Tweet
Tweet

Slide 1

Slide 1 text

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

Slide 2

Slide 2 text

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.

Slide 3

Slide 3 text

a.a IDENTITY

Slide 4

Slide 4 text

λ JS I = a => a I := a.a

Slide 5

Slide 5 text

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

Slide 6

Slide 6 text

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

Slide 7

Slide 7 text

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

Slide 8

Slide 8 text

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

Slide 9

Slide 9 text

id 5 == 5

Slide 10

Slide 10 text

?

Slide 11

Slide 11 text

a.a FUNCTION SIGNIFIER

Slide 12

Slide 12 text

a.a FUNCTION SIGNIFIER PARAMETER VARIABLE

Slide 13

Slide 13 text

a.a FUNCTION SIGNIFIER PARAMETER VARIABLE RETURN EXPRESSION

Slide 14

Slide 14 text

a.a FUNCTION SIGNIFIER PARAMETER VARIABLE RETURN EXPRESSION LAMBDA ABSTRACTION

Slide 15

Slide 15 text

No content

Slide 16

Slide 16 text

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

Slide 17

Slide 17 text

λ JS →

Slide 18

Slide 18 text

VARIABLES x x (a) (a)

Slide 19

Slide 19 text

f a f(a) f a b f(a)(b) (f a) b (f(a))(b) f (a b) f(a(b)) APPLICATIONS

Slide 20

Slide 20 text

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

Slide 21

Slide 21 text

((a.a)b.c.b)(x)e.f β-REDUCTION

Slide 22

Slide 22 text

((a.a)b.c.b)(x)e.f β-REDUCTION

Slide 23

Slide 23 text

((a.a)b.c.b)(x)e.f β-REDUCTION

Slide 24

Slide 24 text

((a.a)b.c.b)(x)e.f β-REDUCTION

Slide 25

Slide 25 text

((a.a)b.c.b)(x)e.f β-REDUCTION

Slide 26

Slide 26 text

((a.a)b.c.b)(x)e.f β-REDUCTION

Slide 27

Slide 27 text

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

Slide 28

Slide 28 text

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

Slide 29

Slide 29 text

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

Slide 30

Slide 30 text

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

Slide 31

Slide 31 text

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

Slide 32

Slide 32 text

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

Slide 33

Slide 33 text

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

Slide 34

Slide 34 text

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

Slide 35

Slide 35 text

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

Slide 36

Slide 36 text

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

Slide 37

Slide 37 text

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

Slide 38

Slide 38 text

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

Slide 39

Slide 39 text

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

Slide 40

Slide 40 text

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

Slide 41

Slide 41 text

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

Slide 42

Slide 42 text

((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

Slide 43

Slide 43 text

f.ff MOCKINGBIRD

Slide 44

Slide 44 text

λ JS M = f => f(f) M := f.ff

Slide 45

Slide 45 text

λ JS M(I) === ? M I = ? M := f.ff

Slide 46

Slide 46 text

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

Slide 47

Slide 47 text

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

Slide 48

Slide 48 text

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

Slide 49

Slide 49 text

λ JS M(M) === ? M M = ? M := f.ff

Slide 50

Slide 50 text

λ JS M(M) === M(M) M M = M M M := f.ff (f.ff)g.gg = (g.gg)g.gg

Slide 51

Slide 51 text

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

Slide 52

Slide 52 text

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 = …

Slide 53

Slide 53 text

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

Slide 54

Slide 54 text

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

Slide 55

Slide 55 text

((a.a)bc.b)(x)e.f β-REDUCTION, again

Slide 56

Slide 56 text

((a.a)bc.b)(x)e.f β-REDUCTION, again

Slide 57

Slide 57 text

((a.a)bc.b)(x)e.f β-REDUCTION, again

Slide 58

Slide 58 text

((a.a)bc.b)(x)e.f β-REDUCTION, again

Slide 59

Slide 59 text

((a.a)bc.b)(x)e.f β-REDUCTION, again

Slide 60

Slide 60 text

((a.a)bc.b)(x)e.f β-REDUCTION, again

Slide 61

Slide 61 text

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

Slide 62

Slide 62 text

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

Slide 63

Slide 63 text

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

Slide 64

Slide 64 text

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

Slide 65

Slide 65 text

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

Slide 66

Slide 66 text

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

Slide 67

Slide 67 text

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

Slide 68

Slide 68 text

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

