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as.js A FL O C K of FU N C T I O N S COMBINATORS, LAMBDA CALCULUS, & CHURCH ENCODINGS in JAVASCRIPT

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

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a.a IDENTITY

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λ JS I = a => a I := a.a

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λ JS I = a => a I := a.a

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λ JS I = a => a I := a.a

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λ JS I = a => a I := a.a

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λ JS I = a => a I := a.a

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

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

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

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

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id 5 == 5

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?

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a.a FUNCTION SIGNIFIER

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a.a FUNCTION SIGNIFIER PARAMETER VARIABLE

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a.a FUNCTION SIGNIFIER PARAMETER VARIABLE RETURN EXPRESSION

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a.a FUNCTION SIGNIFIER PARAMETER VARIABLE RETURN EXPRESSION LAMBDA ABSTRACTION

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

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λ JS →

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VARIABLES x x (a) (a)

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

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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

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((a.a)b.c.b)(x)e.f β-REDUCTION

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((a.a)b.c.b)(x)e.f β-REDUCTION

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((a.a)b.c.b)(x)e.f β-REDUCTION

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

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

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

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

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

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

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

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((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

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f.ff MOCKINGBIRD

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λ JS M = f => f(f) M := f.ff

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λ JS M = f => f(f) M := f.ff

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λ JS M = f => f(f) M := f.ff

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λ JS M = f => f(f) M := f.ff

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λ JS M(I) === ? M I = ? M := f.ff

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

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

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λ JS M(M) === ? M M = ? M := f.ff

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

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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

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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

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

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((a.a)bc.b)(x)e.f β-REDUCTION, again

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

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

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

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

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

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ab.a KESTREL

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

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

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

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

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

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

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

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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

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const 7 2 == 7

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

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

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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

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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

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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

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ab.b KITE

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

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

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

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

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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

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?

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

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No content

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?

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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

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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

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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

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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

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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

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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 )

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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CARDINAL fab.fba

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

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

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

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

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

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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

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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

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flip const 1 8 == 8

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-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

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TM

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EVERYTHING CAN BE FUNCTIONS

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… TRUE FALSE NOT AND OR BEQ

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λ JS const result = bool ? exp1 : exp2

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

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

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

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

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

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

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

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

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

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

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

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λ JS !p ! p

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λ JS not(p) NOT p

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λ JS not(p) NOT p F T F T

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λ JS not(T) NOT T F T F T

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λ JS not(F) NOT F F T F T

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λ JS p F T F T p ( ) ( )

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λ JS ( ) ( ) T F T F T T K K KI KI

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λ JS ( ) ( ) F F T F T F KI KI K K

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λ JS const not = p => p(F)(T) NOT := p.pFT

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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

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λ JS not(T) === F not(F) === T NOT T = F NOT F = T

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λ JS not(K) === KI not(KI) === K NOT K = KI NOT (KI) = K ab.a ba.a ba.a ab.a

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

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

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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

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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

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λ JS const and = ? AND := ?

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

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

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

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

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

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

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pq.p F q

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pq.p F q

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pq.p p q

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pq.p p q

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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

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λ JS const or = ? OR := ?

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

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

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

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

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

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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

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pq.ppq

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( pq.ppq ) xy = ?

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( pq.ppq ) xy = xxy

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( pq.ppq ) xy = xxy ( ? ) xy = xxy

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( pq.ppq ) xy = xxy ( ? ) xy = xxy

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( pq.ppq ) xy = xxy M xy = xxy

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( pq.ppq ) xy = xxy M xy = xxy

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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

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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

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

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

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

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

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

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

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

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

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pq.p ( ) T F q q

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( ) pq.p q NOT q

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

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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

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

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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))

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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))

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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))

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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))

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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))

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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))

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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))

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… ZERO ONE TWO THREE SUCC

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… ZERO ONCE TWICE THRICE SUCC

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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))

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λ JS n1(not)(T) = not(T) = ? N1 NOT T = NOT T = ? N1 := fa.fa N1 = f => a => f(a)

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λ JS n1(not)(T) = not(T) = F N1 NOT T = NOT T = F N1 := fa.fa N1 = f => a => f(a)

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λ JS n2(not)(T) = not(not(T)) = ? N2 NOT T = NOT (NOT T) = ? N2 := fa.f(fa) N2 = f => a => f(f(a))

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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))

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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)))

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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))

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λ JS n0(not)(T) = ? N0 NOT T = ? N0 := fa.a N3 = f => a => a

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λ JS n0(not)(T) = T N0 NOT T = T N0 := fa.a N3 = f => a => a

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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

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+ 1 ?

