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™
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(x) === ? I x = ? I := a.a I = a => a
6. λ JS I(x) === x I x = x I := a.a I = a => a
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
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
15. ConsenSys Crash Course
 Edition™
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
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
49. λ JS M(M) === ? M M = ? M := f.ff
50. λ JS M(M) === M(M) M M = M M M := f.ff
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
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
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
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
85. λ JS K(I)(x)(y) === I(y) K I x y = I y K := ab.a K = a => b => a
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
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. ab.b KITE
91. λ JS KI = a => b => b KI = K(I) KI := ab.b = K I
92. λ JS KI(M)(K) === ? KI M K = ? KI := ab.b KI = a => b => b
93. λ JS KI(M)(K) === K KI M K = K KI := ab.b KI = a => b => b
94. λ JS KI(M)(K) === K KI(K)(M) === ? KI M K = K KI K M = ? KI := ab.b KI = a => b => b
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
96. ?
97. SCHÖNFINKEL CURRY SMULLYAN Identitätsfunktion Konstante Funktion verSchmelzungsfunktion verTauschungsfunktion Zusammensetzungsf. I
 K
 S
 C
 B Idiot
 Kestrel
 Starling
 Cardinal
 Bluebird Ibis?
99. ?
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
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
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
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
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
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™
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
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
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
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
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
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
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
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
114. CARDINAL fab.fba
115. λ JS C = f => a => b => f(b)(a) C := fab.fba
116. λ JS C(K)(I)(M) === ? C K I M = ? C := fab.fba C = f => a => b => f(b)(a)
117. λ JS C(K)(I)(M) === M C K I M = M C := fab.fba C = f => a => b => f(b)(a)
118. λ JS C(K)(I)(M) === M C K I M = M C := fab.fba C = f => a => b => f(b)(a)
119. λ JS KI(I)(M) === M KI I M = M C := fab.fba C = f => a => b => f(b)(a)
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
121. flip const 1 8 == 8
122. so?
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
124. TM
125. EVERYTHING CAN BE FUNCTIONS ConsenSys Crash Course
 Edition™: "but note,
 not everything IS or
 SHOULD be functions"
126. !x == y || (a && z)
127. !x == y || (a && z)
128. how‽
129. λ JS const result = bool ? exp1 : exp2
130. λ JS const result = bool ? exp1 : exp2 // true
131. λ JS const result = bool ? exp1 : exp2 // false
132. λ JS const result = bool ? exp1 : exp2 result := ?
133. λ JS const result = bool ? exp1 : exp2 result := bool ? exp1 : exp2
134. λ JS const result = bool ? exp1 : exp2 result := bool ? exp1 : exp2
135. λ JS const result = bool ? exp1 : exp2 result := bool exp1 exp2
136. λ JS const result = bool (exp1) (exp2) result := func exp1 exp2
137. λ JS result := func exp1 exp2 const result = bool (exp1) (exp2) // true
138. λ JS result := func exp1 exp2 const result = bool (exp1) (exp2) // false
139. λ JS const result = bool (exp1) (exp2) result := func exp1 exp2 TRUE FALSE
140. λ JS const result = bool (exp1) (exp2) result := func exp1 exp2 K KI
141. λ JS const T = K const F = KI TRUE := K FALSE := KI = C K CHURCH ENCODINGS: BOOLEANS
142. λ JS p
143. λ JS !p
144. λ JS !p ! p
145. λ JS not(p) NOT p
146. λ JS not(p) NOT p F T F T
147. λ JS C(K) (chirp)(tweet) === tweet C(KI)(chirp)(tweet) === chirp C K = KI C (KI) = K
148. λ JS C(T) (chirp)(tweet) === tweet C(F) (chirp)(tweet) === chirp C T = F C F = T
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
150. λ JS const and = ? => ? AND := ?.?
151. λ JS const and = p => q => ? AND := pq.?
152. λ JS const and = p => q => p(?)(¿) AND := pq.p?¿
153. λ JS const and = p => q => p(?)(¿) AND := pq.p?¿ F F
154. λ JS const and = p => q => p(?)(F) AND := pq.p?F
155. λ JS const and = p => q => p(?)(F) AND := pq.p?F T T
156. λ JS const and = p => q => p(q)(F) AND := pq.pqF
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
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)
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))
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))
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))
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))
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))
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))
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))
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))
167. QUESTION how many combinators are needed to form a basis?
168. QUESTION how many combinators
 are needed to form a basis? 20? 10? 5?
169. STARLING · KESTREL S := abc.ac(bc) K := ab.a
170. THE SK COMBINATOR CALCULUS
171. I = ?
172. I = S K K
173. I = S K K V = ?
174. I = S K K V = (S(K((S((S(K((
 S(KS))K)))S))(KK)))) ((S(K(S((SK)K))))K)
175. IOTA ι := f.(f abc.ac(bc))ab.a I := ιι K := ι(ι(ιι)) S := ι(ι(ι(ιι)))
176. seriously though, why?
181. f.(x.f(xx))(x.f(xx)) THE Y FIXED-POINT COMBINATOR
182. f.(x.f(v.xxv))(x.f(v.xxv)) THE Z FIXED-POINT COMBINATOR
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™
184. “functorial?” “pseudorecursive?”
185. RECURSIVE fact = 
 n => 
 (n > 0) ? n * (n - 1) : 1 fact
186. 
 (n > 0) ? n * (n - 1) : 1 “PSEUDO-RECURSIVE” pseudoFact = 
 n => f f => replaced recursive call with parameter
187. RECURSIVE VS.PSUEDO-RECURSIVE rec = a => ___ rec(x)
 
 pseudoRec = step => a => ___ step(x) rec = Z(pseudoRec) rec(input) -> result
188. so, what is the use case?
189. RECURSION DECORATORS LOGGING
 LIMITING DEBUGGING MEMOIZING
190. DEMO
191. … ADDENDUM
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 (.) . (.)
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
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
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
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
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
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