toshMar 31

Combinators

https://tinyapl.rubenverg.com/docs/info/combinators

Combinators - TinyAPL

A combinator is a function or operator that only refers to its arguments and operands without modifying them in any way. Symbol APL expression Bird(1) TinyAPL Diagram I \mathrm I I y Identity ⊣/⊢ K \mathrm K K x Kestrel ⊣ κ \kappa κ y Kite ⊢ W \mathrm W W y F y Warbler ⍨ C \mathrm C C y F x Cardinal ⍨ B \mathrm B B F (G y) Bluebird ∘/⍤/⍥ Q \mathrm Q Q G (F y) Queer ⍛ B 1 {\mathrm B}_1 B1​ F (x G y) Blackbird ⍤ Ψ \Psi Ψ (G x) F (G y) Psi ⍥ S \mathrm S S y F (G y) Starling ⟜/⇽ Σ \Sigma Σ (F y) G y Violet Starling ⊸/⇾ D \mathrm D D x F (G y) Dove ∘/⟜ Δ \Delta Δ (F x) G y Zebra Dove ⍛/⊸ Φ \Phi Φ (F y) G (H y) Phoenix «» Φ 1 \Phi_1 Φ1​ (x F y) G (x H y) Pheasant «» D 2 {\mathrm D}_2 D2​ (F x) G (H y) Dovekie ⊸ + ⟜ P \mathrm P P (y G x) F (x G y) Parrot(2) ⸚ N \mathrm N N x F (x G y) Eastern Nicator ⇽ ν \mathrm \nu ν (x F y) G y Western Nicator ⇾ Additionally, some other primitives have combinator-like behavior: APL expression TinyAPL Diagram n ⍨ n ⍨ F y ⁖ x G y ⁖ Footnotes Some combinators have bird names, originating from To Mock a Mockingbird by Raymond Smullyan. Some of the bird names are taken from the Uiua combinator page. I made this one up.

Show threadsiruwastakenMar 31
Could somebody provide a bit of context on what exactly this is? It seems interesting, but I have no idea what I am looking at.
Show threadleethompMar 31

Many primitives in array languages match the behaviour of certain combinators in combinatory logic. The page shows (left to right) the symbol for a certain combinator, its effective operation in APL syntax where x and y are left and right arguments (APL operators are either infix or single-parameter prefix) and F and G are similarly left and right function arguments, the 'bird' is a sort of colloquial name for a particular combinator, 'TinyAPL' is the operator that matches the combinator in the author's APL implementation, and the diagram is a way of explaining how the combinator works visually

BQN, another array language has a page of documentation describing the same concept for their language with a bit more explanation for the combinator newcomer: https://mlochbaum.github.io/BQN/tutorial/combinator.html

BQN Tutorial: Combinators

Show threadgeneral_reveal
Can we solve for x and y? All I see is algebra here, is my intuition wrong?
Mar 31 at 2:05pmWeb
Show threadZhylMar 31

It's more like a recipe (for functions).

The first example, I, is an identity function. It takes y and returns y.

The second, K, is a constant which takes X and y and returns x.

This gets more complicated as you go along. The idea is that you get rid of a lot of the syntax for composition and have it all be implicit by what you put next to each other (given APL programs are usually one long line of a bunch of different symbols all representing functions).

Show threadseanhunterMar 31

The intuition here is that combinators are higher order functions which take functions and combine them together in various ways. So for a simple example "fix" is a combinator in regular maths where

Fix f = {f(x): f(x) = x for all x in the domain of f}

So if f is a function or a group action or whatever, the fixed-point set of f is all points x in the domain of f such that f(x)=x. ie the points which are unchanged by x. So if f is a reflection, the points which sit on the axis of reflection.

The fixed-point combinator is of particular relevance to this site because it's often called the y combinator.

Show threadskydhashMar 31
Combinators can be a bit sill for values. The usefulness come when you use them as a meta language for functions.