n k f . n ( k f) = B = f g a . f ( g a) -EQUIVALENCE
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
243. λ JS pow = n => k => ? POW := nk.? POW N2 N3 = N8
244. λ JS pow = n => k => ? POW := nk.? POW N2 N3 = N2 × N2 × N2
245. λ JS pow = n => k => ? POW := nk.? POW N2 N3 = N2 ∘ N2 ∘ N2
246. λ JS pow = n => k => ? POW := nk.? POW N2 N3 = N3 N2
247. λ JS pow = n => k => k(n) POW := nk.kn POW N2 N3 = N3 N2
248. POW N2 N3 = N3 N2 = THRICE TWICE = TWICE ∘ TWICE ∘ TWICE = EIGHTFOLD
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
250. THRUSH af.fa
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 (.) . (.)
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 (.) . (.)
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
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
255. … ISZERO
256. IS0 N0 = T IS0 N1 = F IS0 N2 = F …
257. λ JS is0 = n => ? IS0 := n.?
258. λ JS is0 = n => n(func)(arg) IS0 := n.n func arg
259. λ JS is0 = n => n(func)(arg) IS0 := n.n func arg N0
260. λ JS is0 = n => n(func)(T) IS0 := n.n func T N0
261. λ JS is0 = n => n(func)(T) IS0 := n.n func T N1 F
262. λ JS is0 = n => n(func)(T) IS0 := n.n func T N2 F
263. λ JS is0 = n => n(func)(T) IS0 := n.n func T N > 0 F
264. λ JS is0 = n => n(K(F))(T) IS0 := n.n (KF) T
265. λ JS is0 = n => n(K(F))(T) IS0 := n.n (KF) T IS0 N3 = KF(KF(KF(T))) = F
266. λ JS is0 = n => n(K(F))(T) IS0 := n.n (KF) T IS0 N3 = KF(KF(KF(T))) = F
267. λ JS is0 = n => n(K(F))(T) IS0 := n.n (KF) T IS0 N3 = KF(KF(KF(T))) = F
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
269. + × ^
270. – 1 ?
271. PRED := n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0
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
273. … PAIR FST SND PHI
274. VIREO abf.fab
275. λ JS V = a => b => f => f(a)(b) V := abf.fab
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)
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)
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)
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
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 (.) . (.)
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
283. FST := p.pK SND := p.p(KI)
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
285. λ JS phi = p => pair (snd(p)) (succ(snd(p))) PHI := p.V (SND p) (SUCC (SND p))
286. Φ := p.PAIR (SND p) (SUCC (SND p)) Φ (M, N7) = ?
287. Φ := p.PAIR (SND p) (SUCC (SND p)) Φ (M, N7) = ( , )
288. Φ := p.PAIR (SND p) (SUCC (SND p)) Φ (M, N7) = (N7, )
289. Φ := p.PAIR (SND p) (SUCC (SND p)) Φ (M, N7) = (N7, N8)
290. Φ := p.PAIR (SND p) (SUCC (SND p)) Φ (M, N7) = (N7, N8) Φ (N9, N2) = (N2, N3)
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
292. Φ := p.PAIR (SND p) (SUCC (SND p)) Φ (M, N7) = (N7, N8) Φ (N9, N2) = (N2, N3) Φ (N0, N0) = (N0, N1)
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)
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)
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) = ?
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)
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
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
299. λ JS pred = n => fst(n(phi)(pair(n0)(n0))) PRED := n.FST (n Φ (PAIR N0 N0))
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
301. … SUB LEQ EQ GT
302. λ JS sub = n => k => k(pred)(n) SUB := nk.k PRED n
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
304. λ JS leq = n => k => is0(sub(n)(k)) LEQ := nk.IS0 (SUB n k)
305. λ JS eq = n => k => and(leq(n)(k))(leq(k)(n)) EQ := nk.AND(LEQ n k)(LEQ k n)
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
307. λ JS eq = n => k => not(leq(n)(k)) GT := nk.NOT (LEQ n k)
308. λ JS eq = n => k => not(leq(n)(k)) GT := nk.NOT (LEQ n k)
309. λ JS eq = B(not)(leq) ??? GT := B NOT LEQ ???
310. λ JS eq = B(not)(leq) GT := B NOT LEQ
311. λ JS eq = n => k => not(leq(n)(k)) GT := nk.NOT (LEQ n k)
312. BLACKBIRD fgab.f(gab)
313. λ JS eq = B1(not)(leq) GT := B1 NOT LEQ B1 = fgab.f(gab)
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
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
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 (.) . (.)
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)
318. QUESTION how many combinators
 are needed to form a basis? 20? 10? 5?
319. STARLING · KESTREL S := abc.ac(bc) K := ab.a
320. THE SK COMBINATOR CALCULUS
321. I = S K K
322. I = S K K V = (S(K((S((S(K((
 S(KS))K)))S))(KK)))) ((S(K(S((SK)K))))K)
323. seriously though, why?
328. … ADDENDUM
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 (.) . (.)
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
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
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
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
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
335. f.M(x.f(Mx)) THE Y FIXED-POINT COMBINATOR
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)
337. f.M(x.f(v.Mxv)) THE Z FIXED-POINT COMBINATOR
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