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 |
Reference Notes
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
Papers
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 |