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A specific captivated forest is occupied by talking birds.
Combinatory reasoning shows that even without anything as crude as digits and operators, math and calculation can emerge. Combinatory Forests are populated by talking birds, which speak words delimited by parentheses. We constantly listen to the left-most bird.
T▲ (◆ ● ) -◆ ● ▲ C( WK) x( yz)
▲ The Bekimet Forest
Within the Bekimet forest, we discover the B, KI, M, and T birds. Each bird listens to a message, and responses with another. The very first bird that we observe is the Mockingbird, which duplicates the very first word.
M▲ ◆ ● -▲ ▲ ◆ ● W( WK) xyz
The 2nd bird that we observe is the Kite, which leaves out the very first word.
KI▲ ◆ ● -◆ ● K( WK) xyz
The 3rd bird that we observe is the Thrush, which swaps the very first and 2nd words.
T▲ ◆ ● -◆ ▲ ● C( WK) xyz
We can’t rather construct the 4th bird, however we observe that it is the Bluebird, which is likewise discovered in the next forest, so let’s continue.
| Mockingbird | x y z | Duplicates the very first word |
|---|---|---|
| Kite | y z | Discards the very first word |
| Thrush | y x z | Swaps the very first and 2nd words |
◆ The Becekew Forest
Within the Becekew forest, we discover the B, C, K, and W birds. The very first bird that we observe is the Warbler, which duplicates the 2nd word. This bird’s tune resembles the Mockingbird, which duplicates the very first word.
W▲ ◆ ● -▲ ◆ ◆ ● Wxyz
The 2nd bird that we hear is the Kestrel, which leaves out the 2nd word. This bird’s tune resembles the Kite, which leaves out the very first word. Here is the Kestrel, followed by the Warbler:
K W▲ ◆ ● - W◆ ● -◆ ● ● KWxyz
The 3rd bird that we hear is the Cardinal, which swaps the 2nd and 3rd words. This bird’s tune resembles the Thrush, which swaps the very first and 2nd words. Here is the Cardinal, followed by the Kestrel:
C K▲ ◆ ● - K◆ ▲ ● -◆ ● CKxyz
The evasive Bluebird takes 3 words, and parenthesizes the 3rd word into the 2nd word.
B▲ (◆ ◆ )(● ● ) -▲ (◆ ◆ (● ● ))Bx( yy)( zz)
| Warbler | x y z | Duplicates the 2nd word |
|---|---|---|
| Kestrel | x z | Discards the 2nd word |
| Cardinal | x z y | Swaps the 2nd and 3rd words |
| Bluebird | x (y z) | Composition of x and y used to z |
● The Sekei Forest
Within the Sekei forest, we discover the S, K, and I birds. The easiest of these birds, appears to have no intelligence at all, therefore it is called the Idiot bird given that its action to hearing any word, is that very same word.
I(▲ ◆ ) -(▲ ◆ )( WK)( xy)
The Kestrel is likewise discovered in this forest, however we’re currently knowledgeable about this one. Let’s take a minute to see how words can be organized by parentheses and changed as if they were a single word. The Kestrel leaves out the 2nd group of parenthesized words:
K▲ (◆ ◆ )(● ● ) -▲ (● ● )Kx( yy)( zz)
The Starling does parenthesising, reordering, and duplication. Applied to 3 combinators, it uses each of the very first 2 combinators to the last, then uses the very first outcome of that to the 2nd:
S▲ ◆ ● -▲ (● )(◆ (● ))B( B( BW) C)( BB) xyz
| Idiot | x | x | Returns its word |
|---|---|---|---|
| Starling | xyz | xz (y z) | Returns x( z) used to y( z). |
Conversion
Some birds can reproduce the habits of others:
| Bekimet-Becekew | Becekew-Sekei | Sekei-Becekew | |||
|---|---|---|---|---|---|
| KI | K( WK) | B | S( KS) K | S | B( B( BW) C)( B B) |
| M | W( WK) | C | S( S( K( S( KS) K)) S)( KK) | K | K |
| T | C( WK) | K | K | I | WK |
| W | SS( SK) | ||||
Continue deeper into the forest. To see a list of popular birds, see the Birdwatching Manual
The Ornithomicon is the notorious grimoire of Avian Arithmetic.
Before advancing much deeper into the captivated forest, you need to recognize with the 9 birds of the Bekimet, Becekew and Sekei forests.