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PEANO NUMBERS SUCC N1 = N2 SUCC N2 = N3 SUCC (SUCC N1) = N3

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λ JS succ = n => ? SUCC := n.? SUCC N1 = N2

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λ JS succ = n => ? SUCC := n.? SUCC fa.fa = fa.f(fa)

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λ JS succ = n => f => a => f(n(f)(a)) SUCC := nfa.f(nfa) SUCC fa.fa = fa.f(fa)

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λ JS succ = n => f => a => f(n(f)(a)) SUCC := nfa.f(nfa) SUCC fa.fa = fa.f(fa)

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λ JS succ = n => f => a => f(n(f)(a)) SUCC := nfa.f(nfa) SUCC fa.fa = fa.f(fa)

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SUCC N2 = (nfa.f(nfa)) N2 = fa.f(N2 f a) = fa.f(f(f a) = N3

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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

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BLUEBIRD fga.f(ga)

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λ JS B = f => g => a => f(g(a)) B := fga.f(ga)

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λ JS B(not)(not)(T) === ? B NOT NOT T = ? B := fga.f(ga) B = f => g => a => f(g(a))

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λ JS B(not)(not)(T) === T B NOT NOT T = T B := fga.f(ga) B = f => g => a => f(g(a))

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odd = not . even

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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

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λ JS succ = n => f => a => f(n(f)(a)) SUCC := nfa.f(nfa)

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λ JS succ = n => f => B(f)(n(f)) SUCC := nf.Bf(nf)

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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

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… ADD MULT POW

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λ JS add = n => k => ? ADD := nk.? ADD N3 N5 = SUCC (SUCC (SUCC N5))

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λ JS add = n => k => ? ADD := nk.? ADD N3 N5 = (SUCC ∘ SUCC ∘ SUCC) N5

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λ JS add = n => k => ? ADD := nk.? ADD N3 N5 = N3 SUCC N5

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λ JS add = n => k => n(succ)(k) ADD := nk.n SUCC k ADD N3 N5 = N3 SUCC N5

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ADD N3 N5 = N3 SUCC N5 = THRICE SUCC FIVEFOLD = SUCC (SUCC (SUCC FIVEFOLD))) = EIGHTFOLD

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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

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λ JS mult = n => k => ? MULT := nk.? MULT N2 N3 = N6

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λ JS mult = n => k => ? MULT := nk.? MULT N2 N3 f a = N6 f a

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λ JS mult = n => k => ? MULT := nk.? MULT N2 N3 f a = (f ∘ f ∘ f ∘ f ∘ f ∘ f) a

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λ JS mult = n => k => ? MULT := nk.? MULT N2 N3 f a = ((f ∘ f ∘ f) ∘ (f ∘ f ∘ f)) a

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λ JS mult = n => k => ? MULT := nk.? MULT N2 N3 f a = ((N3 f) ∘ (N3 f)) a

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λ JS mult = n => k => ? MULT := nk.? MULT N2 N3 f a = N2 (N3 f) a

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λ JS mult = n => k => ? MULT := nk.? MULT N2 N3 f a = N2 (N3 f) a

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λ JS mult = n => k => ? MULT := nk.? MULT N2 N3 f = N2 (N3 f)

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λ JS mult = n => k => n(k(f)) MULT := nkf.n(kf) MULT N2 N3 f = N2 (N3 f)

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MULT N2 N3 f = N2 (N3 f) = TWICE (THRICE f) = (f ∘ f ∘ f) ∘ (f ∘ f ∘ f) = SIXFOLD f

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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

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λ JS MULT := nkf.n(kf) MULT N2 N3 f = N2 (N3 f) mult = n => k => n(k(f))

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λ JS MULT := nkf.n(kf) MULT N2 N3 f = (N2 ∘ N3) f mult = n => k => n(k(f))

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λ JS MULT := nkf.n(kf) MULT N2 N3 f = (N2 ∘ N3) f mult = n => k => n(k(f))

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λ JS MULT := nkf.n(kf) MULT N2 N3 = N2 ∘ N3 mult = n => k => n(k(f))

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λ JS MULT := nkf.n(kf) MULT N2 N3 = B N2 N3 mult = n => k => n(k(f))

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λ JS MULT := nkf.n(kf) MULT N2 N3 = B N2 N3 mult = n => k => n(k(f))

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λ JS MULT := nkf.n(kf) MULT = B mult = n => k => n(k(f))

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MULT := B

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Mult := n k f . n ( k f) = B = f g a . f ( g a) -EQUIVALENCE

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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

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λ JS pow = n => k => ? POW := nk.? POW N2 N3 = N8

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λ JS pow = n => k => ? POW := nk.? POW N2 N3 = N2 × N2 × N2

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λ JS pow = n => k => ? POW := nk.? POW N2 N3 = N2 ∘ N2 ∘ N2

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λ JS pow = n => k => ? POW := nk.? POW N2 N3 = N3 N2

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λ JS pow = n => k => k(n) POW := nk.kn POW N2 N3 = N3 N2

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POW N2 N3 = N3 N2 = THRICE TWICE = TWICE ∘ TWICE ∘ TWICE = EIGHTFOLD

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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

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THRUSH af.fa

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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 (.) . (.)

