TECHNOLOGY

Logic

A sure enchanted forest is inhabited by talking birds.

Combinatory Forests are inhabited by talking birds, which speak phrases represented right here by shapes, or groups of shapes delimited by parentheses. The next avian conversations level to that even with out anything else as coarse as digits and operators, arithmetic and computation can emerge. We continuously listen to the left-most fowl first.

T()  C(WK)x(yz)

The Bekimet Woodland

At some level of the Bekimet forest, we uncover the B, KI, M, and T birds. Every fowl listens to a message, and solutions with one other. The first fowl that we look is the Mockingbird, which repeats the principle notice.

M W(WK)xyz

The 2nd fowl that we look is the Kite, which omits the principle notice.

KI Good passable(WK)xyz

The third fowl that we look is the Thrush, which swaps the principle and 2nd phrases.

T C(WK)xyz

We are in a position to no longer rather manufacture out the fourth fowl, nonetheless we look that it’s the Bluebird, which is additionally veil within the next forest, so let’s continue.

Mockingbird x x y z Duplicates the principle notice
Kite y z Discards the principle notice
Thrush y x z Swaps the principle and 2nd phrases

The Becekew Woodland

At some level of the Becekew forest, we uncover the B, C, Good passable, and W birds. The first fowl that we look is the Warbler, which repeats the 2nd notice. This fowl’s tune is simply just like the Mockingbird, which repeats the principle notice.

W Wxyz

The 2nd fowl that we hear is the Kestrel, which omits the 2nd notice. This fowl’s tune is simply just like the Kite, which omits the principle notice. Right here is the Kestrel, followed by the Warbler:

Good passableW W KWxyz

The third fowl that we hear is the Cardinal, which swaps the 2nd and third phrases. This fowl’s tune is simply just like the Thrush, which swaps the principle and 2nd phrases. Right here is the Cardinal, followed by the Kestrel:

CGood passable Good passable CKxyz

The elusive Bluebird takes three phrases, and parenthesizes the third notice into the 2nd notice.

B()()  (()) Bx(yy)(zz)
Warbler x y y z Duplicates the 2nd notice
Kestrel x z Discards the 2nd notice
Cardinal x z y Swaps the 2nd and third phrases
Bluebird x (y z) Composition of x and y utilized to z

The Sekei Woodland

At some level of the Sekei forest, we uncover the S, Good passable, and I birds. Basically the easiest of these birds, seems to construct no longer bask in any intelligence at all, and so it known because the Idiot fowl since its response to hearing any notice, is that identical notice.

I()  () (WK)(xy)

The Kestrel is additionally veil in this forest, nonetheless we’re already conscious of this one. Let’s take dangle of a 2nd to uncover about how phrases can even be grouped by parentheses and remodeled as within the event that they were a single notice. As an example, the Kestrel omits the 2nd community of parenthesized phrases:

Good passable()()  () Kx(yy)(zz)

The Starling copies the third notice after the principle, and parenthesizes the final two/ I do know, it’s loads. In varied phrases, it applies the principle and 2nd phrases to the final, then applies the principle outcome the 2nd outcome:

S ()(()) B(B(BW)C)(BB)xyz
Idiot x x Returns its notice
Starling xyz xz (y z) Returns x(z) utilized to y(z).

Conversion

Some birds can replicate the behaviors of others:

Bekimet-Becekew Becekew-Sekei Sekei-Becekew
KI Good passable(WK) B S(KS)Good passable S B(B(BW)C)(B B)
M W(WK) C S(S(Good passable(S(KS)Good passable))S)(KK) Good passable Good passable
T C(WK) Good passable Good passable I WK
W SS(SK)

Continue deeper into the forest. To appear a list of eminent birds, look the Birdwatching Handbook.

The Ornithomicon is the spoiled grimoire of Avian Arithmetic.

Ahead of progressing deeper into the enchanted forest, you may be conscious of the 9 birds of the Bekimet, Becekew and Sekei forests.

Avian Booleans

The Kestrel takes two phrases and continuously selects the principle, right here is the Avian Binary for Correct:

Good passable Kxy

The Kite takes two phrases and continuously selects the 2nd, right here is the Avian Binary for Incorrect:

KI KIxy

Avian Bitwise

The Cardinal takes two phrases and flips them, right here is the Avian Bitwise for NOT:

C(KI)  C(Kxy)

The Mockingbird takes 2 booleans and returns Correct if in any case one in every of them is Correct, right here is the Avian Bitwise for OR:

M(KI)(Good passable) W(WK)(KI)(Good passable)xy

To return Correct, when both phrases are Correct, we exhaust a combination of Starlings and a Kestrel to invent AND:

SSGood passable(KI)(Good passable) SSK(KI)(Good passable)xy

Since Correct selects the principle of 2 phrases, and Incorrect selects the 2nd, we want a manner to pass the 2nd and third phrases to the principle one. As a consequence of this fact, the Idiotbird is the a lot just like the Avian Binary for IfThenElse:

I(KI) I(KI)xy

Avian-Human Numerals

The Kite is the both the boolean Incorrect, and the number zero, the Idiotbird is the #1. These are the Church Encoded numerals, which talking birds fully exhaust to speak numbers to humans who enter 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 Chook Combinator
λabc.a(bc) B Bluebird S(KS)Good passable
λabcd.a(bcd) B1 Blackbird BBB
λabcde.a(bcde) B2 Bunting B(BBB)B
λabcd.a(b(cd)) B3 Becard B(BB)B
λabc.acb C Cardinal S(BBS)(KK)
λabcd.ab(cd) D Dove BB
λabcde.abc(de) D1 Dickcissel B(BB)
λabcde.a(bc)(de) D2 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.advert(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 Good passable Kestrel (Correct)
λab.b KI Kite (Incorrect) KI
λab.a(bb) L Lark CBM
λa.aa M Mockingbird SII
λab.ab(ab) M2 Double Mockingbird BM
λab.b(ab) O Owl SI
λabc.b(ac) Q Weird Chook CB
λabc.a(cb) Q1 Quixotic Chook BCB
λabc.b(ca) Q2 Quizzical Chook C(BCB)
λabc.c(ab) Q3 Quirky Chook BT
λabc.c(ba) Q4 Quacky Chook 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 W1 Communicate Warbler CW
λa.a(λa) Y Why Chook (aka Narrative Chook) SLL
λab.ab I* Identity Chook As soon as Eradicated S(SK)
λabc.abcc W* Warbler As soon as Eradicated BW
λabcd.abdc C* Cardinal As soon as Eradicated BC
λabcd.acdb R* Robin As soon as Eradicated C*C*
λabcd.adcb F* Finch As soon as Eradicated BC*R*
λabcd.acbd V* Vireo As soon as Eradicated C*F*
λabc.abc I Identity Chook Twice Eradicated  
λabcd.abcdd W Warbler Twice Eradicated B(BW)
λabcde.abced C Cardinal Twice Eradicated BC*
λabcde.abdec R Robin Twice Eradicated BR*
λabcde.abedc F Finch Twice Eradicated BF*
λabcde.abecd V Vireo Twice Eradicated BV*
λab.bb KM Konstant Mocker KM
λab.aa C(KM) Crossed Konstant Mocker C(KM)

    Incoming: forth thue 2022

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