Slide 69

Slide 69 text

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

Slide 70

Slide 70 text

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

Slide 71

Slide 71 text

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

Slide 72

Slide 72 text

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

Slide 73

Slide 73 text

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

Slide 74

Slide 74 text

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

Slide 75

Slide 75 text

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

Slide 76

Slide 76 text

ab.a KESTREL

Slide 77

Slide 77 text

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

Slide 78

Slide 78 text

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

Slide 79

Slide 79 text

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

Slide 80

Slide 80 text

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

Slide 81

Slide 81 text

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

Slide 82

Slide 82 text

const 7 2 == 7

Slide 83

Slide 83 text

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

Slide 84

Slide 84 text

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

Slide 85

Slide 85 text

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

Slide 86

Slide 86 text

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

Slide 87

Slide 87 text

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

Slide 88

Slide 88 text

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

Slide 89

Slide 89 text

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

Slide 90

Slide 90 text

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

Slide 91

Slide 91 text

ab.b KITE

Slide 92

Slide 92 text

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

Slide 93

Slide 93 text

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

Slide 94

Slide 94 text

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

Slide 95

Slide 95 text

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

Slide 96

Slide 96 text

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

Slide 97

Slide 97 text

?

Slide 98

Slide 98 text

SCHÖNFINKEL CURRY SMULLYAN Identitätsfunktion Konstante Funktion verSchmelzungsfunktion verTauschungsfunktion Zusammensetzungsf. I
 K
 S
 C
 B Idiot
 Kestrel
 Starling
 Cardinal
 Bluebird Ibis?

Slide 99

Slide 99 text

No content

Slide 100

Slide 100 text

?

Slide 101

Slide 101 text

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

Slide 102

Slide 102 text

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

Slide 103

Slide 103 text

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

Slide 104

Slide 104 text

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

Slide 105

Slide 105 text

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

Slide 106

Slide 106 text

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

Slide 107

Slide 107 text

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

Slide 108

Slide 108 text

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

Slide 109

Slide 109 text

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

Slide 110

Slide 110 text

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

Slide 111

Slide 111 text

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

Slide 112

Slide 112 text

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

Slide 113

Slide 113 text

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

Slide 114

Slide 114 text

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

Slide 115

Slide 115 text

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

Slide 116

Slide 116 text

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

Slide 117

Slide 117 text

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

Slide 118

Slide 118 text

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

Slide 119

Slide 119 text

CARDINAL fab.fba

Slide 120

Slide 120 text

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

Slide 121

Slide 121 text

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

Slide 122

Slide 122 text

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

Slide 123

Slide 123 text

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

Slide 124

Slide 124 text

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

Slide 125

Slide 125 text

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

Slide 126

Slide 126 text

flip const 1 8 == 8

Slide 127

Slide 127 text

so?

Slide 128

Slide 128 text

-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

Slide 129

Slide 129 text

TM

Slide 130

Slide 130 text

EVERYTHING CAN BE FUNCTIONS *though not everything IS or
 SHOULD BE functions **but maybe more than you expect

