Combinator Birds

Function Abstraction Symbol Bird Construction SK-only
λabc.a(bc) B Bluebird S(KS)K S(KS)K
λabcd.a(bcd) B1 Blackbird BBB S(KS)(S(KK)(S(KS)K))
λabcde.a(bcde) B2 Bunting B(BBB)B S(KS)(S(KK)(S(KS)(S(KK)(S(KS)K))))
λabcd.a(b(cd)) B3 Becard B(BB)B S(S(KS)(S(KK)(S(KS)(S(KK)(S(KS)K)))))(K(S(KS)K))
λabc.acb C Cardinal S(BBS)(KK), RRR S(S(KS)(S(KK)S))(KK)
λabcd.ab(cd) D Dove BB S(K(S(KS)))(S(KK))
λabcde.abc(de) D1 Dickcissel B(BB) S(K(S(K(S(KS)))))(S(K(S(KK))))
λabcde.a(bc)(de) D2 Dovekies BB(BB) S(K(S(K(S(KS)))))(S(K(S(K(S(KK)))))(S(KS)K))
λabcde.ab(cde) E Eagle B(BBB) S(K(S(KS)))(S(K(S(KK)))(S(K(S(KS)))(S(KK))))
λabcdefg.a(bcd)(efg) Ê Bald Eagle B(BBB)(B(BBB)) S(K(S(K(S(K(S(KS)))))))(S(K(S(K(S(K(S(KK)))))))(S(K(S(K(S(K(S(KS)))))))(S(K(S(K(S(K(S(KK)))))))(S(KS)(S(KK)(S(KS)K))))))
λabc.cba F Finch ETTET, CV S(K(S(S(KS)(S(K(S(SKK)))K))))(S(KK)K)
λabcd.ad(bc) G Goldfinch BBC S(S(KS)(S(KK)(S(KS)(S(KK)S))))(K(S(KK)))
λabc.abcb H Hummingbird BW(BC) S(S(KS)(S(KS)))(KK)
λa.a I Identity Bird (aka Idiot) SKx, CKx, WK SKK
λabcd.ab(adc) J Jay B(BC)(W(BC(B(BBB)))) S(S(KS)(S(K(S(KS)))(S(K(S(KK)))(S(K(S(KS)))(S(KK))))))(S(KK)(S(S(KS)(S(KK)S))(KK)))
λab.a K Kestrel (True) K K
λab.a(bb) L Lark CBM, BWB, B(TM)B S(S(KS)K)(K(S(SKK)(SKK)))
λa.aa M Mockingbird SII, WI, LI S(SKK)(SKK)
λab.ab(ab) M2 Double Mockingbird BM, WS, SSI SS(SKK)
λab.b(ab) O Owl SI, BW(CB) S(SKK)
λabc.b(ac) Q Queer Bird CB S(K(S(S(KS)K)))K
λabc.a(cb) Q1 Quixotic Bird BCB S(S(KS)(S(KK)(S(KS)K)))(K(S(K(S(SKK)))K))
λabc.b(ca) Q2 Quizzical Bird C(BCB) S(K(S(S(KS)K)))(S(KK)(S(K(S(SKK)))K))
λabc.c(ab) Q3 Quirky Bird BT S(K(S(K(S(SKK)))))(S(KK))
λabc.c(ba) Q4 Quacky Bird F*B S(K(S(K(S(SKK)))))(S(K(S(KK)))(S(K(S(SKK)))K))
λabc.bca R Robin BBT, CC S(K(SS))(S(KK)K)
λabc.ac(bc) S Starling S, BBBW(BBC) S
λab.ba T Thrush CI S(K(S(SKK)))K
λab.b(aab) U Turing LO S(K(S(SKK)))(S(SKK)(SKK))
λabc.cab V Vireo (aka Pairing) BCT, CF S(S(KS)(S(KK)(S(KS)(S(K(S(SKK)))K))))(KK)
λab.abb W Warbler C(BMR), SS(SK), ST, CSI SS(SK)
λab.baa W1 Converse Warbler CW S(S(KS)(S(K(S(SKK)))K))K
λa.a(λa) Y Why Bird (aka Sage Bird) SLL ((SS)K)((S(K((SS)(S((SS)K)))))K)
λab.ab I* Identity Bird Once Removed BI S(KS)
λabc.abcc W* Warbler Once Removed BW S(S(KS)(S(KS)))(K(SK))
λabcd.abdc C* Cardinal Once Removed BC S(S(KS)(S(K(S(KS)))(S(K(S(KK)))(S(KS)))))(K(KK))
λabcd.acdb R* Robin Once Removed C*C* S(S(KS)(S(KK)(S(KS)(S(KS)))))(K(S(KK)K))
λabcd.adcb F* Finch Once Removed BC*R* S(S(KS)(S(KK)(S(KS)(S(K(S(KS)))(S(S(KS)(S(KK)S))(KK))))))(K(S(KK)K))
λabcd.adbc V* Vireo Once Removed C*F*, B(BC)C S(S(KS)(S(K(S(KS)))(S(K(S(KK)))(S(K(S(KS)))(S(S(KS)(S(KK)S))(KK))))))(K(KK))
λabc.abc I** Identity Bird Twice Removed BCC SS(SK)(S(KS)K)(S(S(KS)(S(KK)S))(KK))
λabcd.abcdd W** Warbler Twice Removed B(BW) S(S(KS)(S(K(S(KS)))(S(K(S(KS))))))(K(K(SK)))
λabcde.abced C** Cardinal Twice Removed BC* S(S(KS)(S(K(S(KS)))(S(K(S(K(S(KS)))))(S(K(S(K(S(KK)))))(S(K(S(KS))))))))(K(K(KK)))
λabcde.abdec R** Robin Twice Removed BR* S(S(KS)(S(K(S(KS)))(S(K(S(KK)))(S(K(S(KS)))(S(K(S(KS))))))))(K(K(S(KK)K)))
λabcde.abedc F** Finch Twice Removed BF* S(S(KS)(S(K(S(KS)))(S(K(S(KK)))(S(K(S(KS)))(S(K(S(K(S(KS)))))(S(S(KS)(S(K(S(KS)))(S(K(S(KK)))(S(KS)))))(K(KK))))))))(K(K(S(KK)K)))
λabcde.abecd V** Vireo Twice Removed BV* S(S(KS)(S(K(S(KS)))(S(K(S(K(S(KS)))))(S(K(S(K(S(KK)))))(S(K(S(K(S(KS)))))(S(S(KS)(S(K(S(KS)))(S(K(S(KK)))(S(KS)))))(K(KK))))))))(K(K(KK)))
λab.b KI Kite (False) KI, SK, CK SK
  Ω Omega MM S(SKK)(SKK)(S(SKK)(SKK))
λab.bb KM Konstant Mocker KM K(S(SKK)(SKK))
λab.aa C(KM) Crossed Konstant Mocker C(KM), SBK S(S(KS)K)K
  Θ Theta YO ((((SS)K)((S(K((SS)(S((SS)K)))))K)(S((SK)K))))

 

Notes on the Combinator Birds:
  1. The combinatory birds were borrowed from To Mock A MockingBird, by Raymond Smullyan.
  2. Some additional information about combinator birds can be found in To Dissect a Mockingbird by David C Keenan.
  3. Some of the SK Combinatory terms were first reduced using the Combinatory Logic Tutorial by Chris Barker.
Based on the original work by Chris Rathman (Chris.Rathman@gmail.com), salvaged from angelfire.com/tx4/cus/combinator/birds.html (Dec 14, 2025)