Why do I write? Two reasons!
First, writing is thinking:
“You think you know when you can learn, are more sure when you can write, even more when you can teach, but certain when you can program.”—Alan J. Perlis
Second, reading feedback like this warms my soul […]
functional.cafe
Original post on functional.cafe
![export const times = b;
export const plus = construction([ [cons, b], cons, s ]);
export const exponent = i;
export const succ = construction([ [z1], plus ]);
export const pred = construction([
// initialization
[[z0], [z0]],
// [[__], [Zn]] => [Zn], [++Zn]
[i, nip, dup, [succ], cons],
// dig the quoted parameter up: [C] [B] [A] => [B] [A] [C]
dig,
// apply it, counting up: [z0], [z0] => [z0], [z1] => [z1], [z2] => [z2], [z3], ...
i,
// discard the rightmost quoted number, revealing the quoted predecessor
zap,
// apply it
i
]);
test("pred", () => {
expect((rewrite([[[]], [unit], [z0], pred]))).toEqual(rewrite([[[]], [unit], z0]));
expect((rewrite([[[]], [unit], [z1], pred]))).toEqual(rewrite([[[]], [unit], z0]));
expect((rewrite([[[]], [unit], [z2], pred]))).toEqual(rewrite([[[]], [unit], z1]));
expect((rewrite([[[]], [unit], [z3], pred]))).toEqual(rewrite([[[]], [unit], z2]));
expect((rewrite([[[]], [unit], [z4], pred]))).toEqual(rewrite([[[]], [unit], z3]));
});](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:tdu2ua5n5twpnw3w3nzqe6pq/bafkreigam5tt37az5nilpbazhck52dw442x3tm37wf53cyl6drktdqxnxe)
![// [C] [B] [A] s == [[C] B] [C] A
export const s = combinator((C: Quotation, B: Quotation, A: Quotation) => [ [C, ...B], C, ...A ]);
export const starling = s;
// [B] [A] k == A
export const k = combinator((B: Quotation, A: Quotation) => [ ...A ]);
export const kestrel = k;](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:tdu2ua5n5twpnw3w3nzqe6pq/bafkreiewblb5opxs26itxj3mpszjaqyic6qmei2u7xic7zofmw7ujli27a)
![test("zn", () => {
expect(z0([ [z0], [z1] ])).toEqual([ z0 ]);
expect(z1([ [z0], [z1] ])).toEqual([ [ z0 ], z1 ]);
expect(z2([ [z0], [z1] ])).toEqual([ [ [ z0 ], z1 ], z1 ]);
expect(z3([ [z0], [z1] ])).toEqual([ [ [ [ z0 ], z1 ], z1 ], z1 ]);
expect(z4([ [z0], [z1] ])).toEqual([ [ [ [ [ z0 ], z1 ], z1 ], z1 ], z1 ]);
expect(z5([ [z0], [z1] ])).toEqual([ [ [ [ [ [ z0 ], z1 ], z1 ], z1 ], z1 ], z1 ]);
expect(z6([ [z0], [z1] ])).toEqual([ [ [ [ [ [ [ z0 ], z1 ], z1 ], z1 ], z1 ], z1 ], z1 ]);
expect(z7([ [z0], [z1] ])).toEqual([ [ [ [ [ [ [ [ z0 ], z1 ], z1 ], z1 ], z1 ], z1 ], z1 ], z1 ]);
expect(z8([ [z0], [z1] ])).toEqual([ [ [ [ [ [ [ [ [ z0 ], z1 ], z1 ], z1 ], z1 ], z1 ], z1 ], z1 ], z1 ]);
expect(z9([ [z0], [z1] ])).toEqual([ [ [ [ [ [ [ [ [ [ z0 ], z1 ], z1 ], z1 ], z1 ], z1 ], z1 ], z1 ], z1 ], z1 ]);
});](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:tdu2ua5n5twpnw3w3nzqe6pq/bafkreiapiw4qcccf7ewmttazpi2to3256jdzxj7zdcoe7fodxrwtqy2a7y)
![// [B] [A] cons == [[B] A]
export const cons = combinator((B: Parameter, A: Parameter) => {
if (!isProgram(A)) return Fail;
if (!isProgram(B)) return Fail;
if (A.length === 0) return Fail;
if (B.length === 0) return Fail;
return [ [B, ...A] ];
});](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:tdu2ua5n5twpnw3w3nzqe6pq/bafkreicgxmuu5kwdyqocdlvbzzrpniq3cw4abij4wgin4bice4fxudygga)
![### the list interpretation of balanced parentheses
Chaining cons cells together by the `cdr` pointer creates a list. For example, this is a list of five holes —`()()()()()`—diagrammed to show the links:
```mermaid
graph LR
EmptyPair@{ shape: dbl-circ, label: "()" }
Five(["()()()()()"]) --> Four(["()()()()"])
Four --> Three(["()()()"])
Three --> Two(["()()"])
Two --> EmptyPair
```
We don't call it a list of empty pairs. The pairs we write in text notation are containers, and if the pairs contain empty, than we say this is a list of five holes.
A list of five empty pairs looks like this:
```mermaid
graph TD
EmptyPair@{ shape: dbl-circ, label: "()" }
Five(["(())(())(())(())(())"]) --o EmptyPair
Five --> Four(["(())(())(())(())"])
Four --o EmptyPair
Four --> Three(["(())(())(())"])
Three --o EmptyPair
Three --> Two(["(())(())"])
Two --o EmptyPair
Two --> One(["(())"])
One --o EmptyPair
```](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:tdu2ua5n5twpnw3w3nzqe6pq/bafkreiep2bm4ce7kdxdz2k7bg6k4t7rkjzcuuksbjqmd4xo5lkls3kruhe)
![### the cons cell interpretation of balanced parentheses
An empty pair of parentheses is a cons cell with both its `car` and `cdr` pointing to the empty string (`ε`). The `car` pointer terminates in a circle, the `cdr` pointer terminates in an arrow.
```mermaid
graph LR
A(["()"]) -- car --o E:::transparent@{ shape: circ, label: "ε" }
A -- cdr --> E
classDef transparent stroke:#ffffff, fill: #ffffff
```
Either or both can point to another cons cell:
```mermaid
graph LR
B(["(())"]) -- car --o A
B(["(())"]) -- cdr --> E
A(["()"]) -- car --o E:::transparent@{ shape: circ, label: "ε" }
A -- cdr --> E
classDef transparent stroke:#ffffff, fill: #ffffff
```
```mermaid
graph LR
C(["()()"]) -- car --o E
C(["()()"]) -- cdr --> A
A(["()"]) -- car --o E:::transparent@{ shape: circ, label: "ε" }
A -- cdr --> E
classDef transparent stroke:#ffffff, fill: #ffffff
```
```mermaid
graph LR
D(["(())()"]) -- car --o A
D(["(())()"]) -- cdr --> A
A(["()"]) -- car --o E:::transparent@{ shape: circ, label: "ε" }
A -- cdr --> E
classDef transparent stroke:#ffffff, fill: #ffffff
```](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:tdu2ua5n5twpnw3w3nzqe6pq/bafkreib3puribiy3hpgw4ftnvnfqqdctplnk7ray53czr4ankecrg4hk3q)
