Combinator Table

Combinators, Birds, Spellings & More

Function Abstraction	Symbol	AN⁴	Bird	CR¹	CH²	Elem³	Haskell	APL	BQN	Intro
`λa.a`	I	-	Identity	`SKK`	`SKK`	✅	`id`	`⊣⊢`	`⊣⊢`	Sch24
`λab.a`	K	-	Kestrel	`K`	`K`	✅	`const`	`⊢`	`⊢`	Sch24
`λab.b`	κ	-	Kite	`KI`	`KI`			`⊣`	`⊣`
`λab.abb`	W	1y2	Warbler	`C(BMR)`	`CSK`	✅	`join`	`⍨`	`˜`	Cur29
`λabc.acb`	C	2y2	🐦 Cardinal	`S(BBS)(KK)`	`B(ΦBS)KK`	✅	`flip`	`⍨`	`˜`	Sch24
`λabc.a(bc)`	B	1y11	Bluebird	`S(KS)K`	`S(KS)K`	✅	`.`	`∘⍤⍥ 2T`	`∘○ 2T`	Sch24
`λabcd.a(bcd)`	B₁	2y12	Blackbird	`BBB`	`DB`		`.:`	`⍤ 2T`	`∘ 2T`	Cur30
`λabcde.a(bcde)`	B₂	3y13	Bunting	`B(BBB)B`	`DB₁`					Cur30
`λabcd.a(b(cd))`	B₃	4y14	Becard	`B(BB)B`	`BDB`					Cur30
`λabc.ac(bc)`	S	1y21	Starling	`S`	`S`		`<*> / ap`		`⟜`	Sch24
`λabc.a(bc)c`	Σ	1y12	Violet Starling	-	?		`=<<`		`⊸`
`λabcd.ab(cd)`	D	2y21	Dove	`BB`	`BB`			`∘`	`⟜`	Smu85
`λabcd.a(bc)d`	Δ	2y12	Zebra Dove	-	?				`⊸`
`λabcd.a(bd)(cd)`	Φ	1y121	Phoenix	-	`B₁SB`		`liftA2`	`3T`	`3T`	Cur31
`λabcd.a(bc)(bd)`	Ψ	2y21	Psi	-	`B(SΦCB)B`		`on`	`⍥`	`○`	Cur31
`λabcde.abc(de)`	D₁	3y31	Dickcissel	`B(BB)`	`BD`
`λabcde.a(bc)(de)`	D₂	2y211	Dovekie	`BB(BB)`	`DD`					Smu85
`λabcde.ab(cde)`	E	3y22	🦅 Eagle	`B(BBB)`	`BB₁`					Smu85
`λabcde.a(bcd)e`	ε	3y22	Golden Eagle	-	?
`λabcde.a(bde)(cde)`	Φ₁	2y222	Pheasant	-	`BΦΦ`			`3T`	`3T`	Cur31
`λabcdefg.a(bcd)(efg)`	Ê	4y222	Bald Eagle	`B(BBB)(B(BBB))`	`D₂D₂D`					Smu85

1 - CR: Chris Rathman (combinator spellings taken from his site)

2 - CH: Conor Hoekstra (combinator spellings generated from C++ program)

3 - Elementary combinators: Listed in Chapter 5 of Combinatory Logic: Volume I

4 - Arity Notation: AyBC = An A-arity yielding binary function (because BC is length two) where the arguments of the function have arity B and C

Code	Author	Year	Paper
Sch24	Moses Schönfinkel	1924	On the building blocks of mathematical logic
Cur29	Haskell Curry	1929	An Analysis of Logical Substitution
Cur30	Haskell Curry	1931	Grundlagen der Kombinatorischen Logik (The Foundations of Combinatory Logic)
Cur31	Haskell Curry	1931	The universal quantifier in combinatory logic
Cur48	Haskell Curry	1948	A Simplification of the Theory of Combinators
Cur58	H. Curry & R. Feys	1958	Combinatory Logic: Volume I
Tur78	David Turner	1979	Another algorithm for bracket abstraction
Smu85	Raymond Smullyan	1985	To Mock a Mockingbird
Iv89	K. Iverson & E. McDonnell	1989	Phrasal Forms
Loc12	Marshall Lochbaum	2012	Added hook, backhook