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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 (.) . (.)

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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

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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

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… ISZERO

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IS0 N0 = T IS0 N1 = F IS0 N2 = F …

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λ JS is0 = n => ? IS0 := n.?

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λ JS is0 = n => n(func)(arg) IS0 := n.n func arg

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λ JS is0 = n => n(func)(arg) IS0 := n.n func arg N0

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λ JS is0 = n => n(func)(T) IS0 := n.n func T N0

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λ JS is0 = n => n(func)(T) IS0 := n.n func T N1 F

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λ JS is0 = n => n(func)(T) IS0 := n.n func T N2 F

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λ JS is0 = n => n(func)(T) IS0 := n.n func T N > 0 F

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λ JS is0 = n => n(K(F))(T) IS0 := n.n (KF) T

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λ JS is0 = n => n(K(F))(T) IS0 := n.n (KF) T IS0 N3 = KF(KF(KF(T))) = F

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λ JS is0 = n => n(K(F))(T) IS0 := n.n (KF) T IS0 N3 = KF(KF(KF(T))) = F

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λ JS is0 = n => n(K(F))(T) IS0 := n.n (KF) T IS0 N3 = KF(KF(KF(T))) = F

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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

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+ × ^

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– 1 ?

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PRED := n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0

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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

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… PAIR FST SND PHI

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VIREO abf.fab

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λ JS V = a => b => f => f(a)(b) V := abf.fab

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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)

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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)

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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)

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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

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No content

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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 (.) . (.)

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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

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FST := p.pK SND := p.p(KI)

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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

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λ JS phi = p => pair (snd(p)) (succ(snd(p))) PHI := p.V (SND p) (SUCC (SND p))

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Φ := p.PAIR (SND p) (SUCC (SND p)) Φ (M, N7) = ?

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Φ := p.PAIR (SND p) (SUCC (SND p)) Φ (M, N7) = ( , )

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Φ := p.PAIR (SND p) (SUCC (SND p)) Φ (M, N7) = (N7, )

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Φ := p.PAIR (SND p) (SUCC (SND p)) Φ (M, N7) = (N7, N8)

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Φ := p.PAIR (SND p) (SUCC (SND p)) Φ (M, N7) = (N7, N8) Φ (N9, N2) = (N2, N3)

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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

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Φ := p.PAIR (SND p) (SUCC (SND p)) Φ (M, N7) = (N7, N8) Φ (N9, N2) = (N2, N3) Φ (N0, N0) = (N0, N1)

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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)

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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)

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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) = ?

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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)

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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

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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

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λ JS pred = n => fst(n(phi)(pair(n0)(n0))) PRED := n.FST (n Φ (PAIR N0 N0))

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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

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… SUB LEQ EQ GT

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λ JS sub = n => k => k(pred)(n) SUB := nk.k PRED n

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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

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λ JS leq = n => k => is0(sub(n)(k)) LEQ := nk.IS0 (SUB n k)

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λ JS eq = n => k => and(leq(n)(k))(leq(k)(n)) EQ := nk.AND(LEQ n k)(LEQ k n)

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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

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λ JS eq = n => k => not(leq(n)(k)) GT := nk.NOT (LEQ n k)

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λ JS eq = n => k => not(leq(n)(k)) GT := nk.NOT (LEQ n k)

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λ JS eq = B(not)(leq) ??? GT := B NOT LEQ ???

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λ JS eq = B(not)(leq) GT := B NOT LEQ

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λ JS eq = n => k => not(leq(n)(k)) GT := nk.NOT (LEQ n k)

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BLACKBIRD fgab.f(gab)

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λ JS eq = B1(not)(leq) GT := B1 NOT LEQ B1 = fgab.f(gab)

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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

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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

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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 (.) . (.)

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B C K I KI = K I = C K B1 = B B B Th = C I V = B C Th = B C (C I)

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

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

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THE SK COMBINATOR CALCULUS

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I = S K K

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

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seriously though, why?

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… ADDENDUM

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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 (.) . (.)

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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

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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

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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

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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

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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

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f.M(x.f(Mx)) THE Y FIXED-POINT COMBINATOR

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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)

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

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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