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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 formerly @ currently*@ presenting @ *Views and opinions in this presentation are my own and do not represent those of my employer.

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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(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

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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 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 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

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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

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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) M I = I 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 = M M = ? M := f.ff

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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

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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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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 β-REDUCTION, 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 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 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

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((a.a)bc.b)(x)e.f = (bc.b) (x)e.f = (c.x) e.f = x 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(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) === ? K I x = ? K := ab.a K = a => b => a

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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) 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) && 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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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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?

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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 PÉTER

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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 PÉTER

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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 PÉTER

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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 PÉTER

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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 PÉTER

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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 ) PÉTER

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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 PÉTER

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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 PÉTER

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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 PÉTER

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PEANO FREGE RUSSELL SCHÖNFINKEL VON NEUMANN CURRY CHURCH GÖDEL TURING KLEENE ROSSER 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 PÉTER

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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 PÉTER

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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 PÉTER

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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 PÉTER

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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 PÉTER

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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 PÉTER

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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 PÉTER

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combinator?

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COMBINATORS functions with no free variables

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COMBINATORS functions with no free variables b.b combinator b.a not a combinator

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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

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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 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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so?

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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 *though in FP, not everything IS or SHOULD BE functions **but maybe more than you expect

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… TRUE FALSE NOT AND OR BEQ

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

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

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how‽

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λ JS bool

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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

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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 := 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

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λ JS !p

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λ JS !p ! p

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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(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

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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 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 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 not( ) NOT F T F T p p

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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

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λ JS ( ) ( ) T F T F T T K K KI KI

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λ JS p F T F T p ( ) ( )

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λ JS ( ) ( ) F F T F T F KI KI

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λ JS ( ) ( ) F F T F T F KI KI K K

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λ JS p F T F T p ( ) ( )

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λ JS p F T F T p ( ) ( ) p . p =>

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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 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 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 = ? => ? 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?¿

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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(?)(¿) 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(?)(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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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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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 = ?

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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

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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

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

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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 SK

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THE SK COMBINATOR CALCULUS SKI

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I = ?

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I = S K K

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I = S K K V = ?

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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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BUT ACTUALLY… 1

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IOTA ι := f.(f abc.ac(bc))ab.a I := ιι K := ι(ι(ιι)) S := ι(ι(ι(ιι)))

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

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f.(x.f(v.xxv))(x.f(v.xxv)) 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