# Lambda calculus (no ontology)

## Basic

### Identity, "Idiot"

Always returns its parameter.

• Symbol: I

### Mockingbird

Gives the same result as applying its parameter to itself.

• Symbol: ω

### Omega

The result of applying the Mockingbird to itself. Also the result of applying the Why bird to the Identity bird. Note that this can't take any parameter, since its evaluation results in an infinite loop. Usually treated as "undefined" and is said to be related to infinity.

• Symbol: Ω

### Why

A fixed-point combinator. Returns a value which remains the same when transformed under the original parameter.

• Symbol: Y

### Kestrel

Always returns the first of two parameters (and ignores the second). Usually treated as "true".

• Symbol: K

### Kite

Always returns the second of two parameters (and ignores the first). Usually treated as "false".

• Symbol: KI (because it is the result of applying the Kestrel to the Identity).

### Identity once removed

By definition, it gives the same result as applying the first parameter to the second. Alternatively, when called with one parameter it returns that parameter... unless I'm missing something. So what would make it any more worthwhile using than the ordinary Identity bird?

• Symbol: I* or CT (because it is technically the crossed Thrush, see below).

• Symbol: T

• Symbol:

### Crossed Konstant Mocker

• Symbol: C(Kω)

• Symbol: L

• Symbol: O

• Symbol: W

• Symbol: CW

### Turing

Applying the Turing bird to itself gives you the Theta bird, which is a fixed-point combinator similar to the Why bird.

• Symbol: U

### Cardinal

Returns the "crossed" version of its parameter, hence why "crossed" birds are usually written as Cx.

• Symbol: C

### Bluebird

Returns the composition of its two parameters a and b, i.e. when the result is applied to a further parameter c, its result is the same as applying b to c and then applying a to that. (Notice that the order of the parameters may seem backwards.)

• Symbol: B

• Symbol: V

• Symbol: R

### Starling

All birds can be derived from some combination of Kestrels and Starlings. Roughly, the Kestrel "destroys", while the Starling "creates".

• Symbol: S

### Jay

Similar to the Starling. All birds, except those which ignore some of their parameters, can be derived from some combination of Idiots and Jays.

• Symbol: J

## Starling-Kestrel constructions

```I = SKK
B = S(KS)K
C = S(BBS)(KK)

O = SI
W = CSI

V = BCT
```

## Jay-Idiot constructions

```T = JII
R = JT
C = RRR
B = C(JIC)(JI)
ω = C(C(C(BJT)T)T)T
W = C(BωR)
S = B(BW)(BBC)
```

## Application of one parameter

• Each table cell is the result of applying the bird at the top of its column to the bird at the start of its row.
YTLOWUBVRSJ
I ΩI*ωωωωI*RI**O?
ω Ω?Ω????????
K ???????????
Y ???????????
T ??????????R
L ???????????
O ???????????
W ???????????
U ???????????
B ???????????
V ???????????
R ???????????
S ???????????
J ???????????

## Logic

```true = K
false = KI
not = V false true
implies = R true
and = R false
or = T true
equiv = CS not
```

## Arithmetic

```zero = I
succ = V false
pred = T false
isZero = T true
```