Avian Booleans
The Kestrel takes 2 words and constantly chooses the very first, this is the Avian Binary for True:
K▲ ◆ -▲ Kxy
The Kite takes 2 words and constantly picks the 2nd, this is the Avian Binary for False:
KI▲ ◆ -◆ KIxy
Avian Bitwise
The Cardinal takes 2 words and turns them, this is the Avian Bitwise for NOT:
C( KI▲ ◆ ) -▲ C( Kxy)
The Mockingbird takes 2 booleans and returns True if a minimum of among them holds true, this is the Avian Bitwise for OR:
M( KI)( K)▲ ◆ -▲ W( WK)( KI)( K) xy
To return True, when both words hold true, we utilize a mix of Starlings and a Kestrel to develop AND:
S S K( KI)( K)▲ ◆ -◆ SSK( KI)( K) xy
Since True chooses the very first of 2 words, and False picks the 2nd, we require a method to pass the 2nd and 3rd words to the very first one. The Idiotbird is the comparable to the Avian Binary for IfThenElse:
I( KI)▲ ◆ -◆ I( KI) xy
Avian-Human Numerals
The Kite is the both the boolean False, and the number absolutely no, the Idiotbird is the number 1. These are the Church Encoded characters, which talking birds just utilize to interact numbers to people who go into the forest.
| 0 | KIfx | x | 5 | SB( B( WB)( WB)) fx | |
|---|---|---|---|---|---|
| 1 | Ifx | fx | 6 | B( WB)( SB( WB)) fx | |
| 2 | WBfx | f( fx) | 7 | SB( B( WB)( SB( WB))) fx | |
| 3 | SB( WB) fx | f( f( fx)) | 8 | B( WB)( B( WB)( WB)) fx | |
| 4 | B( WB)( WB) fx | f( f( f( fx))) | 9 | SB( B( WB)( B( WB)( WB))) fx |
Reference of combinator birds.
Color-coded for the Bekimet, Becekew and Sekei forests.
| Abstraction | Symbol | Bird | Combinator |
|---|---|---|---|
| λabc.a( bc) | B | Bluebird | S( KS) K |
| λabcd.a( bcd) | B 1 | Blackbird | BBB |
| λabcde.a( bcde) | B 2 | Bunting | B( BBB) B |
| λabcd.a( b( cd)) | B 3 | Becard | B( BB) B |
| λabc.acb | C | Cardinal | S( BBS)( KK) |
| λabcd.ab( cd) | D | Dove | BB |
| λabcde.abc( de) | D 1 | Dickcissel | B( BB) |
| λabcde.a( bc)( de) | D 2 | Dovekies | BB( BB) |
| λabcde.ab( cde) | E | Eagle | B( BBB) |
| λabcdefg.a( bcd)( efg) | Ê | Bald Eagle | B( BBB)( B( BBB)) |
| λabc.cba | F | Finch | ETTET |
| λabcd.ad( bc) | G | Goldfinch | BBC |
| λabc.abcb | H | Hummingbird | BW( BC) |
| λa.a | I | Idiot( Identity) | SKK |
| λabcd.ab( adc) | J | Jay | B( BC)( W( BC( B( BBB)))) |
| λab.a | K | Kestrel (True) | |
| λab.b | KI | Kite (False) | KI |
| λab.a( bb) | L | Lark | CBM |
| λa.aa | M | Mockingbird | SII |
| λab.ab( ab) | M 2 | Double Mockingbird | BM |
| λab.b( ab) | O | Owl | SI |
| λabc.b( a/c) | Q | Queer Bird | CB |
| λabc.a( cb) | Q 1 | Quixotic Bird | BCB |
| λabc.b( ca) | Q 2 | Quizzical Bird | C( BCB) |
| λabc.c( ab) | Q 3 | Quirky Bird | BT |
| λabc.c( bachelor’s degree) | Q 4 | Quacky Bird | F B |
| λabc.bca | R | Robin | BBT |
| λabc.ac( bc) | S | Starling | S |
| λab.ba | T | Thrush | CI |
| λab.b( aab) | U | Turing | LO |
| λabc.cab | V | Vireo (aka Pairing) | BCT |
| λab.abb | W | Warbler | C( BMR) |
| λab.baa | W 1 | Converse Warbler | CW |
| λa.a( λa) | Y | Why Bird (aka Sage Bird) | SLL |
| λab.ab | I * | Identity Bird Once Removed | S( SK) |
| λabc.abcc | W * | Warbler Once Removed | BW |
| λabcd.abdc | C * | Cardinal Once Removed | BC |
| λabcd.acdb | R * | Robin Once Removed | C C * |
| λabcd.adcb | F * | Finch Once Removed | BC R * |
| λabcd.acbd | V * | Vireo Once Removed | C F * |
| λabc.abc | I | Identity Bird Twice Removed | |
| λabcd.abcdd | W | Warbler Twice Removed | B( BW) |
| λabcde.abced | C | Cardinal Twice Removed | BC * |
| λabcde.abdec | R | Robin Twice Removed | BR * |
| λabcde.abedc | F | Finch Twice Removed | BF * |
| λabcde.abecd | V | Vireo Twice Removed | BV * |
| λab.bb | KM | Konstant Mocker | KM |
| λab.aa | C( KM) | Crossed Konstant Mocker | C( KM) |
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