Slide 131

Slide 131 text

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

Slide 132

Slide 132 text

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

Slide 133

Slide 133 text

how‽

Slide 134

Slide 134 text

λ JS const result = bool ? exp1 : exp2

Slide 135

Slide 135 text

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

Slide 136

Slide 136 text

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

Slide 137

Slide 137 text

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

Slide 138

Slide 138 text

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

Slide 139

Slide 139 text

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

Slide 140

Slide 140 text

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

Slide 141

Slide 141 text

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

Slide 142

Slide 142 text

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

Slide 143

Slide 143 text

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

Slide 144

Slide 144 text

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

Slide 145

Slide 145 text

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

Slide 146

Slide 146 text

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

Slide 147

Slide 147 text

λ JS p

Slide 148

Slide 148 text

λ JS !p

Slide 149

Slide 149 text

λ JS !p ! p

Slide 150

Slide 150 text

λ JS not(p) NOT p

Slide 151

Slide 151 text

λ JS not(p) NOT p F T F T

Slide 152

Slide 152 text

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

Slide 153

Slide 153 text

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

Slide 154

Slide 154 text

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

Slide 155

Slide 155 text

λ JS const and = ? => ? AND := ?.?

Slide 156

Slide 156 text

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

Slide 157

Slide 157 text

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

Slide 158

Slide 158 text

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

Slide 159

Slide 159 text

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

Slide 160

Slide 160 text

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

Slide 161

Slide 161 text

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

Slide 162

Slide 162 text

λ JS const and = p => q => p(q)(p) AND := pq.pqp

Slide 163

Slide 163 text

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

Slide 164

Slide 164 text

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

Slide 165

Slide 165 text

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

Slide 166

Slide 166 text

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

Slide 167

Slide 167 text

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

Slide 168

Slide 168 text

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

Slide 169

Slide 169 text

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

Slide 170

Slide 170 text

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

Slide 171

Slide 171 text

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

Slide 172

Slide 172 text

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

Slide 173

Slide 173 text

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

Slide 174

Slide 174 text

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

Slide 175

Slide 175 text

( ) pq.p ( ) T T F F q q BOOLEAN EQUALITY

Slide 176

Slide 176 text

pq.p ( ) T F q q

Slide 177

Slide 177 text

( ) pq.p q NOT q

Slide 178

Slide 178 text

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

Slide 179

Slide 179 text

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

Slide 180

Slide 180 text

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

Slide 181

Slide 181 text

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))

Slide 182

Slide 182 text

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))

Slide 183

Slide 183 text

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))

Slide 184

Slide 184 text

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))

Slide 185

Slide 185 text

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))

Slide 186

Slide 186 text

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))

Slide 187

Slide 187 text

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))

Slide 188

Slide 188 text

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))

Slide 189

Slide 189 text

WHAT ELSE CAN WE INVENT? numbers arithmetic data structures type systems recursion Sorry… can't fit them all in today! See part II online

Slide 190

Slide 190 text

QUESTION how many combinators are needed to form a basis?

Slide 191

Slide 191 text

QUESTION how many combinators
 are needed to form a basis? 20? 10? 5?

Slide 192

Slide 192 text

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

Slide 193

Slide 193 text

THE SK COMBINATOR CALCULUS

Slide 194

Slide 194 text

I = ?

Slide 195

Slide 195 text

I = S K K

Slide 196

Slide 196 text

I = S K K V = ?

Slide 197

Slide 197 text

I = S K K V = (S(K((S((S(K((
 S(KS))K)))S))(KK)))) ((S(K(S((SK)K))))K)

Slide 198

Slide 198 text

IOTA ι := f.(f abc.ac(bc))xy.x I := ιι K := ι(ι(ιι)) S := ι(ι(ι(ιι)))

Slide 199

Slide 199 text

seriously though, why?

Slide 200

Slide 200 text

No content

Slide 201

Slide 201 text

No content

Slide 202

Slide 202 text

No content

Slide 203

Slide 203 text

No content

Slide 204

Slide 204 text

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

Slide 205

Slide 205 text

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

Slide 206

Slide 206 text

… ADDENDUM

Slide 207

Slide 207 text

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 (.) . (.)

Slide 208

Slide 208 text

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

Slide 209

Slide 209 text

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

Slide 210

Slide 210 text

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

Slide 211

Slide 211 text

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

Slide 212

Slide 212 text

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

Slide 213

Slide 213 text

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